Chap 18-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 18 Introduction to Quality Statistics for Business and Economics

Chap 18-1Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chapter 18

Introduction to Quality

Statistics for Business and Economics

6th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-2

Chapter Goals

After completing this chapter, you should be able to:

Describe the importance of statistical quality control for process improvement

Define common and assignable causes of variation Explain process variability and the theory of control

charts Construct and interpret control charts for the mean and

standard deviation Obtain and explain measures of process capability Construct and interpret control charts for number of

occurrences


The Importance of Quality

Primary focus is on process improvement Data is needed to monitor the process and to insure the

process is stable with minimum variance Most variation in a process is due to the system, not the

individual Focus on prevention of errors, not detection Identify and correct sources of variation Higher quality costs less

Increased productivity increased sales higher profit


Variation

A system is a number of components that are logically or physically linked to accomplish some purpose

A process is a set of activities operating on a system to transform inputs to outputs

From input to output, managers use statistical tools to monitor and improve the process

Goal is to reduce process variation


Sources of Variation

Common causes of variation also called random or uncontrollable causes of variation causes that are random in occurrence and are inherent in all

processes management, not the workers, are responsible for these causes

Assignable causes of variation also called special causes of variation the result of external sources outside the system these causes can and must be detected, and corrective action

must be taken to remove them from the process failing to do so will increase variation and lower quality


Process Variation

Variation is natural; inherent in the world around us

No two products or service experiences are exactly the same

With a fine enough gauge, all things can be seen to differ

Total Process Variation

Common Cause Variation

Assignable Cause Variation= +






People Machines Materials Methods Measurement Environment

Variation is often due to differences in:



Common cause variation naturally occurring and expected the result of normal variation in

materials, tools, machines, operators, and the environment





Special Cause Variation

Special cause variation abnormal or unexpected variation has an assignable cause variation beyond what is considered

inherent to the process





Stable Process

A process is stable (in-control) if all assignable causes are removed variation results only from common causes


Control Charts

The behavior of a process can be monitored over time Sampling and statistical analysis are used Control charts are used to monitor variation in a

measured value from a process

Control charts indicate when changes in data are due to assignable or common causes


Overview

Process Capability

Tools for Quality Improvement

Control Charts

X-chart for the means-chart for the standard deviationP-chart for proportionsc-chart for number of occurrences


X-chart and s-chart

Used for measured numeric data from a process

Start with at least 20 subgroups of observed values

Subgroups usually contain 3 to 6 observations each

For the process to be in control, both the s-chart and the X-chart must be in control


Preliminaries

Consider K samples of n observations each

Data is collected over time from a measurable characteristic of the output of a production process

The sample means (denoted xi for i = 1, 2, . . ., K) can be graphed on an X-chart

The average of these sample means is the overall mean of the sample observations

K

1ii/Kxx


Preliminaries

The sample standard deviations (denoted si for i = 1, 2, . . . ,K) can be graphed on an s-chart

The average sample standard deviation is

The process standard deviation, σ, is the standard deviation of the population from which the samples were drawn, and it must be estimated from sample data

/KssK

1ii

(continued)


Example: Subgroups

Sample measurements:Subgroup measures

Subgroup number

Individual measurements

(subgroup size = 4) Mean, x Std. Dev., s

1

2

3

…

15

12

17

…

17

16

21

…

15

9

18

…

11

15

20

…

14.5

13.0

19.0

…

2.517

3.162

1.826

…Average subgroup

mean =

Average subgroup std. dev. = s x


Estimate of Process Standard Deviation Based on s

An estimate of process standard deviation is

Where s is the average sample standard deviation c4 is a control chart factor which depends on the

sample size, n Control chart factors are found in Table 18.1 or in

Appendix 13 If the population distribution is normal, this estimator

is unbiased

4/csσ ˆ


Factors for Control Charts

n c4 A3 B3 B4

2 .789 2.66 0 3.27

3 .886 1.95 0 2.57

4 .921 1.63 0 2.27

5 .940 1.43 0 2.09

6 .952 1.29 0.03 1.97

7 .959 1.18 0.12 1.88

8 .965 1.10 0.18 1.82

9 .969 1.03 0.24 1.76

10 .973 0.98 0.28 1.72

Selected control chart factors (Table 18.1)


Process Average

Control Charts and Control Limits

UCL = Process Average + 3 Standard Deviations LCL = Process Average – 3 Standard Deviations

UCL

LCL

+3σ

- 3σ

time

A control chart is a time plot of the sequence of sample outcomes

Included is a center line, an upper control limit (UCL) and a lower control limit (LCL)


Control Charts and Control Limits

sAx

)n/(cs3x

n/σ3x

Deviations Standard 3 AverageProcess

3

4

ˆ

The 3-standard-deviation control limits are estimated for an X-chart as follows:

(continued)

Where the value of is given in Table 18.1 or in Appendix 13nc

3A

4

3


X-Chart

The X-chart is a time plot of the sequence of sample means

The center line is

The lower control limit is

The upper control limit is

sAxLCL 3X

xCLX

sAxUCL 3X


X-Chart Example

You are the manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For seven days, you collect data on five deliveries per day. Is the process mean in control?


X-Chart Example: Subgroup Data

Day Subgroup Size

SubgroupMean

Subgroup Std. Dev.

1

2

3

4

5

6

7

5

5

5

5

5

5

5

5.32

6.59

4.89

5.70

4.07

7.34

6.79

1.85

2.27

1.28

1.99

2.61

2.84

2.22

These are the xi values for the 7 subgroups These are the si values

for the 7 subgroups


X-Chart Control Limits Solution

5.8137

6.796.595.32

K

xx i

2.1517

2.222.271.85

K

ss i

2.73751)(1.43)(2.15.813)s(AxLCL

8.88951)(1.43)(2.15.813)s(AxUCL

3X

3X

A3 = 1.43 is from Appendix 13


X-Chart Control Chart Solution

UCL = 8.889

LCL = 2.737

0

24

68

1 2 3 4 5 6 7

Minutes

Day

x = 5.813__

Conclusion: Process mean is in statistical control


s-Chart

The s-chart is a time plot of the sequence of sample standard deviations

The center line on the s-chart is

The lower control limit (for three-standard error limits) is

The upper control limit is

Where the control chart constants B3 and B4 are found in Table 18.1 or Appendix 13

sBLCL 3s

sCL

sBUCL 4s


s-Chart Control Limits Solution

5.8137

6.796.595.32

K

xx i

2.1517

2.222.271.85

K

ss i

0(0)(2.151)sBLCL

4.49651)(2.09)(2.1sBUCL

3 s

4s

B4 and B3 are found in Appendix 13


s-Chart Control Chart Solution

UCL = 4.496

0

2

4

1 2 3 4 5 6 7

Minutes

Day

LCL = 0

s = 2.151_

Conclusion: Variation is in control


Process Average

Control Chart Basics


UCL

LCL

+3σ

- 3σ

Common Cause Variation: range of expected variability

Special Cause Variation: Range of unexpected variability

time


Process Average

Process Variability


UCL

LCL

±3σ → 99.7% of process values should be in this range

time

Special Cause of Variation: A measurement this far from the process average is very unlikely if only expected variation is present


Using Control Charts

Control Charts are used to check for process control

H0: The process is in control i.e., variation is only due to common causes

H1: The process is out of control i.e., assignable cause variation exists

If the process is found to be out of control,

steps should be taken to find and eliminate the

assignable causes of variation


In-control Process

A process is said to be in control when the control chart does not indicate any out-of-control condition Contains only common causes of variation

If the common causes of variation is small, then control chart can be used to monitor the process

If the variation due to common causes is too large, you need to alter the process


Process In Control

Process in control: points are randomly distributed around the center line and all points are within the control limits

UCL

LCL

time

Process Average


Process Not in Control

Out of control conditions:

One or more points outside control limits

6 or more points in a row moving in the same direction either increasing or decreasing

9 or more points in a row on the same side of the center line


Process Not in Control

One or more points outside control limits

UCL

LCL

Nine or more points in a row on one side of the center line

UCL

LCL

Six or more points moving in the same direction

UCL

LCL

Process Average

Process Average

Process Average


Out-of-control Processes

When the control chart indicates an out-of-control condition (a point outside the control limits or exhibiting trend, for example) Contains both common causes of variation and

assignable causes of variation The assignable causes of variation must be identified

If detrimental to the quality, assignable causes of variation must be removed

If increases quality, assignable causes must be incorporated into the process design


Process Capability

Process capability is the ability of a process to consistently meet specified customer-driven requirements

Specification limits are set by management (in response to customers’ expectations or process needs, for example)

The upper tolerance limit (U) is the largest value that can be obtained and still conform to customers’ expectations

The lower tolerance limit (L) is the smallest value that is still conforming


Capability Indices

A process capability index is an aggregate measure of a process’s ability to meet specification limits

The larger the value, the more capable a process is of meeting requirements


Measures of Process Capability

Process capability is judged by the extent to which

lies between the tolerance limits L and U

Cp Capability Index Appropriate when the sample data are centered between the

tolerance limits, i.e.

The index is

A satisfactory value of this index is usually taken to be one that is at least 1.33 (i.e., the natural rate of tolerance of the process should be no more than 75% of (U – L), the width of the range of acceptable values)

σ3x ˆ

σ6

LUCp ˆ

U)/2(Lx


Measures of Process Capability

Cpk Index Used when the sample data are not centered between

the tolerance limits Allows for the fact that the process is operating closer to

one tolerance limit than the other The Cpk index is

A satisfactory value is at least 1.33

(continued)

σ3

Lx,

σ3

xUMinCpk ˆˆ


You are the manager of a 500-room hotel. You have instituted tolerance limits that luggage deliveries should be completed within ten minutes or less (U = 10, L = 0). For seven days, you collect data on five deliveries per day. You know from prior analysis that the process is in control. Is the process capable?

Process CapabilityExample


Process Capability:Hotel Data

Day Subgroup Size

Subgroup

Mean

Subgroup Std. Dev.

1

2

3

4

5

6

7

5

5

5

5

5

5

5

5.32

6.59

4.89

5.70

4.07

7.34

6.79

1.85

2.27

1.28

1.99

2.61

2.84

2.22


Process Capability:Hotel Example Solution

0.6100.847,0.610Min

3(2.228)

05.813,

3(2.228)

5.81310Min

σ3

Lx,

σ3

xUMinCpk

ˆˆ

0.940c 2.151s 5.813X 5n 4

2.2880.940

2.151

c

sσ Estimate

4

ˆ

The capability index for the luggage delivery process is less than 1. The upper specification limit is less than 3 standard deviations above the mean.


p-Chart

Control chart for proportions Is an attribute chart

Shows proportion of defective or nonconforming items Example -- Computer chips: Count the number of

defective chips and divide by total chips inspected Chip is either defective or not defective Finding a defective chip can be classified a

“success”


p-Chart

Used with equal or unequal sample sizes (subgroups) over time Unequal sizes should not differ by more than ±25%

from average sample sizes Easier to develop with equal sample sizes

Should have large sample size so that the average number of nonconforming items per sample is at least five or six

(continued)


Creating a p-Chart

Calculate subgroup proportions

Graph subgroup proportions

Compute average of subgroup proportions

Compute the upper and lower control limits

Add centerline and control limits to graph


p-Chart Example

Subgroup number, i

Sample size

Number of successes

Sample

Proportion, pi

1

2

3

…

150

150

150

15

12

17

…

.1000

.0800

.1133

…Average sample

proportions = p


Average of Sample Proportions

The average of sample proportions = p

where: pi = sample proportion for subgroup i K = number of subgroups of size n

If equal sample sizes:

K

pp

K

1ii


Computing Control Limits

The upper and lower control limits for a p-chart are

The standard deviation for the subgroup proportions is

UCL = Average Proportion + 3 Standard Deviations LCL = Average Proportion – 3 Standard Deviations

n

)p)(1p(σp

ˆ


Computing Control Limits

The upper and lower control limits for the p-chart are

(continued)

n

)p(1p3pUCL

n

)p(1p3pLCL

p

p

Proportions are never negative, so if the calculated lower control limit is negative, set LCL = 0


p-Chart Example

You are the manager of a 500-room hotel. You want to achieve the highest level of service. For seven days, you collect data on the readiness of 200 rooms. Is the process in control?


p Chart Example:Hotel Data

# NotDay # Rooms Ready Proportion

1 200 16 0.0802 200 7 0.0353 200 21 0.1054 200 17 0.0855 200 25 0.1256 200 19 0.0957 200 16 0.080


p Chart Control Limits Solution

.08647

.080.035.080

K

pp

K

1ii

.1460200

.0864).0864(13.0864

n

)p(1p3pUCL

.0268200

.0864).0864(13.0864

n

)p(1p3pLCL

p

p


p = .0864

p Chart Control Chart Solution

UCL = .1460

LCL = .02680.00

0.05

0.10

0.15

1 2 3 4 5 6 7

P

Day

Individual points are distributed around p without any pattern. Any improvement in the process must come from reduction of common-cause variation, which is the responsibility of management.

_

_


c-Chart

Control chart for number of defects per item Also a type of attribute chart Shows total number of nonconforming items

per unit examples: number of flaws per pane of

glass

number of errors per page of code

Assume that the size of each sampling unit remains constant


Mean and Standard Deviationfor a c-Chart

The sample mean number of occurrences is

K

cc i

The standard deviation for a c-chart is

cσc ˆ

where: ci = number of successes per item K = number of items sampled


c-Chart Center and Control Limits

c3cUCL

c3cLCL

c

c

The control limits for a c-chart are

cCLc The center line for a c-chart is

The number of occurrences can never be negative, so if the calculated lower control limit is negative, set LCL = 0


Process Control

Determine process control for p-chars and c-charts using the same rules as for X and s-charts

Out of control conditions: One or more points outside control limits

Six or more points moving in the same direction

Nine or more points in a row on one side of the center line


c-Chart Example

A weaving machine makes cloth in a standard width. Random samples of 10 meters of cloth are examined for flaws. Is the process in control?

Sample number 1 2 3 4 5 6 7

Flaws found 2 1 3 0 5 1 0


Constructing the c-Chart

The mean and standard deviation are:

1.71437

0150312

K

cc i

1.30931.7143c

2.2143(1.3093)1.7143c3cLCL

5.6423(1.3093)1.7143c3cUCL

The control limits are:

Note: LCL < 0 so set LCL = 0


The completed c-Chart

The process is in control. Individual points are distributed around the center line without any pattern. Any improvement in the process must come from reduction in common-cause variation

UCL = 5.642

LCL = 0

Sample number1 2 3 4 5 6 7

c = 1.714

6

5

4

3

2

1

0


Chapter Summary

Reviewed the concept of statistical quality control

Discussed the theory of control charts Common cause variation vs. special cause variation

Constructed and interpreted X and s-charts Obtained and interpreted process capability

measures Constructed and interpreted p-charts Constructed and interpreted c-charts

Documents

Chap 18-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 18 Introduction to Quality Statistics for Business and Economics