Seven Tools of QC Seven Tools of QC Improvements CourseImprovements Course

Seven Tools of QC ImprovementSeven Tools of QC Improvement

GraphsGraphs

Pareto ChartsPareto Charts

HistogramHistogram

Cause and Effect DiagramCause and Effect Diagram

Check SheetCheck Sheet

Flow DiagramFlow Diagram

Scatter DiagramScatter Diagram

GraphsGraphsBar Chart Line Chart Pie Chart

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Use a Bar line chart for the purpose of comparison through visual representation of the data collected

Show the percentage an item contributes to the whole

Pareto ChartsPareto Charts

Number of units investigated: 5.000

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Pareto Chart helps to highlight “the Vital Few” in contrast to “The Trivial Many” .

Pareto Chart is based on “80-20” rule for instance, 80% of the problem result from 20% of the cause.

A: CrackB: ScratchC: Stain

D: Strain E: GapF: Pinhole

Pareto AnalysisPareto Analysis

PPareto analysis is a ranked comparison of areto analysis is a ranked comparison of factors related to a factors related to a quality problemquality problem. It . It helps a quality improvement project team to helps a quality improvement project team to identify and focus on the vital few factors. identify and focus on the vital few factors.

CONCEPTCONCEPT

Pareto analysis gets its name from the Italian – Pareto analysis gets its name from the Italian – born economist Vilfredo Pareto (1848 – 1923) born economist Vilfredo Pareto (1848 – 1923) who observed that a relative few people held the who observed that a relative few people held the majority of the wealth. Pareto developed majority of the wealth. Pareto developed logarithmic mathematical models to describe this logarithmic mathematical models to describe this non-uniform distribution of wealth, and the non-uniform distribution of wealth, and the mathematician M.O. Lorenz developed graphs to mathematician M.O. Lorenz developed graphs to illustrate it.illustrate it.

Historical EvolutionHistorical Evolution

Dr. Joseph Juran was the first to point out that what Dr. Joseph Juran was the first to point out that what Pareto and others had observed was a “universal” Pareto and others had observed was a “universal” principle – one that applied in an astounding Varity principle – one that applied in an astounding Varity of situations and appeared to hold without of situations and appeared to hold without exception in problems of quality.exception in problems of quality.In the early 1950s. Juran noted the “universal” In the early 1950s. Juran noted the “universal” phenomenon that he has called the Pareto phenomenon that he has called the Pareto Principle; that in any group of factors contributing to Principle; that in any group of factors contributing to a common effect, a relative few account for the bulk a common effect, a relative few account for the bulk of the effect. Juran has also coined the terms “ vital of the effect. Juran has also coined the terms “ vital few” and “useful many” to refer to those few few” and “useful many” to refer to those few contributions which account for a smaller proportion contributions which account for a smaller proportion of the effect.of the effect.

The Pareto PrincipleThe Pareto Principle

As experienced managers and professionals, we As experienced managers and professionals, we intuitively recognize the Pareto principle and the intuitively recognize the Pareto principle and the concepts of the vital few and useful many, for we concepts of the vital few and useful many, for we see them in operation in everyday business see them in operation in everyday business situations. For example, we might observe that;situations. For example, we might observe that;

The top 15% of our customers account for 68% of our The top 15% of our customers account for 68% of our total revenues.total revenues.Our top 5 products or services account for 75% of our Our top 5 products or services account for 75% of our total sales.total sales.A few employees account for the majority of absences.A few employees account for the majority of absences.In a typical meeting, a few people tend to make the In a typical meeting, a few people tend to make the majority of comments, while most people are relatively majority of comments, while most people are relatively quiet.quiet.

The principle of the vital few and useful many also The principle of the vital few and useful many also applies to quality improvement opportunities. Each applies to quality improvement opportunities. Each quality effect that we can observe (for example; quality quality effect that we can observe (for example; quality costs, defects, rework, customer dissatisfaction, costs, defects, rework, customer dissatisfaction, revenues, complaints, etc.) results from numerous revenues, complaints, etc.) results from numerous contributors to that effect. When we look at the many contributors to that effect. When we look at the many individual contributors, we find that a few account for individual contributors, we find that a few account for the majority of the total effect on quality.the majority of the total effect on quality.For example, when we gather the facts, we might find For example, when we gather the facts, we might find that;that;

In a 25 – step-manufacturing process, 5 of the operations In a 25 – step-manufacturing process, 5 of the operations account for 65% of the total scrap generated.account for 65% of the total scrap generated.Of the 12 unique services that our company offers, 3 of the Of the 12 unique services that our company offers, 3 of the services account for 82% of the customer complaints.services account for 82% of the customer complaints.Of the 18 items of information that must be filled in on an order Of the 18 items of information that must be filled in on an order form, 4 of the items generate 86% of the errors found on these form, 4 of the items generate 86% of the errors found on these forms.forms.

In these typical cases, the few (steps, services, items) In these typical cases, the few (steps, services, items) account for the majority of the negative impact on quality. account for the majority of the negative impact on quality. If we focus our attention on these vital few, we can get the If we focus our attention on these vital few, we can get the greatest potential gain from our quality improvement greatest potential gain from our quality improvement efforts.efforts.The Pareto principle is so obvious and so simple that you The Pareto principle is so obvious and so simple that you might wonder what all the fuss is about. After all, might wonder what all the fuss is about. After all, everybody knows that, don’t they ? But if everybody knows everybody knows that, don’t they ? But if everybody knows it already, why do we so often hear mangers complaining it already, why do we so often hear mangers complaining that they are faced with dozens of problems in their that they are faced with dozens of problems in their organization? and why do we so often see company task organization? and why do we so often see company task forces listing dozens of problems and setting out to solve forces listing dozens of problems and setting out to solve all of them simultaneously and with equal vigor ?all of them simultaneously and with equal vigor ?If we really understood the simple but profound Pareto If we really understood the simple but profound Pareto principle, our first step when faced with a host of problems principle, our first step when faced with a host of problems would be to gather data and facts to identify the vital few. would be to gather data and facts to identify the vital few. We could then focus our attention and improvement We could then focus our attention and improvement efforts on those few things that would give us the greatest efforts on those few things that would give us the greatest improvement in quality.improvement in quality.

Pareto Diagrams and TablePareto Diagrams and Table

Pareto diagrams and tables are presentation Pareto diagrams and tables are presentation techniques used to show the facts and separate techniques used to show the facts and separate the vital few from the useful many. They are the vital few from the useful many. They are widely used to help quality improvement teams widely used to help quality improvement teams and steering committees make key decisions at and steering committees make key decisions at various points in the quality improvement or various points in the quality improvement or problem-solving sequence.problem-solving sequence.

Regardless of the form chose, well-constructed Regardless of the form chose, well-constructed Pareto diagram and tables include three basic Pareto diagram and tables include three basic elements;elements;

You will notice that Pareto diagram presents the You will notice that Pareto diagram presents the result of stratifying a problem by one particular result of stratifying a problem by one particular variable. The contributors to the effect are the variable. The contributors to the effect are the categories for that stratification variable.categories for that stratification variable.

A look at the following example of how to A look at the following example of how to construct and use Pareto diagrams and tables construct and use Pareto diagrams and tables will illustrate and further explain these three will illustrate and further explain these three basic elements.basic elements.

The contributors to the total effect, ranked by the The contributors to the total effect, ranked by the magnitude of their contribution.magnitude of their contribution.The magnitude of the contribution of each expressed The magnitude of the contribution of each expressed numerically.numerically.The cumulative-percent-of-total effect of the ranked The cumulative-percent-of-total effect of the ranked contributors.contributors.

The “Out of The “Out of OrderOrder” Orders” Orders

A quality improvement team was chartered to A quality improvement team was chartered to improve the quality of order forms coming in with improve the quality of order forms coming in with errors from field sales offices to the home office. errors from field sales offices to the home office. There were 18 items on the order form, which There were 18 items on the order form, which we will designate here as items A to R. The we will designate here as items A to R. The team developed a check sheet which it used to team developed a check sheet which it used to collect the frequency errors on the forms for a collect the frequency errors on the forms for a week. The results of the team’s study, in the week. The results of the team’s study, in the form of a Pareto table, form of a Pareto table, are shown in Figure 1are shown in Figure 1

FIGURE 1FIGURE 1: PARETO TABLE OF ERRORS ON : PARETO TABLE OF ERRORS ON ORDER FORMSORDER FORMS

Order – Form Item Number of Errors Percentage Cumulative Percentage

G 44 29 29J 38 25 54M 31 21 75Q 16 11 86B 8 5 91D 5 3 95C 3 2 97A 1 0.67 98O 1 0.67 98R 1 0.67 99N 1 0.67 99L 1 0.66 100I 0 0 100E 0 0 100H 0 0 100K 0 0 100F 0 0 100P 0 0 100

Total 150 100

Note that the Pareto table contains the three basic elements Note that the Pareto table contains the three basic elements described above. The first column lists the contributors, the described above. The first column lists the contributors, the 18 items, not in order of their appearance on the form, but 18 items, not in order of their appearance on the form, but rather, in order of the number of errors detected on each rather, in order of the number of errors detected on each item during the study. The second and third columns show item during the study. The second and third columns show the magnitude of contribution – the number of errors the magnitude of contribution – the number of errors detected on each item and the corresponding percentage of detected on each item and the corresponding percentage of total errors on the form. The fourth column gives the total errors on the form. The fourth column gives the cumulative-percent of total. This column is the key to Pareto cumulative-percent of total. This column is the key to Pareto analysis.analysis.““Cumulative –Percent of – form item J, the cumulative-Cumulative –Percent of – form item J, the cumulative-percent of total is 29% + 25%, or 54%. At Q it is 29% + 25% percent of total is 29% + 25%, or 54%. At Q it is 29% + 25% + 21% + 11% , or 86%.+ 21% + 11% , or 86%.In other words, the first four items, G, J, M and Q account for In other words, the first four items, G, J, M and Q account for 86% of the total errors detected in the study. These are the 86% of the total errors detected in the study. These are the “Vital Few”. “Vital Few”. A Pareto diagram of the same data is shown in Figure 2. A Pareto diagram of the same data is shown in Figure 2. Again, note the three basic elements that make up the Again, note the three basic elements that make up the diagram.diagram.

Pareto Diagram

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FIGURE 2:FIGURE 2: PARETO DIAGRAM OF ERRORS ON PARETO DIAGRAM OF ERRORS ON ORDER FORMSORDER FORMS

On the Pareto diagram, the 18 items on On the Pareto diagram, the 18 items on the order form are listed on the horizontal the order form are listed on the horizontal axis in the order of their contribution to the axis in the order of their contribution to the total. The height of each bar relates to the total. The height of each bar relates to the left vertical axis, and shows the number of left vertical axis, and shows the number of errors detected on that item. The line errors detected on that item. The line graph corresponds to the right vertical graph corresponds to the right vertical axis, and shows the cumulative-percent of axis, and shows the cumulative-percent of total Note how the slope of the line graph total Note how the slope of the line graph begins to flatten out after the first four begins to flatten out after the first four contributors (the vital few) account for 86% contributors (the vital few) account for 86% of the total.of the total.

Both the Pareto table and the Pareto Diagram are widely Both the Pareto table and the Pareto Diagram are widely used, but the diagram form generally tends to convey used, but the diagram form generally tends to convey much more information at a glance than the table much more information at a glance than the table numbers.numbers.

The implications of the Pareto analysis for the quality The implications of the Pareto analysis for the quality improvement team described above are profound. If the improvement team described above are profound. If the team can find remedies that will prevent errors on the team can find remedies that will prevent errors on the four vital few information items, they can significantly four vital few information items, they can significantly improve the quality of order forms coming in from the improve the quality of order forms coming in from the sales offices. This is an important point; without the facts sales offices. This is an important point; without the facts and without a Pareto analysis, the team would be faced and without a Pareto analysis, the team would be faced with the much larger and more costly task of trying to find with the much larger and more costly task of trying to find ways to prevent errors from occurring on all 18 items. ways to prevent errors from occurring on all 18 items. We can clearly see from the Pareto table or diagram that We can clearly see from the Pareto table or diagram that a significant improvement can be achieved with a much a significant improvement can be achieved with a much smaller, but more precisely focused, effort.smaller, but more precisely focused, effort.

SUMMARYSUMMARY

Pareto analysis leads a quality improvement team Pareto analysis leads a quality improvement team to focus on the vital few problems or causes of to focus on the vital few problems or causes of problems that have the greatest impact on the problems that have the greatest impact on the quality effect that the team is trying to improve. In quality effect that the team is trying to improve. In Pareto analysis, we gather facts and attempt to Pareto analysis, we gather facts and attempt to find the highest concentration of quality find the highest concentration of quality improvement potential in fewest projects or improvement potential in fewest projects or remedies. These offer the greatest potential gain remedies. These offer the greatest potential gain for the least amount of managerial and for the least amount of managerial and investigative effort.investigative effort.

HOW TO INTERPRET PARETO ANALYSISHOW TO INTERPRET PARETO ANALYSIS

Let us now summarize what we have said Let us now summarize what we have said about using and interpreting Pareto about using and interpreting Pareto analysis.analysis.

Separating the Vital Few and Useful ManySeparating the Vital Few and Useful Many

Our objective in Pareto analysis is to use the facts to find Our objective in Pareto analysis is to use the facts to find the highest concentration of quality improvement the highest concentration of quality improvement potential in the fewest number of projects or remedies. potential in the fewest number of projects or remedies. These offer the greatest potential gain for the least These offer the greatest potential gain for the least amount of managerial and investigative effort – the amount of managerial and investigative effort – the highest return on investment. Vital Few Useful Many86highest return on investment. Vital Few Useful Many86

The goal of Pareto is, therefore, to separate the The goal of Pareto is, therefore, to separate the numerous problems or causes of problems into two numerous problems or causes of problems into two categories; the vital few and the useful many. The categories; the vital few and the useful many. The easiest way to do this is to look for a “break point” in the easiest way to do this is to look for a “break point” in the slope of the cumulative-percent-of total line graph on the slope of the cumulative-percent-of total line graph on the Pareto diagram for example See the following Figure Pareto diagram for example See the following Figure

Pareto Diagram

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Note that the slope of the cumulative-percent of total Note that the slope of the cumulative-percent of total line-graph for the fifth category, item B, is substantially line-graph for the fifth category, item B, is substantially “flatter” than the lope of the graph for the fourth category, “flatter” than the lope of the graph for the fourth category, item Q. This substantial change in slope represents a item Q. This substantial change in slope represents a “break point” on the cumulative graph, and this break “break point” on the cumulative graph, and this break point identifies the boundary between the vital few and point identifies the boundary between the vital few and the useful many.the useful many.

The above discussion is a bit oversimplified – reality is The above discussion is a bit oversimplified – reality is often not as clear and simple as we have portrayed it often not as clear and simple as we have portrayed it here. Sometimes there is not a clear break point here. Sometimes there is not a clear break point between the vital few and the useful many. In reality, between the vital few and the useful many. In reality, there is a third category that lies between the vital few there is a third category that lies between the vital few and useful many _ what J.M. Juran called “awkward and useful many _ what J.M. Juran called “awkward zone” in his classic book, zone” in his classic book, Managerial break thoughManagerial break though (McGraw-Hill: New York, 1964, pgs. 53 - 54) Figure 3 (McGraw-Hill: New York, 1964, pgs. 53 - 54) Figure 3 illustrates this “awkward zone”.illustrates this “awkward zone”.

FIGURE 3:FIGURE 3: THE PARETO DISTRIBUTION THE PARETO DISTRIBUTION AND THE “AWKWARD ZONE”AND THE “AWKWARD ZONE”

Most Dollars Are In the “Vital Few” Categories

The “Awkward Zone”Few Dollars Are In The ‘Useful Many”

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Determining the break point is not an exact Determining the break point is not an exact science. In practice, a quality improvement team science. In practice, a quality improvement team faced with interpreting a Pareto diagram that does faced with interpreting a Pareto diagram that does not show a clear break point usually takes the not show a clear break point usually takes the following approach:following approach:

Identify those few contributors, which account for about 60% of the Identify those few contributors, which account for about 60% of the quality effect.quality effect.Call these the “vital few” and begin the diagnostic journey.Call these the “vital few” and begin the diagnostic journey.When the diagnostic and remedial journeys are complete for these vital When the diagnostic and remedial journeys are complete for these vital few, repeat the Pareto analysis. The contributors that were in the few, repeat the Pareto analysis. The contributors that were in the awkward zone may now be among the vital few.awkward zone may now be among the vital few.Repeat steps 1 through 3 as long as profitable projects can be Repeat steps 1 through 3 as long as profitable projects can be identified.identified.

By dealing with those contributors, which are By dealing with those contributors, which are clearly among the vital few, we often gain a better clearly among the vital few, we often gain a better understanding of what to do with those in the understanding of what to do with those in the awkward zone.awkward zone.

HISTOGRAMSHISTOGRAMS

AA histogram is a graphic summary of histogram is a graphic summary of variation in a set of data. The variation in a set of data. The pictorial nature of the histogram pictorial nature of the histogram enables us to see patterns that are enables us to see patterns that are difficult to see in a simple table of difficult to see in a simple table of numbers.numbers.

CONCEPTCONCEPT

The case of “Couldn’t Hear”The case of “Couldn’t Hear”

We have stressed the importance of using data and facts in our We have stressed the importance of using data and facts in our problems - solving and quality improvement efforts. But sometimes problems - solving and quality improvement efforts. But sometimes the data can seem overwhelming or of little value to us as we tackle the data can seem overwhelming or of little value to us as we tackle the problem at hand. Consider the following example.the problem at hand. Consider the following example.

A manufacturer of electronic telecommunications equipment was A manufacturer of electronic telecommunications equipment was receiving complaints from the field about low volume sound on long receiving complaints from the field about low volume sound on long distance connections. Aunt Millie in California couldn’t hear Cousin distance connections. Aunt Millie in California couldn’t hear Cousin Bill in Florida.Bill in Florida.

A string of amplifiers manufactured by the company was used to A string of amplifiers manufactured by the company was used to boost the signal at various points along the way in these long boost the signal at various points along the way in these long connections.connections.

The boosting ability of the amplifiers (engineers call it the “gain”) The boosting ability of the amplifiers (engineers call it the “gain”) was naturally the prime suspect in the case.was naturally the prime suspect in the case.

The design of the amplifiers has called for a gain of 1 0 decibels The design of the amplifiers has called for a gain of 1 0 decibels (d B). (d B).

This means that the output from the amplifier should be about This means that the output from the amplifier should be about ten times stronger than the input signal. This amplification ten times stronger than the input signal. This amplification makes up for the natural fading of the signal over the long-makes up for the natural fading of the signal over the long-distance connection. Recognizing that it is difficult to make distance connection. Recognizing that it is difficult to make every amplifier with a gain of exactly 10 dB the design allowed every amplifier with a gain of exactly 10 dB the design allowed the amplifiers to be considered acceptable if the gain fell the amplifiers to be considered acceptable if the gain fell between 7.75 dB and 12.25 dB. Theses permissible minimum between 7.75 dB and 12.25 dB. Theses permissible minimum and maximum values are sometimes called the specification (or and maximum values are sometimes called the specification (or spec) limits. The expected value of 10 dB is the nominal value. spec) limits. The expected value of 10 dB is the nominal value. Since there were literally hundreds of amplifiers boosting the Since there were literally hundreds of amplifiers boosting the signal in series on a long connection. Low gain amplifiers signal in series on a long connection. Low gain amplifiers should have been balanced out by high gain amplifiers to give should have been balanced out by high gain amplifiers to give an acceptable volume level.an acceptable volume level.

The quality improvement team investigating the ‘Couldn’t Hear” The quality improvement team investigating the ‘Couldn’t Hear” condition arranged to have gain testing performed on 120 condition arranged to have gain testing performed on 120 amplifiers. The results of the tests are listed in Figure 1.amplifiers. The results of the tests are listed in Figure 1.

Figure 1: Data on amplifier gainFigure 1: Data on amplifier gain

Gain of 120 Tested Amplifiers

8.1 10.4 8.8 9.7 7.8 9.9 11.7 8.0 9.3 9.0

8.2 8.9 10.1 9.4 9.2 7.9 9.5 10.9 7.8 8.3

9.1 8.4 9.6 11.1 7.9 8.5 8.7 7.8 10.5 8.5

11.5 8.0 7.9 8.3 8.7 10.0 9.4 9.0 9.210.79.3 9.7 8.7 8.2 8.9 8.6 9.5 9.4 8.8 8.3

8.4 9.1 10.1 7.8 8.1 8.8 8.0 9.2 8.4 7.8

7.9 8.5 9.2 8.7 10.2 7.9 9.8 8.3 9.0 9.6

9.9 10.6 8.6 9.4 8.8 8.2 10.5 9.7 9.1 8.0

8.7 9.8 8.5 8.9 9.1 8.4 8.1 9.5 8.7 9.3

8.1 10.1 9.6 8.3 8.0 9.8 9.0 8.9 8.1 9.7

8.5 8.2 9.0 10.2 9.5 8.3 8.9 9.1 103 8.4

8.6 9.2 8.5 9.36 9.0 10.7 8.6 10.0 8.8 8.6

This table of data is certainly formidable; there are 120 This table of data is certainly formidable; there are 120 numbers to examine. More importantly. Since the gain of all the numbers to examine. More importantly. Since the gain of all the amplifiers fell within the specification limits, the team was amplifiers fell within the specification limits, the team was tempted to conclude, based on a quick glance at the numbers, tempted to conclude, based on a quick glance at the numbers, that the data was of little value. The testing and data gathering that the data was of little value. The testing and data gathering done by the team obviously represented a dead end in their done by the team obviously represented a dead end in their investigation of the case. Or did it?investigation of the case. Or did it?

The team decided to construct a histogram to give them a The team decided to construct a histogram to give them a better “picture” of the 120 data points. They divided the better “picture” of the 120 data points. They divided the specification range into nine intervals of 0.5 dB each and specification range into nine intervals of 0.5 dB each and counted the number of data points that fell in each interval. counted the number of data points that fell in each interval. They found that there ere 24 amplifiers whose gain reading fell They found that there ere 24 amplifiers whose gain reading fell between 7.75 dB and 8.24 dB, 28 amplifiers between 8.25 dB between 7.75 dB and 8.24 dB, 28 amplifiers between 8.25 dB and 8.74dB, and so on.and 8.74dB, and so on.

The histogram of the data is shown in Figure 2. The height of The histogram of the data is shown in Figure 2. The height of each bar on the histogram represents the number of amplifiers each bar on the histogram represents the number of amplifiers with gain readings which fell within the dB range that the bar with gain readings which fell within the dB range that the bar covers on the horizontal axis. For example, the histogram covers on the horizontal axis. For example, the histogram indicates that 19 amplifiers had a gain reading between 9.25 dB indicates that 19 amplifiers had a gain reading between 9.25 dB and 9.74 dB.and 9.74 dB.

Figure 2: Histogram of amplifier gain dataFigure 2: Histogram of amplifier gain data

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The histogram of the data gave the team a very different view of The histogram of the data gave the team a very different view of the situation. While all the amplifiers fell within the specification the situation. While all the amplifiers fell within the specification limits, the readings were certainly not evenly distributed around limits, the readings were certainly not evenly distributed around the nominal 10dB value. Most of the amplifiers had a lower –than the nominal 10dB value. Most of the amplifiers had a lower –than – nominal value of gain. This pattern was hard to see in the table – nominal value of gain. This pattern was hard to see in the table of data. But the histogram clearly revealed it.of data. But the histogram clearly revealed it.

If most of the amplifiers in the series on a long – distance If most of the amplifiers in the series on a long – distance connection boost the signal a little bit less than expected. The connection boost the signal a little bit less than expected. The result will be a low volume level – Anunt Millie in California won’t result will be a low volume level – Anunt Millie in California won’t be able to hear Cousin Bill in Florida.be able to hear Cousin Bill in Florida.

The histogram gave the team a clearer and more complete The histogram gave the team a clearer and more complete picture of the data. Their testing, data gathering, and analysis picture of the data. Their testing, data gathering, and analysis efforts were not a dead end. They could now concentrate their efforts were not a dead end. They could now concentrate their investigation in the factory to find out why the manufacturing line investigation in the factory to find out why the manufacturing line was not producing more amplifiers closer to the nominal value.was not producing more amplifiers closer to the nominal value.

Histograms in Problem SolvingHistograms in Problem Solving

As this example illustrates, the histogram is a simple but As this example illustrates, the histogram is a simple but powerful tool for elementary analysis of data. Let us look powerful tool for elementary analysis of data. Let us look again at the example and summarize some key concepts again at the example and summarize some key concepts about data and the use of histograms in problem solving.about data and the use of histograms in problem solving.Concept 1:Concept 1: Values in a set of data almost always show Values in a set of data almost always show variation. Although the amplifiers were designed for a variation. Although the amplifiers were designed for a nominal value of 10dB gain, very few of them actually had a nominal value of 10dB gain, very few of them actually had a measured gain of 10dB. Furthermore, very few amplifiers measured gain of 10dB. Furthermore, very few amplifiers had exactly the same gain. This variation is due to small had exactly the same gain. This variation is due to small differences in literally hundreds of factors surrounding the differences in literally hundreds of factors surrounding the manufacturing process, the exact values of the component manufacturing process, the exact values of the component parts, the nature of the handling that each amplifier parts, the nature of the handling that each amplifier receives, the accuracy and repeatability of the test receives, the accuracy and repeatability of the test equipment, even the humidity in the factory on the day that equipment, even the humidity in the factory on the day that the amplifier was made.the amplifier was made.Variation is everywhere. It is inevitable in the output of any Variation is everywhere. It is inevitable in the output of any process manufacturing, service or administrative. It is process manufacturing, service or administrative. It is impossible to keep all factors in a constant state all the time.impossible to keep all factors in a constant state all the time.

Consider these examples of variation. Will the Consider these examples of variation. Will the measurement be a constant, or will there be some measurement be a constant, or will there be some variation in the data?variation in the data?The height of 10 year old boys.The height of 10 year old boys.The number of pieces of candy in a one pound bag.The number of pieces of candy in a one pound bag.The exact weight of a 2’ x 2’ piece of sheet steel.The exact weight of a 2’ x 2’ piece of sheet steel.The exact volume of product in a container.The exact volume of product in a container.The time required to repair an appliance for a customer.The time required to repair an appliance for a customer.The number of passengers on a 747 airplane.The number of passengers on a 747 airplane.The number of minutes require4d to process an invoice.The number of minutes require4d to process an invoice.

In each case, the measurement will show some In each case, the measurement will show some variation; few values will be exactly the same.variation; few values will be exactly the same.In the space below, list other examples of variation In the space below, list other examples of variation that occur in your organization.that occur in your organization.

Concept 2:Concept 2: variation displays a pattern. In the variation displays a pattern. In the amplifier example, the pattern of variation shown in amplifier example, the pattern of variation shown in Figure 2 had a number of characteristics, for example:Figure 2 had a number of characteristics, for example:All values fell within the specification limits.All values fell within the specification limits.Most of the values fell between the nominal and the lower specification Most of the values fell between the nominal and the lower specification limit.limit.The values of gain tended to bunch up near the lower specification limit.The values of gain tended to bunch up near the lower specification limit.More values fell in the range of 8.25 dB category decreased uniformly More values fell in the range of 8.25 dB category decreased uniformly for values of gain greater than 8.75 dB.for values of gain greater than 8.75 dB.

Different phenomena will have different variation, but Different phenomena will have different variation, but there is always some pattern to the variation. For there is always some pattern to the variation. For example, we know that the height of most 10-year old example, we know that the height of most 10-year old boys will be close to some average value and that it boys will be close to some average value and that it would be relatively unusual to fined an extremely tall or would be relatively unusual to fined an extremely tall or extremely short boy. If we gathered the data on the extremely short boy. If we gathered the data on the time required to repair an appliance for a customer, or time required to repair an appliance for a customer, or the time required to process paperwork, or the time the time required to process paperwork, or the time required to complete a transaction at a bank, we would required to complete a transaction at a bank, we would expect to see some similar pattern in the numbers.expect to see some similar pattern in the numbers.

Four our purposes, we simply want to point out that there are usually dissemble Four our purposes, we simply want to point out that there are usually dissemble patterns in the variation, and these patterns often tell us a great deal about the patterns in the variation, and these patterns often tell us a great deal about the cause of a problem. Identifying and interpreting these patterns are the most cause of a problem. Identifying and interpreting these patterns are the most important topics in this chapter. There are three important characteristics of a important topics in this chapter. There are three important characteristics of a histogram:histogram:Its centerIts centerIts widthIts widthIts shapeIts shape

Concept 3:Concept 3: patterns of variation are difficult to see in simple tables patterns of variation are difficult to see in simple tables of numbers. Again, recall the amplifier example and the table of of numbers. Again, recall the amplifier example and the table of data in Figure 1. Looking at the table of numbers, we could see data in Figure 1. Looking at the table of numbers, we could see

that no values fall outside the specification limits, but we cannot that no values fall outside the specification limits, but we cannot see much else. While there is a pattern in the data, it is difficult for see much else. While there is a pattern in the data, it is difficult for

our eyes and minds to see it. It is easy to conclude erroneously, as our eyes and minds to see it. It is easy to conclude erroneously, as the team almost did, that the data represents a “dead end” in our the team almost did, that the data represents a “dead end” in our

problem-solving efforts.problem-solving efforts.

Concept 4:Concept 4: patterns of variation are easier to see when the data patterns of variation are easier to see when the data are summarized pictorially in a histogram. The histogram in Figure are summarized pictorially in a histogram. The histogram in Figure

2 gave the team more insight into ho9w to improve the quality of 2 gave the team more insight into ho9w to improve the quality of long-distance telecommunications service. The histogram made it long-distance telecommunications service. The histogram made it

easier for the team to draw conclusions.easier for the team to draw conclusions.

Summary of ConceptSummary of ConceptThey histogram is a useful tool when a team is faced They histogram is a useful tool when a team is faced

with the task of analyzing data that contain with the task of analyzing data that contain variation. We know intuitively that the variation will variation. We know intuitively that the variation will usually follow some pattern, but the pattern is often usually follow some pattern, but the pattern is often hard to see from the table of numbers. Because it hard to see from the table of numbers. Because it is a “picture” of the data, a histogram enables us to is a “picture” of the data, a histogram enables us to see this pattern of variation.see this pattern of variation.

How to interpret HistogramsHow to interpret HistogramsLet us now summarize what we have been saying Let us now summarize what we have been saying

about using and interpreting histograms.about using and interpreting histograms.

Identifying and explaining patterns of variationIdentifying and explaining patterns of variation

We know that the values in any set of data will vary. That variation We know that the values in any set of data will vary. That variation will display some pattern. The goal of our analysis of a histogram is will display some pattern. The goal of our analysis of a histogram is

to;to;Identify and classify the pattern of variation.Identify and classify the pattern of variation.

Develop a plausible and relevant explanation for the pattern.Develop a plausible and relevant explanation for the pattern.

We will present some typical patterns of variation to help you We will present some typical patterns of variation to help you classify histograms (step 1 of the analysis). We will also give some classify histograms (step 1 of the analysis). We will also give some general advice on possible explanations for the patterns (step 2 of general advice on possible explanations for the patterns (step 2 of the analysis). But there is no magic set of rules that you can use to the analysis). But there is no magic set of rules that you can use to

explain the patterns precisely in every situation. The explanation explain the patterns precisely in every situation. The explanation must be based on the team’s knowledge and observation of the must be based on the team’s knowledge and observation of the

specific situation. And must be confirmed through additional specific situation. And must be confirmed through additional analysis. The histogram is just a tool; a team must use experience analysis. The histogram is just a tool; a team must use experience

and knowledge of the process and problem to use the tool and knowledge of the process and problem to use the tool effectively.effectively.

Typical Patterns of VariationTypical Patterns of VariationFigure 3 shows common patterns of variation. General Figure 3 shows common patterns of variation. General explanations of each type and suggestions for further analysis are explanations of each type and suggestions for further analysis are given on the following pages.given on the following pages.

Figure 3: Common Histogram PatternsFigure 3: Common Histogram Patterns

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Bell-Shaped

The Bell-Shaped Distribution; a symmetrical shape with apeak in the middle of the range of the data. This is the normal, natural distribution of data from a process. Deviations from this bell-shape may indicate the presence of complicating factors or outside influences. While deviations from a bell-shape should be investigated, such deviations are not necessarily bad. As we will see below, some non-bell distributions are to be expected in certain cases.

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Double-Peaked

The Doubled-Peaked Distribution; a distinct valley in the middle of the range of the data with peaks on either side. This pattern is usually a combination of two bell-shaped distributions and suggests that two distinct processes are at work.Try various stratification schemes to isolate the distinct processes or conditions. (There are other possible interpretations; see Interpretation Exercise 4.).

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Plateau

The Plateau Distribution; a flat top with no distinct peak, and slight tails on either side. This pattern is likely to be the result of many different bell-shaped distributions with centers spread evenly throughout the range of the data.Diagram the flow and observe the operation to identify the many different processes that are at work. An extreme case occurs in organizations that have no defined processes or training – everyone does the job his or her own way. The wide variability in process leads to the wide variability observed in the data. Defining and implementing standard procedures will reduce this variability.

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Comb

The Comb Distribution; high and low values alternating in a regular fashion. This pattern typically indicated measurement error, errors in the way the data were grouped to construct the histogram, or a systematic bias in the way the data was rounded off. This might also be a type of plateau distribution, but the regularity of alternating highs and lows is a warning of possible errors in data collection or in histogram construction.Review the data-collection procedures and the construction of the histogram before considering possible process characteristics that might cause the pattern.

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Skewed

The Skewed Distribution; an asymmetrical shape in which the peak is off-center in the range of data and the distribution tails off sharply on one side and gently on the other. The illustration in Figure 3 is called a “Positively Skewed” distribution because the long tail extends rightward, toward increasing values. A “Negatively Skewed” distribution would have a long tail extending leftward decreasing values.

The skewed pattern typically occurs when a practical limit, or a The skewed pattern typically occurs when a practical limit, or a specification limit, exists on one side and is relatively close to the specification limit, exists on one side and is relatively close to the nominal value. In these cases, there simply are not as many values nominal value. In these cases, there simply are not as many values available on one side as there are on the other side. Practical limits available on one side as there are on the other side. Practical limits occur frequently when the data consists of time measurements or occur frequently when the data consists of time measurements or counts of things.counts of things.For example, tasks that take a very short time can never be completed For example, tasks that take a very short time can never be completed in zero or less time. So those occasions when the task takes a little in zero or less time. So those occasions when the task takes a little longer than average to complete create a positively skewed tail on the longer than average to complete create a positively skewed tail on the distribution of task time.distribution of task time.The number of weaving defects per 100 yards of fabric can never be The number of weaving defects per 100 yards of fabric can never be less than zero. If the process averages about 0.7 defects per 100 yard, less than zero. If the process averages about 0.7 defects per 100 yard, then sporadic occurrences of 3 or 4 defects per 100 yards will result in then sporadic occurrences of 3 or 4 defects per 100 yards will result in a positively skewed distribution.a positively skewed distribution.One-sided specification limits (a maximum or minimum value only) also One-sided specification limits (a maximum or minimum value only) also frequently give rise to skewed distributions.frequently give rise to skewed distributions.Such skewed distributions are not inherently bad. But a team should Such skewed distributions are not inherently bad. But a team should question the impact of the values in the long tail. Cold they cause question the impact of the values in the long tail. Cold they cause customer dissatisfaction (e.g., long waiting times)? Could they lead to customer dissatisfaction (e.g., long waiting times)? Could they lead to higher costs (e.g., overfilling containers)? Could the extreme values higher costs (e.g., overfilling containers)? Could the extreme values cause problems in downstream operations? If the long tail has a cause problems in downstream operations? If the long tail has a negative impact on quality. The team should investigate and determine negative impact on quality. The team should investigate and determine the causes for those values.the causes for those values.

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Truncated

The truncated Distribution; an asymmetrical shape in which the peak is at or near the edge of the range of the data, and the distribution ends very abruptly on one side and tails off gently on the other. The illustration in Figure 3 shows truncation on the left side with a positively skewed tail. Of course, you may also encounter truncation on the right side with a negatively skewed tail. Truncated distributions are often smooth, bell-shaped distributions with a part of the distribution removed, or truncated, by some external force such as screening, 100% inspection, or review process. Note that these truncation efforts are an added cost and are, therefore, good candidates for removal.

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Isolated-Peaked

The Isolated Peaked Distribution; a small, separate group of data in addition to the larger distribution. Like the double peaked distribution, this pattern is a combination, and suggests that two distinct processes are at work. But the small size of the second peak indicates an abnormality, something that doesn’t happen often or regularly.

Look closely at the conditions surrounding the data in the small peak to see if you can isolate a particular time, machine, input source, procedure, operator, etc. such small isolated peaks in conjunction with a truncated distribution may result from the lack of complete effectiveness in screening out defective items. It is also possible that the small peak represents errors in measurements or in transcribing the data, re-check your measurements and calculations.

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Edge-Peaked

The Edge-Peaked Distribution; a large peak is appended to an otherwise smooth distribution. This shape occurs when the extended tail of the smooth distribution has been cut off and lumped into a single category at the edge of the range of the data. This shape very frequently indicated inaccurate recording of the data (e.g., values outside the “acceptable” range are reported as being just inside the range).

SUMMARY: SUMMARY: HOW TO CONSTRUCT A HISTOGRAMHOW TO CONSTRUCT A HISTOGRAM

1) Obtain the table of raw data and determine the high value, 1) Obtain the table of raw data and determine the high value, low value, and range.low value, and range.Range = high value – low valueRange = high value – low value

2) Decide on the number of cells. Use the following guide;2) Decide on the number of cells. Use the following guide;

Data Points Number of Cells

20* - 50 651 – 100 7101 – 200 8201 – 500 9501 – 1000 10Over 1000 11-20

Less than 40 only as result of stratification.Less than 40 only as result of stratification.

3) Calculate the approximate cell width.3) Calculate the approximate cell width.Approx. cell width = range / number of cellsApprox. cell width = range / number of cells

4) Round the cell width to a convenient number.4) Round the cell width to a convenient number.Make the cell width 1. 2. Or 5; 0.1. 0.2. or 0.5; 10. 20. Or 50; etc.Make the cell width 1. 2. Or 5; 0.1. 0.2. or 0.5; 10. 20. Or 50; etc.

5) Construct the cells by listing the cell boundaries.5) Construct the cells by listing the cell boundaries.The first cell should include the lowest data value.The first cell should include the lowest data value.Cell boundaries should be one more significant digit than data.Cell boundaries should be one more significant digit than data.

6) Tally the number of data points in each cell 6) Tally the number of data points in each cell Check that total tally marks equal number of data points.Check that total tally marks equal number of data points.

7) Draw and label the horizontal axis.7) Draw and label the horizontal axis.Go one cell width beyond the lowest and highest cell.Go one cell width beyond the lowest and highest cell.Provide numeric labels and a caption to describe the measurement and its Provide numeric labels and a caption to describe the measurement and its units.units.

8) Draw and label the vertical axis.8) Draw and label the vertical axis.Label the axis from 0 to a multiple of 5 that is greater than the largest tally in Label the axis from 0 to a multiple of 5 that is greater than the largest tally in any cell.any cell.Provide a caption of “number” or “percent”. Provide a caption of “number” or “percent”.

9) Draw in the bars to represent the number of data points in 9) Draw in the bars to represent the number of data points in each cell.each cell.The height of The height of thethe bars should be equal to the number of data points bars should be equal to the number of data points in that cell as measured on the vertical axis.in that cell as measured on the vertical axis.

10) Title the chart, indicate the total number of data points, and 10) Title the chart, indicate the total number of data points, and show nominal values and limits ((If applicable)).show nominal values and limits ((If applicable)).

11) Identify and classify the pattern of variation.11) Identify and classify the pattern of variation.Refer to Figure 3 and accompanying text.Refer to Figure 3 and accompanying text.

12) Develop a plausible and relevant explanation for the 12) Develop a plausible and relevant explanation for the pattern.pattern.Refer to Figure 3 and accompanying text.Refer to Figure 3 and accompanying text.

Use your team’s knowledge and observation.Use your team’s knowledge and observation.

Confirm your theories through additional analysis.Confirm your theories through additional analysis.