quality tools for improvment

1. Seven Tools of QCSeven Tools of QC
Improvements CourseImprovements Course

2. 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

3. GraphsGraphs
Bar Chart Line Chart Pie Chart
0
1
2
3
4
5
6
A B C D E
0
1
2
3
4
5
6
7
8
A B C D E
B
13%
A
7%
D
27%
C
20%
E
33%
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

4. Pareto ChartsPareto Charts
Number of units investigated: 5.000
0
20
40
60
80
100
120
140
160
180
200
D B F A C E Others
NumberofDefectiveunits
0
20
40
60
80
100
120
CumulativePercentage
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: Crack
B: Scratch
C: Stain
D: Strain
E: Gap
F: Pinhole

5. Pareto AnalysisPareto Analysis
PPareto analysis is a ranked comparison ofareto analysis is a ranked comparison of
factors related to afactors related to a quality problemquality problem. It. It
helps a quality improvement project team tohelps a quality improvement project team to
identify and focus on the vital few factors.identify and focus on the vital few factors.

6. 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 thewho observed that a relative few people held the
majority of the wealth. Pareto developedmajority of the wealth. Pareto developed
logarithmic mathematical models to describe thislogarithmic mathematical models to describe this
non-uniform distribution of wealth, and thenon-uniform distribution of wealth, and the
mathematician M.O. Lorenz developed graphs tomathematician M.O. Lorenz developed graphs to
illustrate it.illustrate it.
Historical EvolutionHistorical Evolution

7. Dr. Joseph Juran was the first to point out that whatDr. 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 Varityprinciple – one that applied in an astounding Varity
of situations and appeared to hold withoutof 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 Paretophenomenon that he has called the Pareto
Principle; that in any group of factors contributing toPrinciple; that in any group of factors contributing to
a common effect, a relative few account for the bulka common effect, a relative few account for the bulk
of the effect. Juran has also coined the terms “ vitalof the effect. Juran has also coined the terms “ vital
few” and “useful many” to refer to those fewfew” and “useful many” to refer to those few
contributions which account for a smaller proportioncontributions which account for a smaller proportion
of the effect.of the effect.
The Pareto PrincipleThe Pareto Principle

8. As experienced managers and professionals, weAs experienced managers and professionals, we
intuitively recognize the Pareto principle and theintuitively recognize the Pareto principle and the
concepts of the vital few and useful many, for weconcepts of the vital few and useful many, for we
see them in operation in everyday businesssee 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 ourThe top 15% of our customers account for 68% of our
total revenues.total revenues.
Our top 5 products or services account for 75% of ourOur 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 theIn a typical meeting, a few people tend to make the
majority of comments, while most people are relativelymajority of comments, while most people are relatively
quiet.quiet.

9. The principle of the vital few and useful many alsoThe principle of the vital few and useful many also
applies to quality improvement opportunities. Eachapplies to quality improvement opportunities. Each
quality effect that we can observe (for example; qualityquality effect that we can observe (for example; quality
costs, defects, rework, customer dissatisfaction,costs, defects, rework, customer dissatisfaction,
revenues, complaints, etc.) results from numerousrevenues, complaints, etc.) results from numerous
contributors to that effect. When we look at the manycontributors to that effect. When we look at the many
individual contributors, we find that a few account forindividual 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 findFor example, when we gather the facts, we might find
that;that;
In a 25 – step-manufacturing process, 5 of the operationsIn 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 theOf 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 orderOf 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 theseform, 4 of the items generate 86% of the errors found on these
forms.forms.

10. 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 theIf we focus our attention on these vital few, we can get the
greatest potential gain from our quality improvementgreatest potential gain from our quality improvement
efforts.efforts.
The Pareto principle is so obvious and so simple that youThe 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 knowseverybody knows that, don’t they ? But if everybody knows
it already, why do we so often hear mangers complainingit already, why do we so often hear mangers complaining
that they are faced with dozens of problems in theirthat they are faced with dozens of problems in their
organization? and why do we so often see company taskorganization? and why do we so often see company task
forces listing dozens of problems and setting out to solveforces 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 ParetoIf we really understood the simple but profound Pareto
principle, our first step when faced with a host of problemsprinciple, 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 improvementWe could then focus our attention and improvement
efforts on those few things that would give us the greatestefforts on those few things that would give us the greatest
improvement in quality.improvement in quality.

11. Pareto Diagrams and TablePareto Diagrams and Table
Pareto diagrams and tables are presentationPareto diagrams and tables are presentation
techniques used to show the facts and separatetechniques used to show the facts and separate
the vital few from the useful many. They arethe vital few from the useful many. They are
widely used to help quality improvement teamswidely used to help quality improvement teams
and steering committees make key decisions atand steering committees make key decisions at
various points in the quality improvement orvarious points in the quality improvement or
problem-solving sequence.problem-solving sequence.
Regardless of the form chose, well-constructedRegardless of the form chose, well-constructed
Pareto diagram and tables include three basicPareto diagram and tables include three basic
elements;elements;

12. You will notice that Pareto diagram presents theYou will notice that Pareto diagram presents the
result of stratifying a problem by one particularresult of stratifying a problem by one particular
variable. The contributors to the effect are thevariable. 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 toA look at the following example of how to
construct and use Pareto diagrams and tablesconstruct and use Pareto diagrams and tables
will illustrate and further explain these threewill illustrate and further explain these three
basic elements.basic elements.
The contributors to the total effect, ranked by theThe contributors to the total effect, ranked by the
magnitude of their contribution.magnitude of their contribution.
The magnitude of the contribution of each expressedThe magnitude of the contribution of each expressed
numerically.numerically.
The cumulative-percent-of-total effect of the rankedThe cumulative-percent-of-total effect of the ranked
contributors.contributors.

13. The “Out ofThe “Out of OrderOrder” Orders” Orders
A quality improvement team was chartered toA quality improvement team was chartered to
improve the quality of order forms coming in withimprove 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, whichThere were 18 items on the order form, which
we will designate here as items A to R. Thewe will designate here as items A to R. The
team developed a check sheet which it used toteam developed a check sheet which it used to
collect the frequency errors on the forms for acollect the frequency errors on the forms for a
week. The results of the team’s study, in theweek. 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

14. 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 29
J 38 25 54
M 31 21 75
Q 16 11 86
B 8 5 91
D 5 3 95
C 3 2 97
A 1 0.67 98
O 1 0.67 98
R 1 0.67 99
N 1 0.67 99
L 1 0.66 100
I 0 0 100
E 0 0 100
H 0 0 100
K 0 0 100
F 0 0 100
P 0 0 100
Total 150 100

15. Note that the Pareto table contains the three basic elementsNote that the Pareto table contains the three basic elements
described above. The first column lists the contributors, thedescribed above. The first column lists the contributors, the
18 items, not in order of their appearance on the form, but18 items, not in order of their appearance on the form, but
rather, in order of the number of errors detected on eachrather, in order of the number of errors detected on each
item during the study. The second and third columns showitem during the study. The second and third columns show
the magnitude of contribution – the number of errorsthe magnitude of contribution – the number of errors
detected on each item and the corresponding percentage ofdetected on each item and the corresponding percentage of
total errors on the form. The fourth column gives thetotal errors on the form. The fourth column gives the
cumulative-percent of total. This column is the key to Paretocumulative-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 forIn other words, the first four items, G, J, M and Q account for
86% of the total errors detected in the study. These are the86% 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 theAgain, note the three basic elements that make up the
diagram.diagram.

16. Pareto Diagram
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
G J M Q B D C A O R N L I E H K F P
Order Form Item
NumberofErrors
0
20
40
60
80
100
120
CumulativePercentofTotal
Vital Few
Useful
Many
2
2
1
3
3
FIGURE 2:FIGURE 2: PARETO DIAGRAM OF ERRORS ONPARETO DIAGRAM OF ERRORS ON
ORDER FORMSORDER FORMS

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

18. Both the Pareto table and the Pareto Diagram are widelyBoth the Pareto table and the Pareto Diagram are widely
used, but the diagram form generally tends to conveyused, but the diagram form generally tends to convey
much more information at a glance than the tablemuch more information at a glance than the table
numbers.numbers.
The implications of the Pareto analysis for the qualityThe implications of the Pareto analysis for the quality
improvement team described above are profound. If theimprovement team described above are profound. If the
team can find remedies that will prevent errors on theteam can find remedies that will prevent errors on the
four vital few information items, they can significantlyfour vital few information items, they can significantly
improve the quality of order forms coming in from theimprove the quality of order forms coming in from the
sales offices. This is an important point; without the factssales offices. This is an important point; without the facts
and without a Pareto analysis, the team would be facedand without a Pareto analysis, the team would be faced
with the much larger and more costly task of trying to findwith 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 thatWe can clearly see from the Pareto table or diagram that
a significant improvement can be achieved with a mucha significant improvement can be achieved with a much
smaller, but more precisely focused, effort.smaller, but more precisely focused, effort.

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

20. HOW TO INTERPRET PARETO ANALYSISHOW TO INTERPRET PARETO ANALYSIS
Let us now summarize what we have saidLet us now summarize what we have said
about using and interpreting Paretoabout using and interpreting Pareto
analysis.analysis.

21. Separating the Vital Few and Useful ManySeparating the Vital Few and Useful Many
Our objective in Pareto analysis is to use the facts to findOur objective in Pareto analysis is to use the facts to find
the highest concentration of quality improvementthe 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 leastThese offer the greatest potential gain for the least
amount of managerial and investigative effort – theamount 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 theThe goal of Pareto is, therefore, to separate the
numerous problems or causes of problems into twonumerous problems or causes of problems into two
categories; the vital few and the useful many. Thecategories; the vital few and the useful many. The
easiest way to do this is to look for a “break point” in theeasiest way to do this is to look for a “break point” in the
slope of the cumulative-percent-of total line graph on theslope of the cumulative-percent-of total line graph on the
Pareto diagram for example See the following FigurePareto diagram for example See the following Figure

22. Pareto Diagram
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
G J M Q B D C A O R N L I E H K F P
Order Form Item
NumberofErrors
0
20
40
60
80
100
120
CumulativePercentofTotal
Vital Few
Useful Many
86

23. Note that the slope of the cumulative-percent of totalNote that the slope of the cumulative-percent of total
line-graph for the fifth category, item B, is substantiallyline-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 aitem 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 andpoint identifies the boundary between the vital few and
the useful many.the useful many.
The above discussion is a bit oversimplified – reality isThe above discussion is a bit oversimplified – reality is
often not as clear and simple as we have portrayed itoften not as clear and simple as we have portrayed it
here. Sometimes there is not a clear break pointhere. 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 fewthere is a third category that lies between the vital few
and useful many _ what J.M. Juran called “awkwardand 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”.

24. FIGURE 3:FIGURE 3: THE PARETO DISTRIBUTIONTHE 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”
Categories
Cumulative
Percent of
Quality Costs
Categories
1
0
N
50
100

25. Determining the break point is not an exactDetermining the break point is not an exact
science. In practice, a quality improvement teamscience. In practice, a quality improvement team
faced with interpreting a Pareto diagram that doesfaced with interpreting a Pareto diagram that does
not show a clear break point usually takes thenot show a clear break point usually takes the
following approach:following approach:
Identify those few contributors, which account for about 60% of theIdentify 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 vitalWhen the diagnostic and remedial journeys are complete for these vital
few, repeat the Pareto analysis. The contributors that were in thefew, 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 beRepeat steps 1 through 3 as long as profitable projects can be
identified.identified.
By dealing with those contributors, which areBy dealing with those contributors, which are
clearly among the vital few, we often gain a betterclearly among the vital few, we often gain a better
understanding of what to do with those in theunderstanding of what to do with those in the
awkward zone.awkward zone.

26. HISTOGRAMSHISTOGRAMS
AA histogram is a graphic summary ofhistogram is a graphic summary of
variation in a set of data. Thevariation in a set of data. The
pictorial nature of the histogrampictorial nature of the histogram
enables us to see patterns that areenables us to see patterns that are
difficult to see in a simple table ofdifficult to see in a simple table of
numbers.numbers.

27. CONCEPTCONCEPT
The case of “Couldn’t Hear”The case of “Couldn’t Hear”
We have stressed the importance of using data and facts in ourWe have stressed the importance of using data and facts in our
problems - solving and quality improvement efforts. But sometimesproblems - solving and quality improvement efforts. But sometimes
the data can seem overwhelming or of little value to us as we tacklethe 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 wasA manufacturer of electronic telecommunications equipment was
receiving complaints from the field about low volume sound on longreceiving complaints from the field about low volume sound on long
distance connections. Aunt Millie in California couldn’t hear Cousindistance 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 toA string of amplifiers manufactured by the company was used to
boost the signal at various points along the way in these longboost 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.

28. The design of the amplifiers has called for a gain of 1 0 decibelsThe 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 aboutThis means that the output from the amplifier should be about
ten times stronger than the input signal. This amplificationten 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 makedistance connection. Recognizing that it is difficult to make
every amplifier with a gain of exactly 10 dB the design allowedevery amplifier with a gain of exactly 10 dB the design allowed
the amplifiers to be considered acceptable if the gain fellthe amplifiers to be considered acceptable if the gain fell
between 7.75 dB and 12.25 dB. Theses permissible minimumbetween 7.75 dB and 12.25 dB. Theses permissible minimum
and maximum values are sometimes called the specification (orand 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 theSince there were literally hundreds of amplifiers boosting the
signal in series on a long connection. Low gain amplifierssignal in series on a long connection. Low gain amplifiers
should have been balanced out by high gain amplifiers to giveshould 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 120condition 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.

29. 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 9.2
10.7
9.3 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

30. This table of data is certainly formidable; there are 120This table of data is certainly formidable; there are 120
numbers to examine. More importantly. Since the gain of all thenumbers to examine. More importantly. Since the gain of all the
amplifiers fell within the specification limits, the team wasamplifiers 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 gatheringthat the data was of little value. The testing and data gathering
done by the team obviously represented a dead end in theirdone 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 aThe team decided to construct a histogram to give them a
better “picture” of the 120 data points. They divided thebetter “picture” of the 120 data points. They divided the
specification range into nine intervals of 0.5 dB each andspecification 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 fellThey found that there ere 24 amplifiers whose gain reading fell
between 7.75 dB and 8.24 dB, 28 amplifiers between 8.25 dBbetween 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 ofThe histogram of the data is shown in Figure 2. The height of
each bar on the histogram represents the number of amplifierseach bar on the histogram represents the number of amplifiers
with gain readings which fell within the dB range that the barwith gain readings which fell within the dB range that the bar
covers on the horizontal axis. For example, the histogramcovers on the horizontal axis. For example, the histogram
indicates that 19 amplifiers had a gain reading between 9.25 dBindicates that 19 amplifiers had a gain reading between 9.25 dB
and 9.74 dB.and 9.74 dB.

31. Figure 2: Histogram of amplifier gain dataFigure 2: Histogram of amplifier gain data
0
10
20
30
Number
. 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13
Gain in dB
Results of Amplifier Gain Testing
Nominal
10 dB
Spec Limit
7.75 dB
Spec Limit
12.25 dB

32. The histogram of the data gave the team a very different view ofThe histogram of the data gave the team a very different view of
the situation. While all the amplifiers fell within the specificationthe situation. While all the amplifiers fell within the specification
limits, the readings were certainly not evenly distributed aroundlimits, the readings were certainly not evenly distributed around
the nominal 10dB value. Most of the amplifiers had a lower –thanthe 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 – distanceIf most of the amplifiers in the series on a long – distance
connection boost the signal a little bit less than expected. Theconnection boost the signal a little bit less than expected. The
result will be a low volume level – Anunt Millie in California won’tresult 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 completeThe histogram gave the team a clearer and more complete
picture of the data. Their testing, data gathering, and analysispicture of the data. Their testing, data gathering, and analysis
efforts were not a dead end. They could now concentrate theirefforts were not a dead end. They could now concentrate their
investigation in the factory to find out why the manufacturing lineinvestigation 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.

33. Histograms in Problem SolvingHistograms in Problem Solving
As this example illustrates, the histogram is a simple butAs this example illustrates, the histogram is a simple but
powerful tool for elementary analysis of data. Let us lookpowerful tool for elementary analysis of data. Let us look
again at the example and summarize some key conceptsagain 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 showValues in a set of data almost always show
variation. Although the amplifiers were designed for avariation. Although the amplifiers were designed for a
nominal value of 10dB gain, very few of them actually had anominal value of 10dB gain, very few of them actually had a
measured gain of 10dB. Furthermore, very few amplifiersmeasured gain of 10dB. Furthermore, very few amplifiers
had exactly the same gain. This variation is due to smallhad exactly the same gain. This variation is due to small
differences in literally hundreds of factors surrounding thedifferences in literally hundreds of factors surrounding the
manufacturing process, the exact values of the componentmanufacturing process, the exact values of the component
parts, the nature of the handling that each amplifierparts, the nature of the handling that each amplifier
receives, the accuracy and repeatability of the testreceives, the accuracy and repeatability of the test
equipment, even the humidity in the factory on the day thatequipment, 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 anyVariation is everywhere. It is inevitable in the output of any
process manufacturing, service or administrative. It isprocess 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.

34. Consider these examples of variation. Will theConsider these examples of variation. Will the
measurement be a constant, or will there be somemeasurement 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 someIn 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 variationIn the space below, list other examples of variation
that occur in your organization.that occur in your organization.

35. Concept 2:Concept 2: variation displays a pattern. In thevariation displays a pattern. In the
amplifier example, the pattern of variation shown inamplifier 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 specificationMost 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 uniformlyMore 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, butDifferent phenomena will have different variation, but
there is always some pattern to the variation. Forthere is always some pattern to the variation. For
example, we know that the height of most 10-year oldexample, we know that the height of most 10-year old
boys will be close to some average value and that itboys will be close to some average value and that it
would be relatively unusual to fined an extremely tall orwould be relatively unusual to fined an extremely tall or
extremely short boy. If we gathered the data on theextremely short boy. If we gathered the data on the
time required to repair an appliance for a customer, ortime required to repair an appliance for a customer, or
the time required to process paperwork, or the timethe time required to process paperwork, or the time
required to complete a transaction at a bank, we wouldrequired 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.

36. Four our purposes, we simply want to point out that there are usually dissembleFour 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 thepatterns 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 mostcause of a problem. Identifying and interpreting these patterns are the most
important topics in this chapter. There are three important characteristics of aimportant topics in this chapter. There are three important characteristics of a
histogram:histogram:
Its centerIts center
Its widthIts width
Its shapeIts shape
Concept 3:Concept 3: patterns of variation are difficult to see in simple tablespatterns of variation are difficult to see in simple tables
of numbers. Again, recall the amplifier example and the table ofof numbers. Again, recall the amplifier example and the table of
data in Figure 1. Looking at the table of numbers, we could seedata in Figure 1. Looking at the table of numbers, we could see
that no values fall outside the specification limits, but we cannotthat no values fall outside the specification limits, but we cannot
see much else. While there is a pattern in the data, it is difficult forsee 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, asour 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 ourthe 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 datapatterns of variation are easier to see when the data
are summarized pictorially in a histogram. The histogram in Figureare summarized pictorially in a histogram. The histogram in Figure
2 gave the team more insight into ho9w to improve the quality of2 gave the team more insight into ho9w to improve the quality of
long-distance telecommunications service. The histogram made itlong-distance telecommunications service. The histogram made it
easier for the team to draw conclusions.easier for the team to draw conclusions.

37. Summary of ConceptSummary of Concept
They histogram is a useful tool when a team is facedThey histogram is a useful tool when a team is faced
with the task of analyzing data that containwith the task of analyzing data that contain
variation. We know intuitively that the variation willvariation. We know intuitively that the variation will
usually follow some pattern, but the pattern is oftenusually follow some pattern, but the pattern is often
hard to see from the table of numbers. Because ithard to see from the table of numbers. Because it
is a “picture” of the data, a histogram enables us tois 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 Histograms
Let us now summarize what we have been sayingLet us now summarize what we have been saying
about using and interpreting histograms.about using and interpreting histograms.

38. Identifying and explaining patterns of variationIdentifying and explaining patterns of variation
We know that the values in any set of data will vary. That variationWe 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 iswill 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 youWe will present some typical patterns of variation to help you
classify histograms (step 1 of the analysis). We will also give someclassify histograms (step 1 of the analysis). We will also give some
general advice on possible explanations for the patterns (step 2 ofgeneral advice on possible explanations for the patterns (step 2 of
the analysis). But there is no magic set of rules that you can use tothe analysis). But there is no magic set of rules that you can use to
explain the patterns precisely in every situation. The explanationexplain the patterns precisely in every situation. The explanation
must be based on the team’s knowledge and observation of themust be based on the team’s knowledge and observation of the
specific situation. And must be confirmed through additionalspecific situation. And must be confirmed through additional
analysis. The histogram is just a tool; a team must use experienceanalysis. The histogram is just a tool; a team must use experience
and knowledge of the process and problem to use the tooland knowledge of the process and problem to use the tool
effectively.effectively.
Typical Patterns of VariationTypical Patterns of Variation
Figure 3 shows common patterns of variation. GeneralFigure 3 shows common patterns of variation. General
explanations of each type and suggestions for further analysis areexplanations of each type and suggestions for further analysis are
given on the following pages.given on the following pages.

39. Figure 3: Common Histogram PatternsFigure 3: Common Histogram Patterns
10
20
30
40
35
25
15
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.

40. 45
65
90
75
45
20
35 40
80
90
50
15
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.).

41. 75
80
100
120
90 80
110 120
100 110
90 80
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.

42. 50 60
45
65 55
80
55 65
45 50
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.

43. 40 50
100 90 80 70 60 60
40
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.

44. The skewed pattern typically occurs when a practical limit, or aThe skewed pattern typically occurs when a practical limit, or a
specification limit, exists on one side and is relatively close to thespecification limit, exists on one side and is relatively close to the
nominal value. In these cases, there simply are not as many valuesnominal value. In these cases, there simply are not as many values
available on one side as there are on the other side. Practical limitsavailable on one side as there are on the other side. Practical limits
occur frequently when the data consists of time measurements oroccur 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 completedFor 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 littlein zero or less time. So those occasions when the task takes a little
longer than average to complete create a positively skewed tail on thelonger 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 beThe 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 inthen 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) alsoOne-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 shouldSuch skewed distributions are not inherently bad. But a team should
question the impact of the values in the long tail. Cold they causequestion the impact of the values in the long tail. Cold they cause
customer dissatisfaction (e.g., long waiting times)? Could they lead tocustomer dissatisfaction (e.g., long waiting times)? Could they lead to
higher costs (e.g., overfilling containers)? Could the extreme valueshigher costs (e.g., overfilling containers)? Could the extreme values
cause problems in downstream operations? If the long tail has acause problems in downstream operations? If the long tail has a
negative impact on quality. The team should investigate and determinenegative impact on quality. The team should investigate and determine
the causes for those values.the causes for those values.

45. 100 90 80
65 55
40 50 40 40
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.

46. 30 40 50
70
100
70 60
30
0
30 40 30
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.

47. 30 40 50 60
75
60
40
100
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).

48. 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 6
51 – 100 7
101 – 200 8
201 – 500 9
501 – 1000 10
Over 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

49. 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 cell6) 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 itsProvide 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 inLabel 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 in9) Draw in the bars to represent the number of data points in
each cell.each cell.
The height ofThe height of thethe bars should be equal to the number of data pointsbars 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.

50. 10) Title the chart, indicate the total number of data points, and10) 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 the12) 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.