Reid & Sanders, Operations Management © Wiley 2002 Statistical Quality Control 6 C H A P T E R.

Reid & Sanders, Operations Management© Wiley 2002

Statistical Quality Control 6C H A P T E R

Page 2Reid & Sanders, Operations Management© Wiley 2002

Learning Objectives

• Describe quality control methods• Understand the use of statistical process

control • Describe & apply control charts• Distinguish x-bar, R, p and c-charts• Define process capability • Describe & apply capability indexes• Define six-sigma capability


Quality Control Methods

• Descriptive statistics:– Used to describe distributions of data

• Statistical process control (SPC):– Used to determine whether a process is

performing as expected

• Acceptance sampling:– Used to accept or reject entire batches by

only inspecting a few items


Descriptive Statistics

• Mean (x-bar):– The average or central tendency of a data set

• Standard deviation (sigma):– Describes the amount of spread or observed

variation in the data set

• Range:– Another measure of spread – The range measures the difference between the

largest & smallest observed values in the data set


The Normal Distribution


Equations

• Mean:

• Standard deviation:

n

xx

n

ii

1

1

1

2

n

Xxn

ii


Impact of Standard Deviation


Skewed Distributions (One Form of Non-Normal Distribution)


SPC Methods

• Control charts– Use statistical limits to identify when a

sample of data falls within a normal range of variation


Setting Limits RequiresBalancing Risks

• Control limits are based on a willingness to think something’s wrong, when it’s actually not (Type I or alpha error), balanced against the sensitivity of the tool - the ability to quickly reveal a problem (failure is Type II or beta error)


Types of Data

• Variable level data:– Can be measured using a continuous scale– Examples: length, weight, time, &

temperature

• Attribute level data:– Can only be described by discrete

characteristics– Example: defective & not defective


Control Charts for Variable Data

• Mean (x-bar) charts– Tracks the central tendency (the average

value observed) over time

• Range (R) charts:– Tracks the spread of the distribution over

time (estimates the observed variation)


x-Bar Computations

xx

xx

x

n

zxLCL

zxUCL

nn

xxxx

...21


Example

• Assume the standard deviation of the process is given as 1.13 ounces• Management wants a 3-sigma chart (only 0.26% chance of alpha error)• Observed values shown in the table are in ounces

Time 1 Time 2 Time 3

Observation 1 15.8 16.1 16.0

Observation 2 16.0 16.0 15.9

Observation 3 15.8 15.8 15.9

Observation 4 15.9 15.9 15.8

Sample means 15.875 15.975 15.9


Computations

• Center line (x-double bar):

• Control limits:

92.153

9.15975.15875.15

x


2nd Method Using R-bar

RAxLCL

RAxUCL

n

RRRR

x

x

n

2

2

21 ...


Control Chart Factors

Factor for x-ChartA2 D3 D4

2 1.88 0.00 3.273 1.02 0.00 2.574 0.73 0.00 2.285 0.58 0.00 2.116 0.48 0.00 2.007 0.42 0.08 1.928 0.37 0.14 1.869 0.34 0.18 1.82

10 0.31 0.22 1.7811 0.29 0.26 1.7412 0.27 0.28 1.7213 0.25 0.31 1.6914 0.24 0.33 1.6715 0.22 0.35 1.65

Factors for R-ChartSample Size (n )


Example

Time 1 Time 2 Time 3

Observation 1 15.8 16.1 16.0

Observation 2 16.0 16.0 15.9

Observation 3 15.8 15.8 15.9

Observation 4 15.9 15.9 15.8

Sample means 15.875 15.975 15.9

Sample ranges 0.2 0.3 0.2


Computations

22.1433.273.092.15

62.1733.273.092.15

33.23

2.03.02.0

2

2

RAxLCL

RAxUCL

R

x

x


Example x-bar Chart

X-bar Chart

12

13

14

15

16

17

18

1 2 3 4 5 6 7 8 9 10

Time

Ou

nc

es

UCL

LCL

CL


R-chart Computations(Use D3 & D4 Factors: Table 6-1)

0033.2

71.628.233.2

33.23

2.03.02.0

3

4

DRLCL

DRUCL

R

R

R


Example R-chart

R Chart

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9 10 11

Time

Ou

nc

es

UCL

LCL

CL


Using x-bar & R-charts

• Use together• Reveal different

problems


Control Charts for Attribute Data

• p-Charts:– Track the proportion defective in a sample

• c-Charts:– Track the average number of defects per

unit of output


Process Capability

• A measure of the ability of a process to meet preset design specifications:– Determines whether the process can do what we

are asking it to do

• Design specifications (a/k/a tolerance limits):– Preset by design engineers to define the

acceptable range of individual product characteristics (e.g.: physical dimensions, elapsed time, etc.)

– Based upon customer expectations & how the product works (not statistics!)


Measuring Process Capability

Compare the width of design specifications & observed process output


Capability Indexes

• Centered Process (Cp):

• Any Process (Cpk):

6 widthprocess

ion widthspecificat LSLUSLC p

3

;3

minLSLUSL

C pk


Example

• Design specifications call for a target value of 16.0 +/-0.2 microns (USL = 16.2 & LSL = 15.8)

• Observed process output has a mean of 15.9 and a standard deviation of 0.1 microns


Computations

• Cp:

• Cpk:

66.06.0

4.0

1.06

8.152.16

6

LSLUSL

C p

33.033.0or 1min3.0

1.0or

3.0

3.0min

1.03

8.159.15or

1.03

9.152.16min

3or

3min

LSLUSLC pk


Three Sigma Capability

• Until now, we assumed process output should be modeled as +/- 3 standard deviations

• By doing so, we ignore the 0.26% of output that falls outside +/- 3 sigma range

• The result: a 3-sigma capable process produces 2600 defects for every million units produced


Six Sigma Capability

• Six sigma capability assumes the process is capable of producing output where +/- 6 standard deviations fall within the design specifications (even when the mean output drifts up to 1.5 standard deviations off target)

• The result: only 3.4 defects for every million produced


3-Sigma versus 6-Sigma


The End

Copyright © 2002 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United State Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.

Reid & Sanders, Operations Management © Wiley 2002 Statistical Quality Control 6 C H A P T E R.

Documents