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

APPLICATIONS

Chapter 3

Visual Representation of Data: Charts and Graphs for Six Sigma

Chapter 3: Visual Representation of Data: Charts and Graphs for Six Sigma 2

Chapter Outline

Describing Data through Charts and Graphs Background Histograms Using MINITAB Construct a Default Histogram Editing the Default Histograms Histogram Options (Other Type of Histograms) A Histograms with a Fitted Normal Curve A Percent and a Density Histogram Construct a Density Histogram Histogram with Fit and Groups Histogram with Outline and Groups Other Graph Options Graphical Summary of Data Stem-and-leaf Plots Constructing Box-Plots One Y-Simple One Y-with Groups Multiple Y’s-Simple Multiple Y’s-with Groups Dotplots One Y-Simple One Y-with Groups One Y-Stack Groups Multiple Y’s-Simple Multiple Y’s-Stack Y’s Multiple Y’s-with Groups Multiple Y’s-Stack Groups Character Graphs Character Box Plot Character Dot Plot Character Stem-and-leaf Plot Character Histogram Bar Charts A Simple Bar Chart Bar Chart of Categorical Variables Bar Chart as a Function of a Variable A Cluster Bar Chart Pie Charts Pie Chart of Categorical Data Scatter Plots A Simple Scatter Plot Scatter Plot with Regression Line Scatter Plot with Groups Scatter Plot with Regression and Groups Scatter Plot with Connected Line Scatter Plot with Connect and Groups Interval Plots Interval Plot with Multiple Y’s Interval Plot: Multiple Y’s with Groups Individual Value Plot Time Series Plots A Simple Time Series Plot Using Stamp as Time/Scale A Multiple Time Series Plot Graphing Empirical Cumulative Density Function (CDF) Probability Plots Probability Plot: Example 1 Probability Plot: Example 2 Probability Plot: Example 3 Matrix Plot Matrix of Plots: Simple Matrix of Plots: With Groups Matrix of Plots: With Smoother Matrix Plots: Each Y versus each X Matrix Plots: Each Y versus each X with Smoother Marginal Plot 3D Scatter Plot 3D Scatter Plot with Groups 3D Scatter Plot with Projected Lines 3D Scatter Plot/Wireframe Plot Surface Plot Contour Plot Summary of Some Plots and Their Application Hands-on Exercises


Describing Data using Charts and Graphs: Introduction

The graphical techniques described in this chapter are useful in the following ways:

The techniques will help you gain insight into the way the variable or variables seem to

behave.

The graphical techniques enable one to understand how the values of a random variable

under study are distributed.

The shapes produced using the graphical techniques help select an appropriate

theoretical distribution for the random variable in question.

The charts and graphs help us visualize the important characteristics of data which are

usually not apparent from the raw data.

Some of the visual representation of the data provide excellent means of comparing

data from processes, checking the variation, and taking corrective actions when the

deviation from stable conditions occur.

In this chapter, we have presented numerous graphical techniques. We assume that

you are familiar with many of them; therefore, we will not discuss the theory behind them in detail. Instead, we focus on applications. We explain how to construct these graphs and charts using the computer and explain their important characteristics.

Charts and Graphs in this Chapter

Histograms with outline and group

Graphical summary of data

Histogram with normal curve

Stem‐and‐leaf, box plots, and dot plots

Character graphs (to construct stem‐and‐leaf, box plot and dot plot)

Widely used plots including bar charts, pie charts, scatter plots, interval plots, and time series plots

Empirical cumulative density function (CDF), probability plots, matrix plots, marginal plots, 3D scatter plots, 3D surface plots, contour plots, and others.


Some Examples of Graphs & Charts useful in Six Sigma Analysis (The data files to construct the charts below are available with the book)

Constructing a Default Histogram To construct a default histogram, follow the instructions in Table 3.1 below.

Table 3.1

CONSTRUCTING A Open the worksheet Demand.MTW then select DEFAULT HISTOGRAM Graph &Histogram

Click on Simple then click OK. Complete the dialog box by selecting or typing the response shown: Graph variables: C1 or Demand Click on Scale Box Click on Y‐scale type and select Frequency Click OK Click Labels In the Title box, type: Histogram of Demand Data Click on Data Labels and select Use Y‐value label Click OK Click on Data View and check Bars, then click OK

You will be back to the Histogram‐simple dialog box. Click OK. The histogram shown in

Figure 3.1 will be displayed on the graphics window.

645648403224

25

20

15

10

5

0

Demand

Freq

uenc

y

4

65

10

15

23

16

11

6

22

Histogram of Demand

Figure 3.1: A Default Histogram

: : :


Other examples: Histogram with Fit and Groups This option can be used to compare the mean and variability of two sets of data. Suppose you want to compare the variability in diameter of the shafts produced by two manufacturers. A sample of 124 shafts from manufacturer 1 and a sample of 200 shafts from manufacturer 2 were measured. The data are in the file ShaftDia.MTW. Table 3.6 shows the instructions. Table 3.6

Open the worksheet ShaftDia. MTW Select Graph & Histogram : : Click Labels and type Comparing the Variability in the Title box Click OK.

The graph in below will be displayed.

Histogram with Fit and Groups

Other Graph Options Using the command sequence, Stat &Basic Statistics & Display Descriptive Statistics and selecting the Graphs in the Display Descriptive Statistics dialog box, provides options for graphs, such as, histogram, histogram of data with normal curve, individual value plot, and box plot.

Graphical Summary of Data This option provides useful statistics of the data along with graphs. To produce a graphical summary of the data, follow the steps in Table 3.8.

Data

Freq

uenc

y

75.0675.0375.0074.9774.9474.91

50

40

30

20

10

0

Mean StDev N75.02 0.01000 12575.00 0.02984 200

VariableShaft Dia 1Shaft Dia 2

Comparing the VariabilityNormal


Table 3.8 Open the worksheet ShaftDia.MTW Select Stat &Basic Statistics &Graphical Summary :

Figure 3.10 shows the graphical summary of the data.

75.0875.0575.0274.9974.9674.93

Median

Mean

75.00074.99874.99674.99474.992

1st Q uartile 74.977M edian 74.9963rd Q uartile 75.017M aximum 75.077

74.992 75.000

74.991 75.000

0.027 0.033

A -Squared 0.22P -V alue 0.834

M ean 74.996S tDev 0.030V ariance 0.001Skew ness -0.0524329Kurtosis -0.0708460N 200

M inimum 74.906

A nderson-Darling Normality Test

95% C onfidence Interv al for M ean

95% C onfidence Interv al for Median

95% C onfidence Interv al for S tDev9 5 % Confidence Inter vals

Summary for Shaft Dia 2

Figure 3.10: Graphical Summary of the Shaft Diameter Data (Note: when the graph is displayed, you can double click any where on the bars and edit to change color and fill of the bars).

Stemandleaf Plots Stem‐and‐leaf plots are very efficient way of displaying data, checking the variation and shape of the distribution. Stem‐and‐leaf plots are obtained by dividing each data value into two parts, stem and leaf. For example, if the data are two‐digit numbers, e.g., 34, 56, 67, etc., then the first number (the tens digit) is considered the stem value, and the second number (the ones digit) is considered the leaf value. Thus, in data value 56, 5 is the stem and 6 is the leaf. In a three digit data value, the first two digits are considered as the stem and the last digit as the leaf. To construct a stem‐and‐leaf plot, follow the steps in Table 3.9. Table 3.9

Open worksheet DEFECTS.MTW From the main menu, select Graph & Stem‐and‐Leaf For Graph variables, select No. of Defects Click OK

The stem‐and‐leaf plot of number of defects will be displayed on the session window.


The plot is shown in Figure 3.11.

Stem‐and‐Leaf Display: No. of Defects (out of 1000) Stem‐and‐leaf of No. of Defects(out of 1000) N = 200 Leaf Unit = 1.0 5 5 23333 6 5 4 12 5 666777 26 5 88888889999999 48 6 0000000000011111111111 76 6 2222222333333333333333333333 (30) 6 444444444444445555555555555555 94 6 6666666666677777777777777777 66 6 888888888888889999999 45 7 00000000000001111111 25 7 2222222222333 12 7 44444555 4 7 667 1 7 8

Figure 3.11: Stem‐and‐Leaf Plot of Number of Defects

Another Example on Stem-and-leaf plot

Stem-and-Leaf Display: Moisture Content Stem-and-leaf of Moisture Content N = 55 Leaf Unit = 0.10 1 8 0 2 8 8 3 9 4 4 9 7 9 10 02334 16 10 5667888 26 11 1223333444 (5) 11 56677 24 12 24444 19 12 5567 15 13 2333 11 13 578 8 14 01 6 14 6 5 15 23 3 15 56 1 16 1

Stem‐and‐leaf Plot of Moisture Content in Samples of Clay (in percent)


BoxPlots The box‐plot displays the smallest and the largest values in the data along with the three quartiles: Q1, Q2, and Q3. The display of these five numbers (known as five measure summary) is used to study the shape of the distribution. There are different types of box plots you can do including: (a) One Y ‐ Simple (b) One Y ‐ With groups (c) Multiple Y’s ‐ Simple (d) Multiple Y’s ‐ With Groups

(b) One Y ‐ With groups This plot is useful when there is one y‐variable (diameter in this case) that you want to monitor by assessing several days of production. You would like to plot the box plot for each day of production. To do this plot, follow the steps in Table 3.12. Table 3.12

The box‐plot in Figure 3.14 will be displayed.

Open the worksheet DIAMETER.MTW From the main menu, select Graph &Boxplot (you can also select Stat &EDA &Boxplot) Under One Y select With Groups Click OK Select Diameter for Graph Variables For Categorical variables for grouping (1‐4, outermost first), select Day for attribute assignment : : Click OK in all the dialog boxes.


Figure 3.14: Box Plot of Diameter vs. Day

Suppose you want to check the consistency of the diameters of 5 samples with respect to three machine operators. You can construct a box plot to check the consistency. The steps are same as described for the box plot in Figure 3.15. Open worksheet DIAMETER3.MTW, follow the steps above but select Operator for categorical variables grouping and select Operator in the Data View box. The box plot is shown in Figure 3.17. Figures 3.16 and 3.17 are useful in checking the distribution and consistency of critical production parameter with respect to categorical variables such as, machine and operator.

Figure 3.17: Box plot of Samples vs. Operators

Day

Dia

met

er

87654321

75.05

75.04

75.03

75.02

75.01

75.00

74.99

74.98

Day

5678

1234

75.019575.015575.01775.016

75.0275

75.018575.015

75.0115

Boxplot of Diameter vs Day

Dat

a

Operator CBASam

ple 5

Sample

4

Sample

3

Sample

2

Sample

1

Sample

5

Sample

4

Sample

3

Sample

2

Sample

1

Sample

5

Sample

4

Sample

3

Sample

2

Sample

1

75.05

75.04

75.03

75.02

75.01

75.00

74.99

OperatorABC

Boxplot of Sample 1, Sample 2, Sample 3, Sample 4, ... vs Operator


Bar Charts

MINITAB provides several options for bar charts including simple, clustered, and stacked bar charts. There are several variations of these charts.

A Simple bar chart: Suppose you want to plot monthly sales for your company. You have two columns of data; the first column is the categorical variable (month) and the second column contains the sales values. To do a bar chart, follow the steps in Table 3.26.

Table 3.26

Open the worksheet BAR1.MTW From the main menu, select Graph &Bar Chart Below Bars represent, click the down arrow and select Values from a table Make sure Simple is highlighted under One column of values Click OK For Graph variables, select Sales ($) In Categorical variable, select Month Click Labels box then click Titles/Footnotes and enter a title for your graph Click Data Labels tab Click the circle next to Use y‐values labels Click Data View, in the Categorical variables for attribute assignment, type or select Month Click OK in all the dialog boxes.

The bar chart in Figure 3.29 will be displayed.

Figure 3.29: A Bar chart of Monthly Sales

Month

Sale

s ($

)

Septem

ber

Augus

tJul

yJu

neMayApri

l

March

Febr

uary

Janu

ary

300

250

200

150

100

50

0

Month

MayJuneJulyAugustSeptember

JanuaryFebruaryMarchApril

Chart of Sales ($) vs Month


Bar Chart of Categorical Variables

The worksheet BAR1.MTW contains a categorical variable: Causes of Failure. This column contains various causes of failure for machined parts. We want to know the frequency of each type of failure.

Figure 3.30: A Bar chart of Categorical Data

A Cluster Bar Chart

Suppose you want to compare the quarterly sales for the past four years. A good way is to plot the sales of each quarter using a bar chart. We can use the cluster option to group the four quarters of each of the four years.

Figure 3.32: A Cluster Bar Chart to Compare Quarterly Sales

Causes of Failure

Coun

t

Measu

remen

t Erro

rs

Inter

nal F

laws

Fatig

ue Fa

ilures

Inco

rrect

Dimen

sion

Machin

ing Er

rors

40

30

20

10

0

Causes of Failure

Measurement Errors

Fatigue FailuresIncorrect DimensionInternal F lawsMachining Errors

899

17

36

Chart of Causes of Failure

Sale

s

YearQuarter

43214321432143214321

200

150

100

50

0

Year1234

170160

150140

175

150

200

180

150

170160

130135

105

170

150

Chart of Sales vs Year, Quarter


Pie Charts A pie chart is used to show the relative magnitudes of parts to a whole. In this chart relative frequencies of each group of data are plotted. A circle is constructed and is divided into distinct sections. Each section represents one group of data. The area of each section is determined by multiplying the relative frequency of each section by the angle of a circle. Since there are 3600 in a circle, each section is multiplied by 3600 to obtain the correct number of degrees for each section.

The data file PIE.MTW contains the causes of failures in machined parts. To construct a pie chart of this data, follow the steps in Table 3.30.

Table 3.30

Open the worksheet PIE.MTW From the main menu, select Graph &Pie Chart In the Pie Chart dialog box, click on the circle next to Chart values from a table In the Categorical variable box, type or select Failures In the Summary variables box, type or select Count Click on Pie Chart Options tab Click on the circle next to Decreasing volume Click OK Click on Labels then click on Titles/Footnotes and type a title for your plot Click on Slice Labels and check Frequency and Percentage Click OK in all the boxes

The pie chart in Figure 3.33 will be displayed. You may double click on the pie chart click custom and select a pattern from the Type box.

Figure 3.33: Pie Chart of Failure Data

15, 4.1%22, 6.1%

25, 6.9%

30, 8.3%

33, 9.1%

40, 11.0%

48, 13.3%

65, 18.0%

84, 23.2%

C ategory

Internal F law sMeasurement Erro rsMechanical Erro rsC MM Erro rsF atigue F ailures

Incorrect D imensionsDamaged PartsMach in ing Erro rsDraw ing Erro rs

Pie Chart of Cause of Failures


Scatter Plots Scatter plots are helpful in investigating the relationship between two variables. One of these variables is considered as a dependent variable and the other an independent variable. The data value is thought of as having an x value and a y value. Thus, we have (xi,yi), i=1,2,3,.....,n pairs. If we are interested in the relationship between the two variables, one of the easiest way to investigate this relationship is to plot the (x,y) pairs. This type of plot is known as a scatter plot. Several options are available for scatter plot. We have demonstrated some below. In the example given below, we have plotted the summer temperature and the amount of electricity used by customers (in millions of kilowatts). The data is shown in SCATTER1.MTW. Follow the steps in Table 3.32 to do a scatter plot of the data. A simple Scatter Plot

Scatter Plot with Regression Line

Summer Temperature

Elec

tric

ity

Used

1051009590858075

32

30

28

26

24

22

20

S 0.944325R-Sq 89.8%R-Sq(adj) 89.4%

Fitted Line PlotElectricity Used = - 2.630 + 0.3134 Summer Temperature

Summer Temperature

Elec

tric

ity

Used

1051009590858075

32

30

28

26

24

22

20

Scatterplot of Electricity Used vs Summer Temperature


Fitted Line Plot with Regression Equation

Interval Plots

The interval plot displays means and/or confidence intervals for one or more variables. This plot is useful for assessing the measure of central tendency and variability of data. The default confidence interval is 95% however; this can be changed by using the command sequence Editor &Edit Interval Bar � Options. We will demonstrate the interval plot using the data in INTERVAL2.MTW. This data file contains the amount of beverage in 16 oz. cans from 5 different production lines. The operations manager suspects that the mean content of the cans differs from line to line. He randomly selected 5 cans from each line and measured the contents. The data is shown in INTERVAL2.MTW. The interval plot below is created from this data.

Summer Temperature

Elec

tric

ity

Used

1051009590858075

32

30

28

26

24

22

20

Scatterplot of Electricity Used vs Summer Temperature


Time Series Plots A time series plots the data over time. The graph plots the (xi,yi) pairs of points and connects these plots through a straight line where the x values are time. The plot is helpful in visualizing a trend or pattern in a data set. In the example below, a time series plot of demand data over time is explained. The data file TIMESERIES1.MTW shows weekly demand data for five quarters. Each quarter is divided into 13 weeks.

A Simple Time Series Plot To do a simple time series plot of the demand data, follow the steps in Table 3.44. Table 3.44

Open the worksheet TIMESERIES1.MTW In the Time Series dialog box, click on Simple then click OK Type or select Demand for Series Click on the Time/Scale box Under Time Scale, click on the circle next to Index Click OK in all dialog boxes.

A simple time series plot shown in Figure 3.48 will be displayed. Note that the Index under Time Scale is used to label the x‐axis with integer values starting from 1. If you want x‐axis label as Week, double click on Index when the graph is created, and change the index to week in the dialog box that is displayed.

Sample

Pist

on R

ing

Dia

2019181716151413121110987654321

45.04

45.03

45.02

45.01

45.00

Sample

3456789

101112

1

1314151617181920

2

Interval Plot of Piston Ring Dia vs Sample95% CI for the Mean


Figure 3.48: A Simple Time Series Plot of Demand Data A Simple Time Series Plot using Stamp as Time/Scale If you want to show quarter and week on the x‐axis, use the stamp option under Time/Scale. The Stamp option is used to label x‐axis with values from one or more stamp columns. This option is explained below. A time series plot shown in Figure 3.49 will be displayed.

Figure 3.49: A Simple Time Series Plot of Sales Data

Sale

s

QuarterWeek,T

5544332211160544842363024181261

100

90

80

70

60

50

40

30

Time Series Plot of Sales

Index

Dem

and

60544842363024181261

400

350

300

250

200

150

Time Series Plot of Demand


A Multiple Time Series Plot

Use this option to plot two or more series of data (multiple values). For example, suppose you want to plot the actual sales and the forecast on the same plot.

Table 3.46

Open the worksheet TIMESERIES2.MTW

In the Time Series dialog box, click on Multiple then click OK

Type or select Sales and Forecast for Series

Click on the Time/Scale box

Click on the circle next to Stamp under Time Scale

Click on Stamp columns (1‐3, innermost first) then select Period,T

Click OK in all dialog box.

A multiple time series plot shown in Figure 3.50 will be displayed.

Figure 3.50: A Multiple Time Series Plot Showing Sales and Forecast

Probability Plots Probability plots are used to determine if a particular distribution fits sample data. The plot allows us to determine if a distribution is appropriate and also, to estimate the parameters of fitted distribution. We have also seen that an empirical cdf can be used to fit the distribution to the data. The curve in cdf (Figure 3. 51) is not convenient to use and may be misleading sometimes. The probability plots are a better alternative and are achieved by the use of probability paper.

Period,T

Dat

a

60544842363024181261

100

90

80

70

60

50

40

30

Variab leSalesF o recast

Time Series Plot of Sales, Forecast


MINITAB provides individual probability plots for the selected distribution for one or more variables. The steps to probability plotting procedure are

1. Hypothesize the distribution: select the assumed distribution that is likely to fit the data 2. Order the observed data from smallest to largest. Call the observed data x1, x2,x3,.....,xn 3. Calculate the cumulative percentage points or the plotting position (PP) for the sample of size n (i=1,2,3,....,n) using the following

( 0 . 5 )1 0 0iP P

n

Tabulate the xi values and the cumulative percentage (probability values or PP). Depending on the distribution and the layout of the paper, several variations of cumulative scale are used. 4.Plot the data using the graph paper for the selected distribution. Draw the best fitting line through these points. 5. Draw your conclusion about the distribution. MINITAB provides the plot based on the above steps. To test the hypothesis, an Anderson‐Darling (AD) goodness‐of‐fit statistic and associated p‐value can be used. These values are calculated and displayed on the plot. If the assumed distribution fits the data:

the plotted points will form a straight line (or approximate a straight line) the plotted points will be close to a straight line the Anderson‐Darling (AD) statistic will be small, and the p‐value will be larger

than the selected significance level, a ( commonly used values are 0.05 and

0.10).

Probability Plot: Example 1 To demonstrate the probability plot, we will use the data in PROBABILITY Plot. MTW. The file contains the length of 15 cast iron tubes from a manufacturing process. We want to use the probability plot to check if the data follow a normal distribution. Follow the steps in Table 3.48. Table 3.48

Open the worksheet PROBABILITY Plot. MTW From the main menu select, Graph &Probability Plot Click on Single then click OK For Graph Variables, type or select Length (Cm) Click on Distribution and select Normal from drop down menu in the distribution box (Leave Historical Parameters box blank) Click OK in all boxes.

The probability plot is shown in Figure 3.52. From the plot we can see that the cumulative percentage points approximately form a straight line and the points are close to the straight line. The calculated p‐

value is 0.508. At a 5% level of significance (a=0.05), p‐value is greater than a so we cannot reject the null hypothesis that the data follow a normal distribution. If you place the cursor on the middle line, you can see the calculated cumulative percentage points and also the lower and the upper bounds for the points.


Figure 3.52: Example of a Probability Plot of Length Data

Probability Plot: Example 2

Matrix Plot

A matrix plot can be used to investigate the relationships between pairs of variables by creating an array of scatterplots.

One of the options under matrix plot is Each Y versus each X. You can specify y and x variables

Service Time

Perc

ent

100.010.01.00.1

99

90

807060504030

20

10

5

3

2

1

Mean 5.498N 30AD 0.459P-Value 0.531

Probability Plot of Service TimeExponential - 95% CI

Time to Failure

Perc

ent

1.000

0E+0

5

1000

0.000

0

1000

.0000

100.0

000

10.00

00

1.000

0

0.100

0

0.010

0

0.001

0

0.000

1

0.000

0

0.000

0

99

90807060504030

20

10

5

32

1

Shape

0.246

0.3450Scale 98.62N 30AD 0.461P-Value

Probability Plot of Time to FailureWeibull - 95% CI


and create a plot for each possible xy pairs. This is useful when the relationships between certain pairs of variables are of interest. Using Each Y versus each X, you can create a simple, with groups, and with smoother plots. Some examples of matrix plot are given below.

Matrix of Plots: Simple

Chapter 3 contains numerous charts/graphs used in Six Sigma, Lean, and Design for Six Sigma. The chapter contains detailed computer instruction with several examples of each plot.

To buy chapter 3 or Volume I of Six Sigma Quality Book, please click on our

products on the home page.

A vg. T emp.

A ge of Furnace

Heating Cost

House Size

5

3

1

10

5

0

50250

400

200

0531 1050

Matrix Plot of Avg. Temp., House Size, Age of Furnace, Heating Cost

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