Quality Tools for Six Sigma

Six Sigma Quality: Concepts & Cases‐ Volume I STATISTICAL TOOLS IN SIX SIGMA DMAIC PROCESS WITH MINITAB® APPLICATIONS

Chapter 5

Chapter 5: Quality Tools for Six Sigma

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 2

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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.

Chapter 5: Quality Tools for Six Sigma

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

Histogram of Ring Dia: Run 2 4.95 5 5.05 18

8 Frequency 6

0 4.950 4.965 4.980 4.995 5.010 5.025 5.040 Ring Dia: Run 2 (a) Figure 5.1(a) Most values within specification, some assignable causes may be present

Histogram of Ring Dia: Run 3 4.95 5 5.05

Frequency 10

0 4.950 4.965 4.980 4.995 5.010 5.025 5.040 Ring Dia: Run 3

(c)

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

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Hi stogram of Ring Dia: Run 5 Hi stogram of Ring Dia: Run 6 4.95 5 5.05 4.95 5 5.05 25 20

15 15

Fr e que ncy10 Fr e que ncy

5 5

0 0 4.94 4.96 4.98 5.00 5.02 5.04 4.950 4.965 4.980 4.995 5.010 5.025 5.040 Ring Dia: Run 6 Ring Dia: Run 5

(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)

Histogram of Ring Dia: Run 9 4.95 5 5.05 14

6 Frequency

0 4.953 4.966 4.979 4.992 5.005 5.018 5.031 5.044 Ring Dia: Run 9

(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 5

Chapter 5: Quality Tools for Six Sigma

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

A Run Chart Showing a Stable Process 3.5

3.0

2.5 Process 1 2.0

1.5 1 10 20 30 40 50 60 70 80 90 Observation

Number of runs about median: 53 Number of runs up or down: 56 Expected number of runs: 46.00000 Expected number of runs: 59.66667 Longest run about median: 4 Longest run up or down: 5 Approx P-Value for Clustering: 0.93111 A pprox P-Value for Trends: 0.17721 A pprox P-Value for Mixtures: 0.06889 A ppro x P-Value fo r O scillatio n: 0.82279

A Run Chart Showing a Stable Process

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A Run Chart Showing a Trend 10

6 Process 2 5

3 1 5 10 15 20 25 30 35 40 45 50 55 60 65 Observation

Number of runs about median: 18 Number of runs up or down: 43 Expected number of runs: 33.96970 Expected number of runs: 43.66667 Longest run about median: 12 Longest run up or down: 4 Approx P-Value for Clustering: 0.00004 Approx P-Value for Trends: 0.42178 A pprox P-Value for Mixtures: 0.99996 A p p ro x P -Value fo r O scillatio n: 0.57822

Run Chart Showing a Trend

A Run Chart Showing a Shift in the Process 10

6 Process 3 5

3 1 5 10 15 20 25 30 35 40 45 50 55 60 Observation

Number of runs about median: 19 Number of runs up or dow n: 42 Expected number of runs: 32.96875 Expected number of runs: 42.33333 Longest run about median: 13 Longest run up or down: 4 A pprox P-Value for C lustering: 0.00021 A pprox P-Value for Trends: 0.46007 A pprox P-Value for Mixtures: 0.99979 A p p ro x P -Value fo r O scillatio n : 0.53993

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)

Pareto Chart of Defects in Machined Parts

90 80 70 60 50 40 Count 30 20 10 0 Failure Cause s ts rs s r s r s s rs r s n r o w o o re o o io a r la r r u r r s P r F r r il r r n d E l E E a E E e e g a t l F h g m g n n n a s n i a i r e ic e i i D in e m n us in w t m h t e a - F a c a c In r h n e r e D a u c I c D rr M s e a o a M rf c e u In M S Count84 55 40 33 25 21 17 15 12 Percent 27.8 18.2 13.2 10.9 8.3 7.0 5.6 5.0 4.0 Cum % 27.8 46.0 59.3 70.2 78.5 85.4 91.1 96.0 100.0

Pareto Chart of Defect Data with No Cumulative Points Plotted

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Pareto Chart of Defects in Machined Parts

300 100

250 80 200 60 150 Count

40 Percent 100

50 20

0 0 Failure C ause s ts rs s rs rs s rs rs n r o w o o re o o io a r la r r u r r s P r F r r il r r n E l E E a E E e d g a t l F h g m e n n n ca is n i g i r e i se n i D a in e m n i w t m h t e a -u F a c a c In r h n e r e D a u c I c D rr M s e a o a M rf c e u In M S

Count 84 55 40 33 25 21 17 15 12 Percent 27.8 18.2 13.2 10.9 8.3 7.0 5.6 5.0 4.0 Cum % 27.8 46.0 59.3 70.2 78.5 85.4 91.1 96.0 100.0

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.

Pareto Chart of Defects in Machined Parts Shift = Day 100 120 100 80

80 60 60 Count

40 Percent 40 20 20 0 0 Types of Failure rs n r s s rs s ts r s o io o w o re r o r s r l a r u a r r n r F r il P r E e E l E a d E g m t a h F e g n i n n is g n i D e r n se a i in t m e i u w h c e nt F - m a c e r I e n a r a rr u c I D D M o s a c a r f n e u I M S Count 37 24 22 14 11 9 7 4 Percent 28.9 18.8 17.2 10.9 8.6 7.0 5.5 3.1 Cum % 28.9 47.7 64.8 75.8 84.4 91.4 96.9 100.0

(a)

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Pareto Chart of Defects vs. Shift

Pareto Chart of Defects in Machined Parts

s n rs r io o ro s s rr r r n E ts E s s ro e t r s h e r r im n a w is r ro E e P l a in ilu r g D m d F a E in ct e e l F F g in e ur g a ce in h rr s a rn a se c o a m e r f u w a c e a t u - r a M I n M D I n S I n D

Shift = Day Shift = Morning Types of Failure 120 Machining Errors 90 Incorrect Dimension 60 Measurement Errors 30 Damaged Parts 0 Shift = Night Internal Flaws

Count 120 Surface Finish Errors 90 In-use Failures 60 Drawing Errors 30 0 s s s s s s s r n r t r e r ro io ro ar w ro r ro r ns r P la r ilu r E e E F E a E g t d l h F g n im n e a is n i D e g rn n e i in t m a e i u s w ch c e m t F - r a a e r a In e n r r su D c I D M co a fa n e ur I M S Ty pe s of Fa ilure

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.

A Cause-and-Effect Diagram of Scrap and Rework

Measurements Material Personnel

Gage Problem Defective from Insufficient Vendor Training Caliberation Wrong Batch Lack of Problem Experience Error in Composition Variation in Measuring Training Wrong Heat Treatment Variation in Measuring Experience Measurement Casting Voids New Machinist Rework Program Error and

Ventilation Vibration Scrap Wrong Cutting Prameters Heat Assignment Fixture Dust Wrong Job sequence T ool Force Humidity Process Planning Temperature Tempeature T ool wear

Environment Methods Machines

A Cause-and-Effect Diagram of Rework and Scrap

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Example 10: Causeandeffect Diagram (2)

Major Causes of Production Problems

Measurements Material Personnel

Calibration Material quality T raining Supplier Lack of experience Equipment Many suppliers Work instructions Gage problem Lead Time Absenteeism Poor tools Wrong gaging Delivery Lack of motivation Observer errors Internal flaws Communication External flaws Lack of training Personal problems Quality check Health Fatigue Central Stress and fatigue Production Work conditions Analysis Hydraulic problems Distractions Design problem Prev maintenance Noise Technology Exessive downtime Poor maintenance Heat No automation A utomation Dirt Poor sk ills Poor tooling Dust Instructions Technology Humidity Process planing Capability Temperature Work methods Old machines

Env ironm ent Methods Machines

A Cause-and-Effect Diagram of Production Problems

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

Costs of Poor Quality (COPQ)

Prevention Int e r nal

Improve Quality eng Scrap Improve Manufacturing eng Rework Im roving design Repair Cost of quality planning Retest Cost of quality training Failure analysis Special equipments to control Downtime Building supplier relations Variability analyses Cost of yield losses Cost of skilled quality professional Downgrading products COPQ Loss of future business Cost of field audits Loss of market share Customer dissatisfaction Cost of product audits Loss of goodwill Test and inspection records Paperwork processing Corrective action Cost of inspection and testing Settlement costs Legal services and costs Cost of maintaining test Cost of testing Recalls Cost of maintaing materials Customer complains Inspection Supervisors Returned Warranty charges Appraisal External

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Summary and Applications of Quality Tools

Type of Applications Number of Variables Plotted Chart/Graph

Run Chart One variable plotted

Univariate data

Stem-and-leaf : One variable plotted Plot : Univariate data

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

Pareto Chart The chart shows (in descending order) the One variable divided into contribution of the vital few versus the trivial different categories many. Used to identify the few problems, causes, sources, or defects that should be Univariate data considered first in the problem-solving process. 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 13

Chapter 5: Quality Tools for Six Sigma 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

Scatter Plot with Individual Histograms of X and Y Variables

14 13 10 9 9 8 7 7 6 3 2 0 2 120 0 3 4 10 100 6 6 6 10 80 6 6

Profit ($) 13 6 60 4 5 1 40 40 50 60 70 80 90 100 Sales ($)

Scatterplot with Histograms of x and y Variables

Scatterplot with Box Plots:

Scatter Plot with Box Plot of X and Y Variables

120

100

80 Profit ($) 60

40 40 50 60 70 80 90 100 Sales ($)

Scatterplot with Box Plots of x and y Variables

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

Fitted Line Plot Yield = - 1022 + 320.3 Temp. - 1.054 Temp.**2

25000 S 897.204 R-Sq 97.8% R-Sq(adj) 97.7% 20000

15000 Yield 10000

5000

0 50 100 150 200 250 300 Temp.

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 y 1022  320.3xx  1.054 2 .

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 A B C D Totals Means Size Small 60 76 85 68 65 83 91 73 611 76.375 Medium : : 699 87.375 Large 95 102 91 102 109 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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Multi-Vari Chart for Profit by Store Location - Store Size

110 Store Location 1 2 100 3 4

90 Profit

1 2 3 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

Main Effects Plot (data means) for Profit

Store Size Store Location 100

85 Mean of Profit Mean

1 2 3 1 2 3 4

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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Interaction Plot (data means) for Profit

1 2 3 4 110 Store Size 100 1 2 90 3 Store Size 80

70 110 Store Location 100 1 2 90 3 Store Location 4 80

1 2 3

Interaction Plot of Profit vs. Store Size and Store Location

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

Multi-Vari Chart for Strength by Alloy Type - Thickness

Alloy 740 Type 1 735 2 3

730

725

Strength 720

715

710

1 2 3 4 Thickness

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

Multi-Vari Chart for Strength by Thickness - Alloy Type

Thickness 740 1 2 735 3 4

730

725

Strength 720

715

710

1 2 3 A lloy Ty pe

A Multi-Vari Chart for Strength by Thickness and Alloy Type 18

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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.

Interaction Plot (data means) for Strength

1 2 3 740 Thickness 1 2 730 3 4 Thickness 720

710

740 A llo y Type 1 730 2 3 Alloy Type 720

710

1 2 3 4

Interaction Plot of Strength by Thickness and Strength by Alloy Type

Example 19: Multivari Chart for a Threefactor Design

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

1300 1500

1 2 3 Grain 550 Size 5 10 500 15

450

Strength 400

350

300 1300 1500 1300 1500 Temperature 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). 19

Chapter 5: Quality Tools for Six Sigma

 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………..

Interaction Plot (fitted means) for Strength

5 10 15 1300 1500 Material 480 Type 1 2 Material Type 400 3

320 Grain 480 Size 5 Grain Size 10 400 15

320

Temperature

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

Summary and Applications of Plots TYPE OF APPLICATIONS NUMBER OF VARIABLES CHART/GRAPH PLOTTED

In a scatterplot, two variables usually denoted by Two variable plotted X and Y are examined. A sample of n observations (Bivariate data) is collected and the bivariate data (x1, y1), (x2, y2),...., (xn, yn) are plotted. The plot is used to Scatterplot 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 ……..

Marginal Plot

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Mult-vari charts can be used as clue generation Up to four factors can be techniques. The charts have several applications. plotted using MINITAB Multi-vari Chart 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.

Type of Number of Variables Chart/Graph Plotted Applications

Interaction Plot Interaction plots are used to assess the Interaction plot matrix of interaction effects between the variables. In several factors can be the initial stages of data analysis, interaction generated using MINITAB. 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 ……………………

Symmetry Plot Symmetry plots are a quick and easy way to One variable plotted at a time 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 …….

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

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