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In This Chapter

Looking at the advantages of planned experimentation ^ Examining experimental considerations and terminology ^ Exploring the 2* full factorial experiments

The point of Six Sigma is improvement, and you are now at the point in the DMAIC roadmap where you synthesize improvements and/or reconfigure your system or process to be better. Six Sigma offers extremely powerful tools

To aid you in your improvement efforts. Chief among these tools is experimentation. In Six Sigma, you first design an experiment before you carry it out. Then you follow it up with analysis to uncover previously hidden knowledge.

Design of Experiments (or DOE For short) has always been at the technical heart of Six Sigma. As necessity is the mother of invention, the field of DOE

Has matured due to the need to understand, and then improve, the world around you. This chapter gives you the lowdown.

Whij Experiment} The Improvement Porter Of Six Sigma Experiments

How is improvement achieved? The spark of improvement comes from a

Curious mind, trying to figure out what it is that makes things tick.

What is an experiment, amju’auj}

In an Observational study (covered in Chapter 7), you simply act as an outside

Observer, recording data as it happens, trying to glean understanding from

Careful review of the world around you. In these types of studies, you just let

The XS (inputs) of the system or process you are working on take whatever

Values they do. And as this plays out, you record the corresponding process

Output Y Values.

Experiments, On the other hand, are different from observational studies in one fundamental way: In experiments, instead of just letting the Xs of the process you are studying take on whatever values they do, you purposely

Setand control the values that the XS take on. In an experiment, you actively

Control and modify the process being studied.

Experiments offer a greater level of insight and knowledge than observational studies do. Think of the many observational studies performed for decades

In fields like medicine, education, economic policy, diet, and so on. Dozens

Upon dozens of observational studies have only added incremental knowledge to these areas; there is, for example, still a lot of debate about what specific foods are part of a healthy diet. Because you purposefully control the factors, the amount of specific knowledge you get out of an experiment

Almost always exceeds what you gain from observational studies. For that

Reason, designing and analyzing experiments — in spite of the complexity of the topic — has always been one of the core pillars of Six Sigma breakthrough

Improvement.

The purpose of Six Sigma experiments

Every experiment in Six Sigma is targeted at better understanding the Y = f(X) Relational foundation between the inputs and outputs of the process or

System being improved. Better understanding from experimentation includes

Is Knowing which input Xs have a significant effect on the output Y and

Knowing which XS are insignificant.

Formulating and quantifying the mathematical relationship between the

Significant Xs and the output Y.

F Statistically confirming that a change or improvement has been made to a process or system.

Discovering where to set the values of the significant XS so that they combine together to produce the optimal output value of Y.

Few activities in Six Sigma offer as much insight and change horsepower as experiments do. That’s because properly designed experiments reveal, quantify, and confirm the underlying Y = f(X) relationship of a process or system.

Experimenting With Words

The field of planning and analyzing experiments is much older than Six Sigma.

As a result, a few somewhat unique terms are used. Here are some of the interesting terms you need to know — along with their relation to Six Sigma.

Response Is the term used for the output of the process that you investigate in the experiment. In Six Sigma terms, the response is synonymous

With the Y In the Y = F(X) Equation. The whole point of the experiment is to figure out how the Xs combine together to effect the response, or Y.

F The input characteristics, or variables you purposely control during

Theexperiment, are called experimental Factors. Sometimes, they’re also

Called Conditions, variables, Or simply Inputs. In all cases, the experimental factors are the Xs in the Six Sigma Y = f(X) equation.

In your experiment, you choose two or more values for each of the experimental factors. These values are called the Levels For that factor.

Planning your experiment includes deciding how many levels you need to use for each factor.

V Processes and systems have variation. Part of experimentation is

Repeating your whole experiment, or parts of it, to understand how

Much variation there actually is. These types of repetitions are called

Replications. Deciding what part of your experiment needs to be replicated

And deciding how many replications there will be is part of developing

Your experiment plan.

V Every experiment is made up of a series of Runs. Each experimental run consists of a unique, predetermined set of values for each of the factors.

You then conduct the process or system through one cycle with those input values, and the output is recorded. That is a run in an experiment.

The end game of Six Sigma experiments

It has been said, "Knowledge is power." Six Sigma experiments are a confirmation of that statement.

The power of Six Sigma experiments lies in their ability to formulate, quantify and validate the Y = f(X) Relationship of a process or system. Knowing the form and details of Y = f(X) For a system, you literally have a window into the past, present, and — most importantly — the future.

After wrapping up an experiment, you have in your hands a Y = f(X) Equation

That identifies each critical input XAnd quantifies its influence on the output Y. For example, if you are working on a marketing plan to improve brand awareness (that’s the output Y), A Six Sigma experiment provides an equation that tells you which type of advertisements — newspaper, radio, TV, Internet, and so on — and how many of each type to run (the input Xs) to reach a specified

Improvement goal. Or if you are managing the production ofplastic seals that must meet a minimum tear strength requirement (the Y), After proper experimentation, you have an equation that tells you exactly where to set the mold press temperature (X1), how much pigment to add (X2), and the correct operating temperature of the mold press (X3). In all cases, whether they involve

Continuous or attribute data, successful experimentation reveals detailed, specific knowledge of which input Xs influence the output Y — and by how much.

With this level of system or process knowledge, your operational focus immediately switches from passively watching the output and hoping for success to actively monitoring and controlling identified key inputs, knowing that

Your purposeful management and control of these inputs will always lead to the desired process outcome. This is where you open the door to the new world of breakthrough performance.

Look Before \lou Leap: Experimental

Considerations

Trial and error — tinkering with the input knobs of a process or system — is temptingly simple. We all have a desire to jump in and quickly fix a problem. In the long-run, though, careful planning almost universally leads you to a

Quicker and better solution.

Frankenstein should have planned

How should you approach experimentation? Where do you start? The trial-and-error approach

Many people approach experimentation by rolling up their sleeves and jumping into an unstructured exploration of the experimental variables and their resulting output: Tweak the knobs, adjust the settings, and observe the results.

Often, judgment and intuition are the basis for steering the exploration and

Interpreting the findings.

For obvious reasons, however, this unstructured, haphazard approach

Rarely increases knowledge. Every once in a while, you may get lucky, but

Thisapproach, isunreliable.

The one-factor-at-a-time approach

At the other end of the spectrum, there’s a structured approach: Isolate a single input variable and study its effect on the output; carefully hold all other

Factors constant while the selected input variable is incremented across an

Exploratory range of operation. Then repeat this meticulous scan for each of

The input variables.

The downfall of this method is twofold:

The one-factor-at-a-time approach is inefficient and expensive. A scan, conducted one factor at a time, of the possible operating range for each input variable leads to a huge amount of experimental runs. Unless you have only one variable in your system, this approach becomes unwieldy

And wastefully expensive.

The results of one-factor-at-a-time experiments are often misleading.

When you isolate individual variables, you automatically negate the possibility of two or more factors combining together to affect the outcome. But these types of interaction effects are an unavoidable part of reality. Think of baking a cake. A delicious-tasting outcome (the Y) Is a function of several input Xs — like "amount of flour," "number of eggs," "oven temperature," "baking time," and so on. Obviously, the right value for the

Variable of "baking time" depends on the setting for "oven temperature."

How hot the oven is and how long you leave the cake in the oven are two

Input variables that interact with each other. One-factor-at-a-time experiments will never uncover this essential relationship. The danger is that

You draw unfounded conclusions from your experiment — or miss important information altogether.

Use the one-factor-at-a-time approach only when you have a process or

System with a single input variable. This approach works with a single-x

System because there is no possibility of an interaction effect.

The Six Sigma approach — doing more than one thing at a time

Now you know the drawbacks of the haphazard approach and the one-factor-at-a-time approach. Is there a better way? Six Sigma uses a reliable approach

To experimentation that:

Is Efficiently accumulates information about a process or system

Provides valid insights, including knowledge regarding variable interactions

F Quantifies the amount of knowledge discovered about a system as well

As the amount of knowledge that remains unknown

The experimental approach you use in Six Sigma incorporates the best practices from the various disciplines of science. Over the years, scientists have

Developed experiment plans that return a vast amount of knowledge in a very

Efficient way. The key elements of the Six Sigma approach include:

F Planning out the experiment before you conduct it. "Look before you leap" is a mantra of every good experimenter. Careful planning always

Increases the value of your experiment results while minimizing the amount of work and money you have to invest.

F Exploring the effect of more than one input variable at a time. This

Allows you to be efficient while at the same time capturing unsuspected and sometimes hard-to-find interaction effects.

F Minimizing the number of required runs in your experiment. It’s surprising how much you can get out of a small number of properly planned

Experimental runs.

T-" Replicating key experiment conditions to assess variation. A part of every experiment is understanding how much of your system’s or process’s behavior is deterministic and how much is random variation.

Accounting for known and unknown factors that you are not directly including in your experiment. You can never take everything into consideration in your experiment. There are ways, however, to keep these nuisance factors from clouding the results of your experiment.

Simple, sequential, and systematic is best

Rome wasn’t built in a day. Properly planned experiments fit into a larger strategy of iteratively converging to an ideal improvement solution.

The problem ©ith boil-the-ocean super-experiments

The power of designed experiments is intoxicating. Be careful, though, not to

Get carried away. There is a temptation to try to solve everything in one fell

Swoop, using a big, well-designed super-experiment. But putting all your eggs into one experimental basket has some definite drawbacks.

F Creating a single super-experiment based only on the knowledge you

Have Before The experiment begins necessitates that you include all the variables that you suspect are contributing to the situation. This always

Leads to a long list of potential Xs, and consequently, always results in a long, expensive, unwieldy experiment.

As a large super-experiment is carried out over a protracted period of

Time, there is a greater chance of unknown factors creeping in, confounding the experimental conditions and results.

With no prior knowledge, it is difficult to know what values and ranges

To assign to each experimental XInput.

Conducting an experiment takes time and money. If something goes wrong in your one super-experiment or if new information is revealed

That requires a change to your initial assumptions, you will have already

Consumed your experimental budget and resources. The progressive, iterative approach

An efficient and consistently successful approach to experimentation follows

A progressive and iterative approach.

Screening experiments: At this first stage, experiments are designed to handle a large number of factors or variables. When you first start investigating a process or system, you identify all the possible Xs that may be

Influencing the output Y. The whole point of screening experiments is

To quickly verify which of these factors has a significant effect on the output.

Characterizing experiments: When you have screened out the unimportant variables, your experiments focus on characterizing and quantifying the effect of the remaining critical few. These characterization experiments reveal what form and what magnitude the critical factors take in

The Y = f(X) Equation for your process or system.

Optimization experiments: After characterizing your process or system,

The final step is to conduct experiments that teach you what the best settings are for the input variables to meet your desired outcome goal.

Your goal may be to maximize or to minimize the value of the output. Or it may be to hit a certain target level. More often, your goal is simply to

Minimize the amount of variation in the output Y. Optimization experiments find the best settings of the XS to meet your Y Goal.

The purpose of each of these types of experiments — screening, characterizing, and optimizing — are very different. The form and plan of the experiments you conduct at each of these stages, therefore, are necessarily different from

Each other.

Figure 9-1 shows the progressive and iterative approach used in Six Sigma

Experiments.

2k Factorial Experiments

Design and analysis of experiments is a topic large enough for a whole For Dummies Book by itself. To get you quickly up to speed, however, the following section of this book shows you how to plan, conduct, and analyze the most common type of experiment in Six Sigma — the 2" factorial (pronounced two to the k). 2" factorial experiments can be easily adapted to provide screening,

Characterization, or optimization information. Insights into other types of

Experiment designs and variations used in Six Sigma are offered along the way.

Plan your experiment

Like in almost all other endeavors, time spent in planning is rewarded with

Better results in a shorter period of time. Planning 2" factorial experiments

Follows a simple pattern that is outlined in the following sections.

Select the experiment factors

The first thing to do in your planning is to identify the input variables, the

XS, that you will include in your experimental investigation. The factors youinclude should all be potential contributors to the output Y You are investigating.

How many factors you want to include in your experiment guides you in

Choosing the right experimental design. 2" factorial experiments work best

When you have between two and five Xs. But if you have over five Xs in your experiment, full 2" factorial experiments become relatively inefficient and can be replaced with pared down versions called Fractional factorials, Or with

Other screening designs.

One good strategy is to include all potential Xs in a first screening experiment — even the ones you are skeptical about. You then use the analysis of the experiment results to tell you objectively, without any

Guessing, which variables to keep pursuing and which ones to set aside.

Remember, in Six Sigma, you let the data do the talking.

Experience with experiments verifies the Pareto Principle Introduced in Chapter 7 — that even if you include dozens of contributing factors in your experiment, only a small number of these Xs have a significant effect on the output response. When you initially have more than four or five factors, your experiment purpose is to screen out the "trivial many" factors from the "critical few." After that, you then run characterization experiments to provide the

Detailed knowledge about the remaining critical few.

Plac"ettBurman experiment designs Are an advanced method you may hear about for efficiently screening dozens of potential Xs. Although they don’t reveal all the detailed knowledge provided by a 2" factorial design, Plackett-Burman experiments quickly identify which experimental variables are Active In your system or process. You then follow these screening studies up with

More detailed characterization experiments.

Set the factor levels

2" Factorial experiments all have one thing in common — they use only two levels for each input factor. (That’s what the "2" in 2" stands for! The " Represents the number of factors included in your experiment.) For each X In your

Experiment you select a "high" and a "low" value that bounds the scope of your investigation.

For example, suppose you are working to improve an ice cream carton filling

Process. Each filled half-gallon carton needs to weigh between 1,235 and 1,290 grams. Your Six Sigma work up to this point has identified ice cream

Flavor, the time setting on the filling machine, and the pressure setting on the

Filling machine as possible contributing Xs to the Y Output of weight. For each of these three factors, you need to select a "high" and a "low" value for your

Experiment.

With only two values for each factor, you want to select high and low values

That bracket the expected operating range for each variable. For the ice

Cream flavor variable, for example, you may select Vanilla and Strawberry to

Book-end the range of possible ice cream consistencies. Table 9-1 provides a

Summary of the selected experiment variables and their values.

Table 9-1 Variable Values for the Ice Cream _Carton Filler Experiment_

Variable_Symbol_"Low" Setting "High" Setting

Ice cream flavor X Vanilla Strawberry

Fill time (seconds) X2 0.5 1.1

Pressure (psi) X3 120 140

2" experiments are intended to provide knowledge only Within The bounds

Ofyour chosen variable settings. Be careful not to put too much credence on information extrapolated outside these original boundaries.

Experimental codes and the design matrix

With the experiment variables selected and their "low" and "high" levels set, you are now ready to outline the plan for the runs of your experiment. For 2"

Factorial experiments, there will be 2" number of unique runs, where " Is the

Number of variables included in your experiment. For the ice cream carton

Filler example, then, there will be 23 = 2 X 2 X 2 = 8 runs in the experiment,

Because there are three input variables. For an experiment with two variables

There will be 22 = 2 X 2 = 4 runs, and so on.

Each of these 2" experimental runs corresponds to a unique combination of

The variable settings. In a full 2" factorial experiment, you conduct a run or

Cycle of your experiment at each of these unique combinations of factor

Settings. In a two-factor, two-level experiment, the 22 = 4 unique setting combinations are with:

Both factors at their "low" setting

The first factor at its "high" setting and the second factor at its "low" setting

F The first factor at its "low" setting and the second factor at its "high"

Setting

Both factors at their "high" setting

There are no other ways that these two factors can combine with their two levels. For a three-factor experiment, there are eight such unique variable setting combinations.

A quick, shorthand way to create a complete table of an experiment’s unique run combinations is to create a column for each of the experiment variables and a row for each of the 2" runs. Then, using -1s as a code for the "low" variable settings and +1s as a code for the "high" settings, start with the left-most variable column, and fill in the column cells with alternating -1s and +1s.

With the left-most column filled in, move on to the next column to the right

And repeat the process — but this time with alternating Pairs Of -1s and +1s. Fill in the next column to the right with alternating Quadruplets Of -1s and +1s,

And so on, repeating this process from left to right until, in the right-most

Column, you have the first half of the runs marked as -1s and the bottom half listed as +1s. This table of patterned +1s and -1s is called the Coded design matrix. Table 9-2 shows the coded design matrix for a three-factor experiment, such as the ice cream carton filler.

Table 9-2 Coded Design Matrix for a Three-Factor Experiment

RunX, X2 X3

1 -1 -1 -1

2 +1 -1 -1

3 -1 +1 -1

4+1 +1 -1

5-1 -1 +1

6 +1 -1 +1

7 -1 +1 +1

8 +1 +1 +1

Remember that these three factors are coded values in the table; when you see a under the X1 column, it really represents a discrete value, such as "Vanilla" in the ice cream experiment; and a really represents the other

Value, like "Strawberry."

Conduct your experiment

With your experiment well planned, the act of carrying it out is easy — it’s like falling off a log. Now it’s time to roll up your sleeves and get into the scientific trenches.

Randomize: Safeguard against unknown nuisance factors

Despite your best efforts, external factors beyond the control of your selected experiment variables may creep in and influence the outcome of your experiment. These are factors (called Nuisance factors) That you haven’t foreseen,

But they have the potential to blur the clarity of your analysis and insights.

For example, in the ice cream carton filling process discussed in the preceding

Section, a rise in the ambient factory temperature during the duration of the experiment may affect the experiment outcomes and be falsely construed as a

Real effect from your selected experimental factors.

One way to compensate for these unknown nuisance variables is to Randomize The order of your experimental runs. This spreads out the otherwise concentrated or confounding potential for nuisance effects evenly and fairly over all of the experimental runs and preserves the clarity of your results.

Always randomize the order of your experiment runs. This reduces the risk of extraneous variables skewing the results of your analysis.

Randomize materials being used in your experiment, your personnel, or your

Equipment. The idea is to guarantee that only the effect of your selected factors is purposely concentrated during your experiment.

Blocking: Safeguard against known nuisance factors

When you know the source of nuisance variation that is not part of your

Selected experimental factors, you can purposely include this nuisance effect

In All Your experimental runs. In this way, you guarantee that there will be no

Bias on only a portion of your experimental settings.

In the ice cream carton filling example, you may decide to perform each

Experimental run at the same time each day. This way, the influences from

Different times of day are blocked from impacting only some of the experimental runs.

A catchy phrase may help you remember the roles of randomizing and blocking in your experiments: Block what you can and randomize against what you can’t block.

Perform the experiment and gather the data

Running the experiment is the fun part. All you have to do is follow your experimental plan, like the one shown in Table 9-3 for the ice cream carton

Filler project.

Table 9-3 Plan and Results for the Ice Cream _Carton Filler Experiment_

Run OrderX,: FlavorX2: TimeX3: PressureY

1 7-1-1 -1 1,238

2 2 +1 -1 -1 1,252

35-1 +1 -1 1,228

48+1 +1 -1 1,237

53-1 -1 +1 1,223

66+1 -1 +1 1,234

71 -1 +1 +1 1,238

84+1 +1 +1 1,250

In Table 9-3, the coded design matrix is augmented with a column showing the random order in which the experimental runs are conducted. Also, on the

Far right, a column is added to capture the outcome Y Variable for each experimental run. In Table 9-3, recorded values for the ice cream carton filling example experiment are provided.

Analyze your experiment

The purpose of analyzing your experiment is to take the experiment results

And piece together the Y = F(X) Puzzle for your process or system. How much effect does x1 have on Y? What mathematical form does this relationship take on? These are the questions that your analysis will answer.

Visualize and calculate the main effects

A Main effect Is the quantitative influence a single experiment factor has on

The response Y. There will be a main effect for each factor in your experiment.

For example, how much effect does ice cream flavor — going from "Vanilla" to "Strawberry" — have on the resulting filled weight of the carton?

The main effect of the X ice cream flavor factor is the average response of the experiment runs with X! at its "high" or "Strawberry" setting, minus the average response of the experiment runs with X at its "low" or "Vanilla" setting. To find the answer, refer to the captured values in Table 9-3. Runs 2, 4, 6, and 8 are where X is at its "high" setting. Runs 1, 3, 5, and 7 are where X is at its "low" setting. So the main effect of ice cream flavor (called Ј1) can be written mathematically as

E 1 _ 4 4

E _ 1,252 + 1,237 + 1,234 + 1,250 1,238 + 1,228 + 1,223 + 1,238 E 1 _ 4 4

E1 _ 1,243.25 _ 1,231.75

E1 _ 11.5

Figure 9-2 shows the main effect of ice cream flavor graphically. You can see

That as the ice cream flavor changes from "Vanilla" to "Strawberry," the

Carton weight changes by 11.5 grams.

Ј,: Main Effect

12441242124012381236-

Figure 9-2:

Main effect Ј1 On carton weight due to the ice cream flavor.

Vanilla

Strawberry

JT,: Ice Cream Flavor

To calculate the main effect E2 of fill time on the filled carton weight Y you can leverage the coded setting values for factor X2 in Table 9-3. Call these coded values c21, c22, and so on through c28, for each of the experimental runs. Another way to write the equation for the main effect of fill time, then, is

E 2 _ E 2 _

C2,1 Y + C2,2 Y2 + C2,3 Y3 + C2,4 Y4 + C2,5 Y5 + C2,6 Y + C2,7 Y7 + C2,8 Y.

-1)1,238-

-1) 1,252-

4

+1) 1,228-

-1) 1,237 + (-1) 1,223 -

1)1,234

1)1,238

1)1,250

4

-1,238 – 1,252 + 1,228 + 1,237 – 1,223 – 1,234 + 1,238 + 1,250

E2_1.5

Which gives a main effect of fill time of 1.5 grams.

Then using the coded setting values for X3 — C31, c32, c38 — the same procedure can be used to calculate the main effect’ of pressure E3:

E3 E3

1,238 _ 1,252 _ 1,228 _ 1,237 + 1,223 + 1,234 + 1,238 + 1,250

2.5

With the main effect of pressure being -2.5 grams.

In fact, the coded setting values can be leveraged to create a generalized equation to compute Any Effect in a 2* full factorial experiment.

E._ 1 yC. Y

2 / _ 1

Where k is the number of experiment factors and / designates which effect you’re calculating.

Figure 9-3 shows all three main effects on a single plot for comparison.

Figure 9-3:

Graphical – 1244 comparison

Of main

Effects for

The ice

Cream

Carton filling

Example.

=: 1240

■=T 1236

1232

Vanilla

Strawberry

0.5

1.1

120

140

Visually, it is easy to see that X1, the flavor of the ice cream, has the largest

Main effect on the filled weight of the cartons. (See Chapter 5 for a more detailed discussion of main effects plots.)

Visualize and calculate the interaction effects

One input variable interacting with another is always a possibility. Are there any of these type of interaction effects in the ice cream carton filling example? How do you find out?

Call the interaction effect between ice cream flavor (X) and fill time (X2) E12. What you do next is create a new column of coded setting variables that represents the interaction of factors X and X2. You do this by multiplying the coded values of X and X2 together for each experiment run. For example, c121 = c11 x c21, c122 = c12 x c22, and so on up through c128 = c18 x c2 8. Table 9-4 shows the ‘ new coded setting values for the two-variable and the three-variable interactions possible in the 23 ice cream carton filler experiment.

Table 9-4 Interaction Coded Variables for the _Ice Cream Carton Filler Experiment

Rune, c2 c3 c,2 c,3 c23 c123 Y

1 -1 -1 -1 +1 +1 +1 -1 1,238

2 +1 -1 -1 -1 -1 +1 +1 1,252

3-1 +1 -1 -1 +1 -1 +1 1,228

4 +1 +1 -1 +1 -1 -1 -1 1,237

5-1 -1 +1 +1 -1 -1 +1 1,223

6 +1 -1 +1 -1 +1 -1 -1 1,234

7-1 +1 +1 -1 -1 +1 -1 1,238

8 +1 +1 +1 +1 +1 +1 +1 1,250

With the coded values for the interaction effects, you can now use the general formula to calculate each of the possible two-variable interaction effects.

For example, the interaction effect between ice cream flavor (X1) and fill time

(X2) Is calculated as

E12 = 2T-T 2 C kjYJ

_ ( +1)1,238 + (-1)1,252 + (-1) 1,228 + ( +1) 1,237 + ( +1) 1,223 + (-1)1,234 + (-1) 1,238 + ( +1) 1,250

E12 4

1,238 – 1,252 – 1,228 + 1,237 + 1,223 – 1,234 – 1,238 + 1,250

E12 _-4-

E12 _-1.0

Or -1.0 grams effect when the X and the X2 factors are combined together.

Using the same procedure, you can calculate interaction effects for E13 and E23. You should get values of 0.0 grams and 14.0 grams, respectively. Figure 9-4

Shows all three two-variable interaction effects.

V, : Flavor

Figure 9-4:

Two-factor interactions in the ice cream carton filler example.

0.5

Interaction Effects

1.1 120

140

1242

1236

1230

1242

1236

1230

Flavor Vanilla Strawberry

Fill Time

0.5 1.1

.V3 : Pressure

In the grid layout of Figure 9-4 for the X2 – X3 interaction, you can see that the plotted effect lines have very different slopes. This is your graphical clue to know that E23 is very strong. The plotted effect lines for X1 – X2 and X1 – X3, however, have very similar slopes. It is no surprise that their calculated interaction effects, E12 and E13, are rather small.

For a three-factor experiment, there is one more interaction effect you need to compute. It is the possible interaction when all three variables are combined (E123). This may sound tricky, but it’s not because you’re using the

Coded setting values and the same general formula for calculating the effects.

E123 _ TJT-T2 C123,/ Y/ 2 / _ 1

E -1,238 + 1,252 + 1,228 – 1,237 + 1,223 – 1,234 – 1,238 – 1,250

E 123 _ 4

E123_1.5

Or 1.5 grams effect when all three factors are combined.

Which effects are significant}

Even though you can calculate all the main and interaction effects of the variables, are they all significant? Are they all necessary? The Pareto Principle (see Chapter 7) tells you that a relatively small subset of all the possible effects explains the vast majority of the output responses. So how do you

Know which effects to hold on to and which ones to cast aside?

If the factors you select for your experiment have no impact on the outcome

Y the calculated main and interaction effects will just be random — they’ll be normally distributed and centered around zero. But if any one of the effects is

Significant, it will depart from the random cluster of the rest.

The easiest way to detect this departure is graphically, by plotting all the calculated effects against a line representing a normal distribution. If a plotted effect doesn’t fit this line, you know that it is not part of the random noise, but instead is significant.

To create this graph for the ice cream carton filler example, you list all the calculated effects in rank order from smallest to largest and write down the

Rank i next to each effect. In case of ties, like between E2 and E123, you assign

The average rank to the tied effects. You can see this in Table 9-5.

As an intermediate step, you have to calculate the expected probability for

Each rank. This is called P and is in the fourth column of Table 9-5. The formula calculating the P for each row in the table is

So for the E13 effect, its expected probability, P is

PI _ ‘l1^ _ T5 _ °.357

The final step in creating the values of Table 9-5 is looking up the Z value for each intermediate P value. Using a look-up table for Z, you can see that the Z score corresponding to the P of °.357 on E13 Is -°.366.

Having filled in all the values of Table 9-5, you now simply plot the calculated Z value against each of the corresponding effect values. This is shown for the ice cream carton filler example in Figure 9-5.

Looking at Figure 9-5, it is obvious that effects E1 and E23 are very different from the rest of the effects. While E1 and E23 are not centered around zero and clearly don’t fit the expected normal probability line, all the others do.

The more complicated a potential interaction is, the less likely it is to be significant in reality. Very often, for example, two-factor interaction effects are found to be significant. Much less often, three-factor interactions are determined to be important. It is a real rarity to uncover a legitimate interaction

Effect that includes four or more factors. The more complicated an interaction effect is, the more skeptical you should be about it being real.

With just an eight-run experiment, you have determined that there are really

Only two effects that significantly effect the performance of the ice cream

Carton filler. The first is the type or flavor of ice cream being produced. Also,

The combined, interactive effect of filler time and pressure definitely impacts performance. But filler time and pressure, acting by themselves, don’t have a

Significant effect.

This is the power of Six Sigma. Rather than guessing or fumbling in the dark

For the answer, you let the data and the analysis show what is important and

What is not. In return, you look like the hero! The general form of the equation

2* factorial experiments not only reveal which factors effect the output y but they also allow you to understand the form of the Y = F(X) Equation for the system or process you are improving. At the onset, a 2* experiment investigates the possibility of all main and interaction effects being significant. (Subsequent analysis shows you which ones you can safely ignore.)

Picture in your mind a general y = F(X) Equation with a term for each main effect, a term for each interaction effect, and an overall offset effect. For the

Three-factor ice cream carton filler example, this general equation takes the

Form:

Y _ F3 ° + Ј 1 x1 + P 2 X 2 + P 3 X 3 + P12 X1 X 2 + P13 X1 X 3 + /3 23 X 2 X 3 + /3123 X1 X 2 X 3

In this general equation, each combination of the input X Variables is prefixed with a multiplier coefficient represented by the jtfs (the Greek letter beta; pronounced BAY-tah). The little subscripts at the lower right of each jO tell you which effect it corresponds to. In stuffy mathematical terms, these jtfs are called coefficients.

A two-factor system would have a general equation of

Y _ / ° + /1×1 + / 2 x 2 + P12 x1x 2

While a four-factor system would include additional terms for all the three-variable and four-variable interactions.

The A, term in all these equations represents the overall level of the process or system you are working on. No matter what you do to the setting of any of

The system variables, the system will take on at least this value. That’s why it

Is often called an Offset Or Constant Term.

Define §our \t = f(K) equation

For the system or process you are working on, the only terms of the general

Equation you need to hang on to are the ones that correspond to the effects you have found to be significant. For example, in the ice cream carton filler process, only the ice cream flavor x1 and the filler time-pressure interaction x2x3 effects were found to be significant. That leads to a simplified equation form of

Y = P „ + P X X X + P 23 X 2 X 3

But what are the values of the jHs? Again, finding these values is easier than you may think.

The value for the offset ft is simply the computed average for all the 2" Experiment runs. For the ice cream carton filler example, the average output rfor the eight experiment runs is 1,237.5, so

Ft = 1,237.5

The /lvalue for all other significant factors is found by dividing the corresponding effect value in half. That means that

Pi = T = ^ = 5.75 p 23 = ^ = 140 = 7.0

Why are the {i Coefficients half the effect value instead of the full effect value? It’s because the effect value is calculated over a span of +1 to -1 for the variable. That’s an effective distance of two, not one. Therefore, to get back to the right equation coefficient, you have to divide the calculated effect value by two.

With these coefficients calculated, you can write the Y = ice cream carton filler system:

F(X) Equation for the

Y = 1,237.5 + 5.75 X + 7.OX2X3

Armed with this equation, you can now go out to the ice cream production line and immediately correct the problem situation of this example.

4?

4E

Suppose that the weight of the filled ice cream cartons is required to be between 1,225 and 1,280 grams. If you are producing a batch of vanilla ice cream, you can plug that coded value into the equation (X = -1), and then

Plug in various coded values for X2 and X3 to calculate what your fill time and pressure settings should be on the ice cream filler machine. When you switch

Over to making strawberry ice cream, you can then pull out your equation again and know exactly how to alter your fill time and pressure settings to

Maintain the correct filled carton weight.

Be careful to plug only Coded Values into your derived Y = F(X) Equation.

\[ou’Ve Ontu lust Begun — More Topics in Experimentation

2k Full factorial experiments give you a powerful jump start into the world of improvement through DOE. But really, they are just the tip of the iceberg. As

You gain experience, you want to discover how to address more advanced topics.

Curvature: The assumption of 2* experiments is that the effects of your experimental factors is linear. Although this is often a good first approximation, there are many times when a line doesn’t fit your process or

System. For those cases, you need to design your experiment to reveal the curved nature of reality. This is usually done by including more than

Two levels for each of your experimental factors.

Replications: If you repeat your experiment, you get slightly different results. This shouldn’t surprise you. Variation, as always, is a part of everything — including your experiment. Repeating runs of your experiment (called Replications) Allows you to estimate how much of the

Observed variation in your process or system is explained by the

Derived y = F(X) Equation and how much remains unexplained.

Analysis of variance (ANOVA): Almost all experiments involve exploring, investigating, and comparing the sources of observed variations.

ANOVA is an advanced method that allows you to categorize and quantify all the various sources of variation.

F Robustness: The ability of a process or system to perform consistently in the face of variation is called Robustness. Taguchi and other experiment designs allow you to investigate and optimize your process or system so that it is as immune as possible to the ravages of variation.

Response surface methods (RSM) and optimization: The purpose of

Many experiments is to find out the best values to set the input variables at. A whole branch of the field of DOE focuses on designing and

Analyzing experiments to find the local or global optimal operation settings.

Fractional factorial experiments: 2* full factorial experiments can be adapted to more efficiently search through a large number of experimental factors. What you give up in increasing the number of experimental factors is analysis accuracy. Fractional factorial experiments teach how and where to adapt your experiment to get the most out of your search efforts.

Chapter 10

In This Chapter

► Seeing the big picture first

► Aligning Six Sigma with your needs

► Defining projects

► Realizing the benefits

The essence of Six Sigma is to solve problems that are impacting business or personal performance. But before you can solve a problem or improve performance, you have to properly define your goal or objective — that is, you have to define the focus of your Six Sigma project. In fact, defining a project is 50 percent of the improvement game, and finding the right problems is critical to the success of your organization.

The Define stage of the breakthrough strategy (DMAIC) assumes you’ve identified a certain number of problems to be solved, and these problems are then

Converted into Six Sigma projects. A key challenge for all Six Sigma practitioners and management alike is to find these problems in a strategic way that assures maximum benefit from the application of the Six Sigma methodology.

Defining projects is about recognizing problematic areas of the business and

Subsequently creating a clear direction for resolving these problematic areas. It’s akin to the question, "How do you eat an elephant?" The answer? One bite at a time. Problematic areas of the business (like warranty returns, accounts receivable, product yield, and customer satisfaction issues) are the elephant-sized issues of the business. More likely than not, each of these problematic

Areas requires that you engage in more than one Six Sigma project, thereby eating the elephant one bite at a time.

The Sir Sigma Project

Six Sigma progress is obtained the old fashioned way — one project at a time. And progress is not necessarily serial, but often in parallel, as many Black Belts, Green Belts, and Yellow Belts apply the breakthrough strategy throughout an organization (see Chapter 3 for more on Belts). In essence, projects are the unit of measurement, the physical entity, by which most Six Sigma progress is accomplished. Projects represent — and in fact are — the level of granularity needed to manage a single process improvement or a large-scale

Business improvement effort.

The basics of a project

A Six Sigma project starts out as a practical problem that is adversely impacting the business, and ultimately ends up as a practical solution that improves

Business performance. Projects state performance problems in quantifiable terms that define expectations related to desired levels of performance and

Timing, as described in Figure 4-1. A project:

Has a financial impact to EBIT (Earnings Before Income Tax) or NPBIT (Net Profit Before Income Tax) or a significant strategic value

F Produces results that significantly exceed the amount of effort required

To obtain the improvement

F Solves a problem that is not easily or quickly solvable using traditional

Methods

V Improves performance by greater than 70 percent over existing performance levels

The focus of a project is to solve a business problem that is hurting key business performance elements, such as:

Practical Problem

Z

Six Sigma Project

Z

Statistical

Problem

Generally a systemic or chronic problem that is impacting the success of a process or function.

A well defined effort that states the problem in quantifiable terms with known expectations.

Data-oriented problem that is addressed with facts and data

Analysis methods.

Statistical

Solution

Data-driven solution with known confidence/risk levels versus an "I think – solution.

Figure 4-1:

Project

Life cycle.

Control

Plan

Z

Practical Solution

Results

A method of assuring the long-term sustainability of the fix to the problem.

The solution is not complex, expensive, or irrational and is

Readily implement-able.

Tangible results measurable in metrics with quantifiable financial or strategic value.

The problem transformation

When a particular problem is selected to become a potential Six Sigma project, it goes through a critical metamorphosis — first from a practical business problem into a statistical problem, then into a statistical solution, and, finally, into a practical solution. When you state your problem in statistical language, you ensure that you will use data, and only data, to solve it. This forces you to abandon gut feelings, intuition, and best guesses as ways to address your

Problems.

Almost any problem is solvable if you throw enough time and money at it. But this does not qualify as a practical solution, and it’s not the goal of a Six Sigma project. A practical solution is one that is not complex, not difficult to implement, and does not require extensive resources to affect the improvement.

Project responsibilities

There is a management framework and set of responsibilities inherent in the Six Sigma methodology that entails finding problems, defining projects, determining solutions, and implementing improvements. A Six Sigma project goes through an ownership transfer as demonstrated in Figure 4-2.

Project Responsiblities

Figure 4-2:

Project responsibility and ownership.

Define

Measure

Z

Analyze

Improve

Control

Management

"Belt"

"Belt"

"Belt"

Process Owner

In This Chapter

^ Differentiating between the different types of data — attribute and continuous ^ Understanding measurement system capability

Separating the critical few performance influencers from the trivial many ^ Using observational studies as a tunneling tool

N Chapters 4, 5, and 6, many different ways are introduced to identify and

4 Discover all the possible variables — the *s — influencing an important outcome, process, or characteristic — the Y. This chapter covers how to

Start to whittle this large list of potential influencers to a handful of variables

On which to focus your improvement efforts.

Just like an experienced whittler selects a different knife for carving different types of wood, you need to choose among many different Six Sigma tools to narrow your field of suspect variables. What tool you choose depends a lot on what type of data you are using. So it’s important for you to assess what

Type of data you have.

An area of potential influence that is often overlooked in Six Sigma projects is your measurement system itself. In Six Sigma, data forms the foundation of your knowledge and decisions. Imagine what would happen if all your analysis and decisions were based on faulty data. You need to immediately eliminate

The chance that the measurements you are using are creating an illusion.

Reducing a large collection of potential factors down to a smaller area of

Focus allows you to concentrate your limited resources on the items that really will have an impact on improvement. In this whole process, you want

To have your choices guided by the data, rather than opinion or guesses.

Understanding Data Types

All data are not created equal. Before you can do much else, you first need to

Know what type of performance data you have. Just as knowing what the fish

Are biting tells you which lure to use, knowing what kind of data you are dealing with tells you which tools to use.

Attribute or category data

Some data consist of measurements that describe an attribute of the characteristic or process. When these attributes are named categories, the data are called Attribute Or Category Data.

Attribute data are all around you.

Telephone area codes are attribute data.

V S, M, L, XL, XXL clothing sizes are attribute data.

F "Pass" or "fail" judgments pronounced on just-assembled products are attribute data.

F "Good" or "bad" assessments of the output from a process are attribute

Data.

When studying the performance of automobiles, manufacturer names

Like Ford, Chevy, and GM are attribute data.

How do you know whether you are working with attribute data? The telltale test is to ask yourself: Can I meaningfully add or subtract values of this data?

If the answer is "no," what you have is attribute data. For example, what do

You get when you add a S-sized shirt to a M-sized shirt? There is no meaningful answer to this question. Or, if you subtract telephone area code 415 from area code 213 does the resulting area code of 202 mean something? Of course not! And so you know that you’re dealing with attribute data.

What you can do with attribute data is count how many times each category or attribute appears. For example, you may find that a process produces 152 "good" items and 28 "bad" items over a given period of time. You use the results of these type of counting studies as the starting point for many of the analyses discussed in Chapters 8, 9, and 10.

Lord Kelvin said, "When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot

Express it in numbers, your knowledge is of a meager and unsatisfactory kind." Measuring with attribute data is a first step in getting to a satisfactory knowledge of what you are trying to improve. Even though there are limitations to what you can do with it, attribute data is a giant first step into the world of Six

Sigma improvement.

A subset category of attribute data that provides a little more horsepower is called Ordinal Or Rank order Data. Ordinal data is attribute data that can be logically placed in an order from smallest to greatest or in an order of time.

Like the months of the year: January, February, March, and so on. If you have "month" data on a set of last year’s invoices, you can sort them into buckets of occurrence starting with January, and then move on through the year. Or

You may not have actual completion times, but you may have data about

Which employee finished a task first, second, third, and so on. In this case, you have a powerful set of ordinal data that you can use to begin analysis andimprovement.

Continuous or Variable data

Keep in mind the question you asked yourself to find out whether the data you have is attribute data: Can I meaningfully add or subtract values of this data?

If the answer to this question is "yes," the data you are working with is called Continuous Or Variable Data.

Stf^tySt. Both Continuous And Variable Are bad names for this type of data, but for historical reasons, these are the names that have stuck. The name continuous is Pjifw ) meant to convey the idea that this data type can have any value from acon -

\jL/ Tinuous scale, like the reading on a mercury thermometer. Variable is meant to say the same thing — that the measured values can vary anywhere along a given scale. You can get 98.23T or 98.25T or 98.37°F. The problem is that no

Matter how continuous or variable you think your scale of measure is, as soon as you write down a measurement on paper or record it in a computer you always truncate its reading to some fixed length, making it no longer continuous. The important thing to remember is that when you have data that can be meaningfully added or subtracted, you have what is called continuous or variable data. For example, a count of the number of children in each household can only occur in integer values. (You can’t physically have 2.3 children. The scale of measure of children in a household is not continuous at all.) But

You can take the integer measurement from each household and perform mathematical operations to calculate a meaningful average or standard deviation. Being able to mathematically operate on continuous data is what sets it apart from attribute data. (See Table 7-1 for a summary of data types.)

Table 7-1 Summary of Data Types

Data Type Description_Examples

Attribute or Data observations fall Eye color: brown, blue, green

Category into discrete, named Location: Factory 1, Factory 2,

Value categories. Factory 3

No mathematical Inspection result: pass, fail

Operations can be Size: large, medium, small

Performed on the raw Fit check: go, no-go

Data. Questionnaire response: yes, no

You can count the number Attendance: present, absent

Of occurrences you see Employee: Fred, Suzanne, Holly

Of each category. Processing: Treatment A, Treatment B

Continuous Data observations can Bank account balance: dollars

Take on numerical value Length: meters

And are not confined to Time: seconds

Nominal categories. Electrical current: amps

Data values can be Survey response: 1 = disagree,

Meaningfully added and 2 = neutral, 3 = agree

Subtracted.

Examples of continuous data include:

V A numbered GPA scale representing letter grades at school

The temperature in your oven f The amount of money you spend on groceries Is The time it takes to complete a process task

The gas mileage of your car

Avoiding Illusion: Measurement System Capability Analysis

Measurement is critical. It is the foundation of knowledge and subsequent

Improvement. It is the way you verify that you have the right answer, have

Corrected the problem, or have improved the situation.

Measurement is unavoidable. That’s because nothing is observed outside of the filter of some kind of measurement system — your eyes, your brain, your perception; a physical ruler, a stopwatch; or a laser interferometer. Everything comes to you through some kind of measurement system lens. You need to

Know whether your measurement system is giving you a clear picture of reality.

Measurement itself is a process you can never avoid. The act of measuring is a process all by itself. And as with any other process, it has variation within it. Think back to the last time you watched an Olympic ice skating competition. In that situation, expert judges act as the measurement system. Imagine showing a series of judges a recording of an ice skater’s routine. It’s likely

That each judge would give the routine different marks. This is evidence of variation in the measurement system. If you came back a year later with the

Same recording to the same series of judges, you’d almost certainly get a different score from each judge. In fact, the variability in this Olympic example is parallel to how it is with every measurement system.

Everything you see is through an imperfect lens of measurement. Any time

You place something into a category or quantify one of its attributes, you are doing so through an imperfect measurement system. Figure 7-1 illustrates this concept.

Figure 7-1:

Everything you observe has the added variation

Ofyour

Measurement system.

Observed Variation

Measurement System Variation

Actual Variation

Six Sigma teaches that data and measurements are the starting point of knowledge and improvement. Before you get too far down the improvement

Road, however, you need to determine whether your measurement system is

Clouding your observations to the point of illusion — whether what you see

Is not what really is.

2

Sources of measurement system Variation

There are several aspects of a measurement system that affect how much clouding variation it contributes to your observations.

Measurement resolution

Resolution Is a comparison of the smallest increment your measurement

System can provide to the characteristic you are trying to measure. For example, imagine measuring a grain of sand with a tape measure. You’d be kidding yourself if you treated the results of this measurement system seriously. The

X6th-inch increments on the tape measure are not fine enough to discern the much, much smaller grain of sand. What you need instead is a system, like maybe a microscope, that can measure in increments of 1/1,000th of an inch. Then you could trust your measurements of the sand grain sizes.

A good rule of thumb is to use a measurement system with at least ten increments within the specification width you are measuring or within the process variation you are trying to observe. So, as an example, for measuring

The variation of a process that should be completed between 9 and 10 minutes, you want to use a measurement system with increments of no more than 0.1 minutes.

The idea of resolution also applies when you are measuring attribute data. A

Customer survey that allows only responses of "satisfied" and "dissatisfied" offers less resolution than one where the customer can mark "delighted,"

"satisfied," "indifferent," "dissatisfied," and "disgusted."

Measurement accurac§

Accuracy describes how centered your measurement system’s variation is with the actual variation of the process or characteristic. Figure 7-2 shows this visually. In (a), the dots representing the measurement system variability

Are centered on the target that represents the actual variation of the process or characteristic being measured. The measurement system depicted in (a) is accurate. In (b), the measurement system variation is offset, or biased, from the center of the actual process or characteristic variation. The (b) measurement system is not accurate.

Figure 7-2:

How

Centered isyour

Measurement system’s variation?

(a) Accurate

(b) Not accurate

Several conditions can cause accuracy problems in your measurement system. Sometimes a measurement system can have problems with Linearity. Good linearity is when the centering and the magnitude of the measurement system variation is consistent across its range of operation. A measurement system has poor linearity when its centering or magnitude of variation changes across its range of variation. A Stable Measurement system is one that stays centered

And free of offset changes. In an instable measurement system, the location of its variation center bounces around.

Measurement precision

Accuracy and precision are two distinct properties of a measurement system.

Accuracy describes how centered your measurement system is compared to the actual variation. Precision Describes how widely spread the variation of

Your measurement system is compared to the actual variation of the process

Or characteristic you are measuring. Figure 7-3 illustrates the idea of measurement precision. In (a), the dots representing the measurement system

Variability are clustered tightly together compared to the target representing

The actual variation of the process or characteristic being measured. The measurement system depicted in (a) is precise even though it is not accurate in its location. In (b), the measurement system variation varies widely compared to the actual process or characteristic variation. Although the (b) measurement system is accurate in its location, it is not precise.

Figure 7-3:

Measurement system variation.

(a) Precise

(b) Not precise

Measurement system precision is made up of two components that you hear talked about a lot in Six Sigma: repeatability and reproducibility.

Is Repeatability Is the part of measurement variation that occurs when yourepeat measurements with the same item, the same measurement setup, and the same equipment, under the identical conditions. In a way, repeatability can be thought of as the short-term part of measurement system variation.

Is Reproducibility Is the part of measurement variation that occurs when

You repeat measurements with different items and different measurement

Setups, under different environmental conditions. Reproducibility captures all the long-term variation influences in your measurement system.

Together, the metrics of repeatability and reproducibility capture all of your measurement system’s precision. In Six Sigma there’s an acronym (of course!) for repeatability and reproducibility — R&R. You see this acronym used to describe how precise a measurement gauge is, as in its gauge R&R. (When you hear "R&R" don’t think of "rest and relaxation" or of a railroad line. Instead, think of how good your measurement system is.)

Measuring measurements: Measurement system analysis (MSA)

So how do you measure the goodness of your measurement system? There

Are several ways for you to do this, depending on the type of measurement data you are collecting and the type of measurement system you are using. Typical measurement system analyses (MSA) include audits, attribute studies, and gage or continuous variable studies.

Audit measurement system studies

An Audit Is a measurement system study where you compare your measurements to a known, correct standard. For example, you may compare what your computer system says you have in inventory (your measurement system) to what is actually in your physical inventory warehouse. Any differences

Between the two reflect variation in your measurement system.

For example, read the following paragraph as quickly as you can. As you’re

Reading, circle each occurrence of the sixth letter of the alphabet — either

Lower or upper case. Do not go back or reread parts of the paragraph.

The necessity of training Farm Hands for first class farms in the fatherly handling of farm livestock is foremost in the minds of Farm Owners. Since

The forefathers of the farm owners trained the Farm Hands for first class

Farms in the fatherly handling of farm livestock, the Farm Owners feel they should carry on with the family tradition of training Farm Hands of first class farms in the fatherly handling of farm livestock because they believe it is the basis of good fundamental farm management.

Now, count how many fs you found in the paragraph. How many did you find? Going through this paragraph slowly and carefully, you will find exactly 36 Fs.

Reading through the paragraph quickly, circling fs, is a type of measurement system. (It is a lot like the inspections that are placed in a process to verify the quality of what is being produced.) How good was your measurement system? Of the 36 Fs, What percentage did you find? This analysis of how well you did is an audit of your measurement system.

A visual inspection system

A computer disk drive manufacturer in the mid 1980s was experiencing a nagging problem with poor yields. The principle concern was that the sensitive magnetic media coating the disks was in some way defective. As a result, a collection of very demanding standards was put in place and a battery of stringent tests was implemented with the hope of detecting and removing media problems from the system.

At one point, design engineers at the company

Happened to notice some visual defects and spots in the magnetic coating on the disk. They

Concluded that this was the long sought-after

Cause of their persistent yield problems.

Engineering immediately requested that Manufacturing implement a final visual inspection of each disk to be done at the end the

Already onerous test cycle. With the implementation of this new inspection, the disk reject rate jumped from 8 to 10 percent. At $30 a disk, the scrap bill neared $300,000 per month!

With no real improvement evident, Engineering proposed to further tighten the specifications on the magnetic disk media. With mounting assembly and scrap costs, Manufacturing asked that an expert from Engineering audit their test and inspection process one last time before tightening the specs again.

Upon arrival at the manufacturing facility the Engineering expert reviewed the entire test and

Inspection process. He then decided to run some experiments to validate the final visual

Inspection process on the disks. His first experiment was to send a bunch of previously

Rejected disks back through the final inspection process without the inspectors’ knowledge. The

Results were so startling he reran the experiment several times: Each time the previously

Inspected disks were secretly sent back through the final visual inspection process, an

Additional 10 percent of the disks would be

Rejected!

Armed with this new insight, the engineer tried another test. This time he took a bunch of disks

That had already passed the final inspection

Step and secretly reinserted them back into the inspection process. Even with these "passed" disks, the inspectors continued to find ten percent of the previously passed disks to be visually defective.

As a final confirmation, the engineer sent a collection of passed disks and a collection of failed

Disks into the final stages of the assembly

Process. At the end of the assembly process, the disk drives with the rejected media actually

Had a slightly higher final performance yield

Than those with disks passing the visual inspection.

Clearly, this company was living in a measurement system illusion. The visual inspection

System they had added provided no benefit to the company but was costing over $300,000 per month in incorrectly rejected disk media.

Industrial engineers have found human screening or inspection measurement

Systems to be consistently about 80 percent effective. Yet most people act

Under an illusion that the outcome of a screening operation is 100 percent

Correct. Performing a simple data audit tells you how effective your measurement system is.

What can you do to improve the effectiveness of a screening or inspection measurement system?

Divide a large screening task into smaller pieces and assign it among several individuals.

Is Clarify inspection criteria with pictures, examples, and so on.

F Use successive inspectors to incrementally increase the effectiveness of

The inspection.

F Incorporate technology or automation to remove human error.

Pareto Diagrams are a great diagnostic tool for detecting problems with a measurement system. Vilfredo Pareto (1848-1923) was an Italian economist who proposed that 80 percent of an economy’s wealth is held by 20 percent of its

Population. Since Pareto proposed his famous principle, other researchers

Have confirmed that it also applies to many other phenomena, including the

Distribution of measured defects. For example, it has been found that 80 percent of the observed defects on a product or in a process can be attributed to 20 percent of the possible causes. What this means is that when you create a

Bar chart of the observed number of each type of defect on an item, and then sort the order of the bars from most frequently observed defects to the least frequently observed, only the first few defect categories should have a significant contribution to the overall defect count. So if your measurement system divides things up into multiple defect categories and a Pareto diagram shows approximately equal contribution from each category, you should suspect that something is wrong with your measurement system. Instead, a healthy measurement system should show that only a few defect categories make up

The bulk of the observed defects. (Another name for the Pareto Principle Is the 80-20 rule.)

Attribute measurement system studies

When you are measuring attribute data — like pass-fail measurements of an invoice process or categorizing of failed products by failure type — it is

Important to determine whether your ability to put items into correct categories is consistent and reliable. The risk of a poor attribute measurement system is two-fold: You may falsely accept bad items or you may falsely reject good items. Either way, the risk is that you’ll make a decision that is not

Consistent with reality.

Consider a measurement system that categorizes items — whether it be a characteristic or a process — into categories of "pass" and "fail." How would you begin to study the effectiveness of this type of measurement system?

Here’s the answer:

1. Start by setting aside 15 to 30 samples of what is being measured.

You want these samples to represent the full range of variation that is

Typically encountered, with about half of the samples being "passes" and

The other half "fails."

2. Create a "master" standard by designating each of the samples as a "pass" or a "fail."

Use a panel of experts to make these distinctions or some standard that

You know is absolutely correct.

3. Pick two or three inspectors.

Have them review the sample items in a random order and record their conclusions — whether each item is either a "pass" or a "fail."

4. Have each inspector repeat their measurements of the samples after mixing them up into a new random order (or spin the inspectors around in place until they are very dizzy). Record the repeated measurements.

Note: It is important that each inspector’s second measurements are fair, as if they were happening for the first time. You may need to wait for a day before performing the second measurements. Randomizing the samples before the second measurements is critical.

5. For each inspector, calculate the percentage of their first and second measurements that agreed with each other. This is the Repeatability

For each inspector.

You can also calculate an overall measurement system repeatability by averaging the repeatabilities of the individual inspectors.

Note: The calculated repeatability for the individual inspectors needs to be as close to 100 percent as possible. Calculated individual repeatabilities

Less than this mean that the inspector is not consistent in distinguishing between good and bad items. Training helps inconsistent inspectors become consistent in their measurements.

6. For each of the sample items, calculate the percentage of the recorded

Measurements where each of the inspectors agreed with themselves

And all inspectors agreed with each other. This is the Reproducibility For the measurement system.

The calculated measurement system reproducibility tells you how precise the measurement system is over the long-term — over different inspectors, different set-ups, and different environmental conditions.

7. You can also calculate the percent of the time individual inspectors and the group of inspectors agree with themselves And Agree with the

"master" standard created in Step 2.

This tells you how consistently your measurement system detects what your experts have decided really is pass and fail.

As an example, a calculated 63 percent agreement between all inspectors

For all samples with the "master" standard in a measurement system

Study means that there is a 63 percent likelihood that this measurement system will correctly measuring the items and a 37 percent chance of

Error. Clearly, the goal is to achieve a measurement system with as high

An effectiveness as possible, that is as close to 100 percent agreement as

Possible.

StfA-Styfc More sophisticated analysis tools are available for situations when an 3^Јjf\ attribute measurement system has more than two categories. These tools, * ffl W ) like kappa analysis, can be found in advanced statistical analysis software,

Y^jj^/ like Minitab and JMP.

Gauge or continuous Variable measurement system studies

When you are measuring continuous or variable data, there are more tools and analyses at your disposal. In all cases, measurement system studies for

Continuous data mathematically compare the total observed variation to the

Portion of the variation stemming from the measurement system itself.

Recalling Figure 7-1, the total observed variation is made up of two parts, the variation of the actual items or process being measured and the variation imparted by the measurement system itself. This make-up of the overall variation is summarized by the following equation of variances (see Chapter 5 for

A definition of variance):

O Observed = O Measure + O Actual

In an effective measurement system, the contribution of the measurement system itself will be small compared to the overall observed variation. Table

7-2 provides a summary of measurement-to-observed variance ratio scores

And what to do for each situation.

Table 7-2 Measurement-to-Observed Variation

Ratio Values and Interpretation

Calculated Variance Ratio

Diagnosis Prescription

O o

< 0.1

Good measurement system. Use measurement Contribution of the measure- system as it is. Look for ment system to the overall opportunities to simplify observed variation is small or make the measure-enough to enable good ment system less decisions from the expensive or more

Measurements. efficient.

Calculated Variance Diagnosis Prescription

Ratio_

0.1 < °H»"E < 0.3 Marginal measurement Use with caution only if

J Observed System. Contribution of the no better measurement

Measurement system to the alternative exists.

Overall observed variation is Begin to improve the

Beginning to cloud results. measurement system

There is a significant risk of by training operators,

Making a wrong decision standardizing measure -

From the measurements. ment procedures, and

Investigating new mea -

_Surement equipment.

°H»"e > 0.3 Unacceptable measurement Measurement system

J Obsebved System. Guessing is probably needs to be corrected

Just as precise. Do not base before any valid infor -

Important decisions off of mation can be derived

Information from a measure-from the system.

Ment system in this condition. Investigate causes of

Gross inconsistency.

Mathematically comparing measurement system variation to the overall observed variation is not difficult. The difficult part, rather, is obtaining a good estimate of the measurement system variation from which to make the comparison.

Valid estimates of the variance of the measurement system usually involve

Two to three inspectors and five to ten process outputs or characteristics to

Measure. Each inspector also measures each process output or characteristic

Two to three times. From this, advanced statistical analysis tools, like Minitab

Or JMP, can be used to automatically perform the gauge analysis calculations

And you can begin to diagnosis and improve, if required, your measurement system.

Fitting the Funnel

To get a concentrated stream out of the bottom of a funnel, you first must fill the top abundantly. Six Sigma is no different. You start by dispassionately carrying all possible causes into your project. But as you progress, you let your

Analyses of the data itself tell you which variables to keep along for the ride

And which ones you can safely cast aside as excess baggage.

Let the data do the talking

One of the hallmarks of maturity in Six Sigma is an unwavering reliance on data. Data is used to understand what happened in the past. Data is used to decipher and improve the current situation. And data is the basis for predicting how things will perform in the future.

In Six Sigma, data trumps the usual fare of opinion, speculation, guesswork, and politics. Data-driven decision-making is the culture of Six Sigma. People say, "In God we trust; all others must have data."

In a pure application of Six Sigma, you simply "let the data do the talking."

You almost withhold judgment regarding what it is that is wrong or what the

Solution will be and instead quietly listen to what the data is telling you about the situation and what should be done. This new way of operating stems from an acquired confidence in the science and power of Six Sigma — that gathering and querying data from a process more efficiently reveals the real,

Unbiased truth of its performance as well as the most effective and lasting improvement solution.

Cast a big net

A corollary of "let the data do the talking" is exposing yourself to the voice of the data in many different ways. In Chapter 4, you discover tools like processing mapping, fishbone diagrams, X-Y Matrices, and failure mode effects analysis (FMEA). These are very powerful tools for querying a process to discover what potential factors may be contributing to its performance. In the

Remainder of this chapter, you discover graphical tools that you can use to mine data forevidence of factors influencing process or characteristic performance. In Chapter 8, you find out how to employ statistical hypothesis tests; Chapter 9 discusses designed experiments. Figure 7-4 shows how all these tools are used progressively to identify — and then narrow — the field of potential input xs in the equation Y = f(X).

To be successful, it is important to cast a big net, to start your improvement

Effort by capturing as many potential xs as possible. Then, you allow the Six Sigma tools — not your pre-judgment or your opinion — to naturally weed

Out the XS that are not critical and retain those that in fact are. That is one of

The beauties of Six Sigma: Its formulaic application guides you to the solution

Of your improvement task.

All fitissihfn iafwl Xx

Figure 7-4:

Tools for identifying all potential Xs and

Funneling

Them down to a set of critical Xs.

Mining Data for Insight

Data mining Is just what its name implies — it is the labor of digging and sorting through data for clues to where the improvement gems may lie. Sometimes you have to go through a lot of dirt to find the gems. Searching for clues in data is not much different.

Go With ©hat you have: Observational studies

Where do you begin your search for improvement gems? And what are the tools of the trade? Six Sigma practitioners have refined the data mining process

To an efficient, powerful set of tools.

Data, data everywhere

A world of potential data exists all around you:

You fill the gasoline tank of your car up to a different amount two or three times a month.

(•* The number of reams of paper your company uses in its copy center varies from day to day.

There are different numbers of students in each classroom.

Different people work on a single process step depending on their daily assignment.

F The feed rate of a milling machine is adjusted depending on the task.

The list goes on and on.

One way to immediately tap into this cache of information is to simply begin

To observe all the potential input and output variables in your improvement project and record them.

Record the data surrounding your project in tabular form — with a column for each X Or YVariable and a new row for each point of observation, as shown in Table 7-3.

Table 7-3 Observational Study Data Recording Template

Obs. No. Dept. (X,) Hour (X2) System (X3) Processor (XJ Items/

Hour (Y)

1 B 8 Web Sally 43

2 A 5 Web Sally 37

3B4WebBob44

4 B 8 Desktop Sally 35

5B4WebSally42

6 A 5 Web Sally 39

7 B 3 Mainframe Sally 41

8 A 8 Mainframe Joan 36

9 A 1 Web Sally 39

10B4MainframeJoan40

The curious mind: Observational studies

Thinking about, pondering over, and probing your recorded observations is

A proven path to increased understanding. In Six Sigma, these activities are called Observational studies. Observational studies revolve around analyzing

The variation in the observed critical output or outputs and investigating which input variables it is linked to. What you are looking for are potential sources of the variation.

Observational studies are different from planned experiments. In an observational study, you simply investigate the variation and data as it happens naturally — whatever the values may be. In an experiment, however, you

Actively control the variable values to see what the output will do under certain input condition. Experiments provide greater insight and resolution than observational studies do. (You find out about the design and execution of

Experiments in Chapter 9.) But sometimes, it is not possible or ethical to perform a more powerful experiment. For example, it wouldn’t be right to purposely overcrowd a kindergarten classroom with 75 students to see what

The effect on learning would be. Instead, education researchers gather naturally occurring data on classroom size, and then perform observational studies.

Usually the results of your observational study are a list of likely suspects.

This narrowed list of variables is then investigated further for confirmation and for conclusive evidence using the techniques covered in Chapters 8 and

9. Sometimes, however, your observational study immediately reveals the

Real set of culprits. So always be on the lookout.

Digging in: Identifying potential sources of Variation through graphical analysis

To study whether an observed input has an effect on an observed output, you

Create a set of box and whisker plots of the critical output — with each box and whisker plot corresponding to a different condition of the input variable (See Chapter 5 for an explanation of box and whisker plots.) Several computer programs — including Minitab, JPM, or Excel — automatically create these plots (see Chapter 11).

Looking at an example

For example, Table 7-3 is a partial list of the data collected for a transactional process. The key output (Y) is how many items per hour are produced. The

"big net" of possible input variables includes the department performing the

Transaction (X1), the hour of the day in which the transaction was processed (X2), the processing system used (X3), and the actual person performing the transaction (X4). In this example, over 200 historical observations were

Collected.

What effect does the processor (X4) have on the items per hour output (Y)? Figure 7-5 shows a set of box and whisker plots of Y For each condition of the

X4 Input.

Again, does either Bob, Joan, or Sally have much influence on the items transacted per hour? From the graphical view in Figure 7-5, it is clear that the items transacted per hour is about the same for each operator — they have about

The same average level and about the same amount of variation. This tells you

That the processor variable (X4) is not a key contributor to the output variation.

Statisticians using advanced techniques look at the data for Figure 7-5 and numerically compute the variation between the centers of variation for each of the different x4 conditions and call this the Between group variation. They then perform a similar calculation to quantify the average width of variation of all the conditions and call this the Within group variation. If the between group variation is large compared to the within group variation, they conclude that the investigated variable does indeed influence the output. The

Graphical method outlined previously is just a simple, intuitive way to accomplish the same thing while bypassing all math and technicalities.

Returning to the example, what about the department performing the transaction (x3)? Does it contribute to the output? Figure 7-6 is another box and

Whisker plot of the output versus the department doing the transaction.

Graphically, you can quickly see that the difference between the centers of variation from department A and department B is significant compared to

Theaverage width of variation within the departments. This tells you that whichever department performs the transaction does have some influence on the output. This variable will pass through your funnel and be investigated further for conclusive evidence.

Another way to perform observational studies is through correlation calculations (covered in Chapter 8). These give you the same insight, but are not

Graphical, so they’re harder to use and interpret.

Figure 7-6:

The effect of department on the items per hour output.

Considering additional studies

There are many other tools at your disposal when performing observational

Studies, including:

F Multi-variable studies: Multi-vari Studies, as the name is often shortened, allow you to investigate the effect of several input variables at a

Time on a critical output.

Is Main effects plots: Main effects plots Are introduced as a basic graphical technique in Chapter 5. They are an extremely easy and powerful way to

Explore the principle effect of a variable and its different levels on a critical output.

Interaction effects plots: Sometimes, one variable by itself doesn’t have

A major impact on an output. But when you combine it with other variables, it has a significant influence. This is called an Interaction effect. For

Example, adding eggs by themselves to a cake batter doesn’t immediately impact the cake’s texture. But adding eggs Together With oven heat to the batter produces a yummy dessert.

Each of these additional observational studies is available in most off-the-shelf

Six Sigma software packages (see Chapter 11). This makes it much easier today to perform these analyses automatically, giving you a big advantage

Over your predecessors.

In This Chapter

► Understanding specifications and how they relate to defects and errors ^ Knowing how to calculate and interpret measures of yield and defect rate

^ Calculating and interpreting the sigma score (z) of a process or characteristic

► Calculating and interpreting short – and long-term capability indices (CP, CPK, PP, And PPK)

Hen you distill your measurements of a process or characteristic into

WW Statistical metrics, like a mean X And a standard deviation cr, you describe its properties. The process cycles on and on, and the characteristic is created over and over again. But each time a new instance happens, it’s slightly different than the one before. This variation affects the performance

Of the larger system and, ultimately, impacts the customer.

This chapter is about two voices — the voice of the process and the voice of the customer — and the effect each has on the other. In Six Sigma, this relationship is called Capability. Capability is how well the voice of your process or characteristic matches up with the voice of your customer, or how well

Your process performs in meeting customer expectations.

In the DMAIC strategy, you use capability calculations to quantify and communicate the performance of characteristics and processes relative to their

Requirements. These capability metrics allow you to know where to focus your attention and to verify that real improvement has been made.

Specifications: The Voice of the Customer

When you buy a Coca-Cola, you, as a customer, expect a certain experience. You expect it to be the same each time you open a new can. If there were too

Much sugar one time, or not enough secret ingredient another time, you’d

Notice and feel dissatisfied. The Coca-Cola company knows this about you, its customer, and so it very carefully controls the amounts and makeup of each

Ingredient going into its drinks.

The way Coca-Cola controls its product is through Specifications. Each specification represents what the customer requires in order to be satisfied.

0o© close is close enough? Or u’luj specifications?

Before the 1800s, all products were manufactured one at a time by craftsmen. A gunsmith, for example, would shape a single barrel of a gun, and then expertly

Carve a single wooden stock to match the barrel’s dimensions. The pieces fit together because the craftsman adjusted each part to match the other.

A revolution was ignited when specialists were engaged to separately create

Each of the components of a product. Because the specialists each focused

On a smaller area, they became more expert and efficient in producing that piece. The overall result was an economy that produced goods and services much more quickly and at a much lower cost than before.

It was in this environment of economic revolution that specifications were

Born. In order for Billy Bob’s barrels to fit into Cletus’s wooden stocks, there

Had to be some formal coordination. Specifications told the specialists what

Size or shape to make their parts. That way, when all the separate parts were assembled together, they would still fit.

What are specifications?

Specifications are performance values beyond which the performance of a process or characteristic is considered unacceptable. Spending more than

$200 a month on movie tickets is probably considered unacceptable for your personal budget. But what about $100? Or $50? At what dollar value is movie ticket spending acceptable or unacceptable? A Specification Is that value separating acceptable from unacceptable performance. This definition holds for all process or characteristic performance measures.

There are several different types of specifications (see Figure 6-1):

V Specification limit (SL): Any value designating acceptable from unacceptable performance.

One-sided specification: A specification limit that designates only a single transition point from acceptable to unacceptable performance. For example, if you care only that the characteristic or process performance not exceed a certain upper value, that is a one-sided specification.

Two-sided specification: A pair of specification limits creating an interval of acceptable performance between the two limits.

Upper specification limit (USL): A value designating an upper limit

Above which the process or characteristic performance is unacceptable.

V* Lower specification limit (LSL): A value designating a lower limit below

Which the process or characteristic performance is unacceptable.

F Target (T): The single designated value you wish the process or characteristic to perform at. (A specification target is an Ideal. Variation prevents the process or characteristic from exactly hitting the target every time.)

Figure 6-1:

Target and specification limits for a characteristic or

Process.

LSL

Variation

"i-1-1-1-1-1-r

Characteristic Scale of Measure

USL

If set up correctly, specifications represent the range of values a characteristic can be at and still be acceptable to the customer. Often, customers are not

Directly involved in your work. You never directly interact with them or see them. But a characteristic’s specification is always available to you! In this way, specifications are said to represent the Voice of the customer (or VOC

Forshort).

Do §ou do the RUMBA? Creating

Realistic specifications

Specifications should never be arbitrary. Unless they actually represent the values that separate good from bad performance, they become a stumbling block to progress. If a specification is set too loosely, your customer will be dissatisfied or upset with the performance of what you provide — even though it meets its specification. If a specification is set too tightly, you spend more resources than you should to always perform within your overly narrow

Goalpost.

Imagine a specification requiring a delivered pizza to be between 120.4T and 120.6T when it arrives at the customer’s door. To be within this required temperature range, the pizza company would have to take some pretty complicated and expensive actions. Maybe they’d have to use space-age ceramic

Pizza boxes made out of space shuttle tiles. In any case, it would take a lot of

Work and expense to meet that specification.

But do customers really require this extent of control over the temperature of their pizzas? Probably not. A less troublesome specification, and one just as satisfactory to the customer, might be 115°F to 125°F.

A mind-jogging acronym, RUMBA, is used in Six Sigma to help you evaluate the appropriateness of any specification:

F REasonable: Is the specification based on a realistic assessment of the customer’s actual needs? Does the specification relate directly to the

Performance of the characteristic?

F UNderstandable: Is the specification clearly stated and defined so that

There can be no argument about its interpretation?

MEasurable: Can you measure the characteristic’s performance against the specification? If not, there will be a lot of debate between you and

Your customer as to whether the specification has been met or not.

F BElievable: Have you bought into the specification setting? Can you and

Your coworker peers strive to meet the specification?

ATtainable or achievable: Can the level and range of the specification be

Reached?

You need to review each specification to make sure that it passes the RUMBA test. If it falls short in any of the RUMBA categories, begin to develop a plan

To bring the rogue specification back into control.

Very often, an improvement project is fast-tracked or solved through a simple review and adjustment of the involved specifications. The performance of the characteristic or process never has to be changed. Always review the appropriateness of specifications early in your Six Sigma project.

Don’t push that big red button! What happens ©hen §ou exceed a specification

So what happens when a characteristic’s performance exceeds a specification? Does the system involved immediately stop working? Well, that depends. If

The amount of departure from the specification limits is significant, maybe the

System will indeed break. But what if the specification is exceeded only by a teeny bit?

Figure 6-2 shows four different points, each representing a different performance scenario.

LSL

Figure 6-2:

Four

Different performance

Scenarios.

(1) (2) (3) (4)

USL

-—i-1-

Characteristic Scale of Measure

Point 4 in Figure 6-2 is clearly the best; it’s closest to the ideal specified target

Value for the characteristic. Point 1 is well beyond the lower specification limit.

But what about points 2 and 3? Is Point 2 much more likely than Point 3 to create a defective condition? The answer is no. Points 2 and 3 are about equal in their likelihood of causing a problem. Why treat real-world observations like Point 2 differently from observations like Point 3?

A traditional view of quality has been:

Quality = compliance with specifications

This all-or-nothing perspective is flawed. Quality and the cost associated with poor quality rarely behave this digitally. Realistically, problems and associated

Costs increase more and more the farther and farther performance strays

From the specified ideal target. A better definition of quality is:

Quality = on-target performance with as little variation as possible

The target, not the limits, is the most important part of a specification. Getting a characteristic to operate on target with as little variation as possible should

Be the focus of improvement and cost reduction efforts.

Figure 6-3 graphically compares the flawed traditional and the Six Sigma perspectives on specifications and their relation to quality.

Figure 6-3:

The traditional and the Six Sigma views on specifications and the costs caused by poor quality.

Traditional perspective

_\___

Six Sigma

Perspective

USL /

I

"i-1-1-1-r

Characteristic Scale of Measure

The loss curve shown in Figure 6-3 is often called a Taguchi loss function, Due to the pioneering work of Dr. Genichi Taguchi in the areas of optimization and robust design. As the figure shows, the traditional perspective is a hit or miss view, where unnecessary cost is incurred only when specifications are missed. The Six Sigma view is one in which additional cost is incurred as performance moves away from the target.

Capability: Comparing the Voice of the Customer® to the Voice o{ the Process

Creating a specification is one thing. Meeting that specification through your processes and characteristics is another. A central task of Six Sigma is to

Understand how well your processes or characteristics meet their associated customer specifications.

Measuring yield

In the simplest terms, a process or characteristic can either meet or not meet its specification. Just as when you harvest the fruit from an apple tree, the Yield Of a characteristic or process relates to how much good stuff — performance within specifications — you get out.

Traditional yield: Output Versus input

Traditionally, yield is the proportion of correct items (conforming to specifications) you get out of a process compared to the number of raw items you put into it. Figure 6-4 illustrates the idea of traditional yield.

Figure 6-4:

The traditional view of yield and of output compared to input.

Items in

Process -*- Items out

Scrap

In This Chapter

► Sampling distributions and the central limit theorem Establishing confidence intervals for means, variances, and proportions

Understanding correlation and curves

►►Fitting curves

Everal chapters of this book show you how to identify the potential

W variables that influence a critical outcome. Process mapping, fishbone diagrams, brainstorming, SIPOC diagrams, and failure mode effects analysis are only some of the tools that help you identify Xs (inputs) that are influencing the critical Y Or Ys (output). You may even have a very healthy collection of

Potential XS for your improvement project.

Now it is time to begin to prune, to begin to weed out the identified input

Variables that are not worth carrying along. Your resources are limited, so

Setting aside insignificant or trivial variables that don’t allow you the greatest

Leverage helps you focus and get to the results of your improvement sooner.

In this chapter, you discover how to separate and quantify the critical few

Variables from the trivial many.

Six Sigma demands that you use data to make better decisions. So how is it that you go about using data to accomplish this? At the heart of the matter is the ability to distinguish between real and claimed differences. Are Chevy

Trucks really better than Fords? Are last year’s sales figures no different than

This year’s? Six Sigma gives you tools to quantify the real differences between

Factors and to investigate your confidence in these measurements.

Finding the Best Partner

ViUa Las Vegas: The central limit theorem

Imagine flipping a coin ten times and counting the number of heads that you get. The laws of probability say that there’s a 50-50 chance of getting heads on any single toss. So if you toss the coin ten times, you’d expect to get five heads, right?

Go ahead and pull a coin out of your pocket and try this if you’d like. You may not get the expected five heads after flipping the coin ten times. You may only get three heads. Or maybe you get six. If you keep repeating the ten-flip

Experiment over and over again, the distribution of the number of heads that

You get in each set of ten flips will look something like Figure 8-1. After each

Experiment repetition, the number of heads out of the ten flips was counted.

The experiment was repeated 10, then 100, and finally 1,000 times.

After 10 Repetitions

After 100 Repetitions

After 1.000 Repetitions

Figure 8-1:

Results of a repeated, ten-flip coin experiment.

250 200 150 100 50 0

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10

Number of Observed Heads in Each Repetition of the 10-Flip Experiment

This imaginary coin flip experiment is analogous to any other measurement or sampling event that is repeated more than once — like taking a sample of measurements from a process and calculating the average. Two important facts

Arise from Figure 8-1 that can be generalized to any measurement situation:

Repetitions of the measurement event produce different outcome results (variability). Like in the coin-flipping experiment, not every repetition of the ten-flip series produced the expected five heads. The same

Is true if you repeatedly take a five-point average of the thickness of

Paper coming out of a paper mill.

This resulting measurement, or Sampling distribution, Is normally

Distributed. The variation is also centered on the expected outcome. And the more repetitions you make, the closer and closer the sampling

Variation gets to a perfectly normal distribution.

Statisticians call repeated measurements of a characteristic or a process Samples. So the variation that occurs in repeated sampling events they call its

Sampling distribution.

Statisticians have refined and honed technical definitions of what is called the Central limit theorem. Although each definition is equally mysterious, they all say the same thing: When you repeatedly calculate statistics (like the average of a sample) for a process or characteristic, the repeated sample statistics

Have variation themselves. This sampling variation follows a normal distribution centered on the variation of the process or characteristic itself. Further,

The width of the sampling distribution depends on how many measurements

You take.

Although statisticians have a difficult time explaining the central limit theorem (and perhaps we authors do, too!), its power and utility are nevertheless remarkable. The results of the central limit theorem allow you to predict the bounds of the future and to quantify the risks of the past.

Ho© sure are uou> Confidence internals

Confidence intervals use the central limit theorem to tell you how much confidence you can place in any of your measurements or statistical conclusions.

Do not confuse confidence in your measurements, the topic of this chapter, with measurement system capability, the topic of Chapter 7. The measurement confidence we talk about here does not address the capability of your system

For acquiring measurements. Instead, measurement confidence assumes you have a perfect, ideal system for acquiring your measurements. This should

Serve as another reminder to you of how important it is to validate the capability of your measurement system.

For example, say your factory has just produced 5,000 ballpoint pens. You want to know the average diameter of this population. (Population Is a term that means any full set of something, like all the people in your hometown, or

All the pens you’ve produced, or all the invoices sent out over the last year. A

Sample, On the other hand, is any subset of a population.) To determine the average pen diameter, you randomly select 30 pens from the population and measure each of their diameters and calculate the average to be 0.120 inches

(see Chapter 5 for details of calculating averages).

Rushing into your office your boss asks, "What’s the average diameter of our latest pens? Our customer just called and said they will reject the whole batch if the average is higher than 0.125 inches!" Your sweaty boss waits for your response. What do you say? How confident are you in your calculated average?

The central limit theorem says that if you went out and repeated your

30-sample measurement, you’d get a slightly different average. When your customer measures a sample of the delivered pens, they will, too. But how

Different will each calculation of the average be?

Confidence intervals, the subject of this chapter, give you a way of quantifying how much variation there will be in repeated measurements and statistical calculations. Knowing how to create confidence intervals, you’ll be able to respond to your boss, "With 99.7 percent certainty, our average pen diameter

Will be within our customer’s requirement."

Confidence intervals for means

You see averages every day. Very few of them are communicated with a confidence interval.

Ho© biq is it

When your sample size has more than 30 data points, the confidence around

Your calculated sample average X Can be calculated as

+ 7 fn

Where

Is the sigma value corresponding to the desired level of confidence you want to have

Cr is the calculated standard deviation from your sample

N Is the number of data points in your sample.

Figure 8-2 illustrates what this confidence interval is for X.

From Figure 8-2, you can see that most sample calculations will be close to

The real population average. In fact, 68 percent of calculated X‘s will be within

+ fn

Of the real population average. Further, 95 percent of calculated X‘s will be

Within

Of the real population average. And 99.7 percent of calculated X‘s will be

Within

Of the real population average.

99.7% -95%–68%-

This formula works any time you have more than 30 measurements in your

Sample.

Any time you calculate a confidence interval, there is also an associated risk

Of being incorrect. This risk is simply the compliment of the calculated confidence. So for a 95 percent confidence interval, there is also a 5 percent risk

Ofthe actual population average being outside your calculated confidence interval.

The risk of incorrectly concluding that the population average is within your calculated confidence interval when, in reality, it is not, is called Alpha (u) Risk.

When you have only a few data points in your sample, you’re not able to get an accurate estimate of the population standard deviation cr. When your sample has anywhere from two to 30 data points, you have to use a different factor instead of 7. Statisticians call this new factor for small samples t. T Is

More conservative, because your smaller sample size lessens the accuracy of your calculated value for o. In fact, for each desired confidence level, T Is adjusted depending on how many data points are in your sample. Table 8-1 provides values for T For selected confidence percentages and sample sizes.

Table 8-1_- Values_

Confidence n=2 N=5 N=W n=25Z

68%_1.837 1.142 1.059 1.021 1

95%_13.968 2.869 2.320 2.110 2

99.7%235.811 6.6204.0943.3453

Using T, The formula for the confidence interval becomes

Fn

Where the value for T Depends on your desired level of confidence and the

Number of data points in your sample. Which is better)

Very often, you need to determine whether two or more items are different and, if so, by how much. Examples include:

V Is there a difference between operators of a process?

Do two alternative manufacturing processes lead to significantly different outputs?

Is the gas mileage of Car A better than Cars B and C?

Are the marketing collateral materials with color graphics really better

At generating leads than black and white equivalents?

Confidence intervals for sample averages (X‘s) can be used to verify differences between any two or more versions of the same outcome. Here is the

Process for doing this:

1. Take samples and perform measurements of each of the different versions or conditions you are analyzing.

2. Calculate the appropriate confidence interval for each different version or condition of the characteristic.

Remember, if your sample has less than 30 data points, you need to use the – formula to calculate the confidence interval. Also remember to use the same confidence level for each condition or version you are comparing.

3. Graphically or numerically determine if there is any overlap of the confidence intervals of the different versions or conditions.

If there is overlap between any of the confidence intervals, you can say with the decided level of confidence that there is no difference between the overlapping versions.

On the other hand, if there is no overlap, you can know right away that

There is a difference between the different versions of the output.

As an example, Figure 8-3 shows a graphical comparison of the confidence

Intervals for three different types of computer systems used in an invoicing

Process.

44

Figure 8-3: Ј

Graphical -3

Comparison §

Of the °

95 percent G*

Confidence gj

Intervals for "* the average

Items-per-hour output

For three

Types of computer systems.

42

40

38

36

34

32

30

1

+

+

Desktop Mainframe Web

Computer System

Graphically comparing the confidence intervals for the average performance makes it easy to see that there is no overlap in the intervals. So, with 95 percent confidence, you can say that the "Web" computer system is better than the "mainframe" computer system (better on average by 3 items per hour) and that the "Web" computer system is better than the "desktop" computer system (better on average by almost 5 items per hour.) If there had been an overlap between any of the three computer system options, you would have

Concluded (with 95 percent confidence) that there was no significant difference between the overlapping versions.

Confidence intervals (or standard deviations

Not surprisingly, your calculations of the standard deviation a of a process or characteristic have sampling variability in them, just like your calculations of the mean do. That means that confidence intervals can be created for standard deviations, too.

0o© much Variation is there}

To construct a confidence interval around your calculated standard deviation, you have to use a new factor invented by statisticians called (This factor is named after the 22nd letter of the Greek alphabet X And is pronounced kye-squared.) Like the. value used to create confidence intervals for averages, the value of ‘Ј depends on how many data points are in your sample — the more data points in your sample, the more confident your estimate. Another twist is that there are different values of Yj For the lower and upper limits of the confidence interval. Table 8-2 shows upper and lower values of F For common 1-, 2-, and 3-sigma confidence values.

Table 8-2_JfVatoes_

Confidence n=2 n=5 n=W n=25Z

68% 1.987 6.599 13.088 30.833 1

0.040 1.416 4.919 17.169

95% 5.187 11.365 19.301 39.749 2

0.001 0.460 2.628 12.225

Confidence N=2 N=5_n=10_N=25Z

99.7%_10.273 17.800 27.093 50.163 3

0.000 0.106 1.241 8.382

Note: First value listed in each table cell is -/lowe„. Second value listed in each cell is tfumR.

The X2 Values from Table 8-2 are used together with the formula below to calculate the confidence interval for a measured standard deviation.

(n – 1) o2 /(n – 1) o

X Lower

X Upper

As an example, suppose your sample of five data points leads to a standard

Deviation of 3.7. To create a 95 percent confidence interval for this standard

Deviation, you use the values in Table 8-2 corresponding to a 95 percent confidence and N = 5. So SLower = 11-365 and SUpper = 0.460. Plugging these values into the equation, you get

/(5 – 1) 3.72 / V 11.365 ’1′

5-1)3.72

0.460

Or [2.195, 10.907]

Another way to say this is that, with 95 percent confidence you know that the real standard deviation lies somewhere between 2.195 and 10.907.

Confidence intervals for standard deviations are usually very wide unless you have a lot of data points in your sample. This is because an estimate of the

Standard deviation is always less accurate than an estimate of the average. Which has less Variation}

Sometimes, you need to compare the variability of two distributions to find out whether one distribution has more variation than the other. You do this by creating a confidence interval for the ratio of the variances of the two distributions. If the ratio confidence interval includes the value 1 within its limits, you

Know that the two distributions have equal variability. If, on the other hand, the confidence interval doesn’t contain the value of 1 within its limits, you

Know that the two distributions have different amounts of variation.

Constructing a confidence interval around this ratio of variances requires yet another statistical factor. This one is called F By statisticians. Its value depends on three things: the desired level of confidence, the number of data points in the numerator distribution (n1), and the number of data points in the denominator distribution (n2). Table 8-3 is a list of F values for 95 percent confidence intervals and various sample sizes.

The F Values from Table 8-3 are used together with the formula below to calculate the confidence interval for a ratio of variances:

F < u:

1

As an example, suppose you have ten data points from distribution A and their variance sA = 4. Another distribution, called B, has five points and its variance sB = 7.5. The 95 percent confidence interval for the ratio of sA to

!B is calculated as:

-1-HA fMO 5")—

3.633 7.5,5.999 7.5

[ O.147,3.199]

This confidence interval contains the value of 1 within its limits, so all you can say with 95 percent confidence is that there is no evidence that the distributions have different variances.

Confidence intervals for variance ratios are usually very wide unless you have a lot of data points in your sample. This is because an estimate of the

Standard deviation is always less accurate than an estimate of the average.

Four out o( five recommend: Confidence intervals for proportions

When you calculate the number of successes out of a certain number of attempts — like "four out of five dentists recommend sugarless gum" — you can write this proportion mathematically as

N

Where § is the number of successes and n is the total number of attempts or trials.

Calculating a proportion creates yet another sampling distribution. The resulting confidence interval around a calculated proportion is:

§ ± 7 /(§/n)(l – §/n) 7n ± 7V n

So, as an example, if you wanted to be 90 percent sure of the calculated proportion for the four out of five dentists, you would calculate the confidence

Interval in this way:

4 ± ,.64J(i/SIlM = 4 ± O.294

This means that, with 90 percent confidence, the proportion of four out of five dentists really could be as small as one-half or as large as one.

In reality, proportions can never be less than zero or greater than one. So if

Your confidence interval for your proportion exceeds these natural limits,

Just adjust the confidence interval to the natural limit. If you are comparing the Difference Between two proportions

Nr And

The confidence interval for this difference becomes

§i – z± ± 7 (

§i/ni)(l -§I/Ni) (§2/N-§2/N2

To illustrate this confidence interval, imagine you are part of a company with

Two production lines. You suspect that your Toledo plant produces a higher proportion of good items (yield) than your Buffalo plant. You select samples of size nl = n2 = 300 from each plant and find that the number of good items from the Toledo plant (§i) Is 213 while the number from the Buffalo plant (§2) Is 189. That means that a 95 percent confidence interval for the difference

Between the Toledo and the Buffalo yields is

213 189 ± 2 /(213/300)(l – 213/300) | (189/300)(1 – 189/3°o) o ±

300 – 300 ± 2V 300 + 300 =0-08 ±0-076

Or, equivalents [0.004, 0.156]. Because this confidence interval does not include zero, you can conclude — with 95 percent confidence — that the Toledo plant produces, on average, a higher proportion of good items than

The Buffalo plant.

Table 8-4 Confidence Interval Summary

Name_Equation_Look-Up Factor

Average with large (> 30) sample size

Y + 7 -2_

+ fN

7

Average with small

(< 30) sample size

FN

Standard deviation

J(n-1)O2^ j(n-i)O2

Y X Lower Y XUpper

Ratio of variances

C, 1 ^ 2, F (n I, N 2) 2 F(n2, NI) 22 v ‘o2

Proportion

§ + 7 /(^N)(1 - //") N + – n-

7

Difference of

Proportions

N I N 2 Y N I

2/N 2 )(^2 § 2/n2 )

7

F

2

F

Understanding Relationships

Y Is a function of X. To get to the next level of understanding, you need to be able to quantify the relationships between the input variables and the critical outputs.

Correlation

Scatter plots (explained in Chapter 5) are a great way to visually discover and explore relationships between variables — both between Ys and Xs and between Xs and Xs. In a scatter plot, you graph the values of one variable

Against paired values of another variable. As an example, Table 8-5 is a list of paired data for the curb weight (lbs.) of some common automobiles and their corresponding fuel economy (mpg).

A data point for each automobile in the study is plotted in Figure 8-4.

The scatter plot in Figure 8-4 shows that there is a negative relationship between the curb weight of the vehicle and its fuel economy — the heavier

The car, the lower its fuel economy. The scatter plot also shows that the relationship between the two variables is approximately linear, meaning that its shape approximately follows a straight line. Finally, the relationship between the variables is fairly strong, as evidenced by the tight clustering of the plotted data points around the drawn line approximating the relationship.

But how do you quantify this relationship? How do you put numbers to it? Correlation offers such a measure. It tells you how closely the relationship between the two variables follows a linear pattern.

«S»NG.’ Correlation tells you only how linear the relationship between the variables ^/^h, Is. There is a chance it will miss more complicated relationships where the I ) variables follow a non-linear pattern. Always include a graphical scatter plot

W/ when doing a correlation analysis. That way you can visually check to make sure the variable relationship really is linear.

Figure 8-4:

Scatter plot of vehicle curb weight versus fuel economy.

40

35

30

25

20

15

10

5

T-1-1-r

2,000 3,000 4,000 5,000 6,000

Curb Weight (lb)

To quantify how linear the relationship is between two variables, you use the following formula to calculate the Correlation coefficient (r):

1

T S

XI - X

§• - Y

O Y

N – I -f-M O X Where N Is the number of data pairs,

X, And Y, Are the individual x-variable and y-variable measurements,

X And Y Are the averages of the X And Y Measurements, respectively,

Errand crrare the standard deviations of theXAnd YMeasurements, respectively, and

Il is a capital Greek letter telling you to add up all the terms, from 1 to N.

X, – x

YI – y 1

The calculated correlation coefficient will always be between -1 and 1.

The sign of R Tells you the direction of the relationship between the

Variables.

If R Is greater than zero (positive), that means that the variable relationship is Positive; If the value of one variable is increased, the other variable

Also increases.

If r is less than zero (negative), the variable relationship is Negative; If

The value of one variable is increased, the value of the other variable decreases, and vice versa.

F The absolute value of R Tells you how strong the relationship is.

The closer R Gets to -1 or 1, the Stronger The variable relationship is.

An R Equal to 1 or -1 indicates a perfect linear relationship, with all points being exactly on the line.

An R Close to 0 indicates that there is an absence of a linear fit to the data.

For the automobile fuel economy example introduced earlier, the calculated correlation coefficient R Then is

10

-1 § (W)(*r) – 9 < – "»»> «*"

An R Of -0.971 verifies that the relationship between the two variables is indeed linear and is negative. Also, an R Of magnitude -0.971 is very close to 1, telling

You that the relationship is very strong.

Correlation basically just confirms that there is a linear relationship between two variables and it quantifies how linear that relationship is. What correlation does Not Tell you is how much a given change in one variable will change a related variable. To get that kind of information, you need to become

Acquainted with some predictive tools.

Mm Even though two variables are correlated, it does not mean that one Causes The other. For example, studies show that a person’s reading comprehension ability (Y) Is correlated with their height (X). Does this mean that height Causes Reading comprehension? Think about it for a second. Young children

Have not yet developed cognition and reading skills. In the teenage years, physical growth continues along with maturation of mental and reading abilities. By the time you’re a full-grown adult, your brain and mental abilities have fully developed. Clearly, a person’s height does not Cause His or her reading ability. Rather, height is an indirect indicator of overall maturation and growth, including cognitive abilities. So be very careful: Don’t assume

There is a causal link when there is correlation.

Curve fitting

A step beyond correlation is curve fitting. In curve fitting, you actually determine the equation for the curve that best fits your data. Armed with this

Information, you know quantitatively what effect one variable has on another.

You also know which variables are significant influencers and which ones are

Just in the noise. Finally, you know how much of the system behavior your equation does Not Explain.

In some rare cases, the exact details of the Y= f(X) equation relating the Xs to the Y Is known without having to do any curve fitting — either from a very

Mature understanding of the physics of the process or system, or from some

Other source of knowledge. These situations are called Deterministic Because you know with certainty that setting the input Xs to certain values will always lead to the exact same value for the output Y even when the process is

Repeated.

For the vast majority of cases, however, the exact relationship between the Xs and the Y Or Ys is Not Known. This is due to the complexity of the system and the human inability to address all the factors that truly exert influence

On the output. Because of this natural limitation, repetitions of the system with the same input values will not always produce the same output performance. These situations are called Statistical.

The goal of curve fitting is to develop an approximate equation that describes the system or process statistical behavior as much as possible. When you work to create an approximate equation for a system that has a single output Y And a

Single input X, this type of curve fitting is called Simple linear regression. When

You work to create an equation that includes more than one variable it is called

Multiple linear regression.

Finding the tine: Simple linear regression

In simple linear regression, you assume that each observed outputpoint Yi

Can be described by a two-part equation:

Y – // 0 + // ,X + E

The first part of this equation is ySo + yS^. The second part is! Ј. Graphically, the decomposition of this equation is shown in Figure 8-5.

Time warping back to your high school algebra days, you may recall that the // o + /X Part of the equation for Y Is just an equation for a straight line: // o by itself tells you at what value the fitted line crosses the Y Axis; and tells you the line’s slope. The t part of the equation is a normal, random distribution with a center value equal to zero.

In simple linear regression, you mathematically determine values for //0 and //1 so that the resulting line fits your observed X - Y Data as closely as possible with the minimum amount of error. Then you determine how wide the T: Portion needs to be so that it accounts for all the extra r variation that isn’t already captured by the line.

You calculate / 0 and / 1 from the equations:

Zi(x, – x >

Where X, And Y, Are the paired data points and X And Y Are the calculated averages for all the X Points and all the Y Points, respectively.

Going back to the previous automobile weight versus fuel economy example, the calculated value for //1 turns out to be -0.00624, and //0 comes out to be 46.9. So the equation for the line that best fits the data is

Y_ 47.3 – 0.00632X

A

Where the Y With the pointed hat "A" over it, Y, represents the estimate or prediction for y not an actual observed value for Y. You are now armed with

A powerful, predictive tool. If, say, you found that a car you were interested in

Had a curb weight of 5,000 pounds, you can plug that X Value right into your

Equation to predict its fuel economy:

A

Y_ 47.3 – 0.00632(5,000) _ 15.7mpg All without ever test driving the car!

There are a couple of points of caution you need to be aware of in using your

New-found Six Sigma powers:

F Be careful not to extend your predictions very far beyond the range of the data in your study. For example, you don’t want to use your equation to predict the fuel economy of a vehicle weighing 25,000 pounds, like a locomotive. Vehicles in this weight class are so much heavier than the automobiles of the study that they use very different mechanisms and technology and don’t fit the line.

The general rule is not to extrapolate any predictions beyond the range of your study data.

V Remember that the derived equation for the line is missing the L: component. The line predicts only the expected Average Performance. In

Reality, the actual performance level varies from the predicted value.

This is the effect of the T: Component. For some situations, this random variation component dominates, leaving little room for effective predictions. In other cases, your derived line equation gets you very close to actual, real-world values.

Your next step is to understand and quantify the T: Component of your regression equation.

If the shoe fits, ©ear it: Residuals and adequacy of the fitted model

For each of the I Data points in your study, you can calculate an error term E, — how far off the predictive equation line is from the observed data. For example, referring to Table 8-5, the data for the Toyota Camry shows that its curb weight is 3,140 pounds and its fuel economy is 29 mpg. But plugging an X Value of 3,140 pounds into the derived regression equation, you get a predicted fuel economy of

A

Y_ 47.3 – 0.00632 (3,140) _ 27.49mpg

The difference between the observed and the predicted fuel economy is

A

E1 _ Y1 – Y1 _ 29 – 27.49 _ 1.51mpg

Similar error E, Terms can be calculated for each of the nine other data points in your regression study. These E, Terms are called Residuals — or what’s left

Over after using the predictive equation.

The beginning assumption of the predictive linear equation is that there is a

Secondary T: Part of the equation that is a normal, random distribution with a center value equal to zero. This variation is manifest in the residuals and is

The bell-shaped variation identified back in Figure 8-5.

The most efficient way to check the validity of your predictive linear equation is to graphically review the residuals to make sure they are behaving as you’ve assumed. You may need to create up to four different types of graphical checks of the residuals.

F A scatter plot of the residuals e, versus the predicted Y values from the

Derived equation.

F A scatter plot of the residuals e, versus the observed X data.

F Additional scatter plots of the residuals e, versus any other X variables

That you didn’t include in your equation.

Is A run chart of the residuals e, versus the previous residuals eM if you collected your study data sequentially over time.

In each of these graphical residual checks you are looking for:

V The variation has no obvious patterns and is truly random, like a cloud

Of scattered dots.

^ The residual variation is centered around the value of zero.

Figures 8-6, 8-7, and 8-8 are examples of residual-checking plots for the previous automobile weight-fuel economy study.

Figure 8-6:

Residuals plotted versus the corresponding predicted Y Values. Review to verify that

The variation

Is normal over time

With a central value of

Zero.

-4

5 10 15 20 25 30 35 40

V(mpg)

4

0

Figure 8-7:

Residuals plotted

Versus the observed X variable of

Vehicle curb

Weight (lbs.).

-4

2,000 3,000 4,000 5,000 6,000

Observed Jf(Curb Weight in lbs.)

Figure 8-8:

Residuals plotted in the order of the study data.

Review to

Verify that the variation is normal over time with a central value of

Zero.

-4

1 2 3 4 5 6 7 8 9 10 Study Order

4

0

4

0

Figures 8-6, 8-7, and 8-8 appear valid. In each case, the residuals show evidence of being truly random, normal, and centered over time around zero.

You can now conclude that your derived linear predictive equation is valid.

What do residual plots look like when you have data that is not valid? Figure 8-9 shows two examples of an inappropriate simple linear regression model. In (a), the variation of the residuals is not centered on zero. Instead it shows a curved pattern. In (b), the residual variation is not consistent —

Lower values of variable Xproduce larger residual variation then higher

Values of variable X.

Figure 8-9:

Examples of

Residual plots showing invalid behavior.

R

(a)

Observed X

(b)

0

0

There is another way to investigate how good your derived regression model

Is. It involves looking at the variation of the output variable Y. This new assessment is done on a squared error basis.

The total sum of the squared error (5570) in the output variable Y can be

Written as

5570 = |j (Y, – Y )2 where Y are the n observed output values.

In a similar way, the squared error from just the derived regression equation (SSR) can be stated as

5SK = 2(Y, – YJ

A

Where Y, are the predicted estimates for the n data points.

Finally, the squared error from the remaining i: variation (SSE) Can be expressed as

55Ј = 2e2 = 2(Y, – Y, J

Together, these three squared error terms can be related with the simple sum

5570 = 55tf + 55Ј

You can do three important tests using these squared error terms.

1. Calculate the Coefficient of determination,!?, For your predictive model.

The coefficient of determination is simply the ratio of the squared regression error (55tf) and the total squared error (5570), like this

P f = 55tf * 5570

What tf2 tells you is how much of the total observed variation is determined or explained by your linear model. You want this number to be 80 percent or higher. This means that the unexplained variation i: accounts for the remaining 20 percent.

With a high tf2 value, you can know that your predictions will be close — and not dominated by the unexplained variation.

For the automobile curb weight versus fuel economy study introduced earlier in this chapter, the tf2 value is

Tf2 = 42ttt = 0.94 450.1

This is very good. Ninety-four percent of the observed variation is explained by your derived linear model. That leaves only six percent

That is unexplained and left to random chance.

2. Quantify the unexplained L: variation in terms of its standard deviation.

Remember that T: Represents a random, normal distribution centered at a value of zero. This value is an inherent part of your predictive linear equation. But how big is its variation?

An estimate of the standard deviation of the t distribution can be calculated using the surprisingly simple equation

A = P55e E V n – 2

This estimate comes in handy when you want to mimic what may happen in reality. You use your derived linear model to predict the average or expected performance of the output Y, and then add to it a

Random number generated from the! Ј distribution — with mean of zero

And standard deviation equal to a° . In this way, you can simulate what

Would happen if your process or characteristic were repeated over and over again.

For the automobile curb weight versus fuel economy study, you can estimate the standard deviation of the unexplained variation as

A° = / if-T = L80mpg

3. Perform an F Test to quantify your confidence in the validity of your regression model.

Another test of validity for your derived linear equation is to statistically

Compare the variation explained by your regression model to the unexplained variation.

Yet another way to mathematically represent the variation in the regression model is by an estimate of its variance

2

AtfH; = 55tf

You also already know — from Step 2 — that

A 2 55Ј

Zz.

N-2

2

Creating a ratio of AtfH; to Oe Is just like the confidence intervals covered in the "Which has less variation?" section earlier in this chapter for comparing the size of two different distributions. So if

2

> F (2, n – 1)

You can say with 95 percent or 99 percent confidence — whichever level of confidence you select — that your derived predictive model is, in fact,

Valid.

For the automobile curb weight versus fuel economy study, if you want to be 99 percent confident with your n = 10 data points, the F test of the

Variances becomes

4242 = 16.4 > 11.259 = F (2,9) Because the calculated ratio value of 16.4 is greater than the critical

99percent Fvalue, you can conclude with 99 confidence that there is something to your derived model.

Toots for fitting tines

Simple linear regression is becoming a commonplace activity. Anyone with Microsoft Excel, for example, can take data from an X and a Y variable and almost immediately create a scatter plot of the two. Then, with just a couple of clicks, the program will automatically derive the fitted line for your data.

If you have Excel, try this on the automobile weight versus fuel economy study introduced in this chapter.

1. Enter the data from Table 8-5 into Excel as two columns of data — one for the curb weight (X) Data and another for the fuel economy (Y) Data.

2. Select the entered data in the spreadsheet and create what Excel calls an XY (scatter) plot.

3. Right click on the plotted data in the graph and select Add Trendline from the menu that pops up. Select the Linear option and click OK.

The best fit line with the right //„ and //i parameters is automatically added to your graph!

4. If you double-click on this fitted line, you are given options to display the equation for the line and also to display the coefficient of determination R2, If you’d like.

The results of these display options are shown in Figure 8-10.

Minitab, JMP, and other statistical analysis software tools provide tremendous detail and options and make simple linear regression almost fun!

40

Figure 8-10:

A simple linear regression model automatically fitted by Microsoft Excel. Notice that you can also

Choose to

Display the equation

And the calculated

R2 value.

35

30

25

20

15

10

5

= -0.0063x + 47.333 v R2 = 0.9425

2,000 3,000 4,000 5,000 6,000

Curb Weight (lb)

Y

Fancy curi/e fitting: Multiple linear regression

You now understand how to generate and validate a predictive model linking a single X To a Y But what about all the situations where there is more than one X influencing a Y? Surely there must be many more situations like this than there are with just a single influencing variable.

When you generate a Y = F(X) Equation with multiple Xs, like

Y = f(X1, X2, . . Xn),

It is called multiple linear regression.

The general form of the multiple linear regression model is simply an extension of the simple linear regression model. For example, if you have a system

Where X7 and X2 Both contribute to Y the multiple linear regression model

Becomes

There are five different distinct kinds of terms in this equation.

1. P „: This is the Overall effect. It sets the starting level for all the other

Effects, regardless of what the XVariables are set at.

2. P X: These are the Main effects Terms in the equation. Just like in the

Simple linear regression model, these terms capture the linear effect each Xi Has on the output Y. The magnitude and direction of each of these effects is captured in the associated / I Coefficients.

3. P„ X2: These are the Second-order Or Squared effects For each of the Xs.

The effect here will not be linear. Instead it is quadratic, because of the

Variable being raised to the power of two. Again, the magnitude and direction of each of these second-order effects is captured in the associated ft, coefficients.

4. PI2XX2: This called the Interaction effect. This term allows the input

Variables to have an interactive or combined effect on the outcome Y.

Once again, the magnitude and direction of the interaction effect is captured in the / 12 coefficient.

5. T:: This is the term that accounts for all the random variation that can’t be explained by all the other terms, R. Is a normal distribution with its center at zero.

The equation for multiple linear regression can fit much more than a simple line. It can accommodate curves, three-dimensional surfaces, and even abstract relationships in N-dimensional space! Multiple linear regression

Can handle about anything you throw at it.

The process for performing multiple linear regression follows the same pattern you used when doing simple linear regression.

1. Gather the data for the Xs and the Y

2. Estimate the multiple linear regression coefficients.

When you have more than one XVariable, the equations for deriving the

//’s become very complex and very tedious. You definitely want to use a statistical analysis software tool to calculate these automatically for you. The //’s just pop right out. Otherwise, go buy a box of Number 2 pencils

And roll up your sleeves!

3. Check the residual values to confirm that the upfront assumptions of

The multiple linear regression model are met.

Checking that the residuals are normal is critically important. If the variation of the residuals is not centered around zero and if the variation is not random and normal, the starting assumptions of the multiple linear

Regression model haven’t been met and the model is invalid.

4. Perform statistical tests to see which terms of the multiple linear

Regression equation terms are significant (and should be kept in the

Model) and those that are insignificant (and need to be removed).

Some terms in the multiple regression equation will not be significant.

You find out which ones by performing an F Test for each term in the equation. When the variation contribution of an equation term is small

Compared to the residual variation, that term will not pass the FTest and you can remove it from the equation.

Your goal is to simplify the regression equation as much as possible while maximizing the r2 metric of fit. As a general rule, simpler is always

Better. So if you find two regression equations that both have the same R2 Value, you want to settle on the one with the fewest, simplest terms.

Usually, the higher order terms are the first to go. There’s just less chance of a squared term or an interaction term being statistically significant.

Many of the more sophisticated statistical analysis software tools even

Have automated algorithms that will search through the various combinations of equation terms while maximizing R2.

5. Calculate the final coefficient of determination R2 For the multiple linear regression model.

Use the r2 metric to quantify how much of the observed variation your

Final equation explains.

With good analysis software becoming more and more accessible, the power of multiple linear regression is becoming available to a growing audience.

Chapter 9

In This Chapter

^ Implementing a strategy for sustainable results ^ Selecting tools to achieve process control

Examining control charts and statistical process control

^ Mistake-proofing with Poka-Yoke ^ Creating the right level of control

Solution that isn’t sustained over the long term has little value. That kind of solution can make you feel good for a little while, but if the problem doesn’t stay solved, it will end up being a frustrating experience. The

Control phase helps you make sure the problem stays fixed, and, if done properly, provides you with additional data to make further improvements tothe process.

In this chapter, you find an abundance of easy-to-use and readily available tools

And techniques to assure that your problem remains solved for the long-term. These tools range from the use of statistical methods for quantitative control

To documented plans and strategies to the use of common-sense approaches

For managing process performance.

The Need for Control Planning

Six Sigma emphasizes the Control phase. This is because previous attempts at improving quality and business performance have repeatedly demonstrated that process behavior is complex and fragile and that hard-earned gains slip

Away if the process is left to itself.

A process is a system of events, activities, and feedback loops. A well designed process exhibits inherent self control, while a poorly designed process requires

Frequent external control and adjustment to meet requirements. Some people

Use the terminology Tampering with the process To describe such adjustments.

A process with control designed-in acts like the heating and cooling system in a house: The system automatically maintains a comfortable temperature at all times. A process without inherent control is like having to get up from the couch to manually turn on the heat, and then, when it gets too hot, you get

Up again to turn it off. When it’s too cold again, you turn the heat back on, starting the cycle all over again. There is a lot of variation in this type of process — and waste comes from excessive variation.

The Six Sigma act of developing a control plan, and the knowledge (the reference documentation and the organizational memory) that you gain from it,

Virtually guarantees the improved performance you’ve worked so hard to

Achieve will stick.

Before Six Sigma came along, developing process controls tended to be like trying to boil the ocean. Some organizations even went to Herculean efforts to have an exhaustive plan for every process and every detail of every process, regardless of the importance of the process. It was like trying to eat the elephant in one bite!

Six Sigma, however, gives you the ability to pick those processes that are most important and to identify the input and output variables of the process that matter most (effectively eating the elephant one bite at a time). This changes the control effort from a broad-swathed flashlight to a focused laser beam. Developing a control plan for a Six Sigma project is a much easier alternative — and it’s an absolutely essential activity.

VcN-Stye,. There are two aspects to a control plan. Y=f(X) Shows the inputs that must be controlled (see Chapter 2 for details on this equation). The outputs can IS JWJJ Be monitored only to see whether control has or has not been achieved. Accordingly, there are two aspects of a Six Sigma control plan:

Is Process monitoring Of outputs uses a tool called a process management summary. The objective of the process management summary is to enable

The visibility, review, and action for all critical process outputs in an organization.

V Process control Of inputs uses a tool called a process control plan. The

Objective of the process control plan is to create systematic feedback loops and actions to assure the process has inherent, automatic control.

With a good process control plan, you can change out people, equipment, materials, and production rates without significantly altering the performance quality of the process.

The two following sections provide further details.

The process management summary

The Process management summary, Shown in Figure 10-1 is intended to be a collection of all the critical-to-quality outputs, or CTQs, for a process, a department, a division, or even up to an entire company. The summary rolls up to whatever level an organization needs for monitoring, reviewing, and taking action to assure acceptable process and business performance. Each time a Six Sigma project is completed, that project’s CTQs are added to the summary, building a complete process management system.

Figure 10-1:

Six Sigma Process management summary.

PROCESS MANAGEMENT SUMMARY

Division: Department: Revision Level:

Date:

Division Executive:

Process Step (Namel

Process

Step (Numberl

Process

Owner (Namel

Key Output CTQs (Ysl

CTQ Requirements

CTQ Metric & Value

Performance Trend

Links to:

(Process Namel

Process

Owner

Improvement Activities

2

3

The administrative section of the summary (Section 1) provides identification of the organizational areas involved, plus the revision level and date of the information. The main body of the summary (Sections 2 through 6) provides enough information so that anyone can readily see the current status of the CTQs, how they relate to downstream processes, and what actions, if any, are currently being taken.

The process control plan

The Process control plan, Shown in Figure 10-2, is the companion to the process management summary. The process control plan is focused on the Xs, The inputs to the process. The inputs, by definition in the formula of Y = F(X), Are the critical Xs that are determined from the Six Sigma project. But it is okay to place process outputs, the Y (the CTQs), on a process control plan, too. When done correctly, the process control plan creates a complete picture of all possible inputs, outputs, and activities for a single process.

Figure 10-2:

Six Sigma process control plan.

PROCESS CONTROL PLAN

Process Name: Process Owner: Revision Level: Date:

Operational Definition:!

2

The administrative section of the process control plan (Section 1) provides key information for identifying the process, including the owner. It also includes revision control information. Section 2 is a larger space for writing

An operational definition of the control plan. This helps people understand

Why the process exists — that is, what its value proposition is — and helps put the control activities into proper context. Section 3 identifies what is specifically being controlled. Section 4 is the requirements and current performance levels of the requirements. In this section, you can quickly see how well the process is performing and how the control activities are going.

The method of control, and the actions to be taken if the process goes out of

Control, are in Section 5. This is where the rubber meets the road. The control methods may include checklists, mistake-proofing methods, statistical process controls, or any other appropriate procedure. More and more, control

Methods include the use of automated data collection and process execution

Technologies. This section also contains the size of the sample required and the frequency of sampling in order to provide the proper feedback to the process. Feedback is important, just like it is when you are driving a car. If you don’t have the right amount of feedback at the right time to keep your car going down the center of the road, you may suddenly find yourself in the

Ditch. The same is true for keeping a process on track. Section 5 documents the action to be taken if the controlled parameter does not meet its requirements. This is analogous to calling 911 — it’s what will be done and who will do it. A sense of emergency exists, always, when an input X or an output Y (or CTQ) goes out of control.

Section 6 contains the names of any governing documents for the process and any standard operating procedures (SOPs). With a good process control

Plan in place, you’re ready to hand your project off to the process owner and process workers, highly confident that they will be able to sustain the improvement your Six Sigma efforts have made.

Creating a process control plan requires some thinking about what needs to

Be done, but it is worth the effort. Without a good control plan in place, your Six Sigma project has not been completed correctly.

Statistical Process Control

Statistical process control (SPC) involves the use of statistical techniques to

Monitor and control the variation in processes. SPC is used first to stabilize

Out-of-control processes. But it is also used as a follow-on, to monitor the consistency of product and service processes.

The primary SPC tool is the Control chart — a graphical tracking of a process

Input or an output over time. In the control chart, these tracked measurements are visually compared to decision limits calculated from probabilities

Of the actual process performance.

The visual comparison between the decision limits and the performance data

Allows you to detect any extraordinary variation in the process — variation that may indicate a problem or fundamental change in the process. There are several different types of control charts, depending on what type of process measurement you are tracking.

These different types of control charts are separated into two major categories: continuous data control charts and attribute data control charts. Here is a list of some of the more common control charts used in Six Sigma:

Is Continuous data control charts:

• Averages and ranges (X - R)

• Averages and standard deviations (X - S)

• Individual values and moving ranges (I - MR) F Attribute data control charts:

• P Chart

• U Chart

The control chart you choose is always based first on the type of data you have, and then on your control objective. The control chart decision tree in

Figure 10-3 aids you in your decision.

Control charts provide you information about the process measure you are charting in two ways: the Distribution Of the process and the Trending Or change of the process over time. Control charts are used to:

Is Provide a simple, common language for discussing the behavior and performance of a process input or output measure

F Control the performance of a process by knowing when and when not to take action

V Reduce the need for inspection

F Understand and predict process capability based on trends and other

Performance insights

Is Determine whether changes made to the process are having the desired

Result

Is Provide an ongoing, continual view of the performance of the process F Create a repository of data for follow-on improvement activities

Monitoring the Process: Control Chart Basics

What gets measured gets managed. Deciding what to measure and manage in Six Sigma is determined by your define, measure, analyze, and improve project activity (see Chapter 3) before you get to the control phase. Simply stated, they

Are the critical input Xs and the output CTQs (the Ys) You discover in your project. These are the movers and shakers in your process that align to the needs of your customer. In the control phase, you monitor the outputs — the CTQs — and you control the inputs, the critical Xs. When done properly, this monitoring allows you to consistently reap the fruits of your efforts.

Control charts are two-dimensional graphs plotting the performance of a

Process on one axis, and time or the sequence of data samples on the other axis. These charts plot a sequence of measured data points from the process. You can also view the sequence of points as a distribution. Figure 10-4 demonstrates how a distribution can be displayed from a sequence of data points.

Figure 10-4:

Data points and distributions.

Control charts have the following attributes determined by the data itself:

V There is an average or center line for the data: The sum of all the input data divided by the total number of data points.

V There is an upper control limit (UCL): Its typically three process standard deviations above the average.

V There is a lower control limit (LCL): Its typically three process standard deviations below the average.

Understanding control limits

You may ask, "What is the significance of a control limit and where does it come from?" The simplicity of control limits, yet their powerful implications,

Will surprise you.

Control limits come from probability, or the likelihood of an event occurring. Control charts use probability, expressed as control limits, to help you determine whether an observed process measure would be expected to occur (in control) or not expected to occur (out of control), given normal process

Variation.

The likelihood that a specific event or measurement value will occur is the ratio

Of the number of times that event or measurement value occurs compared to

The total number of times all other possibilities occur. This is demonstrated using Figure 10-5, which is a population of data that contains 100 data points, plotted in a Histogram (see Chapter 5).

In Figure 10-5, there are 25 data points out of 100 with a value of 50. You then estimate that the probability of getting an event with a value of 50 is 25 out of 100, or 25 percent. Similarly, the probability of getting an event with a value of 52 is approximately 13 percent, and for values of 55 and above, the probability is much less.

Figure 10-6 takes the data from the histogram in Figure 10-5 and plots the data in a chronological sequence as a control chart for individual measurements.

The upper control limit of 58.7 is 3 standard deviations above the average. The lower control limit of 41.3 is 3 standard deviations below the average. Plus or minus three standard deviations from the mean includes 99.7 percent of all the data in a normally distributed population. Therefore, there is a 99.7 percent probability that a data point will fall between these two limits. That means there is only a 0.3 percent chance that a measurement will be above the UCL or below the LCL.

Figure 10-6:

Control chart for individuals.

I Chart of C1

60

5ft

15-

*0

UCL=S3.70

X=50

LCL =41 JO

1 10 20 30 40 50 60 70 SO 90 1QO Observation

In the early 20th century, Walter Shewhart, one of the founders of the modern quality movement, formalized the ideas used in control charts. He defined

That if a measurement falls within plus or minus three standard deviations

Of its average, it is considered "expected" behavior for the process.

This is known as Common cause variation (see Chapter 5). Common cause variation results from the normal operation of a process and is based on the design

Of the process, process activities, materials, and other process parameters.

However, if a data point falls outside of the control limits, something special has happened to the process. In other words, something out of the ordinary has caused the process to go out of control. This is known as Special cause variation (also discussed in Chapter 5). What this says is, "The probability that a

Process measurement could be that far from the average, based on the behavior of the process up to that point, is less than 0.3 percent." A measurement

With such a low probability of occurrence suggests that there was special circumstances affecting the process. This simple, quantitative approach using

Probability is the essence of all control charts.

Using control charts to keep processes on track

If you apply control charting as a part of your process control plan, you can use the control chart itself to trigger action or to leave things as they are,

Based on what the control chart tells you.

Sample data, Also called Subgroup data, Is collected from the process characteristic (an input or an output) in which you are interested. The process must be allowed to operate normally while you’re taking a sample.

How often you sample depends on how sensitive you want your chart to be to detect trends or other special-cause patterns in the process behavior. At

First, error on the side of taking samples very often, and then, if the process

Demonstrates that is stable and in control, you can take samples less often.

If you treat the process you’re charting with out-of-the-ordinary care or with any special treatment, the information from the control chart is invalid. You must allow the process to act as it normally does while you’re creating a control chart for it.

After you have collected a minimum of 25 subgroups of data (with 2 to 5 measurements in each subgroup), you can calculate the statistics and control

Limits using statistical software like Minitab or JMP (see Chapter 11). If you already have historical data, it is useful to include this data in the analysis to

Form a strong baseline of information.

If the sample observations are normally distributed around the average and lie

Within the control limits then the process is said to be Stable And In control.

Further, the sequence of measurements will not show any trends or shifts in centering. This type of behavior is what is expected from a normally operating process, and that is why it is called common cause variation.

Never confuse control chart limits with specification limits! Specification limits — like the USL and LSL introduced in Chapter 6 — represent the voice of the customer. Control charts, however, represent only the voice of the process, something totally different. Discovering how the process performs naturally, apart from whatever its specifications may be, is the purpose of control charts. Another way of saying this is that control charts only determine whether the process is Stable And Predictable. They do not tell you

Whether the process is capable of meeting customer requirements. To assess

A process’s capability, refer to the capability material in Chapter 6. Always

Resist the temptation to interpret control chart limits as specifications, and avoid overlaying specification limits onto your control charts.

A process should be left as it is, if it is stable and predictable (in control) and if it is capable of meeting customer requirements (see Chapter 6). If special cause variation occurs, however, you must investigate what caused this extraordinary variation and find a way to prevent it from happening again. Some form of action is always required to make a correction and to prevent future occurrences.

Using control charts to detect patterns, shifts, and drifts

Besides control chart points that lie beyond the control limits, there are other visual patterns that tell you that something out-of-the-ordinary is happening to your process. These other patterns also indicate special cause

Variation.

Detecting special cause patterns, shifts, and drifts in a control chart is similar to detecting out-of-the-ordinary behavior in a pair of dice. The probability of rolling a seven with two dice is six in thirty-six, or about 17 percent. That’s because there are six possible ways to roll a seven with two dice, out of a total of thirty-six possible outcomes. What is the probability of rolling a seven two times in a row? The combined probability is 17 percent (0.17) X 17 percent (0.17), or 2.8 percent (0.028). The probability of rolling a seven Three Times in a row is 0.17 X 0.17 X 0.17, or about 0.46 percent. So if you see someone roll a seven three times in a row, that probability is small enough that you can safely conclude there must be something out-of-the-ordinary going on (like loaded dice!) This same thinking is used to detect patterns, trends, and

Shifts in control charts.

Dividing the distance between the control limits and the process average into

Three equal zones, as shown in Figure 10-7, the following rules can be used to detect special causes of variation:

Figure 10-7:

Control chart zones.

Any one point beyond either control limit.

Is Two out of any three consecutive points in Zone A, and all three on the same side of the process average.

F Four out of any five consecutive points in Zone B or A, and all five on the same side of the process average.

V Fifteen points in a row in Zone C, on either side of the process average. Table 10-1 shows a handful of additional rules for visually detecting if there

Are special causes acting on the process characteristic being charted.

Table 10-1 Tests for Special Causes for Rules One through Six

Chart Description Example 1_Example 2_Interpretation

Stable Chart points and pre – do not form dictable a particular pattern and they do lie within the upper and

Lower control limits.

-*—

The process is stable, not changing. Only common-cause variation is affecting the process.

20

0

Beyond One or more

Control chart points limitslie beyond

The upper

And lower control

J* 10 ~~

Alerts you that

A special

Cause has

Affected the

Process. Investigate to

Limits. determine the

Source of the special cause.

20

20

0

Run Chart points are on one

Side of the center line. The number

Of consecutive points

On one side

Is the

"length" of

The run.

Suggests that the process has undergone a permanent change. May

Requireyou to compute new control

Limits for the

Shifted process.

ChartDescriptionExample 1 Example 2Interpretation

TrendA continued rise or fall in a series of chart points.

(Seven or more consecutive points in

The same direction.)

——

Indicates a special cause with a gradual, cumulative effect. Investigate possible special

Cause sources.

CycleChart points show the

Same pattern changes (for

Example,

Rise or fall) over equal periods of

Time.

—

Indicates a special cause with a cyclical, repetitive effect. Investigate possible special cause

Sources.

Hugging Chart points

Are close to the center line or to a control limit

Suggests a possible error in data sub-grouping or selection.

Line. Verify validity of

Sampling plan

And/or investigate possible special cause

Sources.

20

15

0

0

20

20

15

0

20

5

0

When you detect any one of these listed patterns, you know that something out-of-the-ordinary has happened in the process input or output you’re

Charting.

Collecting data for control charts

Data for control charts must be collected in such a way that a distorted or inaccurate view — either overly optimistic or too bleak — of the process

Performance is avoided. Using rational subgroups is a common way to assure

That this does not happen.

A rational subgroup is a small set of measurements in which all the items in

The subgroup are produced under as similar conditions as possible, typically

Within a relatively short time period — a time period short enough that special causes are unlikely to occur within the subgroup. It is in this way that rational subgroups enable you to accurately distinguish special cause variation from common cause variation.

Make sure that your subgroup measurements do not unfairly favor any specific operating condition (meaning that your subgroups are instead randomly

Selected). For example, don’t take subgroups only from the first shift’s production if you are analyzing performance across multiple shifts. Or don’t look at only one vendor’s material if you want to know how the overall process, across all vendors, is really running. Finally, don’t concentrate on a single time of the day, like just before the lunch break, to collect your subgroup measurements.

Rational subgroups are usually small in size, typically consisting of three to

Five measurements. It is important that rational subgroups consist of measurements that were produced as closely as possible to each other, especially if you want to detect patterns, shifts, and drifts. For example, if a machine drills 30 holes a minute and you want to create a control chart of hole size, a good rational subgroup may consist of four consecutively drilled holes.

If your process consists of multiple machines, operators, or other process activities that produce streams of the same process characteristic you want to control, it is best to use separate control charts for each of the process streams.

Control Charts for Continuous Data

Continuous control charts Refer to control charts that display performance

Ofprocess input or output measurements that are continuous data — data where decimal subdivisions have meaning (see Chapter 7 for an explanation

Of data types). When control charts are used to control the input *s to a process, it is properly referred to as Statistical process control, Or SPC.

Continuous control charts can also be used to monitor output CTQs, the

Important process output characteristics. When control charts are used in

This way, it is referred to as Statistical process monitoring, Or SPM.

There are two categories of control charts for continuous data: charts for controlling the location of the process average and charts for controlling the width of the process variation. Generally, the two categories are combined in paired, side-by-side charts.

The typical pairing of continuous control charts used in Six Sigma are

Is Individual and moving range (I – MR) chart

F Averages and ranges (X – R) chart

V Averages and standard deviations (X - 5) chart

Table 10-2 summarizes the important parameters of each type of continuous

Control chart.

Table 10-2 Continuous Data Control Chart Summary

Control Subgroup Centerline Control Limits

Chart Size (n)

Average 2-10 and range

X – R

X = X1 + X 2 + – + Xk

UCLx = X + A 2R LCLx = X – A2R UCLr = D 4 R LCLr = D 3R

Average and

Standard deviation

X – S

> 10

X=

UCLx = X + A 3S LCLx = X – A3S UCLs = S 4S LCLs = S 3S

S

Where k is the number of subgroups and:

Sample A2 A3 B3 B4 D3 D4 E2

Size (n)

2 1.880 2.659 0 3.267 0 3.267 2.659

3 1.023 1.954 0 2.568 0 2.574 1.772

40.7291.62802.26602.2821.457

50.5771.42702.08902.1141.290

60.4831.2870.0301.97002.0041.184

70.4191.1820.1181.8820.0761.9241.109

80.3731.0990.1851.8150.1361.8641.054

90.3371.0320.2391.761 0.1841.8161.010

100.3080.9750.2841.7160.2231.7770.975

The use of control charts for SPC and SPM must be carefully planned and

Managed in order to be successful. The general step-by-step approach for theimplementation is as follows:

1. Define what needs to be controlled or monitored.

2. Determine the measurement system that will supply the data.

3. Establish the control charts.

4. Properly collect data.

5. Make appropriate decisions based on control chart information.

Individuals and moving range chart (1 – MR)

The individuals (I) and moving range (MR) control chart is used when you have continuous data and each subgroup consists of only a single, individual measurement. These charts are very simple to prepare and use. Figure 10-8

Shows the individuals chart, where the individual measurement values are plotted, with the centerline being the average of the individual measurements.

The moving range chart shows the range between two subsequent measurements. The centerline is simply the average of these between-point moving ranges.

Figure 10-8:

Individuals moving

105104103102101 -

100

9998979695-

(/- MR

Chart).

Range chart | 21 0

10

_i_

~~r

15

_i_

20

_i_

3.0SL = 104.5

J(= 100.1

-3.0SL = 95.71

25 Observation _L

3.0SL = 5.418

IWfl= 1.658 -3.0SL = 0.000

There are many situations where opportunities to collect data are limited or when gathering the data into subgroups simply doesn’t make practical sense. Perhaps the most obvious of these cases is when each individual measurement is already a rational subgroup. This may happen when each measurement represents one batch, when the measurements are widely spaced in time, or when only one measurement is available in evaluating the process. Such situations include destructive testing, inventory turns, monthly revenue figures, and chemical tests of a characteristic in a large container of material. All of these situations indicate a subgroup size of one.

The formula to calculate the control limits is based on the average moving

Range, which is the variation from one point to the next. The control limits are estimated statistically from these moving ranges.

The / – MR chart in Figure 10-8 shows the individual measurements in the

Upper chart of the pair and a moving range in the lower half, which allows you to examine the process location and variation width at the same time.

Because the / – MR chart is dealing with individual measurements, it is not as sensitive as the X – R or X - 5 chart in detecting process changes (see the two following sections).

Averages and ranges chart (X – R chart)

An X - R Control chart is used when you have continuous data with subgroups of two to ten measurements each. It is used primarily to monito_r and control the stability of the process characteristic’s average value. The X Chart plots the average values of each of a number of the small-sized subgroups. The averages of the process subgroups are collected in sequential, or chronological, order from the process. The X chart, together with its paired R chart shown in Figure 10-9, is a sensitive method for identifying assignable causes of product and process variation. Because it relies on rational subgroups, it

Provides great insight into the process characteristic’s short-term variation.

Figure 10-9:

Averages and ranges chart (X - R chart).

As with all the paired control charts for continuous data, the X and the R charts are most effective when they’re used together as a matched pair. That’s because each chart individually shows only a portion of the information concerning the process characteristic. The upper chart shows how the process average changes. The lower chart shows how the variation of the process changes.

The R chart must show that the process variation width is stable and in control before you can properly interpret the X chart. That’s because the control limits for the X chart are calculated from the observed variation in the ranges. When the R chart is not in control, the control limits on the X chart will be inaccurate and may falsely indicate an out-of-control condition when there

Really is none.

Averages and standard deviation chart (X – S)

The X – 5 chart is constructed similarly to the X – R chart, but instead of

Ranges, it plots the standard deviation of each subgroup.

The calculation for the control limits on the X - R chart uses only two data point_s, the highest and lowest value. The calculation for the control limits on the X – 5 chart uses all the data. The X – 5 chart is, therefore, a more accurate indicator of process variation. The X – 5 chart is also very sensitive to

Small changes in the process average.

Use the X - 5 chart when the size of your subgroups is ten measurements or greater, and the X – R when they are less than ten. You should consider using

This chart for processes with a high rate of production, when data collection

Is quick and inexpensive, or when you need increased sensitivity to variation.

An X – 5 chart is less sensitive than the X - R chart in detecting special causes of variation that result in only a single value in a subgroup being

Unusual.

Control Charts for Attribute Data

Ylffri&ute data is data that can’t fit into a continuous scale, but instead is chunked into distinct ftudtefe, like small/medium/large, pass/fail, acceptable/not acceptable, and so on (see Chapter 7 for a detailed discussion of attribute and continuous data). Although it is preferable to monitor and control products, services, and processes with more sensitive continuous data, there are

Times when continuous data is simply not available, and all you have is less

Sensitive attribute data. But don’t despair, because certain control charts are

Designed specifically for attribute data to draw out startling information and

Allow you to control the behavior of your process.

With knowledge of only two attribute control charts, you can monitor and

Control process characteristics that are made up of attribute data. The two

Charts are the ° (proportion nonconforming) and the u (non-conformities per unit) charts. Table 10-3 summarizes the important parameters of these charts. Like their continuous counterparts, these attribute control charts help you make control decisions. With their control limits, they can help you

Capture the true voice of the process.

Table 10-3 Attribute Data Control Chart Summary

Control Subgroup Chart Size (n)

Centerline Control Limits

Proportion Variable For individual defective (usually > 50) subgroups:

P chart

Number defective

N,

UCL, = P + 3 LCL, = p – 3

Overall:

Total number defective

P

N, + n 2 +—–+ Nk

Number of Variable For individual subgroups:

Defects per

Unit U, = ■

Number defects

UChart

N

Overall:

Total number defects

USL , = u+3 Jnr LSL = u – 3 Nr

U

N, + n 2 +—–+ Nk

Picture a bowl of soup. If you found a fly in it, you’d deem it unacceptable. What if you found ten flies? You’d still call it unacceptable. Data from cases like this, where something wrong — whether big or small, few or many

Causes you to deem the entire item unacceptable, are called defec-wes. It’s where any one or more things makes the entire thing bad. If you’re charting

Defectives attribute data (pass/fail, go/no-go, acceptable/unacceptable), you

Use a ° chart.

—

Now picture a bowl of soup with three flies in it. This bowl has three defec-s. Some attribute data for control charts is defect data — the number of scratches on a car door, the number of fields missing information on an application form, and so on. If you’re counting and keeping track of the number of defects on an item, you’re using defect attribute data, and you use a u chart to perform statistical process control.

«j«NG/ Although the words sound almost identical, it’s critically important to know ^/~§k\ What type of attribute data you have: whether it’s defectives (pass/fail) data, I (fL\" ) or defect (count) data. If you get this wrong, your subsequent control chart will be completely invalid.

° charts for defectives data are based on a binomial distribution. u charts for

Defects data are based on the Poisson distribution.

The ° chart (or attribute data

The ° chart plots the proportion of measured units or process outputs that are defective in each subgroup. The sequential subgroups for ° charts can be of equal or unequal size. When your subgroups are different sizes, the upper and lower control limits will not be a constant, horizontal value — they will look uneven, as exhibited in Figure 10-10. But the same rules for interpreting the control chart remain — it’s just that the control limits move from subgroup to subgroup.

The proportion of defectives for each subgroup is found by dividing the number of defectives observed in the subgroup by the total number of

Measured in the subgroup.

Proportion of Defectives

Figure 10-10:

P chart for proportion defective.

0.35

0.25

0.15

10

Sample Number

3.0sl = 0.2777

P = 0.2232

-3.0sl = 0.1687

A common application of a ° chart is when you have percentage data, and the subgroup size for each percentage calculation may be different from one

Subgroup to the next. For example, the number of patients that arrive late

Each day for their dental appointment. Or, the number of forms processed each day that have to be reworked due to defects. In both of these examples, the total the size of the subgroups measured could vary from day to day.

R -

° charts are generally used where the probability of a defective is low – usually less than ten percent. So to be effective, the subgroup size needs to

Be large enough to register one or more defectives. It is also important to

Consider the length of time that a subgroup represents: Long periods of time can make it difficult to pinpoint a specific cause.

Remember, just as with continuous control charts, you need to be alert for other indicators of special cause variation in addition to just exceeding the

Control limits. The presence of unusual patterns, such as runs or trends, even

If all the points are within the control limits, can be evidence of instability or an out-of-the-ordinary change in performance.

The u chart (or attribute data

Like with the ° chart, the u chart does not require a constant subgroup size. The control limits on the u chart vary with the subgroup size and, therefore, may not be constant.

Counting the number of distinct defects on a form is a common use of the u chart. For example, errors and missing information on insurance claim forms (defects) are a problem for hospitals. As a result, every claim form has to be

Checked and corrected before being sent to the insurance company.

One particular hospital measured its defects per unit performance by calculating the found number of defects per unit for each day’s processed forms.

Figure 10-11 demonstrates their performance on a u chart.

Figure 10-11:

« chart for insurance claim forms.

Subgroup Subgroup size Defects

Each point on the chart in Figure 10-11 represents the average defects per claim form for that subgroup. Points higher on the chart represent a greater number of defects per unit. The center line, calculated at 1.870, means that there is an overall average process performance of 1.87 defects per form.

Poka-\loke (Mistake-Proofing)

M/sfafo>-proo/ifng or Poka-Yoke (pronounced POH kah YOH kay) as it is known in Japan, is an action taken to remove or significantly lower the opportunity for

An error or to make the error so obvious that allowing it to reach the customer

Is almost impossible. These two approaches are depicted in Figure 10-12.

Figure 10-12:

Applying Poka-Yoke for the prevention and detection of

Defects.

Prevention

Poka-Yoke that focuses here works on mistake prevention or on making

Mistakes impossible.

Detection

Poka-Yoke that focuses here works on mistake detection or on making sure mistakes do not turn into defects.

Poka-Yoke is very consistent with the fundamental aims and philosophy of Six Sigma, and it has wide applicability in manufacturing, engineering, and transactional processes. It is one of the simplest tools to master. It involves the creation of actions that are designed to eliminate errors, mistakes, or defects in everyday activities and processes.

Poka-Yoke starts with an understanding of the cause-and-effect relationship of

A defect. This is followed by the implementation of a remedy that eliminates

The occurrence of the mistakes that lead to that defect. Poka-Yoke solutions

Can include the addition of a simple physical feature, the creation of a checklist, a change in the sequence of operation, a highlighted field on a form, a software message that reminds the operator to complete a task, or any other way of helping to ensure that mistakes will be either totally eliminated or substantially reduced.

You can find a number of everyday examples of Poka-Yoke. Look at the connector for your computer keyboard or mouse. Its shape prevents it from being connected in the wrong place or turned incorrectly, damaging your computer. Or remove the gas cap and look at the gas filler tube on your car. It is designed

So that you can only put the right kind of gas into your car.

Poka-Yoke is also an ideal form of control for transactional processes. Some examples include

F Computer data entry forms will not let you advance until all the information is input correctly.

Checklists are used so items are not inadvertently missed. F Process workflow is automatically routed and executed.

In summary, the objective of the control phase is to establish measurement

Points for the critical *s and other significant parameters of the process to assure that the CTQs are predictable and meet established requirements. Different levels of control have different levels of effectiveness.

The most effective form of process control is sometimes called a 7vpe-7 corrective action. This is a control applied to the process that eliminates the error condition from ever occurring. This is the primary intent of the Poka-Yoke method. The second most effective control is called a 7§pe-2 corrective action. This a control that detects when an error occurs and stops the process

Or shuts down the equipment so that the defect cannot move forward. This is the detection application of the Poka-Yoke method.

In This Chapter

► Understanding the basics of statistics and measurement

► Seeing the difference between short-term and long-term variation

Plotting and graphing data to gain insight

Eveloping a problem statement and forming an objective statement is only your first step to better performance. The second step of the Six

Sigma methodology is to measure performance — and by doing so, to determine the vital few factors that influence the behavior of your process.

Measure is generally the most difficult and time-consuming phase in the DMAIC methodology. But if you do it well — and do it right the first time — you save yourself a lot of trouble later and maximize your chance of improvement. The only way to do this is to measure and observe your critical-to-a (CTX) characteristics (see Chapter 1 for an explanation of CTXs).

When you enter the world of measurement and statistics, you discover the ultimate source of problem-solving power: data. While the idea of data may not be

Exciting to some, it should be very exciting to a Six Sigma practitioner who is tasked with improving a process, an operational unit, or an entire organization.

The 7, 2, 3s of Statistics

Statistics is the distilling of numbers, data, and measurements into knowledge and insight. If you understand a little bit about statistics, you can create a data-leveraging environment in which you gain the utmost value from the information you have.

Whu statistics?

Variation is everywhere, and it diminishes your ability to consistently produce quality results, meet schedules, and stay under budget. This is why organizations have performance problems, and it’s why you define those problems

With a problem and an objective statement for improvement.

So what is the next step? How do you begin to approach the nagging problem of variation and achieve your improvement objective? In 1891, the famous scientist Lord Kelvin provided insight that is valuable today. He said:

When you can measure what you are speaking about, and express it in

Numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind. It may be the beginning of knowledge, but you have scarcely, in your thoughts,

Advanced to the state of science.

In other words, until you include science, measurement, and numbers in yourimprovement efforts, you’re bound to remain in the world of gut feel,

Educated guessing, and marginal improvement power. You may work hard and you may marshal significant resources, but your gains will be meager

Andunsatisfactory.

This is where statistics comes into play. Statistics is the branch of the mathematical sciences used to describe performance with measurements and

Numbers (believe it or not, many people have spent their life’s energy furthering statistical theory, methods, and applications). Statistics is what takes you out of the realm of intuition and guessing and into the realm of objective truth.

When you hear statistics mentioned, do you retreat in terror? If so, you wouldn’t be the first, or the last. Aside from pure fear, statistics are often disdained because of their historical misapplication, summarized succinctly

By A. E. Houseman: "Statistics in the hand of an engineer are like a lamppost to a drunk — they’re used more for support than illumination." And British Prime Minister, Benjamin Disraeli: "There are three kinds of lies: lies, damn

Lies, and statistics."

Others, however, have seen past the terror and misapplication, discerning the true power of statistics. One was H. G. Wells, who predicted that statistical thinking would someday be as essential for citizenship as the ability to read and write. And that is why you are reading this book. Because you, too, recognize (at least enough to pick up this book) that statistics — and their embodiment in Six Sigma — are like vegetables: They’re not always what you want to eat,

But you know they’re good for you.

In fact, the effectiveness of Six Sigma is dependent upon taking accurate and

Appropriate performance measurements, so you can, as Lord Kelvin suggests, advance your improvement efforts to a state of scientific certainty.

Measurement 101

To begin your journey into the world of Six Sigma measurement and statistics,

Suppose you need to find out how long it takes to fill out a certain purchase order form. Each time the form is filled out you record the elapsed time to

The nearest second and plot the result as a dot along a horizontal time scale. The first three measurements — of 41, 50, and 47 seconds — are shown in Figure 5-1.

Figure 5-1:

First three measurements for

Filling out a purchase

Order form.

40 45 50

Time (seconds)

55 60

Notice that the recorded results reveal that variation is inherent in the process. Continuing the study, you take a total of 100 purchase order time measurements. Whenever you encounter a measurement that already has a recording (like 47 seconds), you simply stack another dot on top of the previous dot. The completed chart with all 100 measurements, is shown in Figure 5-2.

Notice in Figure 5-2 that the output values that occur often pile up with multiple dots. For example, 50 seconds is the purchase order completion time

Observed more than any other. Consequently, it has the highest peak in the chart (15 occurrences). Output values that were observed less often have

Lower heights, and output values that were never observed have no dot at all.

Figure 5-2 graphically describes how the measured output is Distributed Along

The time scale. Looking at the chart, you can predict that if you were to measure another cycle of the purchase order process, the elapsed time would

Most likely be around 50 seconds. The chart also shows that times longer or shorter than about 50 seconds would be less likely to occur than those around 50 seconds. A completion time of 30 seconds, for example, is just not

Going to happen. Nor will one of 80 seconds.

Distribution Is the statistical term used to describe the relative likelihood of observing values for a variable factor. Synonymous terms include Probability distribution And Probability density function.

Distributions are critically important in creating problem and objective statements for your Six Sigma project. (See Chapter 4 for details of how to create

Effective problem and objective statements). The reason is because you need to know how your process is performing in terms of its output if you are to

Properly define what needs to be improved.

Understanding distributions is also critical for understanding the behavior of the critical Xs (CTXs) in your process — the few factors or variables that determine the quality and consistency of your expected output. If your output

Metric is "purchase order completion time," then will have certain input factors to Measure, Analyze, Improve, and Control.

What does it mean? Measures o( Sanation location

Suppose you know that the purchase order completion time is a variable with distributed output values. But, following Lord Kelvin’s admonition, you may ask,

"How can I describe this distribution numerically?" Where is the distribution located (central tendency) along the scale of measure?

A distribution can have infinitely many points, and it’s important to fix a location for the distribution so you can then understand the variation around that location. To do this, statisticians have developed three different measures of a distribution’s location: the mode, the mean, and the median.

Mode

The Mode Is the value observed most frequently and is associated with the highest peak of a distribution. If 10 students take an exam, and three score 60, three score 70, and four score 80, 80 is the mode because it occurs more frequently than any other value.

Although it is simple and intuitive, using the mode as the measure of variation location has a drawback: Many distributions don’t have a single clear peak and some have more than one peak of roughly the same height. In these cases, the presentation of a single mode metric by itself does little to deepen your

Knowledge of the variation.

Mean

The most common measure of central tendency is the Mean — widely called the Average. Examples of averages are everywhere — the Dow Jones industrial

Average, grade point average, the average temperature in your hometown and the list goes on.

It’s important to understand that the mean is theoretical rather than real; while the mean may not have actually occurred within your measurements, it is the value most likely to occur Next In a sequence or population of data. The mean, therefore, is a mental model by which the Six Sigma practitioner can make comparisons, make predictions, interpret data, and anchor much of the analytical

Work that is done in order to save money in operations, make customers

More satisfied, and improve products and services.

So how is the mean calculated? It’s really very simple. Imagine having ten paper cups, each holding a different amount of water. What is the average amount of water in a paper cup? How do you determine the answer? Consider combining the contents of each of the ten cups into a large bowl. You’d simply measure the collected amount and divide it by the number of paper cups. This tells you how much water would be in each paper cup if the amounts were forced to be equal — or Average.

Mathematically, this process for calculating the mean is written as where

V x (pronounced ex bar) is the symbol representing the calculated mean. il is the Greek capital letter sigma. In the shorthand of math, it tells you

To sum up (add) all the individual measurements.

V x, Represents each of the individual measurement values. ^ n Is the number of individual measurements in your data set.

So, for the purchasing order example discussed in the preceding section, the

Mean (X) is found by adding up each of the N = 100 time measurements and dividing the result by N = 100. The result is an X Of 49.9 seconds. Computing the mean for any distribution is never any harder than that.

Median

The Median Is the point along the scale of measure where half the data are below and half are above. The median is the preferred measure of variation location when your collected data contains outliers, or extreme data points

Well outside the range of other data. An Outlier Is a recorded observation that

Is well outside the range of variation of the rest of the data.

For example, the median is often used when communicating home prices because it is usually more reflective of the central tendency of the distribution of all prices. Suppose you have a set of homes with prices of $158,000, $200,000, $178,000, $125,000, and $535,000. The average price is $239,200. The median, however, is $178,000. Figure 5-3 shows the raw data of home prices with the mean and median specified. Note that the median represents the

Location of the distributed home prices better than the mean. That’s because

The calculated value of the mean is pulled up away from the more accurate location value by the presence of the outlier — the $535,000 home.

The median is the preferred communicator of variation location when the

Data you are describing contains outliers.

Median Mean

Figure 5-3:

Graphical I I

>mparis0n ’ *

Comparison of the mean and median.

H"*Ґ-1-1-I

$100,000 $200,000 $300,000 $400,000 $500,000 $600,000

Oj»NG/ Beware when someone communicates only the average to you when */~ik\ Describing a distributed variable. Without your knowing, he or she could I I have included an outlier (accidentally or on purpose) that has biased the

Calculated mean value.

Putting all three together

Table 5-1 describes the mode, mean, and median of variation location. But

Although measures of variation location are indispensable, they don’t tell the

Whole story. The mode, mean, and median all fail to communicate the critical

Information of how spread out or how widely or narrowly dispersed the data

Is around its central location point. The following section gives you some additional options.

Table 5-1 Summary of Statistical Measures of Variation Location

Measure of Variation Definition Comment Location

Mode Peak of distribution Problematic, seldom used

Mean (average) _ Y.x, Most common and familiar

Median Point where half of data are Used when data contains

Below and half are above outliers

Ho© much Variation is there!

Two sets of measurements having identical means may contain raw data

Values that are distributed very differently. A second measure is needed in addition to the measure of location. You need to be able to quantify how widely or narrowly dispersed the data are around their central location.

The simplest measure of the spread of your data is its range. The Range Of a

Distribution is defined as the difference between the largest and the smallest

Observed data values. Mathematically, this is written as

R = xMAX – xMIN

Where

F R Is the calculated range

XMAX is the largest observed measurement xMIN is the smallest observed measurement

In the preceding purchase order example, the range is simply the longest recorded time to fill out the form (60 seconds) minus the shortest time (41 seconds), or R = 19 seconds.

Calculating the range works just as well when you only have two recorded measurements as it does when you have 1,000. But obviously, outliers directly affect its calculated value. (By their very nature, outliers end up being the xMAX or xMIN used to calculate the range.)

Is there another way to quantify a distribution’s degree of dispersion that avoids the problem of outliers? Look at any single recorded measurement. How far is it from the central location of the data set? Mathematically this

Problem is written as

X, Represents any single recorded measurement from your set of data V x Is the calculated mean of your collected observations

Xt – X Then acts as a numerical "score" for each data point. Like in golf, the lower the magnitude of the score, the better (the less it varies from the central location).

You’ve probably played around with numbers enough, however, to recognize a problem with this scoring system. When X, Is less than the mean (X), the score (x, – X) ends up being less than zero. That won’t work! A point being above or below the central location doesn’t matter; it’s how far away it is that counts. And negative scores make things too complicated. There needs to be a way to score each data point that looks only at its distance from the central location

Regardless of which direction.

For example

The parentheses and the raised "2" tell you to take the quantity X< – x And multiply it together twice. For example, (2)2 = 2 X 2 and (-3)2 = -3 X -3. Notice that no matter whether the quantity you are multiplying by itself is positive or negative, the resulting answer is always positive: 2 X 2 = 4 and -3 X -3 = 9.

The area of a square is the length of any one of its sides (called I For length) multiplied by itself, that is /2. This is why mathematicians call this operation Squaring And the numerical result a Square. In Six Sigma you hear the term "squares" often. Every time you do, remember the mathematicians and know that some quantity is being multiplied by itself.

A side benefit of squaring each individual score is that it penalizes points that are farther away from the central location disproportionately more than those that are close. Figure 5-4 shows a plot of (x,-x) Output values versus input values for x, – x.

Referring to Figure 5-4, if an individual data point is one unit away from the central location (either above or below), x, – x = 1 and (x, – X)2 = 1. If a different point is twice as far away, however, with x, – x = 2, then (x, – x) = 4, resulting in a score that is not twice as bad, but Four times Worse.

To create an overall, combined score for the entire data set, simply add all the individual squared scores together. Mathematicians write this as

2(x,- – x )2

11 is the Greek capital letter sigma, which tells you to add up all the individual

Squared scores.

Statisticians call this result the Summed squared error, Or SSE for short.

In the field of statistics, Error Doesn’t mean something is wrong. The term simply means a calculated deviation from a comparison value. In this case,

Error is the difference between the mean and the individual observations.

Having totaled up all the individual squared scores, what is the typical (average) squared score? To find out, you divide the summed squared error by the number of Mdependent Data points in our collection, like this:

_2 = Z(x, - X)2

N – 1

Where N Is the total number of data points you have collected.

Statisticians call this averaged squared error score the Variance And give it the symbol cr2.

Right now, one of two things has happened. Either your eyes have glazed

Over, or you are saying, "Hold on a minute. Why did you divide by one less than the number of measurements I collected (n – 1) instead of by the total I collected (n)? That doesn’t seem right."

Assuming your eyes are not glazed over, you’ve asked an outstanding

Question. In the equation for the variance, notice that the mean (X) is

Included. This is where you lose the ,ndependence Of one of your collected measurements.

It’s like having a full five gallon bucket and dividing the contents completely

Into ten new buckets. Even though the amounts you pour into the first nine buckets can vary independently from each other, when you get to the tenth bucket there is no more freedom — what you have is exactly what remains. In the same way, using the mean (x) reduces the number of independent measurements available to compute the variance.

A final problem lingers with the development of a measure of how widely or

Narrowly your collected data is distributed. What are the units associated with the computed variance? In the preceding purchase order example, your measurements have been in seconds. That means the variance comes out as

Seconds2. In the real world, what are seconds2? No one knows (and anyone who thinks they do ought to be avoided!)

The person who originally solved this last issue must have known the answer from the beginning. Notice that the symbol for the variance is cr2. And as you’ve likely guessed, the solution is simply to reverse the squaring done previously

To your measurements. Mathematicians call this reverse-squaring operation

The Square root And give it a special operator symbol ( /").

Applying it vigorously to the variance introduces the greatly anticipated measure sought after, namely, the standard deviation:

A -12(Јi-*I V n - 1

The standard deviation is by far the most commonly used measure of dispersion. Represented by the Greek lower case letter sigma, it occurs throughout statistics and Six Sigma — to which the quality initiative owes

Itsname.

What is the real-world meaning of the standard deviation metric? Its units are exactly the same as your original measurements. So for the purchase

Order example, its units are seconds. The standard deviation represents the

Typical (average) distance from the central location you expect to observe. See Table 5-2.

Armed with two quantities — a measure of location and a measure of spread — you can now describe any type of distribution in scientific terms. Lord Kelvin

Would be proud.

The Long and Short of Variation

Peeling the layers of the onion back, there is another aspect of variation

Youneed to know about: the difference between long – and short-term variation.

Short-term Variation

Suppose you monitor certain characteristics of a process — such as the volume of inbound calls per hour at a customer call center — over an extended period. After each hour, you measure and record the number of calls received. To review what you’ve observed, you graph your collected measurements as a sequence of connected points along an axis representing time, as shown in Figure 5-5.

Figure 5-5:

The observed output behavior of a process over an extended

Period

Volume of

Inbound calls at a customer call center.

0 50 100 150 200 250 300 350 400 450 500

Hour

Although the points graphed in Figure 5-5 represent the number of incoming calls per hour, you should recognize that it could also represent any process characteristic in any type of company. All process characteristics vary from cycle to cycle: the exact length of newly manufactured pencils, the time

Required to fill out an invoice, the number of calls per hour, and so on.

70

S0

50

40

30

20

If you zoom in on a narrow portion of the graph, as shown in Figure 5-6, you can see from the scattered points that the output certainly does vary for each measurement cycle. But you can also notice that the variation is not limitless. It lies within upper and lower boundary limits — represented by the dashed,

Horizontal lines.

In fact, for any selected Short period of time, The process essentially varies within the same rough limits. (Try it for yourself. Pick any short time segment

Of Figure 5-5 and eyeball the vertical variation limits with your thumb and index

Finger. Now, keeping the distance between your fingers fixed, move to a different time section of the graph. Do your eyeballed limits capture the output variation for other short-time segments?)

This natural level of variation is called the Short-term Variation of a process. Often, it is designated with a simple ST Notation.

Short-term variation is purely Random. This means that, like rolling a pair of dice, you cannot predict what the next output value will be. If you could, Las Vegas would be bankrupt in a week!

Short-term variation is caused by the combined effect of all the little things that are too hard to include in your understanding of the process. Even Einstein would find it too difficult to determine exactly how the microscopic textures of the dice contribute to their spin as they contact the felt surface of

The table. Or how the drag of the swirling air on the corner of the airborne

Dice alters their tumble. Yet these factors — and many more — are real and

Do add up to affect the outcome of the roll.

This is the reality of short-term variation in any and all processes, from rolling dice to preparing a meal to writing a memo to launching a rocket: The complete chain of causation is unknown and unknowable. Like rolling dice, your

Ability to understand the full depth of causation for any process is ultimately

Limited.

Because these small forces are present to some degree in all processes, they

Are referred to as Common. Consequently, the short-term variation they cause is sometimes called Common cause variation.

Now that you know what short-term variation is, you need to know how to quantify it. The formula for calculating the standard deviation given back in Table 5-2 does not account for any short – or long-term effects. It just looks at the overall variation in all the measurements. But never fear, hard-working statisticians have developed a way to extract the level of the short-term variation out from the overall variation.

The quickest way to get to the short-term variation is to analyze the separation

Or differences between sequential measurements of a critical characteristic.

The difference between any two sequential measurements can be thought of as a kind of range. For a sequence of measurements

Xl, X2, . . . Xn_x, Xn

The difference or range between the first and second measurements can be written as

R I = |x i – X 2|

In general, the difference between any two sequential measurements is

Ri = \x, – x,+1|

And the average range or difference between sequential points is

R = § R, n – L F-i i

The way to calculate the short-term standard deviation from these sequential, between-point ranges is to take their average and multiply it by a special

Correction factor based on the range between two sequential measurements:

Never try to calculate a characteristic’s short-term standard deviation on anything other than a sequential set of measurements. That is, only perform this

Calculation on a set of measurements that are in the order that the measurements were taken. This is because the calculation of the short-term standard deviation is based upon the natural ranges that occur between the characteristic’s measurements; if the order of the measurements is altered at all, it directly effects the calculated value of the short-term standard deviation.

Shift happens: Long-term Variation

Take another look at the extended process behavior graph in Figure 5-5 in the preceding section. Something else besides pure random variation is going on here. Notice that the range of short-term variation doesn’t stay locked at a single level. Instead, it "shifts and drifts" up and down over time. These bumps and currents — called Disturbances to the process — are emphasized with overlaid lines in Figure 5-7.

Figure 5-7:

Non-random disturbances overlaid on the extended process behavior graph.

0 50 100 150 200 250 300 350 400 450 500

Hour

When these underlying disturbances are added to the natural short-term variation, the overall combination is called the Long-term variation of the process. In many cases, it is written with a simple LT Notation.

As opposed to random, short-term variation, these underlying disturbances are Non-random Over the long-term. You can approximate them with a line, a step, a curve, or a repeated pattern. When gambling in Las Vegas, you know

That the long-term disturbance will result in your losing all your money.

(Note: You can prevent this by quitting while you’re ahead in the short term.) The great thing about long-term variation is that you don’t have to be Einstein

To figure it out. With the proper detection techniques and tools, you can see

What part(s) of your process is affected by non-random forces. If the process is to assemble a proposal, and if the critical output of that process is how long

It takes to create the proposal, you want to look at the variation patterns in the output of the process.

Figure 5-7 shows just the Output variation, Or changes in the number of incoming calls at a call center per hour. If the output metric varies in a non-random way, it is safe to say that some combination of special cause factors has affected

The volume of incoming calls.

When we say Special cause, We mean that the output has varied to an extent that is inconsistent with what you would expect from purely normal, short-term, natural — or random — influences. You know that something non-random has occurred and, therefore, you know that you can find the cause and solve

The problem.

A good way of depicting the difference between short-term and long-term variation in a process is with the use of two "probability distributions," as shown in Figure 5-8. Notice that, over time, the long-term variation is wider

Than the inherent, short-term variation.

Figure 5-8:

Long-term (LT) and short-term (ST) process variation summarized as probability distributions.

0 50 100 150 200 250 300 350 400 450 500

Non-random variation is caused by Special Forces whose effects on the process are readily observed and understood. Consequently, this non-random variation is also called Special cause variation Or Assignable cause variation.

Calculating the long-term variation of a characteristic is identical to calculating its overall variation. Therefore, the overall standard deviation is the formula you use to quantify the level of long-term variation in a characteristic. See Table 5-3.

Hour

Table 5-3 Formulas for Calculating Short-Term _And Long-Term Standard Deviation

Short-Term Standard Deviation_Long-Term Deviation

The calculated short-term variation should always be less than or equal to the calculated long-term variation.

This is the crux of the difference between common cause and special cause

Variation: If you can’t see microscopically enough to understand exactly why some variation occurred, you surely can’t do anything to stop it from occurring again (the dice example). On the other hand, if you can see and understand why variations happen, you have a reasonable opportunity to stop the problematic variations from happening at all.

For any example of special cause variation, notice that you can immediately create solutions to solve the problems. You can conduct routine preventive maintenance on your drills. You can make the maintenance procedure so easy that anyone can understand and adhere to it. You can create redundant

Systems in case equipment breaks down or in the advent of losing a principal

Leader. And so on.

Poka-Yoke Is a Japanese term that means "mistake-proof." It is a fundamental concept behind the practice of making processes so easy and simple to follow that even a child can perform them. Essentially, when you Poka-Yoke a process, you vaccinate it against error (see Chapter 10 for details on Poka-Yoke).

Be alt uou can be: Entitlement

For every process, there is natural, short-term variation happening concurrently with an underlying, long-term, shift-and-drift variation. The short-term

Component comes from the unaccounted common causes rooted within the process. The long-term component is the result of factors you can detect — special causes. Suppose that you want to reduce the overall level of variation

In your process. What do you do? How will your new understanding of short-term and long-term variation guide your approach?

Imagine you identify and remove all of the non-random, special causes affecting your process. You’re left, then, with process that is influenced by only random,

Short-term variation. It’s guaranteed: You find that further shrinking the output

Variation is difficult — very, very difficult. That’s because further shrinking requires discovering what previously was unknown about the inner workings of your process. You need to identify, understand, prioritize, and fix the myriad

Of embedded, common factors jiggling the process output.

This hard wall in the improvement path leads to the idea of entitlement. Entitlement Is the level of variation that is naturally built into a process. It is the amount of variation you can expect from a process under the best conditions — even when all the special causes are identified and eliminated.

(Of course, you can see that this is just another name for short-term variation.)

What’s the difference between short-term and long-term variation?

Short-term variation is synonymous with common cause variation — because all the non-random influencing factors have not had time to express themselves or exert their effect on the outcome. Long-term variation is synonymous with special cause variation, because non-random influencing factors have had time to express themselves and affect the outcome. Therefore, there is no set time period where short-term variation transitions into long-term variation for every process characteristic. The transition point depends totally on the process and the time it takes to sufficiently characterize the process.

Some things can go wrong (assignable causes) In a manufacturing environment:

F Tool wear: The bit that drilled holes in your new assemble-it-yourself desk was too worn down at the time it was employed during the manufacturing process. You, and 500 other people who bought tables from the same production batch, now struggle to assemble your desks while muttering unkind

Words about the manufacturer.

Changes in machine operator: Jack replaces Jill on the printing press but doesn’t do his required maintenance. Print quality suffers,

Customers are unhappy, and the finger -

Pointing begins.

W Differences between raw materials: Print quality also suffers non-randomly when Jill is on her shift, because the quality of the ink is sometimes compromised by the supplier, and at other times the ink is slightly off its target color value.

Some things can also go wrong (also Assignable causes) In a service environment:

Equipment breakdown: Your computer crashes, preventing you from providing great customer service at the call center. The jet

Carrying the express mail needs unscheduled maintenance at the airport, thus making the deliveries late.

External forces: Traffic jam patterns in certain geographical areas negatively impact the productivity of a trucking company. Inventory gets backlogged, and deliveries are late.

Health of service provider: The lead litigator in a very important case becomes ill and cannot perform his duties. His colleagues do not have the depth of knowledge and experience to maintain the momentum created.

Long-term variation is always greater than short-term, or entitlement,

Variation.

Short-term, or entitlement, variation is what you use to compare the capability

Of different processes to meet a specified goal. For example, creating a shaped plastic part using an injection mold machine may have an entitlement variation of ±0.002 inches. The process of cutting plastic with a milling machine, on the other hand, may have an entitlement variation of ±0.0005 inches. In this case,

The milling machine process has the better level of entitlement. It has less inherent, short-term variation.

Clearly, a fundamental task in Six Sigma is to observe processes and understand their levels of short-term variation and long-term variation. The only way to really know the capability of a process is to engage in an effort to gather and understand sufficient data about how the process is working. By doing this, you begin to reach the heart of Six Sigma: measuring the gaps between

Current performance and entitlement performance and addressing those gaps.

A Picture’s Worth a Thousand Words

Crunching numbers and data into statistics — like a mean or a standard deviation — provides numerical insight into the inner workings and outside

Influences of a process. Pictures of data, however, often serve as a more

Intuitive way of gaining the same insights. These pictures — called Graphs Or Plots — are definitely better than numbers at communicating your gained

Insight to others.

Using visual material to communicate data is your best way of getting improvement team members to be integral parts of the Six Sigma breakthrough process. When team members can see the reality of performance for themselves, they

Are more motivated to contribute and participate in measurement and improvement efforts. Also, your visual pictures are an effective and essential prop for communicating your project details to management.

Plotting and charting data

The chief purpose of plotting and charting data is to graphically show the central tendency and the spread of variation in a measured item of interest.

You can do this in a couple of different ways, each with its advantages and disadvantages.

Creating dot plots and histograms

Dot plots and histograms both do the same thing, they show ©here The variation occurs in a critical characteristic. Is the variation all lumped together within a narrow interval? Or is it evenly spread out over a wide range? A dot

Plot or a histogram reveals the answer.

After collecting measurements or data for a characteristic, create a Dot plot, Or Histogram, For it by using the following steps:

1. Create a horizontal line, representing the scale of measure for the characteristic.

This scale can be in millimeters for length, pounds for weight, minutes for time, number of defects found on an inspected part, or anything else that quantifies what it is about the characteristic you’re interested in.

2. Divide the horizontal scale of measure into equal chunks or "buckets"

Along its length.

Select a bucket width that makes it so that there are about 10 to 20 equal

Divisions between the largest and the smallest observed values for the characteristic.

3. For each observed measurement of the characteristic, locate its value along the horizontal scale of measure and place a dot for it in its corresponding "bucket."

If another observed measurement falls into the same "bucket," stack the

Second (or third, or fourth) dot above the previous one.

It is not a requirement to use dots. You can use whatever symbol or

Character is available or easy for you to draw.

4. Repeat Step 3 until all the observed measurements are placed onto

The plot.

To create a histogram (so that you can impress your peers with a graph that

Has a much more complicated-sounding name), replace each of the stacks of dots with a solid vertical bar of the same height as its corresponding stack

Ofdots.

Interpreting dot plots and histograms

A dot plot and its fancy cousin, a histogram, offer ready access to a wealth of

Information about the variation of a characteristic’s performance.

F Variation shape: Is the variation of a characteristic lumped around a single spot? Or is it spread out evenly across a range of values? A dot plot or histogram reveals the answer immediately. Figure 5-9 shows a variation shape that is Normally Distributed or bell shaped. For a normal

Distribution, most of the observed values of the characteristic are close to a central point with fewer and fewer appearing as you get farther

Away from the central tendency. Figure 5-10 shows a Uniformly Distributed variation for a characteristic.

For a uniformly distributed characteristic, the variation is evenly spread out across a bounded range. That is, you’re just as likely to observe a value for a characteristic at one end of the interval as you are at the other, or anywhere in between. Figure 5-11 shows a Skewed Distribution shape. A skewed distribution is a variation shape that is not symmetrical. Either

One side or the other of the distribution extends out farther than the other side.

Variation mode: The mode of a distribution is its most likely value, or in other words, its peak. Usually, the variation in a characteristic has a

Single peak, as seen in Figure 5-12.

Histogram of Uniform

Figure 5-10:

Histogram showing a

Uniformly

Distributed variation of a characteristic.

35 30 25 20 15 10

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Uniform

5

0

A histogram showing two or more distinct peaks is Multi-modal. This means that two or more values dominate the variation. Multiple major peaks is not usual. It typically means that there is a factor affecting the

Characteristic’s performance that causes the entire system to behave

Schizophrenically.

But sometimes, a characteristic displays two or more modes, like shown

In Figure 5-13.

Histogram of Skewed

3 4 Skewed

Histogram of Normal

±

I I I. I I

10.8

12.0

13.2 14.4

Normal

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16.8

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0

2

5

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7

Histogram of Bimodal

12

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Bimodal

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24

27

When you encounter a multi-modal distribution, always dig deeper to discover what factor or factors is causing the characteristic’s schizophrenic behavior.

V Variation average: Without having to crunch any numbers, you can visually estimate a characteristic’s mean or average value from a dot plot or

Histogram.

Hold your index finger up against the horizontal axis of the dot plot or histogram. Move your extended finger back and forth across the horizontal axis until you find the point where the middle knuckle of your finger balances your distribution equally on each side. Voila! The point

Along the horizontal axis where you’ve located your finger is the approximate average value of the variation.

F Variation range: The extent or width of variation present in a characteristic is immediately recognized in a dot plot or histogram. The difference between the greatest observed value xMAX and the smallest observed value xMIN creates what is called the Range Of the distribution. The symbol R always represents the range. The range is calculated by

R = xMAX - xMIN

F Outliers: Outliers are measured observations that don’t seem to fit the grouping of the rest of the observations. They’re either too far to the right

Or too far to the left of the rest of the data to be concluded as coming from the same set of circumstances that created all the other points.

And that is exactly their value. When you see an outlier or outliers on a dot plot or histogram, you immediately know that something is probably different about the conditions that created those points, whether in

The set-up or execution of the process, or in the way the process was measured.

Investigate all outliers. Find out what caused their value to be so different from all the other observed values. Isolating the cause almost always leads to the discovery of what factors are degrading the performance of the characteristic.

If you want to get more quantitative with your dot plots and histograms, you

Can use them to calculate the proportion of observations you’ve measured

Within an interval of interest. Or you can use them to predict the likelihood

Of observing certain values in the future. (Seeing into the future is definitely powerful stuff!)

Suppose you measure a characteristic 50 times. Counting and adding up what’s in each of the buckets of your dot plot or histogram, you observe 17 measurements that occur between the values of, say, 5 and 6. You can conclude, then, that 17 out of 50, or 34 percent, of your measurements ended up between 5 and 6. Now peering into the future, you can also say that if the characteristic

Continues to operate as it did during the time of your measurements, that

34 percent of future observations (that you haven’t even made yet) will end up being between 5 and 6! The casinos of Las Vegas thrive in business because they use Six Sigma in this way to know what will happen in the future, say,

When you sit down for a game of craps. Creating box and Whisker plots

The problem with dot plots and histograms is that they only allow you to effectively look at one characteristic’s performance at a time. When you need to compare distributions, few things are quicker to do or more easy to interpret than a box and whiskers plot. Like putting two people back-to-back to see who is taller, box and whisker plots allow you to directly compare two or

More variation distributions.

Box and whisker plots Are sometimes simply called Box plots. A box and whisker plot is made up of a Box Representing the central mass of the variation and thin lines, called Whiskers, Extending out on either side representing the thinning tails of the distribution. An example of a box plot is shown in

Figure 5-14.

To create a box and whisker plot:

1. Rank the captured set of data measurements for the characteristic.

Reorder the captured data from the least to the greatest values.

2. Determine the median of the data.

Find the observation value in the rank ordered data where half of the

Data lies above and half lies below.

Box Plot of Characteristic

Figure 5-14:

Box and whisker plot example.

5.5

-r

6.0

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6.5

-r

7.0

T

7.5

Characteristic

8.0

T

8.5

-r

9.0

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9.5

When the number of observed points in your data set n is odd, the median is the

N+1′

Th

Value in the rank ordered sequence.

When the number of observed points in your data set n is even, the median is the average of the

Th and the

F+1]th

Values in the rank order sequence.

3. Find the first quartile <?,.

The first quartile is the point in your rank ordered sequence, where 25 percent of the observed data fall below this value.

4. Find the third quartile <<3.

The third quartile is the point in your rank ordered sequence, where

75 percent of the observed data fall below this value.

5. Find the largest observed value XMAX.

6. Find the smallest observed value XMIN.

7. Create a horizontal line, representing the scale of measure for the characteristic.

This scale could be in millimeters for length, pounds for weight, minutesfor time, number of defects found on an inspected part, or

Anything else that quantifies what it is about the characteristic you’re

Interested in.

8. Construct the box.

Draw a box spanning from the first quartile Q1 To the third quartile Q3 and draw a vertical line in the box corresponding to the calculated

Median value.

9. Construct the whiskers.

Draw two horizontal lines, one extending out from the Q1 Value to the smallest observed observation XMm, And another extending out from the Q3 value to the greatest observed value XMAX.

10. Repeat Steps 1 through 9 for each additional characteristic to be plotted and compared against the same horizontal scale.

When you have a large set of data for a characteristic, you may find value in extending the whiskers out only to the 10th and 90th percentiles, or to the 5th and 95th percentiles, and so on. Then when outlier data points fall beyond these ends of the whiskers, you can draw them as disconnected dots or stars. This is a great way of graphically identifying and communicating the presence

Of outliers in your data. Interpreting box and Whisker plots

Box and whisker plots are ideal for comparing two or more variation distributions. These may be before and after views of a process or characteristic. Or they may be several alternative ways of conducting an operation. Essentially, when you want to quickly find out if two or more variation distributions are different (or the same) then you create a box plot. Figure 5-15 is an example of using box plots to compare distributions A, B, and C.

In Figure 5-15, distribution B clearly has the lowest level. But it still overlaps the performance of distribution A, indicating that it may not be that different. Distribution C, On the other hand, has a much higher value and no

Overlap with distributions A and B. It also has a much broader spread to its

Variation.

Box Plot of A, B, C

Figure 5-15:

Graphical comparison of three variation distributions using box and whisker plots.

5.0

7.5

10.0 Data

12.5

15.0

A

B

Things to look for in comparative box plots:

Is Differences or similarities in location of the median

Is Differences or similarities in box widths

V Differences or similarities in whisker-to-whisker spread

F Overlap or gaps between distributions

F Skewed or asymmetrical variation in distributions

The presence of outliers Creating scatter plots

Dot plots, histograms, and box plots chart only one distribution (characteristic) at a time. Often, you need to explore the relationship between two characteristics. To do this, you use a Scatter plot. Scatter plots get their name from their appearance — a scattered cluster of dots on a graph.

The key to creating a scatter plot is in the capturing of the measurement data.

To investigate the relationship between two characteristics, you need to capture measurements from the two characteristics simultaneously. So at each measurement time, you have to take simultaneous measurements for each of

The characteristics you are interested in. If you are interested in exploring the relationship between characteristics X and Y at each point of measurement, you have to collect and record values for X and Y.

The two characteristics being plotted can be two inputs. Or, alternatively, one can be an input and the other can be an output. As long as your measurements are made simultaneously, it doesn’t matter if they are inputs or outputs.

With this simultaneous data collected, you’re now ready to create a scatter plot:

1. Form points from the collected data.

At each of the measurement times, pair the simultaneously measured values for the two characteristics together to form an x-y point that can be plotted on a two-axis graph.

2. Create a two-axis plotting framework.

Create two axes, one horizontal and the other vertical, with each being

Assigned to one of the two characteristics under investigation.

The scale for each axis could be in millimeters for length, pounds for weight, minutes for time, number of defects found on an inspected part,

Or anything else that quantifies what it is about the characteristics you’re

Interested in.

3. Plot each formed point on the two-axis framework.

Figure 5-16 shows a sample set of simultaneous measurement data and a corresponding scatter plot.

Figure 5-16:

Scatter plot example showing output characteristic Y being plotted

Against input characteristic X3.

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1.5

Scatter Plot of YvsX

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8

5 11

7

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4

Scatter plots can also be created when one of the characteristic data types

Isnot measured on a continues scale, but fits into discrete categories. For

Example, the characteristic of sales volume (measured on the continuous dollar scale) can be plotted against marketing plans 1 and 2 (measured by two discrete categories). Figure 5-17 shows an example of this type of category data scatter plot.

Scatter Plot of Sales ($K) vs Marketing Plan

Figure 5-17:

Scatter plot sample for category

Data

15 14 13 12 11 10 9 8 7

Marketing Plan

2

Interpreting scatter plots

A scatter plot tells you graphically how two characteristics are related. They may be strongly related or not related at all. A scatter plot immediately tells you the answer. Correlation Is the word used to quantify how closely related

Two characteristics are to each other. Things to look for in a scatter plot:

Is Amount of correlation: If two characteristics are not related, the scatter

Plot of the two should appear as a random cloud of points, like shown in

Figure 5-18.

When two characteristics are unrelated, there is no pattern or trend or

Grouping among the plotted points. It is instead a random scattering of points.

On the other hand, when two characteristics are related, a pattern, trend, shape, or grouping in the plotted points emerges. For example,

Figure 5-19 shows the earlier scatter plot of Figure 5-16 with an overlaid

Line to highlight the trend.

Scatter Plot of YvsJf,

Figure 5-18:

Scatter plot example of two characteristics

Y and X2 that are not related (note

The lack of

Any pattern

In the plotted

Points).

-0.50

-0.25

0.00

0.25

0.50

0.75

8

7

6

5

4

Scatter Plot of YvsX,

Figure 5-19:

Scatter plot showing correlation between characteristics Y and X3.

1.5 1.6 1.7 1.8 1.9 2.0 2.1

^3

2.2 2.3 2.4

8

7

6

5

4

Whenever you can naturally fit a drawn line to a set of plotted points, as in Figure 5-19, you know that the characteristics are correlated. The Amount Of correlation is determined by how closely or tightly the plotted points fit a drawn line. If a line can only loosely fit the plotted points, there is only a

Slight relationship between the characteristics. If, however, the plotted points

Are tightly clustered around a line, there is a high correlation between the

Characteristics.

Of course, the reason you should be concerned with how closely certain input and output characteristics are related is because you’re trying to find operational leverage. You are looking for the factors or variables that can positively influence your desired performance improvement outcome as defined by your project objective statement. See Chapter 2 for a deeper

Discussion of this concept.

How closely clustered do the scatter plot’s points need to be before there is evidence of significant correlation? A good rule of thumb is the Fat pencil test.

Imagine laying a fat pencil on top of the drawn line fitting the plotted points.

If the fat pencil body covers up the plotted points, it passes the test, and you

Can conclude that there is enough correlation between the two characteristics to call it significant.

V Direction of correlation: Two characteristics are Positively correlated If

The relationship indicates that an increase in one characteristic translates

Into an increase in the other. Figure 5-20 shows a scatter plot with a positive correlation between two characteristics.

Two characteristics are Negatively correlated If the relationship indicates

That an increase in one characteristic translates into a decrease in the

Other, and vice versa. The earlier Figure 5-19 shows a scatter plot with a

Negative correlation between two characteristics.

F Strength of effect: Scatter plots also graphically show the strength or

Magnitude of the effect one characteristic has on the other. Two characteristics may be strongly correlated (that is, tightly clustered around a fitted line). Yet a large change in one characteristic may still lead to only a small change in the other. Alternatively, there are situations where a small change in one characteristic is magnified as a large change in the

Other.

The way to visualize this strength of effect between two characteristics

Is to look at the slope of the line fitted to the scatter plot points. Figure 5-21 shows three scatter plots, one for each of three input characteristics’ effects on an output characteristic Y. The steepness of the slope of

The fitted lines determines how strong an effect the input has on the output. Steep slopes mean strong effect.

Scatter Plot of YvsX

> 6

Figure 5-20:

Scatter plot showing positive correlation between two characteristics.

3.50

3.75

4.00

4.25

4.50

8

7

5

4

Scatter Plot of Yvs X, X2X3

Figure 5-21:

Three scatter plots, one for each input characteristic X„ X2, and X3

Against a single output characteristic Y.

1.5

2.0

2.5

The slope of a line is how steep it is. The slope is quantified mathematically by comparing how much the line climbs up to how much it runs across between two points. This comparison is formed from a ratio of rise to Run.

For example, given two points on a line (xu yd And (x2, y2), the slope is calculated by

If the calculated slope is zero, that means the line is horizontal or flat. A negative slope means that the line slopes down from left to right. And a positive

Slope indicates a line that slopes up from left to right.

As the calculated slope value gets farther away from zero (either positively or negatively), the steepness of the line increases. When you get to a slope of positive infinity or negative infinity, you have yourself a vertical, straight up -

And-down line.

In Figure 5-21, you can compare the slopes of the fitted lines for each input characteristic X,, X2, And X3 To the output characteristic Y The scatter plot

Showing a correlation with the greatest slope indicates the greatest impact or

Effect on the output. So in Figure 5-21, characteristic X3 Has the greatest effect on the output Y.

Another way to say this is that if you wanted to effect a one-unit change in the output Y, Then input characteristic X3 Would have to be modified the least to get that change in the output. X3 Is the largest point of leverage among the

Input characteristics.

When you use a scatter plot to determine the strength of effect one characteristic has on another, it is often called a Main effects plot. You can read a lot more about main effects plots in Chapter 9.

Scatter plots are a simple yet extremely powerful tool you can use to explore and quantify the relationship between two or more characteristics. It really is the start of getting to the fundamental Y = f(X) Relationship at the heart of Six Sigma improvement. Scatter plots start to get at the heart of how certain variables impact other variables, how certain inputs either inhibit or enhance your ability to create your desired outcomes.

Hindsight is 20/20: Behavior charts

Dot plots, histograms, box plots, and scatter plots all ignore a critical element: time. None of these graphical methods takes into account the order in which

The measured data is observed. Time or order are critical factors, especially

When you’re trying to figure out the causes behind variation and changes in

Process behavior.

Creating a characteristic or process behavior chart

To investigate the behavior of a characteristic or process, plot your observed measurements one at a time along an axis representing time or order, in the exact sequence the measurements occurred in real life.

To create a characteristic or process behavior chart:

1. Create a horizontal scale representing time or order.

You usually do this by creating an axis for the order in which the measurements occurred, called their Run order.

2. Create a vertical axis representing the scale of measure for the

Characteristic.

This scale could be in millimeters for length, pounds for weight, minutes for time, number of defects found on an inspected part, or anything else that quantifies what it is about the characteristic you’re interested in.

Set the maximum and minimum values on this vertical scale just slightly larger and slightly lower than the maximum and minimum observed data

Values, respectively.

3. Plot each observation as a dot using its order and measurement.

4. Connect the dots.

Draw a line between each sequential point to emphasize the change that occurs between observations.

Figure 5-22 shows an example of a behavior chart for the completion time of an assembly process.

Interpreting characteristic or process behavior charts

Under normal conditions, a process or characteristic should behave normally. This statement is more profound than it sounds. The performance

Of every process or characteristic has natural variation. A behavior chart graphically shows how that variation plays out over time.

Like in Figure 5-22, a process or characteristic has variation that bounces around a central, horizontal level on the behavior chart. Most of the observed variation will be clustered close to this central level. Also, every now and then, there will be excursions that are farther away from the center. The variation will be completely random over time, without patterns or trends. This

Type of behavior is the definition of Normal, And is analogous to the entitlement level of variation covered in "Be all you can be: Entitlement" section.

20

Figure 5-22:

Characteristic or process behavior chart example. Each observation of the characteristic is

Plotted in the order in which it was measured.

18

16

14

12

10

I

I

20

40

60

80

100

Run Order

8

0

A behavior chart not only allows you to see the normal behavior of a process or characteristic, it also allows you to quickly detect Non-normal Behavior — variation above and beyond the expected normal level. The causes of non-normal behavior are the assignable or special causes spoken of earlier in this chapter that erode and degrade entitlement performance over the long term. Behavior charts form the foundation of detecting and finding the root cause

Of non-normal behavior.

Things to look for in process and characteristic behavior charts:

F Variation beyond expected limits: Outliers are measurement observations that occur beyond the limits of the normal short-term variation you expect out of the process or characteristic.

Outliers are non-normal because you don’t expect to see them. It’s like rolling five doubles in a row with a pair of dice. Five doubles in a row is possible, but when it happens, you suspect that something out of the ordinary is at play, like maybe a loaded pair of dice. ("Loaded" is just another way of saying the dice are acting non-normal.)

Figure 5-23 shows an example of a behavior chart showing evidence of

Variation beyond expected levels.

Figure 5-23:

Behavior chart showing evidence of

Variation

Beyond the expected normal

T-1-^-1-1 J-1-^™-1

0 20 40 60 80 100

Limits.

Run Order

When you see excessive variation like this, use the time scale or run order of the behavior chart as a starting point to discover what conditions or factors are causing the non-normal variation. Go back to that point in time identified by the chart and ask yourself. what was different at this point in time to take the characteristic or process behavior out of its normal course? The answer allows you to identify and manage the factor

Or factors influencing the process or characteristic performance. Typical causes of outliers include worker inattention, measurement

Errors, and other one-time changes to the process’s or characteristic’s environment. For example, there may be a data outlier for purchase order processing due to an emergency in the office where two workers had to leave at the same time — thereby leaving a purchase order in the queue

For an excessive period of time.

In Chapter 10, you find out much more about detecting evidence of this

Type of special-cause variation in the performance of your process or characteristic.

Is Trends: Trend is a steady, gradual increase or decrease in the central tendency of the process or characteristic as it plays out over time. If all the conditions in the system stay constant, the level of performance of the process or characteristic will also stay level. The presence of a trend in a graphical behavior plot is evidence that something out of the ordinary

Has happened to move the location of the process or characteristic behavior. Figure 5-24 shows a sample of a trend in a process behavior chart.

Just like with any other evidence of non-normal behavior, when you see

A trend in a behavior chart, you need to look closer at the system to uncover what is causing the changed performance.

20

Figure 5-24:

Behavior

Chart showing evidence

Of a trend in

The location

Of the variation center over time.

~r

80

100

Run Order

Trends in performance are almost always caused by system factors that gradually change over time, like temperature, tool wear, machine maintenance, rising costs, and so on.

V Runs: Run is a sequence of consecutive observations that are each increasingly larger or smaller than the previous observation. Figure 5-25

Shows an example of two runs, one increasing and one decreasing, within a behavior chart.

Runs can be caused by faulty equipment, calibration issues, and cumulative effects, among other things.

20

Figure 5-25:

Behavior chart with evidence of

A run. A

String of consecutive points that increase or decrease are not normal behavior.

20

40 60 Run Order

80

100

0

0

T-" Shifts: Shifts are sudden jumps, up or down, in the process’s or characteristic’s center of variation. Something in the system changes permanently — a piece of equipment, a new operator, a change in material, a new procedure. Clearly, shifts are non-normal behavior.

Figure 5-26 shows an example of a process or characteristic that has experienced a shift in the center level of its variation.

Figure 5-26:

Behavior chart with

Evidence

Ofa non-normal shift

Affecting the level of

Thecentral

Tendency of the variation.

20

40

60

80

100

Run Order

0

So, for the process of inflating the tires on cars in an assembly line, a study may reveal that of the 352 cars that went through the tire inflation process during a day’s production, 347 were later found to have a pressure within the required specification limits. In this case, the traditional yield is

Y=

Out In

347 352

0.986 or 98.6%

Converting from a proportion like 0.986 to perhaps a more familiar percentage scale is done by simply multiplying the proportion by 100. To go from percentage back to proportion, divide the percentage by 100.

Always perform mathematical operations on proportions, not on percentages.

The traditional calculation of yield is often employed on the last, final inspection step of a process to measure the effectiveness of the overall process.

The Six Sigma perspective: First time yield (FT\l)

The results of calculating yield the traditional way are misleading. Take a closer look at the tire-inflating example process in Figure 6-5.

Items in

(352)

Figure 6-5:

Detailed

View of a tire-inflation

Process.

R

(5)

1

Scrap

Items out

(347)

After inflation, the tire is immediately inspected to make sure it meets the required pressure specification limits. In the example, 103 tires are detected that don’t comply with the pressure specification. Of course, the operators of the process reviewed each of these 103 and corrected (or Reworked) 98 of them, leaving only five that were not able to be brought back within the correct pressure range and had to be scrapped.

With this detailed information, you now know that the proportion of tires going through the inflation process correctly the First time Is

249 0 707 or 70 70/ 352 = 0.707 or 707/0

This calculation of yield is appropriately called First time yield Or FTY for short. First time yield is often much different than traditional yield. That’s because, unlike traditional yield, it captures the harsh reality of the effectiveness of

The process, including inspection and rework. Uncovering the hidden factory

Figure 6-6 shows the tire inflation process again, this time with the previously hidden, but now revealed, part of the process clearly identified.

Items in (352)

Figure 6-6:

The hidden factory caused by inspection

And rework.

I (98)

(103)

_L_

Rework

(5) ±

Scrap

Items out

(347)

The hidden factory

I I

The hidden factory is a natural outgrowth of the inability to correctly comply

With required specifications the first time through the process. Here and there throughout organizations, hidden factories arise and become entrenched as tacit appendages of the standard processes. Measuring yield using the first

Time yield method forces you to objectively review and acknowledge the effectiveness of your processes.

In the case of the example tire inflation process, the hidden factor of in-process inspection and rework accounts for 98.6/ – 70.7/ = 27.9/ of production. All

Together, value-sapping hidden factories within organizations combine to consume valuable resources and time.

Rotted throughput uietd (RT\l)

In reality, individual process steps are strung together to create an overall

Process structure for accomplishing complex tasks. One way Six Sigma quantifies the complexity of a system is to count the number of processes involved.

For example, Figure 6-7 illustrates a purchase order process that is made up of five individual process steps.

Figure 6-7:

Several

Smaller

Process steps link together to create a complex process.

How do you calculate the overall yield for a string of processes? The answer: You multiply the first time yields for each step together, creating what is called the Rolled throughput yield (RTY).

For the purchase order example given in the preceding section, the rolled throughput yield for this five-step process is

RTY = FTYI X FTY-2 X FTY:i X FTY4 x FTY5 RTY = 0.75 X 0.95 X 0.85 X 0.95 X 0.90 RTY = 0.518

That means that the chance of a purchase order going through the process the first time with no rework or scrap is only 51.8 percent! (The last "confirmation"

Step in the process acts as a final test. With this last step having a 90 percent yield, there must be a lot of hidden factory stuff going on to drop the RTY down to 51.8 percent.)

Like a chain that is only as strong as its weakest link, rolled throughput yield can never be greater than the lowest first time yield within the system. To immediately improve the overall system performance, focus first on the individual process step with the lowest first time yield. Then move on to the step

With the next lowest first time yield.

The formula for rolled throughput yield can be simplified as

I – 1 I I

I = \

Where the Greek letter pi (I I) tells you to multiply all the first time yields of the system together. (See Table 6-1 for a summary of yield metrics.)

Even if the first time yields of the individual process steps are high, if the overall process becomes more and more complex (that is, more and more

Process steps), the system rolled throughput yield will continue to erode.

Figure 6-8 charts how complexity degrades rolled throughput yield for different levels of individual first time yield.

For very complex systems — like automobiles, aircraft, data switching systems, enterprise-level business processes, and so on — a very high individual first time yield must be achieved in order to have any hope of an acceptable

Rolled throughput yield.

Yield (RTY) Process stream. Tells

You the likelihood of an item passing through all process steps successfully the first time.

Measuring defect rate

The complimentary measurement of yield is defects. If your yield is 90 percent, there naturally must be 10 percent defects. Measuring defects and calculating the rate or how often they occur is like looking at the flip side

Ofthe yield coin. Defects etfuat failure

When a process or characteristic doesn’t perform within its specifications, it is considered defective. Or in other words, it produces a non-compliant condition, called a Defect.

Defining a defect as a non-compliance with specifications may seem overly simplified. Just because a characteristic exceeds a specification doesn’t necessarily mean that the system it is part of will break or stop functioning. It may or may not. For example, misspelling a customer’s name on a billing statement (a defect or non-compliance with specifications) may or may not turn into a complaint (a failure) that costs money to correct.

But over and over again, experts have verified that product or process failures are directly related to compliance with specifications; the less you are compliant with specifications, the more likely you are to have a failure or

Breakdown.

So given the difficulty in directly linking compliance with specifications to

Product or process performance, the safe thing to do is to make sure you strive

To comply with specifications. The absence or reduction of non-compliance

With specifications will always reduce failures or breakdowns in your customers’ experiences.

Defects per ubiquitous unit (DPU)

Six Sigma applies to all areas of business and productivity — manufacturing,

Design, sales, office administration, accounts receivable, healthcare, finance,

And so on. Each of these areas works on and produces different things —

Products, services, processes, environments, solutions, among others.

To bridge these diverse disciplines, in Six Sigma you call the item you are working on a Unit. A unit may be a discretely manufactured product. Or it may be an invoice that crosses your desk. It may be a month’s worth of continually produced product. It may be a hospital patient or a new design. Whatever it is you do, in Six Sigma it is called a Unit.

A basic assessment of characteristic or process capability is to measure the

Total number of defects that occur over a known number of units. This is then

Transformed into a calculation of how often defects occur on a single unit,

Like this

Dpu_ Number of defects observed

Where DPU stands for defects per unit.

For example, if you process 23 loan applications during a month and find 11 defects — misspelled names, missing prior residence information, incorrect

Loan amounts — then the DPU for your loan application process is

DPU_ 11 _ 0.478

That means that for every two loans that leave your desk, you expect to see

About one defect.

Leveling the field: Defects per opportunity (DPO) and per million opportunities (DPMO)

A DPU of 0.478 for an automobile is viewed very differently than the same

Perunit defect rate on a bicycle. That’s because the automobile, with all its thousands of parts, dimensions, and integrated systems, has many more

Opportunities for defects than the bicycle has. A DPU of 0.478 on an automobile is evidence of a much lower defect rate than the same DPU On a relatively simpler bicycle. It’s just not a fair comparison.

So how do you compare or contrast the defect rates of things that have very different levels of complexity? The key is in transforming the defect rate into terms that are common to any unit, whatever it is or however complex it may be.

The common ground between any different units is opportunity. For any product, process, service, transaction, or environment, an Opportunity Is a specific characteristic that could either turn out as a defect or as a success. Success or failure for the opportunity is defined as compliance to the opportunity’s specification.

Examples of opportunities include:

In a product, the critical dimension of diameter on an automobile axle.

In a transactional process, the applicant’s mailing address on a loan approval form.

F In a hospital, getting the correct medial history records into the patient’s file.

V In the design of a retail store environment, the placement of clearance

Sale racks.

F In a manufacturing process, the tightening of a bolt to the correct torque.

The number of opportunities inherent to a unit, whatever that unit may be, isa direct measure of its complexity. In fact, when you want to know how complex a unit is, you count or estimate how may opportunities there are forsuccess or failure. Individual characteristics that are critical to the

Performance of the system are opportunities. Characteristics that have a

Specification represent opportunities.

The way to level the playing field so you can directly compare the defect rates of systems with very different complexities is to create a per-opportunity defect rate. This measurement of capability is called Defects per opportunity (or DPO)

And is calculated as

Dpn _ Number of defects observed on a unit O number of opportunities on a unit

With a calculated DPO measurement, you can now fairly compare how capable an automobile is to a bicycle. For example, you may observe 158 out-of -

Specification characteristics on an automobile. After some study, you also

Determine that there are 14,550 opportunities for success or failure within that automobile. Its DPO Is then

DPO _ 158 _ 0 011 DPO 14,550 0.011

For a bicycle, on the other hand, you may find only two non-compliant characteristics among its 173 critical characteristics. So its DPO Is

DPO _ _ 0.012

Even though an automobile and a bicycle are two very different items with very different levels of complexity, the DPO Calculations tell you that they both have about the same real defect rate. You only observe more defects on the automobile because there are many more opportunities for defects.

Be careful not to overly estimate the number of opportunities on a unit. You can artificially make the DPO Of your product, process, or service look better than it really is by inflating its number of opportunities. For example, you could count the correct name on a patient record as a single opportunity for

Success or failure; whether the name on the form is right or wrong. Or you

Could say that there’s one opportunity for the correct spelling, another for

The correct font, another for the correct darkness of the printed text, another for the name’s placement within the form box, and so on. Playing opportunity

Counting games only shrinks your ability to make an honest assessment and

Begin to make real improvement.

When the number of opportunities on a unit gets large and the number of observed defects gets small, calculated DPO Measurements become so small

They are hard to work with. For example, two commercial airline crashes (defects) observed out of 6 million flights in a year translates into

DPO _ 6 000 000 _ 0.000000333

0.000000333 is definitely an inconvenient number to work with!

You may also want to estimate out into the future, to know how many defects will pile up after running the process or observing the characteristic for a long time. After all, DPU And DPO Look only at a single unit or a single opportunity.

A simple way to solve both of these problems is to count the number of defects over a larger number of opportunities. For example, how many defects occur

Over a set of one million opportunities? This defect rate measurement is called

Defects per million opportunities (or DPMO) And is used very frequently in Six

Sigma.

When a process is repeated over and over again many times — like an automobile assembly process, or an Internet order process, or a hospital check-in process — DPMO Becomes a convenient way to measure capability. Six Sigma

Is famous for its defect rate goal of 3.4 defects per million opportunities.

When calculating DPMO, You don’t want to actually measure the defects over a million opportunities. That would take way too long. Instead, the way you

Calculate DPMO Is using DPO As an estimate, like this

DPMO = DPO x 1,000,000

This also means you can track backward, going from DPMO to DPO:

DPO_ Tij0MO0

A common alternative form of DPMO Is DPPM— Defective parts per million. DPPMIs often used when assessing the defect rate of a continuous material

Or process where the "part" is the opportunity. Like in ongoing shipments of bolts to a supplier, the cumulative number of defective bolts found compared

To the total number shipped over time can be translated into DPPM. (See

Table 6-2 for a summary of defect rate metrics.)

Table 6-2 Summary of Defect Rate Metrics

Linking yield and defect rate

You can calculate the yield of a process or characteristic. You can also calculate the defect rate of a process or characteristic. Are these two measures related? In fact, they are.

When you have an overall process with a relatively low defect rate, say, a process that produces units with a DPU less than 0.10 (or 10 percent), you can mathematically link the process defect rate to the overall process yield:

RTY _ e ~DPU

Where E In the equation is a mathematical constant equal to 2.718. There will be a function or key for raising E To a number on any scientific calculator or any spreadsheet computer program. (Look for the Ex Key on your calculator.)

The actual value of the constant E Is 2.71828182845905. . . . The decimal digits of E Go on forever, never repeating. But you don’t need to know the details of this curious constant called E To excel at Six Sigma. If, however, you feel yourself compelled to know more, you can proudly claim the title of "math geek." And by all means, find yourself a copy of Calculus For Dummies By Mark Ryan (Wiley) to find out more about the fascinating number E!

The power of this mathematical link between yield and defects, is that if you can only measure or have only measurements of the defect rate of a process, you can still calculate its rolled throughput yield.

A little bit of algebraic contortions provides an equation to calculate DPU Based only upon the rolled throughput yield of a process:

DPU = – ln(RTY)

Where ln is the natural logarithm. (Hint: There’s an ln button on every scientific calculator.)

Sigma (Z) score

From a quality perspective, Six Sigma is defined as 3.4 defects per million opportunities. This is called a Six Sigma level of quality. What is this famous

Sigma level or score? Sigma scores are thrown about so much, you definitely

Need to be comfortable understanding what they are and how they are

Calculated.

Ho© many standard deviations can fit>

Figure 6-9 illustrates a process or characteristic performance distribution

Compared to its one-sided specification.

Figure 6-9:

A characteristic’s performance distribution as defined by its mean X And its

Standard

Deviation cr.

"l-1-1-1-r

Characteristic Scale of Measure

The central tendency of the performance distribution is defined by its mean. The amount of variation in the performance, or the width of the distribution, is defined by its standard deviation cr. The question is, how many standard deviations can you fit between the process or characteristic’s mean and its specification limit SL?

Graphically, in Figure 6-9, you can see that four standard deviations can fit

Between the mean and the specification limit. The exact number can always

Be calculated (even without a graph!) by the formula

Z _ -

O

Mis-

Calculating Z Tells you exactly how many standard deviations can fit between the mean and specification limit of any process or specification. In Six Sigma, you call this value the Sigma score Of the process or characteristic.

Statisticians usually call this same value the Z Score or Normal Score. In Six

Sigma, however, you need to be careful not to confuse the sigma score

(sometimes called a Sigma value, Or even simpler just Sigma) With the standard deviation represented by the Greek letter <r. Z Score, Z Value, Z, Sigma score, sigma value, and sigma are all different names for how many standard

Deviations can fit between the mean and the specification limit. Things get

Confused when practitioners call the standard deviation "sigma." In this

Book, we always call the standard deviation the standard deviation. To avoidthe confusion yourself, whenever you are reading or speaking about

A a, don’t call it "sigma." Instead, call out "standard deviation" for what the symbol always represents.

Use a sigma (z) score only on a characteristic that is approximately normal.

That means its distribution needs to be bell shaped. When the distribution is far from normal, the formula for calculating the sigma score (Z)breaks down.

The quickest way to check whether the distribution is approximately normal is to create a dot plot or histogram. (See Chapter 5 to do this.)

A low sigma (z) score means that a significant part of the tail of the distribution is extending past the specification limit. So the higher the sigma (z) score, the fewer the defects. A process or characteristic gets a good sigma (z) score when the variation distribution is safely away from the edge of the

Specification cliff.

There are three ways a sigma (z) score can change: F The location of the central tendency of the distribution, the mean,

Moves either closer or farther from the specification limit.

V The width of the distribution, as defined by the standard deviation A, Gets either wider or narrower.

F The location of the specification limit SL Moves either closer or farther

From the characteristic or process variation.

Actually, changes to X And a usually happen at the same time, with both simultaneously contributing to a change in the computed sigma (Z) score.

Short-term Versus long-term sigma score

From the mean X And the standard deviation cr, you can calculate a sigma (z) score. A wrinkle here is that you must know what type of standard deviation

You are using to calculate the sigma (Z)score: Is it a short-term standard

Deviation <rST, or is it a long-term standard deviation <%? (To understand the critical differences in short – and long-term standard deviations, and the implications, see Chapter 5.)

If you are using a short-term standard deviation, the sigma (Z)score you

Calculate is a short-term sigma score ZST:

If, however, you have a long-term standard deviation, you can calculate the long-term sigma score Zlt:

Zlt _ Olt

Short-term variation performance, as quantified by the short-term sigma

Score Zst, Represents the best variation performance that you can expect out

Of your currently configured process. It is an Idealistic Measure of capability. It is also the easiest type of data to collect — you just go and quickly grab a relatively small sample of measurements from the process or characteristic.

But in the real world, a process or characteristic doesn’t operate ideally like it does in the short-term. Its performance is degraded by shift, drift, and

Trend influences.

At the heart of Six Sigma is a method that combines the best of both worlds. It allows you to leverage the economy of short-term variation data while projecting realistic, long-term performance versus the process’s or characteristic’s specifications.

Shifty business: Linking short-term capability to long-term performance ©ith the 1.5-sigma shift

Figure 6-10 shows the short-term variation of a process or characteristic and

Its expanded, long-term variation.

Figure 6-10:

A characteristic with short-term variation that complies with specifications, but

With an

Expanded long-term

Variation that creates defects.

Short-term

Long-term

Defects

Characteristic Scale of Measure

The characteristic or process shown in Figure 6-10 stays within specifications

During the short-term. It looks like there aren’t problems. But over the long term, disturbances to the problem cause it to expand and sometimes create defects beyond the specification limit.

One mathematical way to simulate the effect of these degrading, long-term influences is to artificially move the short-term distribution closer to the specification limit until the amount of defects for the short-term distribution

Is the same as that for the long-term distribution. This approach is shown in Figure 6-11.

Early practitioners of Six Sigma proposed that mathematically shifting a characteristic’s or process’s short-term distribution closer to its specification

Limit by a distance of 1.5 times its short-term standard deviation (aST) would approximate the amount of defects occurring in the long term. This breakthrough can be applied directly to the calculation of short-term and long-term

Sigma (Z)scores.

Figure 6-11:

Mathematically shifted

Short-term distribution used to estimate the long-term variation performance.

Characteristic Scale of Measure

Because ZST Represents the number of short-term standard deviations between the variation center and the specification, the sigma (Z) score of the shifted

Distribution is

ZShifted= Zst - 1.5

But with the shifted distribution being equivalent, defect-wise, to the long-term distribution, the preceding equation can be rewritten as

Zn= ZST - 1.5

So what Six Sigma practitioners do is measure the short-term variability of a process or characteristic and calculate its short-term sigma score ZSt. Then they immediately translate this to the expected long-term defect rate performance, using the 1.5 short-term standard deviation shift. This long-term sigma score, Zm Is communicated in terms of defects per million opportunities, DPMO.

Table 6-3 is a look-up table that Six Sigma practitioners carry around in their pockets and use over and over until they have it memorized (or until it is worn out, whichever happens first). They figure the ZST For any process or

Characteristic, and then translate that into a long-term defect rate DPMO. Or,

In reverse, they first find the DPMO, and then translate that back to a short-term sigma score ZST.

Table 6-3_Sigma Score Table: Z — DPMO

Z_DPMO_

0.0_933,193_

0.5 841,345

ZDPMO

1.0 691,462

1.5 500,000

2.0308,538

2.5158,655

3.0 66,807

3.5 22,750

4.06,210

4.51,350

5.0233

5.532

6.0 3.4

Note: Paired table values are long-term for DPMO and short-term for Z (example, a long-term DPMO of 6,210 is the result of a process with a short-term sigma score of 4.0). Add 1.5 to corresponding Z values to obtain short-term equivalents (example, a short-term DPMO of 32 is the result of a process with a short-term sigma score of 4.0).

What’s your sigma, bab§>

As you work in Six Sigma, you may hear someone ask, "What’s the sigma of the process?" And the response you’ll hear back is, "2 sigma" or "3.3 sigma" or such-and-such sigmas. The question these people are really asking is, "What is the short-term sigma score ZST Corresponding to the long-term

Defect rate of the process?"

After only a few times looking up sigma score values in Table 6-3, you begin to get a feel for this famous scale of capability. You may even be able to

Approximate sigma scores for defect rate values that fall between the rows

Of the table. Like a DPMO Of 20,000. Its sigma score is about 3.6, a value just a little larger than the 3.5 corresponding to the DPMO Of 22,750 in the table.

The sigma score can be applied to the performance of anything that has a specification and a defect rate: the performance of the mail system in delivering letters to the correct address, the ability of an automobile manufacturer to produce a door that fits to the body within a required dimensional tolerance,

Or a repeated budgeting process that must be completed within its specified schedule window.

All these sigma scores can be directly compared to see how capable the process or characteristic is. And when you communicate this capability with a sigma score, everyone else in Six Sigma knows exactly what you’re talking

About.

Capability indices

Yet another set of measures exists to quantify the capability of a process or characteristic to meet its specifications. This last set are indices that directly compare the voice of the process to the voice of the customer.

Short-term capability index (CP)

The simplest capability index is called CP. It compares the width of a two-sided specification to the effective short-term width of the process. Determining the width between the two rigid specification limits is easy. It is simply the distance between the upper specification limit USL And the lower specification limit LSL. But with variation that trails out at the tails, how do you determine the width of the Process?

To get over this hurdle, Six Sigma practitioners have defined the effective Limits Of any process as being three standard deviations away from the average level. At this setting, these limits surround 99.7 percent, or virtually all, of the variation in the process. This is shown graphically in Figure 6-12.

Figure 6-12:

The effective width of a process or characteristic is ±3 standard deviations, containing 99.7 percent of the process

Variation.

Standard Deviations

So to compare the width of the specification to the short-term width of the process, you use the formula:

Ce = 6CJst

Where Usl - Lsl Represents the voice of the customer’s requirements and 6crST represents the inherent voice of the process.

A calculated CP Value equal to 1 means that the voice of the customer is equal to the voice of the process. A CP Value less than 1 means that the process is wider than the specification, with defects spilling out over the edges. A CP

Value greater than 1 means that the effective width of the process variation isless than the required specification, with fewer defects occurring.

CP Is a measure of short-term process or characteristic capability. Use only the short-term standard deviation to calculate its value. Using a long-term

Standard deviation in its calculation gives you incorrect results.

Adjusted short-term capability index (C„)

A problem with the short-term capability index CP Is that it only compares the widths of the specification and the process. Figure 6-13 illustrates this

Problem.

Figure 6-13:

Two

Distributions, one centered and one offset

From the

Specification limits.

In Figure 6-13, both the distribution drawn with the solid line and the distribution drawn with the dotted line have the same calculated Cp. That’s because

They both have the same specification width and the same process width. But the are not equally capable. Because it is offset from the center of the specification, the dotted line distribution has many more defects than the solid

Distribution.

You can compensate for this by adjusting the CP Calculation for how far it is offset. To do this, you simply compare the distance from the distribution center X To each of the specification limits with the half-width of the short-term variation that should exist between the center of the distribution and the specification limit, like this

The smallest value you calculate of CPU And CPL Is called the adjusted short-term capability index CPK. So the formula for CPK Can be written as

CPk = Min ( CPu , CP

Where the Min In the equation tells you to choose the smallest of the values

In parentheses.

^Sty* If the characteristic or process variation is centered between its specification

Limits, the calculated value for CPK Will be equal to the calculated value for CP. But as soon as the process variation moves off the specification center, it’s penalized in proportion to how far it is offset.

CPK Is very useful and very widely used. That’s because it compares the width

Of the specification with the width of the process while also accounting for

Any error in the location of the central tendency. This is a much more realistic approach than what the CP Method offers.

Generally, a CPK Greater than 1.33 indicates that a process or characteristic is capable in the short-term. Values less than this tell you that the variation is

Either too wide compared to the specification or that the location of the variation is offset from the center of the specification. Or it may be a combination of both width and location. The only way to know for sure is to create a graph and begin to review the details.

Long-term capability indices 0PP and PPK)

The same capability indices that you calculate for short-term variation, CP

And Cpk, Can also be calculated for long-term variation. To differentiate them

From their short-term counterparts, these long-term capability indices are called PP And PPK. The only difference in their formulas is that you use in place of crST.

Long-term capability indices are important because no process or characteristic operates in just the short term. Every process extends out over time to create long-term performance. Table 6-4 summarizes each of the short – and long-term capability indices.

Table 6-4 Summary of Short – and Long-Term Capability Indices

Index Name Formula_Description_

Short-term CP = crT Compares the width of

Capability index ST the specification to the

Short-term width of the

Process

Index Name Formula_Description_

Adjusted short – CPK = Min (U^_~ *, *~LSL) Compares the width of

Term capability V ST ST ‘ the specification to the

Indexshort-term width of the

Process and accounts for off-centering of the process from the specification

Long-term Pp = US^~~LSL Compares the width of

Capability LT the specification to the

Index long-term width of the

Process

Adjusted long – PPK = Min (US-~ *, *-7rrLSL) Compares the width of

Term capability V LT LT ‘ the specification to the

Index long-term width of the

Process and accounts for off-centering of the process from the

Specification

Prescribing a capability improvement plan

When you know what the short – and long-term capability indices of a process or characteristic are, what do you do? How can you use these four indices to chart out a plan for improvement?

Table 6-5 outlines the various scenarios that may occur when measuring the capability of a process or characteristic. The table also describes an improvement plan for each scenario.

Table 6-5_Prescriptive Capability Improvement Plan

Symptom Diagnosis_Prescription_

CP = CPK Overall, your process or As needed, focus on reducing

And characteristic is centered the long-term variation in

PP = PPK Within its specifications. your process or characteristic

While maintaining on-center performance.

Continued