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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
4t41 . .1
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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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3
7
9
9
9
10
11
12
14
16
17
18
18
2
20
21
22
23
5
25
28
29
3
32
32
32
32
33
35
37
38
4
4
4
41
42
43
43
43
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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.
www smmt co
.
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
4ii41 . .1
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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 .
6
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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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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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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.
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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.
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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
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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.
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'
;
'
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
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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
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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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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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t5
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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.
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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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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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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.
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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.
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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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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
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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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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
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Appendix A Constants for Variables Control Charts
'
'
'
'
'
-'=
'
g
:
'
'
'
'
:
:
=
-'=
'=
c:
8
E
a:
8
8
g
c:
c: c:
f3
8
8
8 8
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i
cs:>
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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
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.m
. ] . ,
Standard deviation ; R/d
1
1
/
/
l :T7
I
A
J
I
J
ll
,-
...
f:l
-
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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
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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
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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
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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
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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
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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
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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
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'
::::>
.g
< 0
e s
~ ~ ~
-' '
'
g
_ . s ~
S:C::
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N
j
:1.
. 2
.
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n uw
'
'
~
c -
.... 5 QJ
. 2 E
~ ' '
'' '
~ ~
:""'
~ i
DD
~
k ~
M
M
+
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,.. _
11
11
.,
.,
:.
~
E
0
c
0
0
a .
::
.
0
:::>
....
DD
k
11
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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
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..
: i : f : Q
E
..c a. '
' ' ' '
0
afdwes
u
Jun
Jad
spa;ap)
n
r:
..
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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
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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
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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
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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
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>
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ffi
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d
UJ
s:
UJ
N
u
=
5
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o;
Q
E
a
.,
Cl) Cl)
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- I
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I
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I
>i
I I
afdwes
o
SJIUn fe ut
saa;ap o ;aqwnu)
J
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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
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0
a
1-
z
0
.)
en
en
w
. )
0
a:
0..
Q>
c:
.,
~
Q;
o;
C
E
..c
:;;.
E
E
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c:
E
Q
Q
0
~
.
,
;;
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g
Q; .:;:,
=
Q
::
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.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
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d
U
a:
f---1---
U
V )
c
.5
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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
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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
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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
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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
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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
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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
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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
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27
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Other publications available
from the S T
Continual Improvement Tools Techniques
A Guide For Business Improvement
Process anagement
A Guide For Business Improvement
Failure M ode And Effects Analysis
A Guide For Business Improvement
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
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