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.
|
Table 5-2 Summary of Statistical Measures of Variation Spread
|
|
Measure of variation spread
|
Definition Comments
|
|
Range
|
R = *MAX~ *MIN
|
Simple. Preferred metric for sets of data with few (2 to 5) members. Drawback: Greatly influenced by outliers.
|
|
Variance
|
A – n – 1
|
Theoretically useful, but lacks direct tie to reality.
|
|
Standard deviation
|
A V n – 1
|
Most commonly used.
|
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.
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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
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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.
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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.
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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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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
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Marketing Plan
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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).
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Scatter Plot of YvsX,
Figure 5-19:
Scatter plot showing correlation between characteristics Y and X3.
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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
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Figure 5-20:
Scatter plot showing positive correlation between two characteristics.
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Figure 5-21:
Three scatter plots, one for each input characteristic X„ X2, and X3
Against a single output characteristic Y.
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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.
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Figure 5-22:
Characteristic or process behavior chart example. Each observation of the characteristic is
Plotted in the order in which it was measured.
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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
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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.
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Figure 5-24:
Behavior
Chart showing evidence
Of a trend in
The location
Of the variation center over time.
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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.
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Figure 5-25:
Behavior chart with evidence of
A run. A
String of consecutive points that increase or decrease are not normal behavior.
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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.
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