Statistical Process Control - A Guide for Business Improvement

7/25/2019 Statistical Process Control - A Guide for Business Improvement

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S

tatistical

P

rocess

Control

Guide

for Business Improvement

The Society of tor Manufacturers and Traders Limited

London 2004 .

SMMT and the S T logo are registered trademarks of SMMT Limited

No part

of

this

publication may

be

reproduced,

stored in

any

information

retrieval

system or

transmitted

in

any

form or

media

without the written

prior permission

of

the SMMT.

www smmt co

.uk

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This

third edition

has

been

prepared

by

a

sub-group of

the

SMMT Quality Panel

Contributors:

Dale Robertson

NISSAN M

OTO

R M ANUF

AC

TURI

NG

UK) LTD

David

Linehan

LYNOAKS LTD

Steve Elvin

SMMT LTD

lt is

based

upon the

work carried out

by

Neville Mettrick

and his colleagues

First

edition

1986 reprinted times)

Second edition 1994

Third

edition 2004

The

Society

of Motor

Manufacturers

and

Traders l imited .

ll

rights reserved

Published in 2004 for

S T

bv Findlav Publications Ltd Horton Kirby Kent DA4 9LL

www

s

mmt co uk


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Section 1

Contributors

Foreword

Section 2 Introduction

2.1 Philosophy

2.2 Information from data

2.3 The uses of charting

2.4 Disturbances state-of-control

2.5 Specifications

2.6 Measures of middle

2.7 Measures of spread

2.8 Other measures of shape

2.9 Using calculators to obtain statistical measures

2.10 A reason for chart sample sizes above one

Section

3 Getting Started

3.1

The people involved

3.2 Executive and management considerations

3.3 Planning for process control

3.4 A summary of charting

Section Control Charts in General

4.1

Purpose

4.2 Chart design

4.3 Chart construction

4.4 Control lines

Sect1on 5

Control

Charts for Variables

5.1 Introduction

5.2

Sample size

5.3 Sample selection

5.4 Special circumstances

5.5 Mean and range chart Cx R)

5.6 Mean and standard deviation chart (x s)

5.7 Median and range chart

Cx R)

Section 6 Control Charts for Attributes

6.1 General

6.2 Sample size

6.3 Sample selection

6.4 p chart for production of detectives

6.5

np

chart for number of detectives

6.6 c chart for number of defects

6.7 u chart for production o defects

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Section

7

Chart

Interpretation

44

71 Introduction 44

72 Examination of charts for variables (x R, x R, x s) 44

73 Examination of charts for attributes p, np, c, u) 45

7 Pattern recognition 45

75 Examples of out-of-control patterns 47

7.6 Other examples of patterns 51

7.7 Unusual patterns without special disturbances 52

78 Dealing with disturbances 52

79 Centring 53

Section

Capability

55

8. 1 Capability statements 55

8.2 Capability indexes

58

8.3 Setting indexes 59

8.4 Interpretation of indexes 60

8.5 Estimation of conforming products 61

8.6 Example Reaction

Plan

following process monitoring 62

Sect1on

9

Summary of the Process

Improvement

Stages 64

Section

1

Top1cs

Related

to

Charting

65

10 1 The normal distribution 65

10.2 Introduct ion to analytical methods 68

Sect1on

Control Charts for

Special

Situation 7

11.1 Moving mean charts 70

11.2 Charts for sample size of one 72

11.3

Charts for short production runs 74

11.4 Standardised charts 76

11.5

Cusum charts 78

Sect1on

2

Capability

Estimations

0

12.1

Probability plots 82

12.2 Distribution information from probability plots 84

12 .3 Snap-shot capability estimations 84

12.4 Estimations for non normal distributions 85

Section 3 Bibliography

9

Section 4

Appendices 92

Section 5

Subject Index

26

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In the current climate the

S T

Quality Panel believes it

is

essential that businesses identify

and

take advantage of

improvement opportunities to drive sustainable

competitiveness .

To this end the family of Business Improvement Guides

are designed to provide much needed support for a whole

variety of businesses whatever their

size

.They focus

on

achieving business success by meeting the needs of the

customer through effective

and

efficient processes

utilising improvement

and

associated tools

and

techniques.

he S T Business Improvement Guides cover

Process Management

Continual Improvement

ools and Techniques

Statistical Process Control

Failure ode and Effects Analysis

The purpose of this guide is to explain Stat istical Process Co ntr

ol

The basic principles contained within this guide will equip the reader

wi th the knowledge to use this technique. However before carrying

out any SPC activity you are advised to check with your customer to

understand if they have any specific requirements.

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.

uk


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2

Introduction

2.1 Philosophy

People

Supp ers

have a responsibility to meet or better customers

expectations.

Customers

are the people or machines at the next and later stages

in any process, they might be in other factories or companies but

they always include the people

who

use the ultimate product.

Objectives

ost companies operate

in

markets where it

is

vital that they are

competitive and profitable. e ing competitive means being better

than competitors

in

quality, costs and delivery. Being profitable

entails operating without waste.

The achievement of competitiveness and profitability requires

effective and efficient processes. Processes can only be effective

when they are properly controlled.

Warning

Statistical

and

other methods are not a panacea, they point only to

opportunities for control and improvement wh ich w ill not happen

unless there

is

a will

to succeed.

2.2

Information

from

data

The ability of a system to obtain control and susta

in

continuous

improvement depends upon in ormation and how that information

is

used.

it is wasteful if information is used only to highlight the need for

rectification, it should

be

used also to adjust the process setting.

The waste that

is

tolerated by end-of-line inspection control includes:

the people, facilities, tools, material s and utilities used to

produce

defective products.

the people, facilities, tools, materials and utilities used to

find defective products.

the people, facilities, tools, materials

and

utilities used to

replace

defecti

ve

products.

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Information about the process is essential to control

process

stability

and

therefore product or service

consistency

If process information is not co

ll

ected

and

used there will

be

the

further waste

o

not being able to identify opportunities for

improvement.

Much informat

ion can be

derived from

numerical data such as

measurements counts or ratings. However many people

are

not

as

adept as they might

be in

extracting information from the data .

Hence these guidelines which describe statistical methods that are

used

in

process control for arranging and interpreti

ng

numerical data.

This part of the guidelines concentrates

on

simple

charting

methods

that

have

wide application

in

commercial

and

manufacturing industries. it offers a framework for practi

cal

training

and

can be

used

as an

on-the-job reference.

2 3

The

Uses of harting

Process control charts can be used t obtain information about

process setting

expressed as the process mean which

is

defined in section 5.5

underlying process

variability

expressed

as

the process spread which is explained

in figure 8.2

the

capability

of a process to produce within tolerance

explained in section 8. 1

process

disturbances

that wi

ll

give product va r

iabi lity and

inconsistency

defined in section 2.4 and illustrated in figures 73 to 710

the effects of

any

process change.

Whatever the information

it

s only o value

i

it gives rise to

appropriate action.

The

importance of training

and

a supportive organisation

is

emphasised

in

section 3.2 some helpful non-statistical methods

are

outlined in section

10 2 and

there is more detail in texts referenced

in

the Bibliography.

1

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2 4

istu

r

bances

&

State-of-Control

Variation among the products o any process is inevitable. lt arises

from causes which create process disturbances that

re

called

common or special.

Common disturbances arise from causes that

re

inherent

in the process and to some degree

affect all products

of

the process.

Examples of causes re variable raw materials, rigid working

methods, equipment limitations. atmospheric conditions and

individuals capabilities. These causes re sometimes called

chance causes, this is misleading because the causes of

speci l disturbances also c n occur by chance.

Processes that suffer only from common disturbances

re

in

a

state of statistical control .

In other words, the results of

the process re predictable.

Charting provides a measure of the effect of common

disturbances.

Special disturbances

arise from causes that

affect only

some products

of the process. They re not inherent in

the process .

Examples o causes re material flaws, non-observance of

instructions, power failures, vandalism and inappropriate

training. These c uses

re

sometimes called assignable

causes, this

is

misleading because causes of common

disturbances also

re

assignable.

Processes that suffer from specia l disturbances

re

out-of

statistical control because the effects of a disturbance

re

not predictable.

Charting highlights the occurrence of special disturbances.

iii I. .M

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Figure 2.1 : Design specificat ions

A few approach i

ng

lower

Perform

a

nce

limit

CUSTOMER S

EX

PECTATION

Most at or nea r to

Economic Optimum

A few approac hing

Upper Performance limit

I

y

I

+

- -

+

t

T

I

_ .

I I

I

I I I I I I I

I

I

I I I I I I I I I I

Tolera nce band

lower

Specification

limit

Nomin

al

Upp

er

Specifica

t

ion

limit

SUPPLIERS

S

TARGET

2 5

Speci fi cations

Engineers design for and customers expect

an

idea

l.

Des igners

specify ideal measurements, as targets or nominals.The value that

is specified should

be

the same as the optimum expected by

customers f igure 2.1

.

There

can be di

ff iculties for process control if t

he

nomina l is not

specif ied because setting or centring the process section 79) can

become subjective.

In the real world, even t he best processes do not resu lt in every

product

be

ing on nomina l. Designers cater for variab il ity by offering a

tolerance. Product det

ail

tolerances are not common

in

cert

ain

industries. Whether or not tolerance is specified, customers wi ll

accept va riabi lity if the risk to them is not unreasonabl

e.

A design tolerance

is

a statement of performance limits or the

measurement range w ithin which the product w ill function

satisfactorily. ost often , nom inal is in the m iddle of thi s range.

At end-of-l in e inspection, performance limits provide the

criteria for product acceptance or reJection.

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For process control, the limits

are

used

as

criteria for process

design and in some methods of expressing process

capability. Product

qua

lity is safeguarded through control

lines on

a

cha

rt section 4.4).

Beware of standard tolerances that have been developed as

a basis for contractual payments to piece-workers and

supp liers rather than as a basis for customer satisfaction.

Figure 2 2: The roles of people

in

SPC

EXECUTIVE/MANAGERS

Nominate co-ord inator/

facilitators

Scrap rework

- CO ORDINATOR

Administration

j

Identifies opportunities

coaches facil itators c j

I

MANAGERS

Promote

employee

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Wm i.J.M

2.6

easures

of Middle

Although diagrams u

sua

lly give the bes t idea of the shape of a

distribution, numbers are necessary for comparisons with other

distributions.

One such number

is an

estimate of the

middle o

a distribution,

sometimes it

is

called the

location or central tendency o

a

distribution.

Three ways of expressing an estimate of the middle of a distribution

are the

mode

the

median

and the

mean.

The fol lowing example is used

in

their descriptions below.

9 people were tested and the number of ma r

ks

per person was

2

5

3 6 4

3

8

5 and 3

ode

The mode is

the value which occurs most often .

lt does not have a standard designation but i is commonly used.

There are three 3s, t

wo

5s and one of each of the other four

numbers therefore the mode is

x

= 3.

Median

The median is

the middle value when the data

is

arranged

in

order of magnitude.

lt is denoted by

x

Rearranging the numbers gives 2

3

3

3

4 5

5

6 and

8

the

middle number

is 4, the refore the median is

x

= 4.

Mean

The mean

is

the

arithmetic average

sample mean

is

denoted by

x,

underlying or population mean

is

denoted by

L

lt is calculated by adding the values and dividing by their number,

x

= 2 +5 + 3 + 6 + 4 + 3 + 8 + 5 + 3 = 39 = 4.33 to

two

places)

9 9

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Asymmetrical dis

tribution

Mode= median=

mean

A

non symmetrical dis

tribution

Mode Median

Mean

The mode, median

and

mean

are

compared above for a symmetrical

and

a non-symmetrical distribution.

For a

symmetrical distribution

such as the normal distribution, all

three occur at the middle of the distribution .

The

effect of a 'tail'

in

a

non symmetrical distribution is

to pull the

median away from the mode

and

the mean even further.

In

both situations the median

has 50

of the distribution,indicated

by 50 of the area under the curve,

on

each side of its value.

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mi .J.M

Although the mean is the most common way of expressing

average. there are times wh en the mode or median are pr eferred.

For example

Des igners usually follow market su rveys. In practice this

amounts to fo llowing the mode.

The median tends to be used in salary negotiations, it seems

easier to ignore the extremes and to talk about a level which

has 50 of people above

and be

l

ow

it.

The median

is

used in some manual

cha

rting appl i

ca

t ions,

partly because it

is

easi ly

ca

lculated and understood

and

partly because it avoids t

he

need for calculator

s.

2.7 Measures

of Spread

The spread of a dist ribution is often more important than its average.

Usually, the setting of or average produced by a machine can

be adjusted.

Spread

w

i

ch

indicates

va

riab ility

is

inherent in the machine

and cannot be changed merely by turning a knob.

Three ways of expressing an est imate of the variabi lity of a

distribution are range, variance and standard deviation .

Range

The range is the m ximum value minus the minimum value . lt is

designated R.

it is easily calculated and

is

widely used. However, it is not a

sa

t isfactory

es

t imate of the

sp

read of a large distribution because it

ca

n be und uly influenced by a si ngle measurement value.

Variance

Va rian

ce

is the mean square difference of the values from the

me n

, sample varian ce is denoted s . underlyi ng or popu lation

variance

is

denoted

2

The wider t

he

spread of measurement

s

the

larger the values of s' and a .

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Standard Deviation

Standard deviation is the square root of the variance

The advantage of using standard deviation

ra

ther than variance

is

that its units are the same as the original data

and

the mean.

If standard deviation

is

doubled then the spread of the data

is

doubl

ed and

if standard deviation

is

halved the spread

is

halved.

For normal distributions the spread of data is about six

standard deviations.

2 8 Other Measures of Shape

Measures of middle and spread together provide a summary of a

distr ibution

w

i

ch

w ill be adequate for most purposes.

However there are situations which require other measures to be

considered in

pa

rticu l

ar

when tests for special disturbances are

necessary. The features

w

i

ch

need to

be

considered are:

Symmet

r

ica

l , =

0

or n

ot

sk

ewe

d

departure from symmetry which is ca lled skewness

Pos itive skew

, is positive

Negative skew

, is negative

c

is a coefficient of skewness that is quantified y some computer programmes

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4i#41 . .1

whether the distribution

is

flat-topped or peaked which is

called

kurtosis

a Greek word meaning bulging or convexity) .

lat-topped (platykurtic)

ck

s

low

Yis

negative

Peaked (leptokurtic)

Ck

s

high

Yis

positive

ck

and Yare

different coefficients of kurtosis that are quantified

by some

computer

programmes

.

ckreflects the

shape

of a distributions tails, Yreflects

its

central

shape and

Y=

0 for a

norma

l distr

ibutio

n.

Yis

the Greek capita/letter upsilon, equivalent

to

U

n

Eng lish.

2 9

Using Calculators

to Obtain Statistical Measures

ost sc ien t if ic

ca

lculators have keys which give the mean and

standard deviation at the press of a key.

Relevant keys are often marked x or the mean and


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The way to get around this difficulty

is

t

make use of a

mathematical ru le

ca

lled the central l m t theorem which says that:

no matter what is the distribution of individual measurements,

the distribution of averages of those measurements will increasingly

approximate to normal as sample size increases.

For

practical purposes, the distribution of means of about 5

individuals wil l approximate to normal if the distribution of

individuals is symmetric

al

for example. only su ffering from

kurtosis.

The same applies if the distribution of individuals has a slight

skew.

Th e means of larger samples are needed as skew gets more

extreme, for example, not less than 16 ind ividua ls for an

exponential distribution.

Illustration of an exponential distribution

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3 Getting Started

3 1 The People Involved

The executive

or directors' role is to support the practice of

statistics in process control, to the extent that they commit

re sources in the form of skills, time and occasionally facilities

all of which mean money

The managers'

role is to ensure that information is obtained

from statistics in process control and is used to the best

advantage

of

the business.

Fact-holders need to be found by the executive and/or by

management. These people are the lynch-pin of statistics in

process control. Their principal role is to coach others in the

methods.

They wil l

have

a knowledge of both statistics and the processes in

the business. Knowing the business

is

the pre-requisite,

knowledge of stat istics can be obtained from educational

institutions, from consultants

and

from related software packages.

They are often called SPC facilitators or co-ordinators but they

might

have

other titles and responsibilities . Whatever the title, it

is

important that facilitators are in touch with the work-teams.

lt

is also important that they have a focus in the shape of a co-

ordinator

who

can

promote good practice

and

provide a special

li

nk

to the executive.

Work-teams are the people at the sharp-end. Their role is to

practise the methods and to provide information for

all

to

use

and improve the business.

In very small companies say

two or three people) one individual

might carry out all the above roles.

In

very large compames say tw nty

or thirty thousand people)

there might

be

a facilitator

in each

work area,

an

overall co-ordinator

and others depending upon geography and diversity of processes.

In-bet een small and large compames the approach will be

somewhere between the extremes.

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3.2

Executive

and

Management onsiderations

The following are abstracts from the experience

of

companies

th t

have achieved considerable success

fter

adopting the use

of statistics in process control

Co ordination

The role of co-ordinator, distinct from facilitato r, might be resourced

from within an organisation. Where this is not immediately

practicable, the executive could consider using an outside consultant.

Strategy

In any learning activi ty it

is

advisable first to 'crawl', then to 'wa lk'

so that running is a natural and easy progression . In other words,

gear activit i

es

to the organisation's ab

il

ity to handle the informat ion

that will become available.

Training must start at the top,

so

that executives recognise the

imp lications and managers understand the information that wi ll arise

from the work-teams.

Strategic targets

As with any aspect o business strategy, the executive should

expect to receive progress reports aga inst targets. Ideal ly the

targets will

have

been set after realistic assessment of the best that

comparable organisations

have

to offer.

When targets are not met, problems often rest with management.

mpowennent

People

can be

discouraged

by

being exposed to information that

leaves them helpless. The remedy is empowerment at all levels in an

organisation, in other words, give people authority to make decisions.

Th is demands an educated wo rk-force and clearly defined process

ownership.

Leadership

A more posit ive response

to

process control and improvement

is

obtained from people

who

work

in

teams

with

a recognised leader,

rather than a 'supervisor'.

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# . .M

3 3

Planning for

Process ontrol

A process

is

often thought of only as something to do with making a

product. In fact, it can be any activity that produces a result such as

a design, a purchase, a sale or a se rvice. Also, it

can be an

individual s activity or a company s activity which

is

made up of

many individuals activities. Whatever the resu lt or sca le, a process

has

input

and

output

elements

Identify process elements

To control a process, it

is

first advisable to identify and record its

scope, its inputs and its outputs .

In

other word

s,

pl

ann

in

g for

process control involves understanding the factors that contr ib ute to

the result.

Th e record is best developed col lectively by everybody involved in

the process. Some simple analytica l methods that will help are

referred to

in

section 10 .2 and advanced techniques can be found by

reference to the Bibliography (section 1

3 .

Identify measures

Ca

re should be taken

to

ensure that the measurements are

appropriate for the business processes to ultimately ensu re that

customer and business requirements are monitored. Effective

monitoring

usually requires objective measurement and measuring

eq uipment must be

ca

librated.

The most informative way of presenting measured or counted data

is to use a su itab

le

cont ro l chart.

ote

: Processes are covered

in

greater depth in the SMMT

publication Process Management A Guide For Business

Improvement . See ins

id

e the back cover.

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3 4 A

Summary

of

Charting

The process o charti

ng

is illustrated in figure 3.1. it is a simple

process but there ca n be pit alls that need to be avoided.

Ultimately control charts will provide the fo llowing benefits

Do

Plan the introduction

Nominate

facilitators and a co-ordinator

Nominate process

owners

Train

everybody involved

Remember th

e purpo se is

proce

ss

improvement

Follow

the sequence in Fgure 2.2

Identify and eliminate all causes

of

disturbances

Recognise successful work-te ams

Don

t

Start unless you are comm

itt

ed

Identify

process

contro

l with

single

ndividuals

Measure success by the number of charts

Use control

lines to

indicate acceptance

lim its

Confuse

being

in-control with capability

Assume that early information tells

the

whole story

A cost effective and powerful tool

in

pr

ocess

con

t

ro

l they

are

simple

and

su pport empowerment of the work team.

Th

e abili ty to dist

in

g

ui

sh between specia l and common

disturbances and provide a common language for communication

of process

be

h

aviour.

Init

ia

lly a m

ea

ns of target ing special disturbances but when the

process is predictable the charts show common disturbances as

a

chal

lenge with greater rewards.

Object ive evidence of the effect of process change ca used by

people materials faci lit ies methods and the environment.

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. .1

Figure 3

:

The application of charting

secti

on 3.3

section 5.3)

section 6.3)

L T ~

Collect

data

Construct

co

ntrol chart

section 4.4

L r ~

section 7.2

secti

on 7.3

[

Pattern in control?

[

Pattern centred?

I

section 7.9

[

=J

s e c t i o n

8.1

ssess capability

I

= :1

s e c t i o n 8.4

rocess capab

l

e?

[

=r=

Jsection 5.4

ontinue charting

=r=

J

section 7

.8

educe common disturbances

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4. Control

Charts

in General

4.1 PURPOS

Control charts are one of many tools used

in

process control.

Process control is

a key way

to

achieve, maintain and improve

quality in products and services.

The stages of process improvement are illustrated

in

section 9

where a customer

is

the next operation, the next factory,

the ultimate product user

and

any people or machines

in

between.

The charts signal the existence of process variation

and

should lead

the process owner to react to adverse situations when the process is

out of control not predictable) or

incapable not able to meet tolerance) or

not centred not set on nominal).

Also, charts

can

help

in

identification

o

causes of variation because

they distinguish between the two types of process disturbance

which

are

special disturbances that affect some products and

common disturbances which affect all products.

When disturbances are identified, the work team w ill use other

techniques to f ind causes and then to take improvement action.

Charts have a further use in monitoring the effectiveness of actions.

Charts add

va

lue even when the process

is

in control, capable and

centred at this stage the opportunity is to delight the customer.

mi .UM

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27/131

N

C )

I

3

"

"'

-2

'

g

a:

Location geography)

Process operation/machine)

Component

I

part number)

Feature

Checking media

Specification

Sample

GRAPH

PAPER

"''"( sjl_:r

h t l l

' ~ ~ t

1 ~ i ~ ~ ~ i w r t r ~

\1

IU

3

X

2

LCL

'

>

UCL

1

=>

0

R

'C'

::>

256

3 2 243

286 28

1 2

77

315

46 59 43 4 38

mean load x) = 302.40

UCLx =x+ a =302.40 +

3(35.46)

=409.16

LCLx =x - a=302.40 -

3(35.46)

=195.64

Un its

422 32

7

292

28

1 3

5 333 29

4

1 7 95 35 24 28 39

mean

range

R)

=40.14

UCL, =D,R=3267(4014) =1

31.15

a =Rd , =

40/1.128

=

35.46

This method is applicable wh n measurements are infrequent

The example below uses

two

measurements to determine range.

There

are

variants that use three or more measurements and

introduce additiona l uncertainties of interpretation.

In

all

cases. the

charts are sometimes called 'individuals and moving range charts'.

The charts can be drawn

on

conventional x&R chart paper see

Appendix B, page 94.

Individual un it measurements are plotted.

Range values are ca lculated and plotted. In the example in figure

11. 3, they are the difference between one unit measurement

and

the ne xt, which means that there is one less range value

than individual me

as

ur

em

en

t

s.

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74/131

Mean and control line positions are calculated from about

20 measurements.

For

the individuals plot, the mean line

is

at the average of the

measurements and upper

and lo

wer control lines

are

drawn at the

mean 3a

u can be calculated from the mean range

s

ee figure 5.3). the constant d,) used in

the calcu lation

is

that for sample size

2.

For the ranges plot, mean and control lines are calculated and drawn

in

the same way

as

for a conventional range chart (see section 5.5).

The

constant (D

.)

used in control line calculation is that for

sa

mple size

2.

Chart interpretation is set out

in

sections

7 1

to 7.9.

Charts r sample size of one must be interpreted with

caution because

range plots are not independent, each measurement after the

first affects

two

range

va

lues and the charts

are

not

as

sensitive

to process change as conventional x R charts.

the mean and control lines should reflect the underlying

distribution, this is possible but not probable wi th much

le

ss

than 125 measurements.

interpretation assumes a normal distribution of data

(see section 10 1

),

thi s is more likely when the data consists of

averages of larger sized samples according

to

a mathematical

rule called the central limit theorem .

Note the central limit theorem states that:

no matter what is the distribution of individual measurements, the

distribution of averages of those measurements wi

ll

increasingly,

approximate to normal as sample size increases.

4i41 . .1

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I. .

11 3 Charts for Short Production Runs

This method

is

applicable to processes that produce several similar

products,

each in

low volume but often

an

overall large quantity.

For example, a simple plate is produced in batches to order, each

w ith a flange height (18mm, 12mm, 6mm, etc) specified by the

customer.

A conventional run chart could look like the actual results

in

Figure

10 .3.

Such a cha rt and the alternati

ve

of a separate chart

for

each

plate would

be

of little use

in

monito ri ng the process.

A soluti

on

to the problem

is

to zero the plate measurements

by

subtracting the nominal for the plate from each measurement.

A plot of these values

is ill

ustrated as the zeroed results

in

figure

11.3.

The control lines shown

in

figure 11.3 are positioned at nominal

3s and s has been calculated from the first 25 zeroed values

- see Appendix C, page 99.

The plots

in

the il lustrations are of indiv

id ual

measurements and

therefore the cont ro l lines could be positioned also by using the

zeroed values and the method described for charts of sample

size one (section

11.2 .

For samples above one, a conventional

x R

chart (section 5.5 is

used with alues that are zeroed sample ~ e n s (means minus

nom inal and of course, the process mean

x) is

zero.

Subject to the limitations applying

to

charts for

sa

mple size of

one (section

11.2 ,

chart interpretation

is

set out in section

71.

7

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25

20

E

E

15

c:

0

v

c:

10

D

r

16.5

18

1 5

11 3:111ustration of a control chart for short production runs

8mm unit 1

2mm

un it _ _

6mm

unit

19 17 17

.5

16

.5

19 5

18

18 18

18 18 18 18

1

0.5

1

5

1 5

0

7 5

1

1 5

2

ctual results

19 5

1

6 5 19

17.5

18

.5

16

18 18 18

18 18 18

1 5

1 5 0

.5 0.5 2

11

12

.5

8 5

15

12 5 13 5

12

1

5 5

0

.5

12 12

0.5 3.5

12

3

4.5

6

12

0.5

1 1 5

1

12

1 5

20

.5

17 20

19

19

.5

18

18 18 18

18

2.5

1

1 5

11

13 13

.5

12

.5

10 5 13

.5

11

.5

12 12 12 12 12

12 12

1

1

1

5

0.5

1

5

1 5 0

.5

7.5 5.5 4

1 5 0 5 2

UCL

17 17

.5

18

.5

18 18 18

1 0.5 0.5

19

17

18 18

1

6

Zeroed results

www.smmt.co.

uk

75


77/131

li5ii UM

11 4

Standardised harts

Standardise d charts are use d to monitor a process when

measurements are influenced

by

factors independent of the

process.

The same items checked

by

different people or using different

facilities often give results that differ according to the person or

facility, even though the item being checked does not cha nge.

Th is method is used to standardise results when it is

impracticable to standardise the people or the facilities

The results from each person or facility are converted onto a scale

whe

re the process mean

is

zero and the control chart LC L

and

UCL

are 3CT and

3CT

respective l

y

The first step is

to

determine the mean and standard deviation

of the first 25 results from each person or facility.

A plot is th

en

made of their actual results minus the mean of

their re sults divided by the standard deviation of their results.

Thi

s plotted

va

lue is kno

wn as

the standardi

se

d deviate or Z value of the

sam

pl

e

ave

rage.

The top picture in figure 11 4 illustrates the combined results of

noise tests on the same product at

two

different locations.

Although the pattern suggests an out-of-control situation (see figure

710), it does not indicate any special disturbances.

In the middle picture, the results have been separated by site, the

mean and standard deviation of each set has been calculated and

the resu lts have been converted to

z

values .

t the bottom is a standa rdi sed chart w here Z values are plotted.

For the first time it

can

be seen that the process aimed at

ach

ieving consistency in product noise suffers from specia l

disturbances.

76 www smmt co

uk


78/131

'

;

'

C

c;

z

u

Q;

0

;:;

'

:s

c;

z

120

100

80

60

40

20

40

20

6

5

Fi

gur

e

11

.4: Ill

us

tr

a

t i

on

of a st

nd r

di

sed cha

rt

UCL

LCL

Combined results

Separated results

W W H

0.09 0.09 0.09 0.95 1.21 0.35 0.09 0.35 0.95 035 0 .35 0.78 0.09 1.21 3.98 0.09 0.35 0.35 0.52 1.21 0.09 0.09 -0

.3

5 1l

.3

5 0.09

- - - - - - - - - - - - - - - - - - ~ ~ - - - - - - - - - - - - - - - - - - -

~

~

2

- 1 ~ . .

V 17

-2

~ . . L L

z - 3 ~ ~

4

- 5

6

Standardised

re

sults

4ifii I. .M

www.smmt.co.uk 77


79/131

Wi#4 1. .1

11 5

Cusum

Charts

Both attribute and variable cusum charts

are

used for monitoring and

for retrospective investigation of processes where changes in mean

values have particular importance, for example:

when any deviation from optimum must be detected.

when the point of any change needs to be identified.

Cusum charts

are

especially useful in relatively stable continuous

processes such

as

motor vehicle paint plants and the petrochemical

industry.

The practical detail of cusum charts and their interpretation is

set out in BS5703 obtainable from

th

British Standards

Institut ion

Of particular interest in the standard

is

the description of masks that help the

identification of changes and patterns on cusum charts.

The illustrations in figure

11 5

compare the appearance of a cusum

chart with that of a conventional run chart for the same data.

Change in the process mean is indicated on the cusum chart

by change in the

slope

of the plot, rather than change in the

level

of the plot

as

on conventional charts.

In

ideal applications, the advantages of cusum charts are:

special disturbances have less influence on indications of change.

the timing of

any change in mean value is usually easier to

es timate.

out-of-control indications often occur with less sample information.

averages over particular sequences can be

read

directly from

the chart.

trends and process cycles are more easily recognised.

The main disadvantages of cusum charts are:

their maintenance demands adept people wi th a high level

of training.

they are not appropriate w n variability is an important matter.

78

www smmt co uk


80/131

25

2

Q

;;

15

;

c

C

1

c::

::::J

5

0

3

2

r=

k- 1

H

0

Week

214

igure 11 5: Illustrations of a cusum chart and a conventional run chart

17

292

119

1 5

2245

1 15

2

Con

venti

onal

run chart

157

118

96 161 1

39 9 1 6 14

3

41

5 6

667

8 6

897

1 3 1146

149 1 8 116 136 169 182

131

135 94 1 2 122 155 168 117

238 2474 2576 2698 2853

3 21 3138

Indications of mean level relative to target

H

orizontal

on

target

Slope down

below

target

25

167

151

153

137

1299 1436

2 5

142

191

128

3329 3457

Slope up

above

target

3

98

84

152

179

3636

157

138

15

133

143

124

1

36 119

1663

1787 1923

2 42

1

87 2

1

174

118 173 197

38

1

3928 41 1 4298

100 -ro-r.-ro-r.-ro-r.-ro-ro-ro-ro-ro-..-,-,-,,-,.-,

Week 1 15

Cusum

chart

2

25

3

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9


81/131

4ii I.UI

12

Capability

Estimations

Capability

is

a measure of how well customers' requirements

are

met.

The topic is explained more fully in sections 8.1

to

8.5.

Figure 121: llustration of

a

probability plot

This illustration shows a straight best-fit line

and

values at points where the best-fit line intersects

with other lines

INTERSECTION

W

ITH

-

50

LSL

=

0 Lf=50

VALUE

- 26

0.05

Th

ese

va l

ues

can

be

used to

ca

lc

ul

ate a 'sna ps hot'

es

t imate of capabi lity (see section 1

2.3

)

- -

- -

- -

Class

Tally chart

I

Li Li

x. -

-

I I I I I

I

I I

140

130

120

110

90 1 99

I

I I I I I I

1

125

100

100

80 1 89

I ll I

I I I I I

8 124 99.2

90

70

1 79

I I I

I I I I

11 116

92.8

80

60 I

69

I I I I

ll

I

I I

23 105

84.0

70

50

I

59

I I I I I I I

////

39 82 65.6

60

40 I 49

HH

I

HH

I

HH

I

HH

I I

I I

21 43

34.4

50

30 I 39

I I I

I I I

I

12 22 17.6

40

20 I 29

I 1 I

I I I I I

7

10 8.0

30

10 19 ll

3 3 2.4

20

10

0

- 10

I I I I I I I

- 20

lo-'

--26

I I I I I I I I

30 -

-

0

0

-

-

5cr

8

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82/131

100

t5


83/131

12.1

robability

lots

Estimat

es

of cap ability need data who

se

distribution is known.

probability plots are a simple way of finding out about data distribution.

Ideally, estimations should be made from about 125

measurements. Th

ey

can be adequate with

as

f w

as

30

measurements, however it is always advisable to confirm resu lts

as more data becomes avai lable.

Data

can

be recorded, arranged, plotted and summarised on a

customised form see secti

on

14) or using a commercial form

such

as

Chartwell ref.5571 only for the plot or without a special

form (see Betteley et

al

referenced

in

the Bibliography, section 1

3).

A data

se

t of measurements is arranged in a tally chart and

cumulative frequencies CL.f ) are calculated.

I is the Greek capital letter sig ma eqiva lent to S in English, here

it means sum of fs

so

far .

Cumulative frequencies are plotted against their class upper

boundary xJ on a probability paper and a best-f it line is drawn

through the plots

as

shown in figure 12.1 which illustrates use of

a normal probability paper.

Probability paper does not allow a plot to be made at Lf = 100,

so to make use of the data, a plot is made at the average of the

two

highest classes x,

and

Lf values.

When the best-fit line through plots

on

normal probab

il

ity paper is

straight, it indicates that the data comes from a normal distribution,

which

is

the case of figure 12 .1 and in figu

re

A on page 83.

T

he

normal distribution is explained on page 65.

82 www.smmt.co. uk


84/131

Figure 12

:

Distribution Information from Probability Plots

Figure A Normal distribution

A straight line

Figure B Two distributions

A kin

ked

or two off-set

lines

from products

off different

machines

that have

been

mixed after

production

Figure D Doubly truncated distribution

Starts vertical and bends

through an

S

sha pe

from data

with

missing high

and

low

values

suc h

as from

a midd le

grade batch

Figure C Truncated distribution

A

diagonal

line t

hat

bends

to the

vertical

from data with

missing high values

such as

from a

sma

ll

grade

batch

Figure E

Skewed

distribu ti on

A

smooth

curve

from

data where

mean

mode

and

median are

different.

See

page s 86 to 89

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85/131

Wi5 1. .1

12 2 Distribution Information from Probability Plots

Figures A to E indicate normal

and

non normal or unusual

distributions of data wh en it

is

plotted on normal probability paper.

Capability statistics

and

control charts for non normal distributions

must

be

interpreted with caution.

In statistics, the term normal refers to a particu l

ar

distribution

(section 10.1) non normal means other distributions, it does not

mean abnormal.

12 3 Snap Shot

Capab

ility

Estimations

Apart from giving a simple picture of data distribution

a probability plot can be used for snap-shot capabi lity estimations.

The method does not readily identify special disturbances and it

gives no idea

of

variation occurring over t ime.

In

the example in figure 12.1 , the plot suggests a not-capable and

not-centred process.

The specification limits, LSL and USL, are 0 and 100 respect ively

therefore TOLERANCE is

100

0 = 100 and NOM INAL = LSL

+

tolerance/2 = 0 + 100/2 = 50

The difference between -5a and 5a is 134 -26) = 160 = 10a

therefore

PROCESS

SPREAD is 160/10 x 6 = 96

u

is the Greek lo

wer

case letter sigma equivalent to s in English, here it

signifies a standard deviation, process spread

is

six standard deviations and

is

illustrated in figure 8.2.

The CAPABILITY INDEX is tolerance/process spread = 100/96 = 104

and the PROCESS MEAN

is

54 which is above the nominal.

See

sections 8.1 to 8.5 for explanation

and

interpretation of capability indexes.

8

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86/131

12.4

Capability

Estimations for

non

normal Distributions

Figure 12 3:111ustration of a distribution truncated t zero

ode

Mean

3

sta

ndard dev iations

r o e s s

sp

read------- - -

If prel imina ry work ind icates that a distribution is non normal, there

are

four approaches which might be adopted.

First and most important

investigate the data more thoroughly.

Many non normal distributions only reflect measurement practice

such

as:

not considering the p

ola

ri

y

of measurements, for example, the

so

called one-sided distributions described

on

page

86

not reporting results above or below particular values.

www smmt co uk

85


87/131

reporting results beyond the precision o the measurement

method.

having differing standards of measurement for example from

sh ift

to

shift.

reporting combined results

o

differently set machines.

The effect of investigations is often to improve process consistency

and

to

determine that the underlying distribution is

in

fact normal.

Second where reasonable,

treat all or part of the distribution as normal.

In

particular for those special cases such

as

ovality taper and run-out

which are often referred to as one-sided distributions and have

nominal at zero.

The mean of the distribution shown

in

figure 12.3

has

little practical

use however the tail to the right of its mode is approximately

norma l

Note: The mode is the value which occurs most often. lt does not

have a standard designation but x is commonly used.

When a distribution is truncated at zero Proce

ss

Spread is three

standard deviations plus the width zero to the mode.

When the mode of a distribution is at zero its Proce

ss

Spread is

effectively half that of a normal distribution

in

other words three

standard deviations.

The mode instead of t he mean and only those measurements n

the approximately normal t ail are used to calculate the

standard deviation.

86 www.smmt.co.uk


88/131

Third if necessary

determine if a distribution other than normal will fit the data.

Probability plotting usually provides the easiest method

of confirming another distribution and of estimating Process Spread.

Several techniques are described

in

various academic texts

see Betteley

et al

and others referenced

in

the Bibliography).

Amongst them

is

the use of probability pape

rs

other than the normal

paper, for example, the paper illustrated

in

figure 12.4 and

in

Appendix J page 124 wi ll give a straight best-fit line if the

distribution is an extreme skew.

When Process Spread is determined for a non normal

distribution, it is the value of the interval between the 0.13 and

99.87 percentile lines which

are

the vertical broken lines

in

figure 12.4.

Horizontal lines are drawn from the vertica l lines/best-f it line

intersections, the Process Spread is the distance between them

on the vertical axis scale w hich is 95.5 84.5 = 110 in figure 12.4.

Finally

if there

is

a very large amount of data that

is

thousands of results),

simply studying a histogram will usually give sufficient information

about Process Spread and its relationship to the tolerance band.

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89/131

Figure 12.4: Illustration of a paper used for extreme sk w distributions

also see AppendixJ , page 124

100

99

99

.

99 99

87

99

90 70

50

30

20

10

98

I

I

97

96

:

- -

- -

- -- - -

r

- - - -

95

94

I

I

93

92

I

I

91

90

89

I

:

I

88

87

:

I

86

85

84

83

I

-

..

I

I

82

81

I

I

88 www.smmt.

co.

uk

10 30

Lf

i.,...o-

_

-

. .

1..-o

50

70 80

90

95 97

9f


90/131


91/131

13 Bibliography

The terms and symbols in this guide are widely accepted

in

manufacturing industr

y.

However, readers should

no

te that the texts

below sometimes use different conventions.

Dietrich, E and Schulze, A 1999) Statistical Procedures for Machine and

Process Qualification, ASQ Quality Press, ISBN 87389 447 2

A comprehensive text for machine and process qualification.

Dietrich, E and Schulze, A 1998) Guidelines for the Evaluation of

Measurement Systems Hanser Publishers, I

SB

N

3

446

19572

6

Explains how to manage the acceptance

o

measurement systems and

production facilities

s

well

s

process evaluation.

Betteley,G, Mettrick,NB, Sweeney,E and W ilson, D 1994) Using

Statistics

in

Industry, NewYork: Prentice

Hall

Comprehensive work place reference text.

Oakland,JS 1984) Statistical process control: A Practical Guide,

Oxford: Heinemann

A brief overview o process capability and the main control charts.

Walpole,RF and Myers,RH 1993) Probability and Statistics for

Engineers and Scientists, 5th edition, New York: Macmi llan

A brief account

o

the main types

o

control chart.

Grant,EL and Leavenworth, RS 1988) Sta tistical Quality Control, 6th

edition, NewYork: McGraw

Hi

Technical details o the main types o control chart.

Montgomery, DC 1985) Introduction to Statistical Quality Control,

New York: Wiley

Detailed treatment

o

process capability and the main control charts.

Mitra,A 1993) Fundamentals of Quality Control and Improvement,

NewYork: MacMillan

Detailed treatment o process capability

9 www.smmt.co.uk


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93/131

14 Appendices

Constants for variables control charts

Control chart forms reduced from A3 size

A worked example

is

shown

on

facing pages for

e ch

form

B Mean and range process control chart

C Mean and standard deviation process control chart

Median and range process control chart

E

p chart for proportion

o

detectives

F

np chart for number

o

defectives

G u chart for proportion of defects

H

c chart for number of defects

Normal probability paper

J Probability paper for extreme skew distribution

9 www smmt co uk

9

94

99

1 2

1 6

11

114

118

122

124


94/131

Appendix A Constants for Variables Control Charts

'

'

'

'

'

-'=

'

g

:

'

'

'

'

:

:

=

-'=

'=

c:

8

E

a:

8

8

g

c:

c: c:

f3

8

8

8 8


95/131


96/131

i

cs:>

CJ1

0>

:

i

'

c c

l

t

SHIFT

DATE

TIME

BY

0>

'

.

.

,

c:

-2

LX

x

R

,,,

I I

1

2

3

r

4

:;

Mean of

x

alues;

x

x hart

UCL

;

x

A

1

R

x

chart

LCL

; x-A

1

R

1: ;.

._ L

I.

I I

I

c

I ,. I

I -'

)_fi

' ,_n

l

I t I 1 I .,,

/I

.Zh

1

.m

. ] . ,

Standard deviation ; R/d

1

1

/

/

l :T7

I

A

J

I

J

ll

,-

...

f:l

-


97/131

t

0 )

i

3

0

i=

A

c:

"'

"'

'

. 2_

MEAN AND

RANGE

(x R) PROCESS

CONTROL CHART

location

(geog

r

aphy)

Process

(ope

r

ation

/machine)

Component

part

number)

Feature

Checking

media

Specification NOM I

NA

L

I

TOLER

ANCE

Sample SIZE

I FREQUE NCY


98/131

i

3

0

C:

'

0

w

l

J

'

: J

-f;

:::

'

i

E:

'

1.\_lC

SHIFT

A

DATE

TIME

BY

il

R

Mean of x

alues =

x

x

h art UCL =x AR

x

hart

LCL

=

x-AR

Standard

devi

at

ion =

Rd

2

..

'

A A

,.

.

'

1

\

f

~ l - f

l:

c.

cc

'

..

/

/

,.....

'

-

";:

'

l

I.

Mean of

R value

s= R

n A,

03

o,

d,

R chart UCL =

D

4

R

2

1.880

0

3.267

1.128

3

1.187

0

2.574

1.693

R chart

LCL

=

D3R

4

0.796

0

2

.28

2 2.059

5

0.691

0

2.114 2.326

n = s mple size


105/131


106/131

i

3

0

c n

Q

g>

Cl

t:

'

c::

SHIFT

D TE

TIME

BY

il

R

Mean

of

x

alues =

x

Mean of

R

values = R

il

chart UCL =

x

A

2

R

Rchart UCL = D

4

R

x hart

LCL

=x-

A

2

R

Rchart

LCL

= D3R

Standard

deviation = Rd

2

n A,

03

o

d,

2 1880 0

3.267 1.128

3

1.187

0

2

574

1.693

4

0.796 0

2

282

2

059

5

069

0

2

114

2

326

n= s mple

size


107/131

0

0>

3

;:;

0

~

'-

E

'

c

c::

c

'-

e

'-

'

"'

,

'

Q

.,:\

r ,.

'

...

:

v x

1 '

1v

1

""

-

p CHART (proportion

of

detectives)

PROCESS

CONTROL

Location (geography) il{o)

Process (operation/

machine) +1111

b.1fk

Component (part

number)

o A . 1 f l { l J . I . \ b t - ( t i ~

Feature

1.11.11. r, -

,tldto1

Checking

media

s

~ - . 1 ~ p t

Specification

1 .

( ttl

' '

Sample TARGET

SIZE

0

\

-

I

l

FREQUENCY

h .,t,.

.,

1--

>

C

C

I

0..

x

m

I

0

0

:1

Ill

0

...

V

0

C

0

c

I

~

0

;.

11


108/131

i

:::

'=

'

0

...J

SHIFT

DATE

TIME

BY

n

sample size)

I faulty

units)

:], l

P(= f+ n)

.

L

Mean of p

values

=

p

Mean

of

n

values

=

n

(

"

t-,

0

;

v

't t,

r ::

\

(, 1 :

I

o.o

10

I

I

t-4-

I

/

/

/

'

'

'

rl l

J

1

j

1C

V

h

' l

Ll t

Upper control line

=

p

+

P (l- p)

I

c

1'.(

I

lower

control

line

=p- 3 p (lii- p)

I

Draw LCL at zero

) :

when this calculation

gives a negative result


109/131

I

3

:"

"

E

"'

< >

"'-

1:

"'-

'

.

; ;

"

'

c

p

CHART

(proportion of defectives)

PROC

E

SS CONTROL

Loca t ion geog raphy)

Proces

s ope ration/

mac

hine)

Component

part

numbe

r)

Feature

Checking

media

Specification

I

Sample

TARGET SIZE

I R

EQUENCY


110/131


111/131

0

3

3

....

i::

Q.

e;

'

'

.s

''

'

,

'

'

...

0

'

Q

e;

'

-

:;

i

w

.

_ ....-

.,0

'\.

'1

3t

1t

(

np CHART number of defectives)

PROCESS CONTROL

location geograp hy)

' .,,

.,

Process (operation/

machine)

l l(pil14

Component (part number)

s

~ - . A t A

Feature

J

' . V ~ U . S

Checking media

s

.

. - . . , \ '

Specification

1

-]; \ ' i t O l \ t J ~ . . f

Sample TA

R

GET SIZ

E

1 A ~ < . C

...

,

}.

.

'

I

I

FREQUENCY

t-Mi\1

-

>

'C

'C

ID

l

c..

)( '

..,

I

l

'C

( )

r

Dl

l

0'

...

z

c

3

C

ID

...

-

r

I ll


112/131

'

::::>

.g

< 0

e s

~ ~ ~

-' '

'

g

_ . s ~

S:C::


113/131

N

j

:1.

. 2

.


114/131

n uw

'

'

~

c -

.... 5 QJ

. 2 E

~ ' '

'' '

~ ~

:""'

~ i

DD

~

k ~

M

M

+

~

,.. _

11

11

.,

.,

:.

~

E

0

c

0

0

a .

::

.

0

:::>

....

DD

k

11


115/131

Appendix G - u Chart for Proportion of Defects

_,

0

a:

z

0

u

'

'

J

u

0

a:

114 www.smmt.co.uk

>-

'-'

z

LU

:::;

c::J

LU

u..

"'

0

c

=

S

' '

..

: i : f : Q

E

..c a. '

' ' ' '

0

afdwes

u

Jun

Jad

spa;ap)

n

r:

..


116/131

a

l. 11

-

I ; f T

SHIFT

D TE

1

' .1 v

TIME

Y

sample size)

1

c faults) '' ' '

u

=c+n)

Meanofuvalues u I :- - I Uppercontrolline V

3ff

I c f


117/131

C l

i

a

< >

0

C:

'

.,

c

'

...

':>.

'

u CHART (proportion

of

defects)

PROCESS CONTROL

Location

geography)

Process (operation/machine)

Component pa rt numbe r)

Feature

Checking media

Specification

I

Sample

TARGET SIZE

1 FREQUE

NCY


118/131

i

C:

'

-....

SHI T

D TE

TIME

BY

n sample size)

c

faults)

I c

+

n

Meanofuvalues =u I I Uppercontrolline =ii 3{ I I

_ I I /u I I raw LCL t zero

Mean of n values ower control line i

\ f f

when th1s calculatiOn

n

g1ves

a

negat1ve

result


119/131

Appendix H c Chart for Number of Defects

0

a:

.....

z

0

' '

Cl)

Cl)

UJ

' '

a:

a

'

a;

ti

c::

.;

...,

E

E

c::

t

.,

.e

E

=

a

i3

E

.,

' '

.,

:c

E

'

X

c

' '

>

-

ffi

::::>

d

UJ

s:

UJ

N

u

=

5

;;;

o;

Q

E

a

.,

Cl) Cl)

I

- I

I

I

I

\

I

]

1

'

I

f\

I

).-

I

I

I

I

I\

I

I

I

I

>i

I I

afdwes

o

SJIUn fe ut

saa;ap o ;aqwnu)

J


120/131

i

3

I

c-

I I

raw

LCL

at zero

lower control line =c - 3\ c / .r

M '

when th s calculatiOn

giVes a negat ve result


121/131

0

a

1-

z

0

.)

en

en

w

. )

0

a:

0..

Q>

c:

.,

~

Q;

o;

C

E

..c

:;;.

E

E

- '

c:

E

Q

Q

0

~

.

,

;;

..

'

g

Q; .:;:,

=

Q

::

. ' ?

.s

c

,

1

c

'

c

a:

0

'

0

c

.,

a.

: :

E

.)

0

0

et

0

.)

120 www

.smmt.co.

uk

c

.,

E

en

c

E

: ;;

s

.,

.,

.:::

.)

>

'- '

z

U

:::>

d

U

a:

f---1---

U

V )

c

.5


122/131

3

a

>

0

SHIFT

D TE

TIME

Y

c faults)

ean of

c

value

s c

I I

Upper

control line

c

3F ~ - - - - - - - - -

c- I I

raw LCLat zero

lower

control line

c 3\

c

when th s

calculatiOn

. . g ves

a negative result


123/131

Appendix I Normal Probability Paper

C P BLITV SSESSMENT

for fe ture it norm l distribution

f

r

r

x

s

the

cl ss upper bound ry

f

f

Class Ta lly chart I

Lf

I I

I I

I I

I I

I

I I

I

I I I

I

I I

I I

I I I

.I

I

I

I I

I

I I I I I

I

I I

I

I I I I

I I

I I I I I

I

t

f

t

f

t

__

5

rr

MEASURED VALUES

1

6

11

16

21

26

31

36 41 46 51

56

61

66

7 76

81

86 91

96

1 1

1 6

111 116 121

2 7 12

17

22 27

32

37 42 47 52 57

62

67

72 JJ

82

87 92

97

1 2 1 7

112 117

122

3 8 13 18

23 28 33

38

43 48

53

58 63

68 73

78

83

93 98 1 3 1 8

113 118 123

4 9 14 19 24 29 34 39 44

9

54

59

64

69 74

79 8 89

94 99

1 4 1 9

114

119 124

5 1

15

2 25 3

35

4 45

5

55 6 65 7 75

8 85

9 95

1

1 5 11 115 12 125

122

www.smmt.co.uk


124/131

4Mii UM

REPORT

Date

t5rr

-

-

-

-

-

-

elete

s

appropriate

- -

CAPABLE

I

NOTCAPABLE

I

SffiiNG

ON NOM INAL I

SETIING OFF

NOMINAL

-

-

- -

-

-

99.87

99

.5 99

98

95 90

80

70 60 50

40 30

20

1

5 2 1.0 0.5 0.13

0.

00

3 -

-

I

I

I

I

0.

13

0.5 1.0 2

1

20

30

40

50

60 70

80 90 95 98 99

99.5

99

.87

99

.

997

l:l

INFORMATION SUMMARY

Uppe r spe cification limit u

Nomin

al

N

L

ower

sp

ec

ifi

ca

tion

lim

it

L

Xu

at line/ +50 intersection

A

Xu at line 5a intersection B

Differ

ence

I= A - B) c

rr estimate (=C/10

rr

Tolerance band(=U L T

Process

sprea d(=

6rr

p

Caoabilitv ind ex(= T P)

c

Pr

ocess mean X

Process settin Q = NI

above

specification

be

l

ow

specificati

on

www.smmt.co.uk 123


125/131

Appendix J Probability Paper for Extreme Skew Distribution

C P BLITY SSESSMENT

.;;

=

f r

fe wre

with norm l

distribution

r

xu

s

the cl ss

upper

bound ry

Class

Ta lly chart I

l:l Lf

I

I

I I I I

I

I

I I

I

I I I I

I

I I

I I I

I I

I I

I

I I I I I I I

I I I

I I I

I

I

I I

I I I I I

I I I I I I I I

I I

1

ME SURED

V LUES

I

6 11 16 21 26

31

36

41

46

51

56 61 66 71 76

81

86 91 96 101

1 6

111

116

121

2 7

12

17

22

27

32 37

42 47

52

57

62 67

72

77

82

87

92

97

1 2 1 7

112 117 2i

3 8

13 18 23 28

33

38

43

48 53 58 63 68 73

78

83 88 93 98 1 3 1 8 113 118

2

:

4 9 14

19

24 29

34 39 44

49

54

59

64 69

74

79 84 89 94 99 1 4 1 9 114 119

12

5

1 15

2 25

3

35 4 45

5 55

6 65 7 75

8 85 9

95

1 1 5

11 115 12

m

124 www smmt co uk


126/131

4ff4 1. .1

EPO

RT

geography)

rocess

or ope ration)

t

or machine)

ro

du

ct or component)

erformed by

D

ate

as appropriate

CAPABL

E

NOT CAPABLE

SETTING

ON NO

MIN

AL

SETTI

NG OFF

NOMIN

AL

t

t

t

r

I

_

9987 99 90 70 50 30

20

10

05 or-olr

0 13 0 05 0 01

I

I

I

I

I

I

I

I

I

1 10

30 50

70

80

90

95 97 98

99

99.5

99.7

99.8

99.87 99.95 99

.9

9

L

l l:f

t

INFOR M

ATION SU MM

AR Y

r

Up

pe r specificatio n limit

u

Nominal

N

Lower specification limit L

Xu

at line /99.87 pe rcentile

A

Xu t line/0.13 pe rcent ile B

Pr

ocess spr

ead

I=A-

B

p

Tolerance band I=U-

L

T

Capability index I=T/PI

C

Proc ess mode

X

Pr

ocess sett

in

g I=x-

NI

above speci fi ca tion

below specificat ion

www.smmt.co.uk 125


127/131

15 Subject Index

Topics and Terms age Topics nd Terms age

AMM 71

customers

9

assignable causes detectives charts 40

attributes charts 28 42 defects charts 40

average movement of the mean

71

distribution

65

c chart 43, 118 disturbances

capability estimation snap-shot)

84 disturbance elimination

52

capa bility indexes

85 executive role

20

capability index interpretation 60 expectation

12

centring

53

facilitators

20

chance causes fact-holders

20

chart design 28 frequency table illustrated)

66

chart for moving mean

71

histogram illustrated)

66

67

chart for sample size of one

7 individuals and moving range chart

7

chart for small batch runs

74

limits

12 , 60

chart pattern interpretation 44

management ro le

20

chart pattern chance occurrence

52 mean

and

range chart

35, 94

chart scales

29 mean and standard deviation chart

37

99

charting purpose

25

mean

14

charting strategy

21

median

and

range chart

38 102

charting summary

23 mode

14

Cm Cp and Pp

58

nominal

12

Cmk, Cpk and Ppk

59 non normal distribution

85

co-ordinators

20 non normal process spread

87

common disturbances

normal

84

control lines

30 normal distribution check for)

67

cusum chart

78

np

chart

43

110

126 www.smmt.co.uk


128/131

Topics and Terms age Topics nd Terms

age

one-sided distribution

86

sigma limits

61

optimum

12

skewed distribution 83

p chart

42 106

special disturbances

performance limits

12

specification limits

12

probability paper

82

standard deviation of process 57

probability plot interpretation 84

standard deviation of

sa

mple 99

probability plots

82

standard tolerances

13

process

capa

bility

55

standardised chart

76

process control 22

standardised deviate

76

process elements 22

statistical control

process spread

84 suppliers

9

R 5

tally chart illustrated) 66

R

R

bar)

6

targets

12

range 5

tolerance

12

s

37

truncated distribution 85

s

s

bar)

37

u chart

43

114

lower case Greek sigma) 57

variab les charts

28 32

sigma circumflex)

57

variab les charts constants

93

sample size for attributes

40

work-teams

20

sample size for variables 32

x

x bar) 5

sample size of one charts 72

x double bar) 6

sampl ing of attributes

40

x x wavy bar or x tilde) 38

sample of variables 32

x

x bar wavy bar) 9

setting 53

Z values

59

setting indexes

59

Z values interpretation

62

short production

run

chart 74

www.smmt.co.uk

27


129/131


130/131

Other publications available

from the S T

Continual Improvement Tools Techniques

A Guide For Business Improvement

Process anagement


Failure M ode And Effects Analysis


To order or find out more, contact:

Publications, The Society of otor Manufacturers Traders Ltd,

Forbes House, Halkin Street, London SW X 70S

Tel

+44 (0)20 7344 1612/1611

Fax +44 (0)20 7344 1603

e-mail: [email protected]


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