Chapter 13Chapter 13Quality Control and ImprovementQuality Control and Improvement

COMPLETE BUSINESS STATISTICS

bybyAMIR D. ACZELAMIR D. ACZEL

&&JAYAVEL SOUNDERPANDIANJAYAVEL SOUNDERPANDIAN

7th edition.7th edition.

Prepared by Prepared by Lloyd Jaisingh, Morehead State Lloyd Jaisingh, Morehead State UniversityUniversity

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

• Using Statistics• W. Edwards Deming Instructs• Statistics and Quality• The x-bar Chart• The R Chart and the s Chart• The p Chart• The c Chart• The x Chart

Quality Control and ImprovementQuality Control and Improvement131313-2

• Determine when to use control charts• Create control charts for sample means, ranges and standard deviations• Create control charts for sample proportions• Create control charts for the number of defectives• Draw Pareto charts using spreadsheet templates• Draw control charts using spreadsheet templates

LEARNING OBJECTIVESLEARNING OBJECTIVES 1313

After studying this chapter you will be able to:After studying this chapter you will be able to:

13-3

A control chart is a time plot of a statistic, such as a sample mean, range, standard deviation, or proportion, with a center line and upper and lower control limits. The limits give the desired range of values for the statistic. When the statistic is outside the bounds, or when its time plot reveals certain patterns, the process may be out of control.

A control chart is a time plot of a statistic, such as a sample mean, range, standard deviation, or proportion, with a center line and upper and lower control limits. The limits give the desired range of values for the statistic. When the statistic is outside the bounds, or when its time plot reveals certain patterns, the process may be out of control.

A process is considered in statistical control if it has no assignable causes, only natural variation.

A process is considered in statistical control if it has no assignable causes, only natural variation.

UCL

LCL

CenterLine

Time

ValueThis point is out of the control limitsThis point is out of the control limits

3

3

13-3 Statistics and Quality13-4

Value

Time

Value

Time

Process is in controlProcess is in control

Process mean varies over time: process is out of control

Process mean varies over time: process is out of control

Control Charts 13-5

Control Charts (Continued)

Time

ValueProcess variance changes over time: process is out of control.

Process variance changes over time: process is out of control.

Time

ValueProcess mean and variance change over time: process is out of control.

Process mean and variance change over time: process is out of control.

13-6

A Pareto diagramPareto diagram is a bar chart of the various problems in production and their percentages, which must add to 100%.

A Pareto diagramPareto diagram is a bar chart of the various problems in production and their percentages, which must add to 100%.

Pareto Diagrams – Using the Template

A Pareto chart helps to identify the most significant problems and thus one can concentrate on their solutions rather than waste time and resources on unimportant causes.

A Pareto chart helps to identify the most significant problems and thus one can concentrate on their solutions rather than waste time and resources on unimportant causes.

13-7

•Finished products are grouped in lots before being shipped to customers.•The lots are numbered, and random samples from these lots are inspected for quality.•Such checks are made before the lots are shipped and after the lots arrive at their destination.•The random samples are measured to find out which and how many items do not meet specifications•A lot is rejected whenever the sample mean exceeds or falls below some pre-specified limit.

•Finished products are grouped in lots before being shipped to customers.•The lots are numbered, and random samples from these lots are inspected for quality.•Such checks are made before the lots are shipped and after the lots arrive at their destination.•The random samples are measured to find out which and how many items do not meet specifications•A lot is rejected whenever the sample mean exceeds or falls below some pre-specified limit.

Acceptance Sampling13-8

• For attribute data, the lot is rejected when the number of defectives or non-conforming items in the sample exceeds a pre-specified limit.

• Acceptance sampling does not improve quality by itself.

• It simply removes bad lots.• To improve quality, it is necessary to control the

production process itself, removing any assignable causes and striving to reduce the variation in the process.

• For attribute data, the lot is rejected when the number of defectives or non-conforming items in the sample exceeds a pre-specified limit.

• Acceptance sampling does not improve quality by itself.

• It simply removes bad lots.• To improve quality, it is necessary to control the

production process itself, removing any assignable causes and striving to reduce the variation in the process.

Acceptance Sampling13-9

• Six Sigma is a further innovation, beyond Deming’s work, in the field of quality assurance and control.

• The purpose of Six Sigma is to push defect levels below a certain specified threshold.

• Six Sigma helps to improve quality.• The key to Six Sigma is a precise definition of the

production process with accurate measurements and valid collection of data.

• Six Sigma is a further innovation, beyond Deming’s work, in the field of quality assurance and control.

• The purpose of Six Sigma is to push defect levels below a certain specified threshold.

• Six Sigma helps to improve quality.• The key to Six Sigma is a precise definition of the

production process with accurate measurements and valid collection of data.

Six Sigma13-10

• It also involves detailed analysis to measure the relationships and causality of key factors in the production process.

• Experimental Design is used to identify these key factors.

• Strict control of the production process is exercised. Any variations are corrected, and the process is further monitored as it goes on line.

• The essence of Six Sigma is the statistical methods described in this chapter.

• It also involves detailed analysis to measure the relationships and causality of key factors in the production process.

• Experimental Design is used to identify these key factors.

• Strict control of the production process is exercised. Any variations are corrected, and the process is further monitored as it goes on line.

• The essence of Six Sigma is the statistical methods described in this chapter.

Six Sigma13-11

Elements of a control chart for the process mean:

Center line:

LCL: UCL:

where: k = number of samples, each of size n = Sample mean for sample i

Ri Range of sample i

R =Rii=1

k

kIf the sample size in each group is more than 10:

LCL = x - 3s / c4

n UCL = x + 3

s / c4

nwhere s is the average of the standard deviations of all groups.

xxii

k

kx A R x A R

xi

1

2 2

Elements of a control chart for the process mean:

Center line:

LCL: UCL:

where: k = number of samples, each of size n = Sample mean for sample i

Ri Range of sample i

R =Rii=1

k

kIf the sample size in each group is more than 10:

LCL = x - 3s / c4

n UCL = x + 3

s / c4

nwhere s is the average of the standard deviations of all groups.

xxii

k

kx A R x A R

xi

1

2 2

n A2 c4

2 1.880 0.7979 3 1.023 0.8862 4 0.729 0.9213 5 0.577 0.9400 6 0.483 0.9515 7 0.419 0.9594 8 0.373 0.9650 9 0.337 0.969310 0.308 0.972715 0.223 0.982320 0.180 0.986925 0.153 0.9896

n A2 c4

2 1.880 0.7979 3 1.023 0.8862 4 0.729 0.9213 5 0.577 0.9400 6 0.483 0.9515 7 0.419 0.9594 8 0.373 0.9650 9 0.337 0.969310 0.308 0.972715 0.223 0.982320 0.180 0.986925 0.153 0.9896

13-4 The X-Bar Chart: A Control Chart for the Process Mean

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• Tests for assignable causes: One point beyond 3 (3s) Nine points in a row on one side of the center line Six points in a row steadily increasing or decreasing Fourteen points in a row alternating up and down Two out of three points in a row beyond 2 (2s) Four out of five points in a row beyond 1 (1s) Fifteen points in a row within 1 (1s) of the center line Eight points in a row on both sides of the center line, all beyond 1 (1s)

• Tests for assignable causes: One point beyond 3 (3s) Nine points in a row on one side of the center line Six points in a row steadily increasing or decreasing Fourteen points in a row alternating up and down Two out of three points in a row beyond 2 (2s) Four out of five points in a row beyond 1 (1s) Fifteen points in a row within 1 (1s) of the center line Eight points in a row on both sides of the center line, all beyond 1 (1s)

The X-Bar Chart: A Control Chart for the Process Mean (Continued)

13-13

Time

Value

112

2

3

3

Test 1: One value beyond 3 (3s)

Test 1: One value beyond 3 (3s)

Time

Value

112

2

3

3

Test 2: Nine points in a row on one sideof the center line.

Test 2: Nine points in a row on one sideof the center line.

Tests for Assignable Causes13-14

Time

Value

112

2

3

3

Test 3: Six points in arow steadilyincreasing ordecreasing.

Test 3: Six points in arow steadilyincreasing ordecreasing.

Time

Value

112

2

3

3

Test 4: Fourteen points ina row alternatingup and down.

Test 4: Fourteen points ina row alternatingup and down.

Tests for Assignable Causes (Continued)

13-15

Time

Value

112

2

3

3

Test 5: Two out of three points in a rowbeyond 2 (2s)

Test 5: Two out of three points in a rowbeyond 2 (2s)

Time

Value

112

2

3

3

Test 6: Four out of fivepoints in a rowbeyond 1 (1s)

Test 6: Four out of fivepoints in a rowbeyond 1 (1s)

Tests for Assignable Causes (Continued)

13-16

Time

Value

112

2

3

3

Test 7: Fifteen points in arow within 1(1s) of the centerline.

Test 7: Fifteen points in arow within 1(1s) of the centerline.

Time

Value

112

2

3

3

Test 8: Eight points in arow on both sidesof the center line,all beyond 1 (1s)

Test 8: Eight points in arow on both sidesof the center line,all beyond 1 (1s)

Tests for Assignable Causes (Continued)

13-17

X-bar Chart: Example 13-1 – Using the Template

13-18

X-bar Chart: Example 13-1(continued) – Using the Template

Note:Note: The X-bar chart cannot be interpreted unless the The X-bar chart cannot be interpreted unless the R or s chart has been examined and is in control.R or s chart has been examined and is in control.

Note:Note: The X-bar chart cannot be interpreted unless the The X-bar chart cannot be interpreted unless the R or s chart has been examined and is in control.R or s chart has been examined and is in control.

13-19

X-bar Chart: Example 13-1(continued) – Using Minitab

Note:Note: The X-bar chart cannot be interpreted unless the The X-bar chart cannot be interpreted unless the R or s chart has been examined and is in control.R or s chart has been examined and is in control.

Note:Note: The X-bar chart cannot be interpreted unless the The X-bar chart cannot be interpreted unless the R or s chart has been examined and is in control.R or s chart has been examined and is in control.

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10.8

10.6

10.4

10.2

10.0

9.8

9.6

Sample

Sam

ple

Mean

__X=10.257

UCL=10.784

LCL=9.731

Xbar Chart of Concentration

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10.8

10.6

10.4

10.2

10.0

9.8

9.6

Sample

Sam

ple

Mean

__X=10.257

UCL=10.784

LCL=9.731

Xbar Chart of Concentration

13-20

n D3 D4 B3 B4

2 0 3.267 0 3.267 3 0 2.575 0 2.568 4 0 2.282 0 2.266 5 0 2.115 0 2.089 6 0 2.004 0.030 1.970 7 0.076 1.924 0.118 1.882 8 0.136 1.864 0.185 1.815 9 0.184 1.816 0.239 1.76110 0.223 1.777 0.284 1.71615 0.348 1.652 0.428 1.57220 0.414 1.586 0.510 1.49025 0.459 1.541 0.565 1.435

n D3 D4 B3 B4

2 0 3.267 0 3.267 3 0 2.575 0 2.568 4 0 2.282 0 2.266 5 0 2.115 0 2.089 6 0 2.004 0.030 1.970 7 0.076 1.924 0.118 1.882 8 0.136 1.864 0.185 1.815 9 0.184 1.816 0.239 1.76110 0.223 1.777 0.284 1.71615 0.348 1.652 0.428 1.57220 0.414 1.586 0.510 1.49025 0.459 1.541 0.565 1.435

Elements of a control chart for the process range:Center line: RLCL: UCL:

where: R =Rii=1

k

k

D R D R3 4

Elements of a control chart for the process standarddeviation:Center line: sLCL: UCL:

where: s =sii=1

k

k

B s B s3 4

13-5 The R Chart and s Chart13-21

R Chart: Example 13-1 using the Template

The process range seems to be in control.The process range seems to be in control.The process range seems to be in control.The process range seems to be in control.

13-22

s Chart: Example 13-1 using the Template

The process standard deviation seems to The process standard deviation seems to be in control.be in control.The process standard deviation seems to The process standard deviation seems to be in control.be in control.

13-23

Example 13-2 using the Template13-24

Example 13-2 using the Template - Continued

13-25

Example 13-2 using the Template - Continued

Based on the x-bar, R, and Based on the x-bar, R, and ss charts, the process charts, the process seems to be in control.seems to be in control.Based on the x-bar, R, and Based on the x-bar, R, and ss charts, the process charts, the process seems to be in control.seems to be in control.

13-26

Example 13-2 using Minitab

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7

6

5

4

3

2

Sample

Sam

ple

Mean

__X=4.8

UCL=7.196

LCL=2.404

Xbar Chart of Delivery Times

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7

6

5

4

3

2

Sample

Sam

ple

Mean

__X=4.8

UCL=7.196

LCL=2.404

Xbar Chart of Delivery Times

13-27

Example 13-2 using Minitab

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6

5

4

3

2

1

0

Sample

Sam

ple

Range

_R=2.342

UCL=6.029

LCL=0

R Chart of Delivery Times

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6

5

4

3

2

1

0

Sample

Sam

ple

Range

_R=2.342

UCL=6.029

LCL=0

R Chart of Delivery Times

13-28

Example 13-2 using Minitab

Based on the x-bar, R, and Based on the x-bar, R, and ss charts, the process charts, the process seems to be in control.seems to be in control.Based on the x-bar, R, and Based on the x-bar, R, and ss charts, the process charts, the process seems to be in control.seems to be in control.

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3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Sample

Sam

ple

StD

ev

_S=1.226

UCL=3.149

LCL=0

S Chart of Delivery Times

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3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Sample

Sam

ple

StD

ev

_S=1.226

UCL=3.149

LCL=0

S Chart of Delivery Times

13-29

Elements of a control chart for the process proportion:Center line: p

LCL: - 3p(1 - p)

n UCL: + 3

p(1 - p)

n

where: n is the number of elements in each sample p is the proportion of defectives in the combined, overall sample

p p

Elements of a control chart for the process proportion:Center line: p

LCL: - 3p(1 - p)

n UCL: + 3

p(1 - p)

n

where: n is the number of elements in each sample p is the proportion of defectives in the combined, overall sample

p p

13-6 The p Chart: Proportion of Defective Items

13-30

13-6 The p Chart: Proportion of Defective Items – Using the Template for Example 13-3

Process is out of control – Two points fall outside the control limit

13-31

13-6 The p Chart: Proportion of Defective Items – Using Minitab for Example 13-3

Process is out of control – Two points fall outside the control limit

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0.4

0.3

0.2

0.1

0.0

Sample

Pro

port

ion

_P=0.1125

UCL=0.2624

LCL=0

1

1

P Chart of Defectives

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0.4

0.3

0.2

0.1

0.0

Sample

Pro

port

ion

_P=0.1125

UCL=0.2624

LCL=0

1

1

P Chart of Defectives

13-32

Elements of a control chart for the number of imperfections per item, c:Center line: c

LCL: - 3 c UCL: + 3 c where: c is the average number of defects or imperfections per item (or area, volume, etc. )

c c

Elements of a control chart for the number of imperfections per item, c:Center line: c

LCL: - 3 c UCL: + 3 c where: c is the average number of defects or imperfections per item (or area, volume, etc. )

c c

13-7 The c Chart: (Defects Per Item)13-33

The c Chart: Example 13-4 using the Template

Observe that one observation is outside the upper control limit, Observe that one observation is outside the upper control limit, indicating that the process may be out of control. The general indicating that the process may be out of control. The general downward trend should be investigated.downward trend should be investigated.

Observe that one observation is outside the upper control limit, Observe that one observation is outside the upper control limit, indicating that the process may be out of control. The general indicating that the process may be out of control. The general downward trend should be investigated.downward trend should be investigated.

13-34

The c Chart: Example 13-4 using Minitab

Observe that one observation is outside the upper control limit, Observe that one observation is outside the upper control limit, indicating that the process may be out of control. The general indicating that the process may be out of control. The general downward trend should be investigated.downward trend should be investigated.

Observe that one observation is outside the upper control limit, Observe that one observation is outside the upper control limit, indicating that the process may be out of control. The general indicating that the process may be out of control. The general downward trend should be investigated.downward trend should be investigated.

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18

16

14

12

10

8

6

4

2

0

Sample

Sam

ple

Count

_C=7.56

UCL=15.81

LCL=0

1

C Chart of Nonconformaties

252321191715131197531

18

16

14

12

10

8

6

4

2

0

Sample

Sam

ple

Count

_C=7.56

UCL=15.81

LCL=0

1

C Chart of Nonconformaties

13-35

13-8 The x Chart

Sometimes we are interested in controlling the process mean, but our observations come so slowly from the production process that we cannot aggregate them into groups. In such case we may consider an x chart. An x-chart is a chart for the raw values of the variable in question.

Sometimes we are interested in controlling the process mean, but our observations come so slowly from the production process that we cannot aggregate them into groups. In such case we may consider an x chart. An x-chart is a chart for the raw values of the variable in question.

The chart is effective if the variable has an approximate normal distribution. The bounds are 3 standard deviations from the mean of the process.

The chart is effective if the variable has an approximate normal distribution. The bounds are 3 standard deviations from the mean of the process.

13-36

13-8 The x Chart for Example 13-3 – Using Minitab

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20

15

10

5

0

-5

-10

Observation

Indiv

idual V

alu

e

_X=4.5

UCL=17.80

LCL=-8.80

I Chart of Defectives

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20

15

10

5

0

-5

-10

Observation

Indiv

idual V

alu

e

_X=4.5

UCL=17.80

LCL=-8.80

I Chart of Defectives

NOTE: The X-Chart Is same as the Individual chart inMinitab

13-37

13-8 The x Chart for Example 13-4 – Using Minitab

252321191715131197531

20

15

10

5

0

-5

Observation

Indiv

idual V

alu

e

_X=7.56

UCL=18.75

LCL=-3.63

I Chart of Nonconformaties

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20

15

10

5

0

-5

Observation

Indiv

idual V

alu

e

_X=7.56

UCL=18.75

LCL=-3.63

I Chart of Nonconformaties

13-38