CHAPTER 5- Sample Volume 1 - QMS Global LLC · Pareto Chart A Pareto chart is very similar to a...

Six Sigma Quality: Concepts & Cases‐ Volume I

STATISTICAL TOOLS IN SIX SIGMA DMAIC PROCESS WITH MINITAB®

APPLICATIONS

1

Chapter 5

Quality Tools for Six Sigma

© 2010-12 Amar Sahay, Ph.D.

Chapter 5: Quality Tools for Six Sigma

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Chapter Outline

Process Histograms Evaluating Process Capability Using Histogram Stem-and-leaf Plot Box Plot Run Chart

Example 1: Constructing a Run Chart Example 2: A Run Chart with Subgroup Size Greater than 1 Example 3: A Run Chart with Subgroup Size Greater than 1 (Data across the Row) Example 4: Run Chart Showing a Stable Process, a Shift, and a Trend

Pareto Chart Example 5: A Simple Pareto Chart Example 6:Pareto Chart with Cumulative Percentage Example 7:Pareto Chart with Cumulative Percentage when Data are in One Column Example 8:Pareto Chart By Variable

Cause-and-Effect Diagram or Fishbone Diagram Example 9:Cause-and-Effect Diagram (1) Example 10: Cause-and-effect Diagram (2) Example 11: Creating other Types of Cause-and-effect Diagram

Summary and Application of Plots Bivariate Data: Measuring and Describing Two Variables Scatter Plots

Example 12: Scatterplots with Histogram, Box-plots and Dot plots Example 13: Scatterplot with Fitted Line or Curve Example 14: Scatterplot Showing an Inverse Relationship between X and Y Example 15: Scatterplot Showing a Nonlinear Relationship between X and Y Example 16: Scatterplot Showing a Nonlinear (Cubic) Relationship between X and Y

Multi-Vari Chart and Other Plots Useful to Investigate Relationships Before Running Analysis of Variance

-Example 17: A Multi-vari Chart for Two-factor Design -Main Effects Plot -Interaction Plot

Example 18: Another Multi-vari Chart for a Two-factor Design -Multi-Vari plot -Box Plots -Main Effects Plot -Interaction Plot


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Example 19: Mult-vari chart for a Three-factor Design -Multi-Vari Chart -Box Plots -Main Effects Plot

Example 20: Multi-vari Chart for a Four-factor Design -Multi-Vari Chart -Box Plots -Main Effects and Interaction Plots

Example 21: Determine a Machine-to-Machine, Time-to-Time variation Part-to-Part Variation in a Production Run using Multi-vari and Other Plots

Symmetry Plot Summary and Applications

This chapter explains the graphical techniques that have been applied as problem-solving tools in quality control. Many graphs and charts are helpful in detecting and solving quality problems. Some of the graphical techniques in this chapter are described in previous chapters. In this chapter we describe and analyze the graphs and charts as they relate to quality control. We also provide examples and specific situations in which these graphical techniques are used in detecting and solving quality problems.

Some examples from the chapter are presented below. The book provides stepwise instructions with data files for each case.


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Histograms

In quality, histograms are useful in

detecting process problems-including a shift in the process evaluating process capability (ability of the process to be within its specification

limits) determining how well centered the process is or how close the data values are to the

target value, and determining the process variation.

Examples

Ring Dia: Run 2

Freq

uenc

y

5.0405.0255.0104.9954.9804.9654.950

18

16

14

12

10

8

6

4

2

0

4.95 5 5.05

Histogram of Ring Dia: Run 2

(a) Figure 5.1(a) Most values within specification, some assignable causes may be present

(c)

Figure 5.1(c) Process variation has reduced compared to (a): a shift to the right

Ring Dia: Run 3

Freq

uenc

y

5.0405.0255.0104.9954.9804.9654.950

25

20

15

10

5

0

4.95 5 5.05



5

Ring Dia: Run 6

Freq

uenc

y

5.045.025.004.984.964.94

25

20

15

10

5

0

4.95 5 5.05


(e) (f)

Figure 5.1(e) The process has shifted to the left; products out of specification (may be calibration problem)

Figure 5.1(f) Process shift to the left; more variation compared to (e)

Ring Dia: Run 9

Freq

uenc

y

5.0445.0315.0185.0054.9924.9794.9664.953

14

12

10

8

6

4

2

0

4.95 5 5.05


(i)

Figure 5.1 (i) Process stable and close to the target (desirable)

The above histograms are useful in examining the possible problems and the causes behind them.

Evaluating Process Capability using Histograms

Histograms can also be used to assess the process capability (the ability of a process to be

Ring Dia: Run 5

Freq

uenc

y

5.0405.0255.0104.9954.9804.9654.950

20

15

10

5

0

4.95 5 5.05



6

within its specifications). The process capability can be determined once the process is stable ……. The next chapter provides the measures of process capability and their relationship to six sigma quality.

Run Charts

A run chart is used in Quality Control to analyze the data either in the development stage of a product or before the state of statistical control.

A run chart can be used to determine if the process is running in a state of control or if special or assignable causes are influencing the process, thereby making the process out of control.

:

:

Example 4: Run Chart Showing a Stable Process, a Shift, and a Trend

Observation

Proc

ess

1

9080706050403020101

3.5

3.0

2.5

2.0

1.5

Number o f runs about med ian:

0.06889

53Expected number o f runs: 46.00000Longest run about median: 4A pprox P-Value fo r C lustering: 0.93111A pprox P-Value fo r M ixtures:

Number of runs up o r down:

0.82279

56Expected number o f runs: 59.66667Longest run up o r down: 5A pprox P-Value fo r Trends: 0.17721A pprox P-Value fo r O scillation:

A Run Chart Showing a Stable Process

A Run Chart Showing a Stable Process


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Observation

Proc

ess

2

65605550454035302520151051

10

9

8

7

6

5

4

3

Number o f runs about med ian:

0.99996

18Expected number o f runs: 33.96970Longest run about median: 12A pprox P -Value fo r C luster ing: 0.00004A pprox P -Value fo r M ixtures:

Number o f runs up o r dow n:

0.57822

43Expected number o f runs: 43.66667Longest run up o r dow n: 4A pprox P -Value fo r Trends: 0.42178A pprox P -Value fo r O scillation:

A Run Chart Showing a Trend

Run Chart Showing a Trend

Observat ion

Proc

ess

3

605550454035302520151051

10

9

8

7

6

5

4

3

Number o f runs abou t med ian:

0.99979

19Expected number o f runs: 32.96875Longest run abou t med ian: 13A ppro x P -Value fo r C lu ster ing : 0.00021A ppro x P -Value fo r M ixtu res:

Number o f runs up o r dow n:

0.53993

42Expected number o f runs: 42.33333Longest run up o r dow n: 4A pp ro x P -Value fo r Trends: 0.46007A pp ro x P -Value fo r O sc illation :

A Run Chart Showing a Shift in the Process

A Run Chart Showing a Shift

A run chart is a quick, easy, and economical way to detect process problems. In the initial stages of a process, or for a new process, run charts provide opportunities for improvement without the implementation of control charts.


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Pareto Chart

A Pareto chart is very similar to a histogram or a frequency distribution of attribute data where the bars are arranged by categories from largest to smallest with a line that shows the cumulative percentage and count of the bars. This chart is widely used in Quality Control to analyze the defect data. Through this chart, the defects that occur most frequently can be identified quickly and easily. This helps to focus the improvement efforts on the largest percentage of defects. A Pareto chart does not identify the most important category; it identifies the categories that occur most frequently. Pareto charts are also used widely in non manufacturing applications.

There are several variations of the Pareto chart. MINITAB provides several options to construct this chart. The data for the chart can be entered in several ways.

:

Example 6: Pareto Chart with Cumulative Percentage (Second Option)

Coun

t

Failure Cause

Count 12Percent 27.8 18.2 13.2 10.9 8.3 7.0 5.6 5.0

844.0

Cum % 27.8 46.0 59.3 70.2 78.5 85.4 91.1 96.0

55

100.0

40 33 25 21 17 15

Draw ing

Erro

r s

S urfac

e Fini

sh Er

rors

In-us

e Fail

ures

Mecha

nical

Er ror s

M eas u

remen

t Er ro

r s

Intern

al Fla

ws

Machin

ing Er

r ors

Damag

ed Pa

rts

Incor

r ect D

imen

sions

90

80

70

60

50

40

30

20

10

0

Pareto Chart of Defects in Machined Parts

Pareto Chart of Defect Data with No Cumulative Points Plotted


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Coun

t

Perc

ent

Failure Cause

Count 12Percent 27.8 18.2 13.2 10.9 8.3 7.0 5.6 5.0

844.0

Cum % 27.8 46.0 59.3 70.2 78.5 85.4 91.1 96.0

55

100.0

40 33 25 21 17 15

Draw ing

Erro

rs

Surfa

ce F ini

sh Er

rors

In-u

se Fail

ures

Mecha

nica l E

rrors

Measu

remen

t Erro

rs

Inte

rna l F

laws

Machin

ing Er

rors

Damag

ed Pa

rts

Inco

rrect

Dimen

sions

300

250

200

150

100

50

0

100

80

60

40

20

0


Pareto Chart of Defect Data with Cumulative Points Plotted

Example 8: Pareto Chart by Variable (Fourth Option)

The data file PARETOCHART.MTW shows the types of failures in manufactured parts and the shifts they produced (columns C3 and C4). Suppose we want to construct Pareto charts for each of the three shifts. This will help visualize the defect data for each shift separately and analyze the causes of defects. The Pareto charts for each Shift vs. Types of Failure are shown below.

Coun

t

Perc

ent

Types of Failure

CountPercent 28.9 18.8 17.2 10.9 8.6 7.0 5.5 3.1Cum %

37

28.9 47.7 64.8 75.8 84.4 91.4 96.9 100.0

24 22 14 11 9 7 4

Drawing

Error

s

Damag

ed Pa

rts

In-us

e Fa il

ures

Sur fa

ce F i

nish E

rrors

Inter

nal F

l aws

Measu

r emen

t Er ro

r s

Incor

r ect D

imen

sion

M achin

ing Er

rors

120

100

80

60

40

20

0

100

80

60

40

20

0

Pareto Chart of Defects in Machined PartsS hift = Day

(a)


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Pareto Chart of Defects vs. Shift

Types of Failure

Coun

t

Dr aw ing

E rror s

I n- us

e F ai l

u res

Surface

Fin ish

Erro

rs

I nter

nal F l aw

s

D amag

e d Pa rts

Me as

ur em

ent E

rr ors

I nco

rrect

D imen

sion

Ma ch

in in g Erro

rs

120

906030

0

D r awing E

rror s

I n-u se

Fail

ures

S urfa ce

Fin ish

Erro

rs

Inte

rna l F

la ws

D ama ged P

ar ts

M e asu rem

ent E

r rors

I nco

r rect

D imens

ion

Mac

h in ing Erro

rs

1209060

300

S hift = D ay S hift = M orning

S hift = N ight

Ty pes o f F ailu re

Internal F law sSurface F in ish Erro rsIn-use F ailu resDraw ing Erro rs

Mach in ing Erro rsInco rrect D imensionMeasurement Erro rsDamaged Parts


Pareto Chart of Defects vs. Shift (all shifts plotted on one graph)

CauseandEffect Diagrams or Fishbone Diagrams

A cause-and-effect diagram is a very useful tool in establishing the relationship between the causes and their effects. Once a problem has been identified, it is necessary to analyze potential cause or causes of the problem. In such cases, the cause-and-effect diagram is a very helpful tool. To construct a cause-and effect diagram:

(a) Identify the problem or effect

(b) Identify the cause or possible causes of the problem through study or experience

:

:

(f) Take necessary action to solve the problem.


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Example 9: Causeandeffect Diagram (1)

The problem is to study the possible causes of a high percentage of scrap and rework produced. The problem (effect) is identified and possible causes are to be explored using a cause-and-effect diagram. The data file CAUSE.MTW shows six major causes for scrap and rework (Machines, Materials, Methods, Measurement, Personnel, and Environment) in columns C1-C6. Each major cause is further classified into categories that are shown below each major cause (see columns C1-C6 of the data file CAUSE.MTW). The problem or the effect is typed in the Effect box when the dialog box is displayed.

Note: MINITAB provides six major areas which are considered the main causes of problems. These are Personnel or people, methods, materials, machines, measurements, and environment. These are provided as the default causes for the cause-and-effect diagram. However, these defaults can be changed as desired to create different cause categories.

ReworkandScrap

Environment

Measurements

Methods

Mater ial

Machines

P ersonnel

New Machinist

Var iation inExper ience

Var iation inT raining

Lack ofExper ience

InsufficientT raining

Vibration

Cutting Prameters

Fixture

T ool Force

T emperature

T ool wear

Casting Voids

Heat T reatment

Composition

Wrong Batch

D efective fromVendor

WrongA ssignment

Wrong Jobsequence

Process P lanning

MeasurementProgram Er ror

WrongMeasur ing

Error inMeasur ing

CaliberationP roblem

Gage P roblem

Ventilation

Heat

D ust

H umidity

T empeature

A Cause-and-Effect Diagram of Scrap and Rework

A Cause-and-Effect Diagram of Rework and Scrap


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Example 10: Causeandeffect Diagram (2)

P r oduction

Env ironm e nt

M e a sure m e nts

M e thods

M ate ria l

M achine s

Pe rsonne l

Str ess and fatigueH ealth

P er sonal problemsC ommunication

Lack of motivationP oor tools

A bsenteeismWor k instr uctions

Lack of exper ience T r aining

H ydraulic pr oblemsP r ev maintenance

Exessive dow ntimeP oor maintenance

A utomationP oor tooling

T echnologyC apability

O ld machines

C entr alQ uality check

External flaw sInter nal flaw s

D eliver yLead T ime

M any supplier sSupplier

M ater ial quality

A nalysis D esign pr oblem

T echnologyNo automation

P oor sk illsInstr uctions

P r ocess planing Wor k methods

Fatigue

Lack of tr aining

O bser ver er r or s

Wr ong gaging

Gage pr oblem

Equipment

C alibr ation

Work conditionsD istr actions

NoiseH eat

D ir tD ust

H umidityT emper atur e

Major Causes of Production Problems

A Cause-and-Effect Diagram of Production Problems

The Cause-and-Effect Dialog Box for Cost of Poor Quality

COPQ

Appraisal

Prevention

External

Internal

Downgrading products

Cost of yield losses

Downtime

Failure analysis

Retest

Repair

Rework

Scrap

Loss of future businessLoss of market share

Customer dissatisfactionLoss of goodwill

Paperwork processingCorrective action

Settlement costsLegal services and costs

Cost of testingRecalls

Customer complainsReturned

Warranty charges

Cost of skilled quality professional

Variability analyses

Building supplier relations

Special equipments to control

Cost of quality training

Cost of quality planning

Imroving design

Improve Manufacturing eng

Improve Quality eng

Cost of field audits

Cost of product audits

Test and inspection records

Cost of inspection and testing

Cost of maintaining test

Cost of maintaing materials

Inspection Supervisors

Costs of Poor Quality (COPQ)


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Summary and Applications of Quality Tools

Type of Chart/Graph

Applications Number of Variables Plotted

Run Chart One variable plotted

Univariate data

Stem-and-leaf Plot

:

:

One variable plotted

Univariate data

Box-plot Plot of five measures: the minimum, first quartile, median, third quartile, and the maximum. Displays the distribution of underlying data.

One variable plotted

Univariate data

Pareto Chart The chart shows (in descending order) the contribution of the vital few versus the trivial many. Used to identify the few problems, causes, sources, or defects that should be considered first in the problem-solving process.

One variable divided into different categories

Univariate data

All the above graphs-except the cause-and-effect diagram-are used to summarize or describe one characteristic at a time, and therefore, describe univariate data or measurements made on one variable.

Bivariate Data: Measuring and Describing Two Variables Bivariate data consist of two variables. A scatter plot is usually used to describe the relationship between two variables. MINITAB provides several options for scatter plots, which were explained in chapter 3. Here we consider some more applications of scatter plots.

Scatter Plots

In a scatterplot, two variables (usually denoted by X and Y) are examined. A sample of n observations is collected and the bivariate data (x1, y1), (x2, y2),...., (xn, yn) are plotted. The plot is used to extract the relationship between x and y. Several of these plots were described in Chapter 3.

Sometimes it is useful to plot histograms, box plots, or dot plots of x and y variables


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along with the scatter plot. These plots are helpful in determining the relationship between x and y, and their distributions.

Example 12: Scatterplots with Histograms, Box plots, and Dot plots

Sales ($)

Prof

it (

$)

100908070605040

120

100

80

60

40

20

6

987

14

7

13

910

3

20

34

10666

1066

136

45

1

Scatter Plot with Individual Histograms of X and Y Variables

Scatterplot with Histograms of x and y Variables

Scatterplot with Box Plots:

Sa le s ($)

Prof

it (

$)

100908070605040

120

100

80

60

40

S ca tte r P lot w ith Box P lot of X a nd Y V ar ia ble s

Scatterplot with Box Plots of x and y Variables


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Example 15: Scatterplot Showing a Nonlinear Relationship between X and Y

Temp.

Yie

ld

30025020015010050

25000

20000

15000

10000

5000

0

S 897.204R-Sq 97.8%R-Sq(adj) 97.7%

Fitted Line PlotYield = - 1022 + 320.3 Temp.

- 1.054 Temp.**2

Scatterplot with Best Fitting Curve

The plot shows a strong relationship between x and y. The equation relating the temperature and the yield can be very useful in predicting the maximum yield or optimizing the yield of the process. Note that the equation of the fitted curve is

21022 320.3 1.054y x x .

In this equation, y is the yield and x is the temperature. The equation relating the two variables is written on the plot.

MultiVari Charts and Other Plots Useful to Investigate Relationships Before Running the Analysis of Variance

This section presents several plots that can be used to present the information in the form of graphs and charts in the preliminary stages of data analysis. These plots can be very helpful in visualizing the data, and provide invaluable information for a formal analysis of variance. The graphs and chart include Multi-vari Chart Box plot

Main effects plot Interaction plot Marginal plot

Some of these plots-for example the box plot and marginal plot-were explained earlier. Here we explain how to construct all of the above plots. Next, we will analyze the resulting graphs to get further insights. The analyses can be helpful in determining the type of statistical tests to be used for the data.


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Example 17: A Multivari Chart for a Twofactor Design

The marketing manager of a departmental store chain is interested in studying the effect of store location and store size on the profit. Four different locations (A, B, C, and D) and three different store sizes (small, medium, and large) were selected for the study. For each store location a random sample of two stores of each size was selected and the monthly profits (in thousands of dollars) were recorded. Table 5.18 shows the data. Note that this problem can be formulated as a two-factor ANOVA (analysis of variance) with Store Size as Factor A with three levels (small, medium and large); Store Locations as Factor B with four levels (A, B, C, and D), and the Profit as the response variable.

Table 5.18

Store Location

Store Size

A B C D Totals Means

Small 60

65

76

83

85

91

68

73

611

76.375

Medium :

:

699

87.375

Large

95

102

102

109

91

95

Totals 580 484

Means 80.83 96.67

Multi-Vari Chart

This chart displays the mean values at each factor level for every factor. For our example, there are two factors: store size and store location. The multi-vari plot will display the average profit for each level of store size: small (1), medium (2), large (3) and each level of store location: A(1), B(2), C(3), and D(4).

The multi-vari chart is shown below.


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Store Size

Prof

it

321

110

100

90

80

70

Store

4

Location123

Multi-Vari Chart for Profit by Store Location - Store Size

A Multi-Vari Chart for Profit by Store Location and Store Size

In this figure, we have plotted the two factors. The solid lines connect the means of factor B (store location) levels (at each level of factor A, store size). The dotted line connects the means of factor A (store size) levels.

:

Main Effects Plot

Mea

n of

Pro

fit

321

100

95

90

85

80

4321

Store Size Store Location

Main Effects Plot (data means) for Profit

Main Effects Plot for Profit

Interaction Plot

Interaction plots are used to assess the interaction effects between the variables. For our example, the response variable is Profit, and there are two factors of interest; Store Size and Store Location. Our objective is to determine how profit is affected by store size and location and also by their combination. In cases where there are two or more factors are involved, the interaction effects are critical.


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Store Size

Store Location

4321110

100

90

80

70

321

110

100

90

80

70

Sto re

3

S ize12

Sto re

34

Location12

Interaction Plot (data means) for Profit

Interaction Plot of Profit vs. Store Size and Store Location

The presence or absence of interaction can be determined from the above interaction plot.

Thickness

Stre

ngth

4321

740

735

730

725

720

715

710

A llo yTy p e

123

Multi-Var i Chart for S trength by Alloy Type - Thickness

A Multi-Vari Chart for Strength by Alloy Type and Thickness

A lloy Type

Stre

ngth

321

740

735

730

725

720

715

710

Thick ness1234

Multi-Vari Chart for Strength by Thickness - Alloy Type

A Multi-Vari Chart for Strength by Thickness and Alloy Type


19

Both the plots above show that alloy type 2 and thickness 2 has the maximum strength. Also, there is an indication of interaction between the alloy type and thickness because the response variable is not a linear function of the combinations of the levels of the two factors.

Thickness

Alloy Type

321

740

730

720

710

4321

740

730

720

710

Thick ness

34

12

A lloy

3

Type12

Interaction Plot (data means) for Strength

Interaction Plot of Strength by Thickness and Strength by Alloy Type

Example 19: Multivari Chart for a Threefactor Design

Temperature

Stre

ngth

15001300

550

500

450

400

350

300

15001300

15001300

1 2 3 GrainSize

51015

Multi-Vari Chart for Strength by Grain Size - Material Type

Panel variable: Material Type

Multi-Vari Chart for Strength by Grain Size, Temperature, and Material Type

In this figure:

Factor 1 is Grain Size, Factor 2 is Temperature, Factor 3 is Material Type

Black lines connect the means of factor 1 levels (at each combination between factor 2 and 3 levels).


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Red lines (dotted lines) connect the means of factor 2 levels (at each level of factor 3).

A green line (the solid line from the center of columns 1, 2, and 3) connects the means of factor 3 levels.

For 3 and 4 factors, the multi-vari chart is paneled. The panel variable is the 3rd factor for a 3 factor chart………..

M a te r ia l T y pe

Gr a in S ize

T e m pe r a tur e

15105 15001300

480

400

320

480

400

320

Material

3

Ty pe12

Grain

15

S ize5

10

Interaction Plot (fitted means) for Strength

Interaction Plot: Material Type & Grain Size, Material Type & Temperature, and Grain Size & Temperature

Summary and Applications of Plots TYPE OF

CHART/GRAPH APPLICATIONS NUMBER OF VARIABLES

PLOTTED

Scatterplot

Marginal Plot

In a scatterplot, two variables usually denoted by X and Y are examined. A sample of n observations is collected and the bivariate data (x1, y1), (x2, y2),...., (xn, yn) are plotted. The plot is used to extract the relationship between x and y.

Types of Scatterplot: (a) scatterplot with histograms, box plots, or dot plots of x and y variables along with the x,………

(b) Scatterplot with Fitted Line or Curve: these scatterplots show if the relationship between x and y is linear or nonlinear. The plot is also used to determine the correlation or degree of association ……..

Two variable plotted (Bivariate data)


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Multi-vari Chart

Mult-vari charts can be used as clue generation techniques. The charts have several applications. They can be used in preliminary stages of data analysis to investigate the relationships between the main factors and their interactions. Two-to four factor analysis of variance data can be plotted using multi-vari chart…….

:

In a discrete manufacturing environment these charts can be used to detect part-to-part variation, within part variation, and also variation between time periods etc. In a continuous manufacturing, the chart can be used to detect shift-to-shift, day-to-day or week-to-week variation in the response variable.

Up to four factors can be plotted using MINITAB

Type of Chart/Graph

Applications

Number of Variables Plotted

Interaction Plot Interaction plots are used to assess the interaction effects between the variables. In the initial stages of data analysis, interaction plots can be used to see the interactions between variables. These plots can visually show the interaction effects without formally running the analysis of variance (ANOVA). However, the interaction plot may be misleading in ……………………

Interaction plot matrix of several factors can be generated using MINITAB.

Symmetry Plot Symmetry plots are a quick and easy way to check if the data follows a symmetrical distribution. The symmetry plot is generated by first calculating the median of the data and then forming the first pair of values that are closest to the median: one above and one below the median. The second pair is formed using the two values that …….

One variable plotted at a time

The detailed treatment of all the above plots with computer instructions and data files can be found in Chapter 5.

To buy chapter 5 or Volume I of Six Sigma Quality Book, please click on our products on the home page.

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