Statistical Process Control - A Guide For Business Improvement

4t41!.!.1

Statistical Process Control A Guide for Business Improvement

The Society of Motor Ma nufacturers and Traders Limited London 2004 . 'SMMT' and the SMMT 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.

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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 A NUFACTURI 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 7 times) Second edition 1994 Third edition 2004

© The Society of Motor Manufacturers and Traders limited. All rights reserved

Published in 2004 for SMMT bv Findlav Publications Ltd, Horton Kirby, Kent DA4 9LL

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Section 1 3 7

Contributors Foreword

Section 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

4

Introduction

Philosophy Information from data The uses of charting Disturbances & state-of-control Specifications Measures of middle Measures of spread Other measures of shape Using calculators to obtain statistical measures A reason for chart sample sizes above one

9 9 9 10 11 12 14 16 17 18 18

Section 3 - Getting Started

20

3.1 3.2 3.3 3.4

20 21 22 23

The people involved Executive and management considerations Planning for process control A summary of charting

Section 4- Control Charts in General

25

4.1 4.2 4.3 4.4

25 28 29

Purpose Chart design Chart construction Control lines

30

Sect1on 5 - Control Charts for Variables

32

5.1 5.2 5.3 5.4 5.5 5.6 5.7

32 32 32 33 35

Introduction Sample size Sample selection Special circumstances Mean and range chart Cx&R) Mean and standard deviation chart (x&s) Median and range chart Cx&R)

37 38

Section 6 - Control Charts for Attributes

40

6.1 6.2 6.3 6.4 6.5 6.6 6.7

40 40

General Sample size Sample selection p chart for production of detectives np chart for number of detectives c chart for number of defects u chart for production of defects

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41 42 43 43 43

Section 7 - Chart Interpretation

44

7.1 72 73 74 75 7.6 7.7 78 79

44 44 45 45 47 51 52 52 53

Introduction Examination of charts for variables (x&R, x&R, x&s) Examination of charts for attributes (p, np, c, u) Pattern recognition Examples of out-of-control patterns Other examples of patterns Unusual patterns without special disturbances Dealing with disturbances Centring

Section 8- Capability

55

8. 1 8.2 8.3 8.4 8.5 8.6

55 58 59 60 61 62

Capability statements Capability indexes Setting indexes Interpretation of indexes Estimation of conforming products Example Reaction Plan following process monitoring

Sect1on 9 - Summary of the Process Improvement Stages Section 10- Top1cs Related to Charting

64 65

10.1 The norma l distribution 10.2 Introduction to analytical methods

65 68

Sect1on 11

70

11.1 11.2 11.3 11.4 11.5

Control Charts for Special Situation

Moving mean charts Charts for sample size of one Charts for short production runs Standardised charts Cusum charts

70 72 74 76 78

Sect1on 12- Capability Estimations 12.1 12.2 12.3 12.4

0

Probability plots Distribution information from probability plots Snap-shot capability estimations Estimation s for non normal distributions

Section 13- Bibliography Section 14 - Appendices Section 15- Subject Index

82 84 84 85

90 92 126

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In the current climate, the SMMT 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 . The SMMT Business Improvement Guides cover: Process Management Continual Improvement Tools and Techniques Statistical Process Control Failure Mode and Effects Analysis

The purpose of this guide is to explain Statistical Process Control. The basic principles contained within this guide will equip the reader with 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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2. Introduction 2.1 Philosophy People Supp "ers have a responsibility to meet or better customers' expectations. Customers are the people or mach ines 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 Most companies operate in markets where it is vital that they are competitive and profitable. Being 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 im provement 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 defective products. www.smmt.co.uk

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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 collected and used, there will be the further waste of not being able to identify opportunities for improvement. Much information 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 interpreting numerica l data. This part of the guidelines concentrates on simple charting methods that have wide application in commercia l and manufacturing industries. it offers a framework for practical training and can be used as an on-the-job reference.

2.3 The Uses of Charting Process control charts can be used to 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 riabi 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 is only of value if 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.

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iii!I.!.M 2.4 Disturbances & State-of-Control

Variation among the products of any process is inevitable. lt arises from causes which create process disturbances that are called common or special. Common disturbances arise from causes that are inherent in the process and to some degree affect all products of the process . Examples of causes are variable raw materials, rigid working methods, equipment limitations. atmospheric conditions and individuals' capabilities. These causes are sometimes called 'chance' causes, this is misleading because the causes of special disturbances also can occur by chance. Processes that suffer only from common disturbances are in a state of statistical control . In other words, the results of the process are 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 are not inherent in the process . Exam ples of causes are material flaws, non-observance of instructions, power failures, vandalism and inappropriate training. These ca uses are sometimes called 'assignable' causes, this is misleading because causes of common disturbances also are assignable. Processes that suffer from specia l disturbances are out-of statistical control because the effects of a disturbance are not predictable. Charting highlights the occurrence of special disturbances.

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Figure 2.1 : Design specificat ions CUSTOMER'S EXPECTATION A few approach ing lower Performance limit

A few approac hing Upper Performance limit

Most at or nea r to Economic Optimum

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2.5 Specifi cations Engineers design for and customers expect an idea l. Des igners specify ideal measurements, as targets or nominals . The value t hat is specified should be th e same as the optimum expected by customers (figure 2.1 ). There can be diff icu lties for process control if t he nom ina l is not specified 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 being on nomina l. Designers cater for variabil ity by offering a tolerance. Product detail tolerances are not common in certain industries. Whether or not tolerance is specified, customers wi ll accept va ri abi lit y if t he risk to them is not unreasonable. A design tolerance is a statement of performance limits or the measurement range w ithin which the product w ill functio n satisfactorily. Most often , nom inal is in the middle of this range. At end-of-l ine inspection, performance li mits provide the criteria for product acceptance or reJection.

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4ii41!.!.1 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 suppliers rath er th an as a basis for custo mer satisfaction. Figure 2.2: The roles of people in SPC

EXECUTIVE/MANAGERS Nominate co-ord inator/ facilitators

Scrap/rework

-

CO-ORDINATOR

~

Identifies opportunities - - Warr~~d coaches facil itators

_j _j

Administration --Etc

I MANAGERS Promote employee

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Wm!i.J.M 2.6 Measures of Middle Although diagrams usua lly give th e best idea of th e shape of a distribution, numbers are necessary for comparisons with other distributions. One such number is an estimate of the middle of a distribution, sometimes it is called the location or central tendency of 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 rks per person was 2, 5, 3, 6, 4, 3, 8, 5 and 3

Mode 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 = 3.

x

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 = 4. middle number is 4, therefore the median is

x

Mean The mean is the arithmetic average, sample mean is denoted by underlying or population mean is denoted by J.L . lt is calculated by adding the values and divid ing by their number,

x= 2 +5 + 3 + 6 + 4 + 3 + 8 + 5 + 3 = 39 = 4.33 (to two places) 9

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x,

A symmetrical distribution

Mode= median= mean

A non symmetrical distribution

Mode Median Mean

The mode, median and mean are compared above for a symmetrica l 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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Wmi!.J.M Although the mean is the most common way of expressing average. there are time s wh en the mode or median are preferred. For example Designers usually follow market su rveys. In practice, this amounts to fo llowing t he 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 below it. The median is used in some manua l cha rting appl ica tions, partly because it is easi ly ca lculated and understood and partly because it avoids t he need for calculators. 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 wh ich indicates va ri ability, is inherent in the machine and cannot be changed merely by turning a knob. Three ways of expressing an estimate of the variabi lit y of a distribution are range , variance and standard deviation .

Range Th e range is the maximum 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 estimate of the spread of a large distribution because it ca n be und uly infl uenced by a si ngle measurement value. Variance Va ria nce is the mean square difference of the values from the mean, sample vari ance is denoted s'. underlyi ng or popu lation variance is denoted
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Standard Deviation Standard deviation is the square root of the variance . The advantage of using standard deviation rather 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 doubled 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 M easures of middle and spread toget her provide a summary of a distribution wh ich w ill be adequate for most purposes. However, there are situations which require other measures to be considered, in pa rticu lar when tests for special disturbances are necessary. The features wh ich need to be considered are: departure from symmetry which is ca lled skewness.

Symmetrica l (c, = 0) or not skewed

Pos itive skew (c, is positive)

Negative skew (c, is negative)

c, is a coefficient of skewness that is quantified by 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) .

ll Flat-topped (platykurtic)

Peaked (leptokurtic) Ck is high 'Yis positive

ck is low 'Yis negative

ck and 'Yare different coefficients of kurtosis that are quantified by some computer programmes. ck reflects the shape of a distributions tails, 'Yreflects its central shape and

'Y= 0 for a normal distribution. 'Yis the Greek capita/letter upsilon, equivalent to U in English.

2.9 Using Calculators to Obtain Statistical Measures Most scientific ca lculators have keys which give the mean and standard deviation at the press of a key.

x

Relevant keys are often marked for the mean and the sample standard deviation (un_ 1 = s). s is an estimate of u and


or s for

xis an estimate of f.L.

Often these calculators are programmed only for single normal distributions. lt is advisable to confirm the distribution before using the statistics produced (see section 12).

2.10 A Reason for Chart Sample Sizes Above One The simple rules for control chart interpretation (section 7) assume that plotted values have a normal distribution (section 10.1 ). There are frequent occasions when this assumption is not correct. especia lly when the va lues are of individual measurements and their distribution is skewed or subject to kurtosis (section 2.8) .

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The way to get around this difficulty is to make use of a mathematical ru le ca lled the central limit theorem which says that: 'no matter what is the distribution of individual measurements, the distribution of averages of those measurements will increasingly approximate to norma l 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 symmetrical, for example. only suffering from kurtosis. The same applies if th e distribution of individuals has a slight skew. The 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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8i5!i.UM

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 facilitiesall 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. The se people are the lynch-pin of statistics in process control. Their principal role is to coach others in the methods. They will have a knowledge of both statistics and the processes in the business. Knowing the business is the pre-requisite, knowledge of statistics 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 coordinator 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 twenty 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 Considerations The following are abstracts from the experience of companies that have achieved considerable success after 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 activit y, it is advisable first to 'crawl', then to 'wa lk' so that "running" is a natural and easy progression . In other words, gear activities to the organisation's abil ity to handle the informat ion that will become available.

Training must start at the top, so that executives recognise the implications and managers understand the information that wi ll arise from the work-teams. Strategic targets As with any aspect of 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 orga nisations have to offer.

When targets are not met, problems often rest with management. Empowennent 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 work-force and clearly defined process ownership. Leadership A more positive response to process control and improvement is obtained from people who work in teams with a recognised leader, rathe r than a 'supervisor'.

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Wi#§!!.!.M 3.3 Planning for Process Control A process is often t hought 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 service. 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 words, plann ing fo r process control involves understanding th e factors that contribute to the result. The record is best developed col lectively by eve rybody 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 Bibliog raphy (section 13). 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 itable con t ro l chart. Note: Processes are covered in greater depth in the SMMT publication 'Process Management- A Guide For Business Improvement'. See inside the back cover.

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3.4 A Summary of Charting The process of charting is illustrated in figure 3.1. it is a simple process but th ere ca n be pitfalls t hat need to be avoided. Ultimately, control charts will provide the fo llowing benefits: Do Plan the introduction Nominate facilitators and a co-ordin ator Nominate process owners Train everybody involved Remember the purpose is proce ss improvement Follow the sequence in Figure 2.2 Identify and eliminate all causes of disturbances Recognise su ccessful work-te ams Don't • • • • • •

Start unless you are committed Identify process contro l with single·individuals Measure success by the number of charts Use control lines to indicate acceptance limits Confuse being in-control with capability Assume that early information tells the whole story

A cost effective and powerful tool in process con tro l, they are simple and su pport empowerment of the work team. The ability to distin guish between specia l and common disturbances and provide a common language for communication of process behaviour. Initia lly, a mea ns of target ing special disturbances but when the process is predictable, t he 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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l ¥!!.!.1 Figure 3.1: The application of charting (section 3.3) (section 5.3) (section 6.3) L------------T------------~

~

Collect data

Construct co ntrol chart

(section 4.4)

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(section 7.9)

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=r= Continue charting

=r= Reduce common disturbances

24

(section 7.2) (section 7.3)

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(sectio n 5.4)

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(section 7.8)

•mi!.UM

4. Control Charts in General 4.1 PURPOSE 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 of 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 find 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.

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4.2 Chart Design There are three groups of control charts: Variables charts which are used for measurements in units of length, mass or time and combinations of these such as volume, energy and acceleration. Attribute charts which are used for counted numbers or proportions such as per unit area, per cent and per sample. Complex charts including variations of the above such as for moving means, and special purpose charts such as cusum and standardised.

There are many versions of each type. Those described in these guidelines are known as Shewhart charts for straight forward variables and attributes. Complex charts are described in texts referenced in the Bibliography. Many charts are drawn by computer and, especially when th ey are used for continuou s or high volume processes, data inputs and sometimes outputs are automated. The descriptions in these guidelines apply in principle to both handdrawn and to computer produced control cha rts. A typical industrial chart is illustrated in figure 4.1. it has four sections: HEADINGS that identify the subject of control, its location and basic in struction s or reminders to people who use the chart, such as Checking Media which li nks to systems of measuring equipment ca libration and Specification which is useful in capability assessments. GRAPH PAPER or grid that is the working part on wh ich plots are made to show variation from time to time in measured or counted data.

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4fiii!l.!.i RAW DATA RECORD with identifications and calculations for the

plots. PARAMETER RECORD with formulae used in their ca lculation. These parameters are explained in following sections of this handbook.

When a chart is actua lly in use, it is often helpful to write remarks on it about relevant actions and subjective observations at the time of recording data. Figure 4.2: Data plotted on a control chart

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10

15

20

25

Sample number

4.3 Chart Construction Data is collected whi lst the process is operating and recorded on the chart.

Sometimes, the data is converted on the chart to another more convenient form such as an average or proportional va lue. lt is then plotted against a sca le shown on the vertical axis, in a sequence shown on the horizontal axis (see figure 4.2).

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Jii41U!.M The vertical scale shou ld be selected so that extreme values can be plotted . Also. to help interpretation (described from section 7.1) there shou ld be about 12 but not less than 8 steps at convenient equal intervals between the control li nes which are explained in the next section. The horizontal scale should accommodate a minimum of 25 plots. This is to allow th e cha rt to give a coherent picture of the process. To help the picture, straight lines are drawn between successive plots. Figure 4.3: Control lines drawn on a control chart

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4.4 Control Lines In some SPC manuals, the lines are cal led limits. The word lines is preferred because people tend to confuse control limits with specification limits.

When there is sufficient data available, a process mean line and control lines are calculated and drawn on th e chart. Means are drawn as broken li nes and control li nes are solid (see figure 4.3) . The lines help interpretation of the chart (see figures 7. 1 to 7.12).

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Sufficient data is usually obtained from about 25 samples each sample consisting of one or more units, batches or groups that are measured or counted . Sample sizes vary according to chart type. Details are in section 5.2 for variables charts and in section 6.2 for attributes charts.

Calculations of the mean and control lines are made from the data using simple formulae that vary according to chart type and sample size. The formulae are set out with the detail chart descriptions.

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5. Control Charts for Variables 5.1 Introduction The charts for variables described in these guidelines are Mean and range (x&R) preferred for manual applications . Median and range Cx&R) the most simple. Mean and standard deviation ( x&s) preferred for automated applications. The symbol - is called a bar and - is a tilde, often called a 'wavy bar'. These charts are used in pairs. one to monitor process setting and the other to monitor variability. lt is usual to plot both on the same screen or sheet of paper. 5.2 Sample Size A sample size of 5 ha s been found to give reliable results and it is the most commonly used for x&R and x&R charts. Sample size must stay constant for any one chart. Smaller sample sizes can be used when necessary fo r x&R charts but a larger sample size (minimum 8) is recommended for x&s charts.

5.3 Sample Selection Samples should be taken periodically at regular intervals to give a picture that relates easily to all aspects of the process. Care must be taken to ensure that regularity does not introduce bias. Rather than specify particular times, the frequency could be about every 500 produced or about every hour or four times a day and so on. Sampling should capture the effects of all likely process disturbances, such as start-up, shut-down, material batches and shift changes.

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Whatever the interval and wherever practicable, the units in a sample should have been produced consecutively. Samples of consecutive units give a better idea of common disturbances than samples where the units are chosen randomly.

5.4 Special Circumstances The first chart The purpose of charting is to get a picture of process setting and variability and it is sensible to do this as quickly and as econom ically as possible.

The guidelines for sample selection allow room for manoeuvre! At the outset, 125 consecut ive units might be taken and then treated as if they were 25 samples of 5. The data might reflect a well trained work team , new facilities , a single material batch and constant environmental conditions. The resulting chart is better than none at all but it is unlikely to represent the true behaviour of th e process. Learn from it and take any action that w ill improve the process . Later charts As charting con tinue s, the picture will become clearer and disturbance s will be more readily re cog nised.

When the picture is stable, sampling frequency can be reduced. Process changes Often, process changes are clearly shown on charts.

Sometimes, changes are not obvious. Whenever there is a change, the chart means and control lines should be recalculated . Recalculation shou ld be from data of about 20 samples after the change.

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4ifi!lolri For the not so obvious changes, completion of a chart provides a timely reminder to check the calculations before a replacement chart is used. Figure 5.1: Data for variables control charts

79 81 77 89 75 76 75 69 72 86 67 84 72 81 77 75 82 72 72 87 86 65 91 78 89 76 76 70 84 76 89 94 82 63 84 85 88 72 82 73 54 75 85 78 65 75 92 82 77 78 88 81 70 85 83 93 87 87 90 75 80 84 86 70 77 76 80 81 79 70 73 72 70 81 72 86 80 72 76 88 86 94 71 89 77 74 67 76 84 76 86 79 68 80 82 71 86 77 79 82 94 68 75.0 78.6 87.4 83.4 80.8 73.6 85.2 77.6 79.6 78.2 79.8 77.4 76.6 73.6 73.6 77.0 81.0 76.0 74.6 83.2 76.6 80.2 75.4 83.2 77. 17 13 21 26 25 24 13 10 11 21 14 14 15 32 15 17 9 21 24 20 12 14 13 22 18 72 76 88 86 82 71 86 77 80 77 82 80 76 76 75 79 80 77 73 86 77 79 76 86 80 6.8 5.0 7.0 10.0 9.3 9.1 4.6 3.5 3.8 7.3 5.1 5.8 5.2 10.8 5.8 6.2 3.2 7.7 7.9 8.2 3.9 4.8 4.5 8.1 6.9

Figure 5.2: A means control chart with control lines based on ranges from the data in figure 5.1 100

~

:;; Q)

80

E Q)

c. ~

~

70

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

Lower control line LCL, =

x- A2R

60

50+-.-.-.-.-,-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-~

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Samp le number

34

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Figure 5.3: A ranges control chart from the data in figure 5.1 40

Upper control line UCLR= D4R

30

"'~ ~

a_ "'

20

~ C/) 10

Lower control line LCLR= D3R (= 0 in this figure) Q

--,

1

2 3 4

5 6

7

I

I

I

y - y - - . . - -1

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Sample number

5.5 Mean and Range Chart (x&R) This chart is preferred for manual use because it avoids complicated calculations. Raw data record The measurements of each unit in the sample are recorded (see figure 5.1 ).

The fol lowing stati stics are ca lculated from the measurements. SAMPLE MEANS

:X

The sum of the individual unit values divided by the sa mple size.

SAMPLE RANGES

R

The difference between the largest and the sma llest unit values in the sample .

Graph Paper The statistics are plotted on a suitabl y scaled graph (see section 4.3) and the plots are joined by straight lines (see figures 5.2 and 53).

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8¥!1.!.1 Parameter Record Two parameters are calculated from th e stati stics and drawn on the chart as broken horizontal lines. PROCESS MEAN

X

The sum of all the sample means divided by the total number of samples.

MEAN RANGE

R

The sum of all the sample ranges divided by the total number of samples.

Four control lines are ca lculated from the parameters using the formulae shown in figures 5.2 and 5.3 and drawn on the chart as solid horizontal li nes. The constants A 2 , D3 and D4 depend upon sample size and are obtained from statistical tables (see Appendix A, page 93). Figure 5.4: A means control chart with control lines based on standard deviations from the data in Figure 5.1 100

90

x

Up per control line UCL, = + A3s

~

:;; 80 "'E

"'

Q_

~

70 Lower control lin e LCL, =

U)

x- A3s

60

5 0 + - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , -, _ . , 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Samp le number

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lii!!.UM Figure 5.5: A standard deviations control chart from the data in figure 5.1 15

Upper control line UC L, = B4s

c: 0

·~

·;;

10

Q)

"C

1"

"' c: "'

"C

~

Q)

Q_

E

"'"' Lower contro l line LCL, = Bis I= 0 in this figure)

0+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-. . 1 2 3 4

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Sample number

5.6 Mean & Standard Deviation Chart (x&s) The x&s chart is best suited to applications where measu rements, calculations and plots can be automated. The alternative x& R chart is preferred for manua l applications because range is easier to calculate than sta ndard deviation. The differences between the x&s chart and the x&R chart are: the standard deviation, instead of the range, is calculated for each sample. -see Appendix C, page 99 .

s

control lines are calcu lated from x and (the mean of the sample sta ndard deviations) using the formu lae shown in figures 5.4 and 5.5. The constants A3 , B3 and B4 depend upon sa mple size- see Appendix A, page 93. Note: they are all different to constants used for x&R charts.

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x&s charts have narrower control lines than x&R charts. However, particularly when sample sizes are above 8, x&s charts are a better ind icator of trends than x&R charts, but less sensitive in detecting some special disturbances. Figure 5.6: A medians control chart from the data in figure 5.1 100 Upper control line UCL,;

x+ A2R

90 w

~

"' c_ "'

80

E

~

(/)

Lower control l ine LCL, ;

x- A2R

60

50+-.-.-.--.-,-,-.-.-.-.-.-.-.-,-,-,,-,-,-,-,-,-,-,-, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Sample number

5.7 Median & Range Chart (x&R) This chart is used widely because it avoids day-to-day arithmetic calcu lations and therefore is more readily accepted than x&R and x&s charts.

The ranges graph is the same as that on an x&R chart (see figure 5.3). The medians graph is effective for detecting changes in th e process mean but it is not an estimator of the process mean. When the actual va lue of the mean is important, more useful information wi ll be obtained from x&R and x&s charts.

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The differences between the

Xchart and the x chart are:

measurements of the un its in a sample are reco rded only as plots in a vertical line on the chart (figure 5.6) rather than as w ritten numbers. the median (or middle) value for each sample is highlighted on the chart, if there are an even number of units in the sample the highlight is placed midway between the two middle values and lines are drawn to join successive medians instead of calculating, plotting and joining means as on the chart.

x

x

control lines a~e calculated from (the mean of the sample medians) and R..:. using the form ulae shown in figure 5.6. The constants A 2, 0 4 and 0 3 depend upon sample size- see Append ix A. page 93. Note:

A2 is used for medians charts and A2 for means charts.

control lines for medians are about 25% further apart than those for means but this is not of any practical significance.

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6. Control Charts for Attributes 6.1 General Sometimes, requirements are subjective and measurement is impossible, for example, a valve functi ons or it does not, a person is absent or is not. A means of monitoring such requirements is to use an attribute chart and data that is counted as 'go' or 'no-go' against an acceptance standard. Attribute charting is often an easy option but wherever practicable, counts against 'go/no-go' criteria and attribute charting should be resisted in favour of measurement and variables charting . One type of attribute chart deals with DEFECTI VES wh ich are units that fail because of defects (p charts in section 6.4 and np charts in section 6.5). A second type deals with the DEFECTS themselves or the reasons why units fail (c eh rts in section 6.6 and u charts in section 6.7). All attribute charts have a similar appearance (figure 6.1) but each type has its own formu lae for calcula ti on of mean and control lines (figure 6 2) . The interpretation of attribute charts is described in sect ion 7.3.

6.2 Sample Size There is one simple guide-line for attribute chart sample sizes, it should be large enough to allow the defect or defective to appear in the majority of samples taken. Sample sizes for attribute charts tend to be la rger than for variables charts. When the objective is to mon itor a f requent ly occurring situation, comparatively small samples (say 10) might be needed.

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• If the objective is to check the effectiveness of action to eliminate a defect th at previously occurred at a rate of about 1 %, the required sample size wou ld be wel l over 1000. Successive sample sizes need not be constant but there is the advantage of less calcu lation with constant sample sizes. np and c charts deal with constant size samples. p and u charts deal with variable size samples.

6.3 Sample Selection Samples shou ld be taken random ly to reflect all likely process disturbance but where practicable, the units should have been produced consecutively. Such samples give better pictures of inherent variability and trends than when the individuals are chosen randomly. Figure 6.1: An attributes control chart

Upper control line (UCL)

Mean

Lower control line ILCL)

10

15

20

25

Sample number

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I

li51!.!.1 6.2: Attribute chart statistics and formulae

p

p

f, + fz + .. + fm

P= n, + nz+ ... + nm f, + fz + .. +fm

f=

np

u

m C1+C2+ ... +Cm

c

C=

u

+ C2 + ... +Cm il= n,+nz+ ... +nm

m C1

Key to symbols c

Defects (fau lty features) in sample Defectives (faulty units) in sample

n

Sample size

m Number of samples

n

Average sample size

= n, + nz + .... +nm

u

Defects per sample unit

=C

p

Proportion of defectives in sample

= f

m n

n ** The lower control line is drawn at zero when the ca lculation gives a negative number. 6.4

p Chart for Production of Detectives Defectives are sometimes cal led non-conform in g or faulty units. The p chart is used: when there is variation in successive sample size and the data is expressed as the fraction or proportion of defectives in the sample. Sample sizes should not vary by more than ± 25 % of the average sample size. Texts in the Bibliography describe alternative but more complex charts that might help when sample sizes are extremely variable.

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4ii4!1.!.1 6.5 np Chart for Number of Detectives The np chart is sim ilar to the p chart, the differences are that: the sample size must be constant and the plotted data can be the actual number of detectives recorded. The p and np charts look exactly the same if plotted for the same data w hen there is constant sample size except that the respective vertical scales wi ll be labelled differently. 6.6 c Chart for Number of Defects The c chart is also used to il lustrate discrete data that is not of attributes such as occurrences of accidents, live births or aircraft movements and it is sometimes ca lled a simple run chart. Defects are sometimes called faults. The c chart is similar to the np chart except that it describes defects rather than detectives. Samples can be : a single unit such as an assembly, a measured production output such as an area of material or a constant sized group of units such as an audit batch. 6.7 u Chart for Production of Defects The u chart is similar to the c chart, the differences are that: the sample size can vary and the plotted data is the average number of defects per unit in the sample. Like the p chart, sample sizes must not vary by more than ± 25 % of the average sample size.

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7. Chart Interpretation 7.1 Introduction A control chart is a pictorial rep resentation of process behaviou r. lt can illustrate: e achievement of customer's targets, e variation from targets, e process stabi lity, e the effect of process change and e the presence of process disturbances. In most practical situations, the picture is worth a thousand words and there is little need to indulge in abstract explanations! Occasionally, the significance of the picture might not be obvious and there is either unnecessary action or there is inaction . The key is to know some basic principles and in extreme cases, who or where to go to next. The following sections set out the principles and some suggestions for a next step are in the Bibliography.

7.2 Examination of Charts for Variables (x&R, x&R, :X&s) The range (R) or standard deviation (s) chart is examined first. lt gives a picture of the process variabi lity. If the R or s chart indicates a state of statistical control (see section 7.4), the process can be judged as being stable. The mean (x) or median (x) chart is examined second. lt gives a picture of process setting. This picture could be misleading if the R or s chart examination indicates that the process is not stable.

x

If the x or chart indicates a state of statistical control (see section 7.4), any changes in setting over a period of t ime probably are the effects of common disturbances. An out-of-control state probably is the effect of one or more special disturbances.

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4i#ii!I.!.M Re-setting of unstable or out-of-control processes, without allowing for special disturbances, could lead to deterioration in product quality

7.3 Examination of Charts for Attributes (p, np, c, u) The charts give a picture of process variability. If their patterns indicate a state of statistical control (see section 7.4), it means they are stable but not improving. Remember that for most attributes, the customer's target is zero defects and a downward trend indicates improved process control. 7.4 Pattern Recognition Chart interpretation amounts to recognition of 'unusua l' patterns plus process knowledge, experience and appreciation of probabil ity.

The starting point is the plot distributions illu strated in figures 7.1 and 7.2 which represent a statistically in-control process. Significant deviation from this ideal are unu sua l if the process is in-control. Exam ples of some unu sual patterns and their interpretation are shown in section s 7.5, 7.6 and 7.7. The illustrated patterns are by no means all that can occur. Anything that looks unu sual should be investigated even if only to confirm an occurrence has no particular cause (section 7.7) or that a mistake has been made in measurement, calculation or plotting.

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f i#41!.!.1 Figure 7.1 : Plot proportions

I

UCL

j

I

Mean

--

2 of 3 in centre third

------1

LCL

19 of 20 in centre two thirds

3 in 1000 - - - - - - - outsicteline s -

I

I

where plots should be for a statisti cal ly in-control process. Figure 7.2: In-control pattern

UCL,------------------------------------------------

Mean

an exa mple of actual plots for a statistica lly in-control process.

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4ffili.UM 7.5 Examples of Out-of-Control Patterns Figure 7.3

Plots above UCL or below LCL are perhaps, a specia l disturbance ca used by: flawe d material. broken tool. operator mistake. power failure. Figure 7.4

2 of 3 consecutive plots in top or bottom sixth of lines are perhaps, a special disturbance caused by : start-up effect. untrained operator. or could be an improvement w hen in the lower ha lf of R, s or attribute charts.

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Figu re 7.5

4 of 5 consecutive plots in top or bottom third of lines are perhaps, a persistent special disturbance with causes similar to those in figure 7.4. Figure 7.6

5 consecutive plots outside centre third of lines are usually, the result of mixed samples from: different people different machines, etc. caused by methods for: measurement. reporting. machine setting, etc.

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4i41!.!.1 Figure 7.7

6 consecutive plots (5 intervals) rising or falling are usua lly, a process cha nge caused by: equipment wear. market shift. seasonal change in weather. or perhaps, a fal ling run is an improvement on R, s or attribute charts. Figure 7.8

9 consecutive plots (8 intervals) in top or bottom half of lines are usually, a process step change caused by: re-setting. management interven t ion. parts or method change. or perhaps, an improvement when in the lower half of R, s or attribute charts.

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Figure 7.9

15 consecutive plots inside the centre third of lines often: indicates the unwitting or deliberate use of fa lse data. or perhaps: indicates a better process when pairs of charts for variables (x&R, x&s, etc) both show the same pattern. Figure 7.1 0

UCL,---------------------------------------------------

LCL1-----------------------------------------------Alternating high/low plots are usually: an extreme example of mixed samples from different people or different machines, etc. caused by unsuitable charts, procedures or standards.

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7.6 Other Examples of Patterns The t wo examples below are not necessarily out-of-control. In other words, they could be entirely predictable. They illustrate situations where more complex process control is necessary. Figure 7.11

Moving means ref lect action to control inherently unstable processes. Figure 7.12

UCL

I

......

Mean

,

....

\,

-----~~~---~~,:---~~~~-----~:--r-/

~v.

"'"

-~

LCL

Cycl ic pattern- the two patterns in figure 7 12 usually: ref lect long-term changes caused by the environment. suggest a need for compensatory action.

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1!4!1.!.1 7.7 Unusual Patterns Without Special Disturbances The patterns shown in section 75 can occur by ch ance without there being a special disturbance. Such occurrences are ran dom but infreque nt, fo r exam ple for charts, they could happen as indicated in figure 7 11. There are similar likelihoods of chance occurrence on other charts but for practical purposes, think of them only as remote possibilities!

x

Figure 7.13: Chance occurrences

I

once in about

--

6.3

Plot above UCL Plots below LCL

740 plots 740 plots

6.4

2 of 3 consecutive plots in top sixth of limits 2 of 3 consecutive plots in bottom sixth of limits

750 plots 750 plots

6.5

4 of 5 consecutive plots in top third of limits 4 of 5 consecutive plots in bottom third of limits

390 plots 390 plots

6.6

5 consecutive plots outside ce ntre third of limits

320 plots

6.7

6 consecutive plots 15 intervals! rising 6 consec utive plots (5 intervals) falling

720 plots 720 plots

6.8

9 consecutive plots (8 intervals) in top half of limits 9 consecutive plots (8 intervals) in bottom half of limits

520 plots 520 plots

15 consecutive plots inside centre third of limits

310 plots

6.9

I

--

--

7.8 Dealing with Disturbances Any disturbance highlighted by a control chart is an opportunity for improvement.

When chart examination shows that there are unusual patterns, the process cou ld be suffering from special disturbances. The cause(s) of special disturbances must be identified and ideally, they should be eliminated.

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lt is easier to deal with common disturbances when special disturbances have been eliminated but unfortunately, it is not always possible. So, there are two possible courses of action If the special disturbance causes have been eliminated, draw a new chart with new data. If the special disturbance causes cannot be eliminated quickly, discard the data that they give rise to, re-calculate the mean and control lines and draw a new chart. Discarding data does not mean ignoring the causes! Whichever course is followed, the chart should picture the underlying variability resulting from inherent problems or common disturbances. The task of the work team is now to identify the causes of common disturbances and take action toward their elimination.

7.9 Centring Customers expect the majority of products to be at or close to nom inal and nominal should be the customer's optimum (section 2.5). Except for one-sided distributions of ovality, taper. run-out, etc chart means for x charts should be on nominal or at the middle of the tolerance band when a target is not specified. One-sided distributions are a special case where the mode, rather than the mean. should be on nominal which usually is zero. The mode of a distribution is its most frequently occuring value. Idea lly, zero is the place for means on R, s, p, np, c and u charts.

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Changes, to re-set or centre an off-nominal process must be based only on statistics of the underlying or inherent variability (section 7.8). Re-setting processes, withou t allowi ng for special disturbances, cou ld lead to deterioration in product quality. Centring should not be attempted f rom data of medians because the X: chart mean does not re lat e to nominal (section 5.7).

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I

I

8. Capability Figure 8.1: Capability in relation to specified tolerance (a) A capable process

LSL

I

LSL

(b) An inherently capable process that is not centred I

USL

(c) An incapable process

LSL

I

USL I

LSL and USL are respectively, the lower and upper specification limits

8.1 Capability Statements A process capability statement is an estimate of the ability of the process to meet customers' requirements. The statement should have t wo components: first, a confirmation that the maJority of products are being produced at or close to nominal and second, a confirmation that very few products are being produced outside tolerance limits when these are specified .

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lif4!1.!.1

There are two ways in which the statement is expressed: In some manufacturing industries, indexes such as Cp and Cpk are used (sections 8.2 and 8.3), they require tolerance to be specified. These indexes often provoke questions as to their practical meaning. The answer is that they indicate the probabilities of products being produced away from nominal and/or outside tolerance. In some industries, indexes are not used. Instead, there is only a statement of the probabilities. This simple approach involves more complex calculations than for indexes but it has the advantage that it can be used when tolerance is not specified. Methods are explained in texts referenced in the Bibliography. However the statement is expressed, it is illustrated in figure 8.1 where diagram (a) shows a capable process, diagram (b) shows an inherently capable process but it is not centred and diagram (c) shows an incapable process. The horizontal lines are measurement scales marked with LSL (lower specification limit), USL (upper specification limit) and nomina l at the middle of their range. The bell shaped curve represents the distribution (see section 10.1) of products at points on the measurement scale. In the diagrams, the top of the curve is at the process mean and most products are at or close to the mean.

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Figure 8.2: Capability indexes LSL

USL

I

I I

I

I I

I I I I I

I

+-- - - - - T olerance band - - - - ---+

CAPAB ILITY INDEX (Cp Pp or Cm) =TOLERANC E BAND PROCESS SPR EAD ' where PROCESS SPREAD = 6 deviation)

a (six times the estimated standard

R

is the mean of sample ranges calculated for an R chart (section 5.5).

s

is the mean of sample standard deviations calculated for an s chart (section 5.6). d2 and c4 are constants that depend upon the chart sample size - see Appendix A, page 93.

u

is the Greek lower case letter sigma equivalent to s in English and the symbol · is a circumflex which here means 'estimate of'.

For purposes of this publication, a standard deviation is a measure of process variability and explanation of how to calculate it is sufficient definition. Full statistical definition can be found by reference to texts in the Bibliography (section 13).

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ii#41!.!.1 8.2 Capability Indexes A capability index is the ratio of al lowable variation to actual variation. The allowable variation is the tolerance band. The actual variation is a quantity ca lled the process spread wh ich is calcu lated from sample measurements. The index can vary according to sam ple selection. Therefore it is symbolised as Cm, Pp or Cp to cover the three most common sampling procedures.

Cm indicating

machine, is used when sampling is over a very short period of time. Often, there wi ll be a single material batch, the best operator, the same method, constant atmospheric conditions and new equi pment. lt is hardly a good measure of a process that might be used for years. Usual ly, it is employed to provide some indication of machine performance during acceptance trials.

Pp indicating preliminary, is used when sampli ng is over a period of time that does not capture all common disturbances likely to be suffered by the process, for example natural variations in materials, people, wea ther and equipment. lt provides a basis for future compa risons but aga in , is not a good measure of capability.

Cp indicating process, is based on data obtained over a long period of time and is li kely to be the best obtainable estimate of capability. All versions of the index can be calcu lated from charted data (see figure 8.2) For most practical purposes, the method shown gives a reasonable estimate but it assumes that the data has a normal distribution (see section 10.1). There are other methods (see the Bibliography) , in particular for when a 'snap-shot' statement is required and for when the data is not norma lly distributed (indicated by a skewed, truncated or onesided distribution curve).

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8.3 Setting Indexes For each capability index CP. PP and Cm, there is a setting index designated c pko ppk and c mk respective ly. lt is the lowest of two values designated Zu and ZL which are ca lculated as shown in figure 8.3.

The descript ion of capability indexes in section 8.2 is equal ly applicable to setting indexes but additionally, a customer would be disappointed if the capability and setting indexes were not the same. the setting index can be used as a capability index for one-sided distributions (arising from some measurements of run-out, ovality, taper, etc) provided th e process mode is at zero which is used instead of ~ in the ca lculations below. The mode of a distribution is its most frequently occurring value. Figure 8.3: Setting indexes USL

LSL

I I I I I

I I I

I I

c pko ppk or c mk is the lowest of ZL = ~- LSL and Zu = USL - ~ 3& 3& ~is the process mean ca lcu lated for an chart (section 5.5). a- is calcu lated in the same way as for c p. pp or c m (figure 8.2).

x

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59

4ii!I.!.M The setting indexes Cpk• Ppk and Cmk: cannot be greater than the capability index (Cp. Pp and Cm respective ly). are equal to the capabi lity index when t he process is centred. are negative when the process mean is outside the tolerance band. 8.4 Interpretation of Indexes Most high volume manufacturers would regard CP = 1 or less as indicative of a low capability process. Targets are likely to be for c p = 1.66 or more and there will be the expectation for c pk to be the same as CP.

A few customers use indexes as a specification or criteria for product acceptance. This is hardly an enlightened approach when the uncertainty in quoted values is taken into account. Better is the approach of some suppliers, who would consider that low values warrant special management attention. Their action could be: to recognise an unreasonable tolerance band and get it widened. to change the process. Frequently, the state-of-the-art is not up to the demands of customers and the only answer is to institute 100% measurement fol lowed by selective use. Examples are in the silicon chip, engine component and egg packing industries.

Note: 100% checking does not negate the need for or value of charting. to re-set and centre the process when Cpk is not the same as Cp. When tolerance is not specified, the indexes cannot be used. However, customers still expect to be informed of expected levels of variation so that risks can be evaluated and appropriate action taken. The alternative to a setting index is simply the deviation of the process mean from nominal. This information is of course crucial in any case, to people who have to re-set processes.

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•

I

Instead of a capability index, the information wi ll be how many products are likely to be outside a particular measurement range. The calcu lation for this information usually involves determining the measurement range which will contain all but about one in a million products. In other w ords, the values of the product measu rement distribution at the process mean plus and minus five standard deviations which are sometimes called the ten sigma limits (section 8.5). The next Section shows how indexes can be used to indicate the approximate quantities of products likely to be outside tolerance.

8.5 Estimation of Conforming Products Be warned that the values given below are crude estimates !

There is an assumption that process data has a normal distribution (Section 10.1 ). Also, no account is taken of the effects of inevitable special disturbances . The true situation is likely to be much worse, especially for higher index values. When the process is centred in other words when nominal is at the middle of the tolerance band and the capability index and the setting index are the same.

The capability index indicates the following very approximate probabilities. The capability index indicates the following very approximate Cp. Pp or Cm index Parts per mil lion outside tolerance sometimes cal led

1.00

1.20

1.30

1.33

13000

300

100

<60

Isixsigma limits

eight sigma limits

1.66

I <1

I

ten sigma limits

2.00 <10"5 twe lvesigma limits

Capability indexes are always greater than zero, they are often between 1 and 2 and rarely are larger than 5.

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4'4!1.!.1 When the process is not centred Z values (figure 8.3) indicate the follow ing very approximate proba biliti es, the total quantity outside tolerance is the sum of ppm or % for ZL and Zu.

high positive ZL/Zuvalues

~

ppm below/ above tolerance

1.00 1500

11.20 150

11.30 50

1.33 ~6 -~1!-2_.o_ o _ __ <30 - ~ <10·5

low positive ZL/Zu values

0.6

0.7

% below/ above tolerance

4

2

negative ZL/Zu values

% below/above tolerance

0.8 1 0.9 1

0.4

-0.1 , -0.2 , -0.3 1 -0.4 1-0.5 1 -0.6 1-0.7 1-0.8 1-0.9 73 82 88 93 96 98 99 99.6

I 62

8.6 Example reaction plan following process monitoring Ongoing Process/Product Monrtoring (Interpretation & Reaction) The most recent point indicate s that the process is in control

Action based on Process output based on historical process capability (Cpk) Less than 1.33

1.33- 1.67

Is in control

100% inspect

Accept produ ct continue to reduce variation

Ha s gone out of control in an adverse direction. All individuals in the sample are within specification

Identify and Correct Special Cause

Has gone out of control in and one or more individuals in the sample are outside spec ification

62

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Greater than 1.67

100% inspect

In spect 100% si nc e la st control point

100% in spect

100% inspect product produced since la st in control sample

Ac ce pt product continue to reduc e variation

• I fill !I.!.M

9. Summary of The Process Improvement Stages

7 Continuous improvement Reduced common disturbances

, I I

6 Identify and action opportunities

I

I

I In control, capable and centred

I

I I

I

I

I

I

--1.

[

Suppliers' responsibilities

I I

~ 5 Set the process In control and capable but not centr ed

I

I

I I I

I

I

I 4 Identify and correct problems In control but not capab le specia l disturbances prese nt

I

~_/ ""'--+1----j-----\--""'""---

// 1 I I

3 Identify and correct problems Out of control and not capable specia l disturbances present

~I 2 Gather data and draw a chart

I 1

64

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1 Find out what the customer wants!

I

•mo.!.M 10. Topics Related to Charting 10.1 The Normal Distribution A set of 125 measurements is shown in figure 10.1. A perceptive ind ividua l mig ht detect the highest and lowest values but overa ll , it is a poor picture of the process that produced the measurements. Figure 10.1: A set of measured data

73 53 50 42 55

63 54 66 82 48

58 52 44 65 27

30 87 15 82 99

54 68 58 56 70

72 31 82 72 12

69 29 51 58 44

79 44 61 63 72

53 54 54 49 46

44 43 29 39 61

68 61 38 28 57

59 42 59 58 53

50 61 72 47 54

56 37 80 26 74

60 59 40 60 66

50 40 43 66 28

81 57 26 89 60

62 46 56 55 83

52 35 33 34 41

52 45 68 63 32

33 65 64 38 51

51 58 55 78 59

71 51 57 37 43

18 48 48 64 68

57 50 78 44 52

Simple calculations can give some process statistics. For example, the mean is about 54 and the range is 97, but the picture is not much better. The data could be charted, perha ps as described in section 5.5. Patterns t hat ca n be interpreted w il l appear bu t the interpretation is based upon t he data havi ng a norma l distribution. The normal distribution and a simple check for a normal distribution are described in this section.

Chart patterns develop as data is collected over a period of time . To check the distribution of the data, it is looked at as a whole in a simple diagram such as a bar chart or histogram (figure 10.2).

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65

I IM!!.i,M Figure 10.2: A histogram of the data in figure 10.1

4

"'

.0

~ z

10

20

30

40

50

60

70

80

90

100

Re corde d val ues

The picture shown in figure 10.2 is not very clea r because the horizontal scale of units is too precise for t he small amount of avai lable data. To improve the picture, the data is put into not less than 8 convenient equal sized groups (figure 10.3). In this example, the convenient group size is 10. Figure 10.3: A frequency table of the data in figure 10.1 Group

Limits

Tally chart

Number

%

0

0

0

below 10

1

10 to 19

Ill

3

2.4

2

-3

20 to 29

ifff/1

7

5.6

30 to 39

ifff ifff If

12

9.6

4

40 to 49

ifff ifff ifff ifff I

21

16.8

ifff ifff ifff ifff ifff ifff ifff IIII

39

31.2

23

18.4

11

8.8

5

50 to 59

6

60 to 69

7

-

70 to 79

8

80to89 -

9

90 to 99

10

-

above 99

- r--ifff ifff ifff ifff Ill ifff ifff I r -ifff Ill I

8

6.4

1

0.8

0

0

The bar chart or histogram is now re-drawn using the grouped data and with its horizontal scale changed from units to tens.

66

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-

-

Figure 10.4: A histogram of percentages from figure 10.3

30

20

10

10 Group

A picture has now emerged which shows that the majority of process measurements are at or close to Group 5 (say between 45 and 65) and occurrences toward the extremes are less frequent. Figure 10.4 shows the distribution of the data in figure 10.1. Figure 10.4 represents the distribution of a sample of only 125 measurements. The next step is to estimate the distribution of all results of the process. If there were many more and much more precise measurements. the histogram groups would be smaller and its steps less pronounced. Theoretically, its outline could become smooth as shown in figure 10.5. Figure 10.5: A theoretical distribution of all process measurements

The curve shown in figure 10.5 is that of a normal distribution. it has a characteristic symmetric bell shape. www.smmt. co.uk

67

I

For most practical purposes, any histogram that has a rough ly bell shaped outline can be assumed to represent a normal distribution of data. More refined checks can be found by reference to the Bibliography. There are many theoretical distributions but the normal distribution occurs most frequently in practical situations involving measurements (variables). Also, it usually approximates sufficiently well to distributions of counted data (attributes) for it to be a reasonable basis for most charting methods.

10.2 Introduction to Analytical Methods The need to col lect and use information was emphasised in section 2.2. Often, these activities are subjective and it is helpful to have the disciplined approach that is offered by some analytical methods. The three methods summarised below are of particular use when deciding what to chart and what data to col lect for charting . Texts referenced in the Bibliography offer other possibilities. Brainstorming

Brainstorming is the industrial name for the universal commonsense activities of obta ining the best possible advice and of involving people. lt is a necessary ingredient in decision making and of many much more complex methods, it should be practised at all levels in an organisation. Cause and effect diagrams

Cause and effect diagrams are sometime s called fishbone diagrams (because of their appearance) or lshikawa diagrams (after the Japanese engineer who promoted them).

68

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I

•mu.u; They are a structured way of reducing the multiple possibilities that usual ly arise from brainstorming to manageable proportions.

Pareto analysis The analys is is a way of separating the vital few from the useful many, in other words, of prioritising actions. lt is named after Vi lfredo Pareto, an Italian economist who first suggested the 80:20 rule,

approximately 80% of total fault incidence is accounted for by approximately 20% of fault types. The method requires data in the form of counted numbers and the description in this brochu re focuses on fault incidences. However, the method and the rule are equally applicable to costs, stock-holdings and other problem criteria.

Note· These analytical methods are covered in greater depth in the SMMT publication 'Continual lmprovementTools & Techniques- A Gu ide For Business Improvement' -see inside back cover.

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69

li#41!.!.1

11. Control Charts for Special Situations 11.1 Moving mean charts Charts for moving means are usef ul where machine wea r is inherent in a process and parts, such as tool bits, are periodically replaced or reset.

The chart indicates when parts need to be replaced and is usually more reliable than theoretical fixed change points. The chart is compa ratively complex and hand drawn examples are unusual. The description on page 71 is more applicable to automated machines where measurement, calcula tions and signals are computerised. Figure 11.1: Moving mean charts

Re-set required ...________

/

~

Upper control line IUCL, ) for re-seHing

/

I>; ~

' ~>--8- ---------

"'~

"'E "'

l IX

Lowe r control line ILCL, ) for re-seHing Re-set carried out

- - - - - - - - One process cycle - - - - - - - - 1

70

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/

4iM!I.!.M Initially, data is collected over at least one process cycle, for example between tool resets or replacements, and is plotted in the same way as for a conventiona l x& R chart. The x&R chart is described in section 5.5. 2

Average and con trol lines are drawn for the ranges plot and provided the ranges are relatively stable (see section 72). a bestfit line is drawn through the means plots.

3

Control lines are drawn ~rallel to the best-fit line at vertical distances from it of ± A,R (see figure 11.1). Values of the constant A, are given in Appendix A, page 93.

4

The best-fit and sloping control lines are used for chart interpretation as if they were conventional horizontal mean and control lines. They must be re-positioned on the chart for each process cycle.

5

After severa l process cycles, conventional horizontal control lines are drawn. They are used to monitor the need for re-setting and are positioned at

-

-

UCL x = x + 0.5(xmax -Xminl + A2R LCL x

-

= x- 0.5

-

(xmax -Xminl- A2R

The quantity ( Xmax- Xminl is the average movement of the mean (AMM).

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71

lfu!!.UM 11.2 Charts for Sample Size of One Figu re 11.2: Example of a control chart for sample size of one. 500

~·•

2400

UCL,

1 ---------------~A---------------------------~

"C

"' .2 C>

"

~

X

300

LCL,

200

"'

C>

UCL

~

"=>

100

0

R

'C' ::>

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

Un its

256 302 243 286 281 277 315 422 327 292 281 305 333 294 46 59 43 4 38 107 95 35 11 24 28 39

mean load (x) = 302.40 UCLx =x + 3a =302.40 + 3(35.46) =409. 16 LCLx =x - 3a =302.40 - 3(35.46) = 195.64

mean range (R) =40. 14 UCL, = D,R =3267(4014) = 131.15 a =R/d, =40/1.128 =35.46

This method is applicable when 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 fi gure 11.3, they are the difference between one unit measu rement and the next, which means that there is one less range value

than individual measurements.

72

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4i41!.!.1 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 lower control lines are drawn at the mean± 3a. u can be calculated from the mean range (see 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 fo 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 with much less than 125 measurements. interpretation assumes a normal distribution of data (see section 10.1 ), t hi 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 individua l measurements, the distribution of averages of those measurements wi ll increasingly, approximate to normal as sample size increases.'

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73

lW!I.!.M 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 alternative of a separate chart for each plate would be of little use in monitori ng the process. A solution 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 nomina l ± 3s and s has been calcu lated from the first 25 zeroed values - see Appendix C, page 99. The plots in the il lustrations are of individ ual measurements and therefore the con t 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 sam ples above one, a conventional x&R chart (section 5.5) is used with values that are zeroed sample ~eans (means minus nom inal) and of course, the process mean (x) is zero.

x

Subject to the limitations applying to charts for sa mple size of one (section 11.2), chart interpretation is set out in section 71.

74

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11.3:111ustration of a control chart for short production runs 25

-

18mm unit

-

12mm un it

_,_ 6mm unit

20

E

E

15

c: 0

·v;

c:

.~

10

D

:r

Actual results

16.5 19 17 17.5 16.5 19.5 18 19.5 16.5 19 17.5 18.5 16 20.5 18 18 18 18 18 18 18 18 18 18 18 18 18 18 ·0.5 0.5 ·2 2.5 ·1.5 ·1 ·0.5 ·1.5 1.5 0 1.5 ·1.5

17 20 19 19.5 17 17.5 18.5 19 17 18 18 18 18 18 18 18 18 18 1.5 ·1 ·0.5 0.5 ·1 ·1

11 12.5 8.5 15 12.5 13.5 11 13 13.5 12.5 10.5 13.5 11 .5 12 12 12 12 12 12 12 12 12 12 12 12 12 ·1 0.5 ·3.5 3 0.5 1.5 ·1 1 1.5 0.5 ·1.5 1.5 ·0.5 7.5 ·1 1.5

5.5 ·2

·0.5

4.5 6 ·1 ·1.5 ·1

7.5 5.5 4 1.5 ·0.5 ·2

UCL

~1---------------------------------------------~L~ CL -6 Zeroed results

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75

li5ii.UM

11.4

Standardised Charts Standardised charts are used 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 sta ndardise the people or the facilities. The results from each person or facility are converted onto a scale where the process mean is zero and the control chart LC L and UCL are -3CT and +3CT respective ly. 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. This plotted va lue is known as th e standardi sed 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 midd le 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 values'.

·z

At 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

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4ifii!I.!.M Figure 11 .4: Illustration of a st andardised chart 120

UCL

100

"'

Q;

"""'" "C

~

·c;

z

80 60 40

LCL

20

Combined results

u; Q; .0

·;:;

::s"' ~

·c;

z

40 20

Separated results

~

W

~

%

~

% W % %

~

~

~

H

~

% %

~

~

~

~

% %

~

0.09 0.09 0.09 0.95 ·1.21 ·0.35 0.09 ·0.35 0.95 ·0.35·0.35·0.78 0.09 ·1.21 3.98 0.09 ·0.35 ·0.35 0.52 ·1.21 0.09 0.09 -0.35-1l.35 0.09

6 5

j +-------------------------~~-------------------U~C~L 2

6~

~

-1

~

-2

z

-3 -4 -5 -6

~..LL.~ V . . . --~.,.

17

~

~L

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

Standardised results

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77

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 the British St andards Institut ion . Of particular interest in the sta ndard 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 estimate. 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 with a high level of training. they are not appropriate when variability is an important matter.

78

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Figure 11.5: Illustrations of a cusum chart and a conventional run chart 250

-

E """

200

Q) Q)

;; 150 0;

c. "C

~ 100 ~

·c::

::::J

50 0 Week

10

15

20

25

30

Conventi onal run chart

170 292

118 410

96 506

161 667

139 806

91 897

157 167 151 157 138 150 133 98 106 143 153 137 84 143 124 136 119 1003 1146 1299 1436 1520 1663 1787 1923 2042

119 149 108 116 136 169 182 105 135 94 102 122 155 168 2140 2245 2380 2474 2576 2698 2853 3021

131 205 142 187 117 191 128 179 174 118 173 3138 3329 3457 3636 3810 3928 4101

21 1 197 4298

300 Indications of mean level relative to target

200

Horizontal on target

Slope down below target

Slope up above target

r=

k-

100

H"

0

- 100+-ro-r.-ro-r.-ro-r.-ro-ro-ro-ro-ro-..-,-,-,,-,.-, Week 10 15 25 20 30 Cusum chart

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79

4ii!I.UI

12. Capability Estimations Capability is a measure of how well customers' requirements are met. The topi c is explained more fully in sections 8.1 to 8 .5 .

Figure 12.1: llustration of a probability plot

This illustration shows a straight best-fit line and values at points where the best-fit line intersects w ith other lines

INTERSECTION WITH

- 50"

LSL = 0

VALUE

- 26

0.05%

Lf% =50

These va lues can be used to ca lcul ate a 'sna p-shot' estimate of capabi lity (see section 12.3) -

Tally chart

Class I

90 80 70 60 50 40 30 20 10

1 99 1 89 1 79 I 69 I 59 I 49 I 39 I 29 19

I

I HH HH HH HH

I

I

I

I

I

I

I

I I I Ill I

I I

I I

I I

I I

I I

I HH I I I I I I I I HH I HH I HH I Ill I I I I HH I HH I HH I HH I HH I HH I ////

HH I HH I HH I HH I I I HH I HH I 11 I I I HH I 1/ I

I I

I I

I I

I

I

I

Ill

I

I

I

I

I

I

I I

I

I

I

I

I

I

I

I

1 8 11 23 39 21 12 7 3

Li

125 124 116 105 82 43 22 10 3

Li%

100 99.2 92.8 84.0 65.6 34.4 17.6 8.0 2.4

x. 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 - 10 - 20 -30 -

-

lo-'"" rr- 5cr

80

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99.~

---26

0.0

•m!!.!.M

ISL; 100

t5
0.2%

134

t51J

__j_ _J__ _J__

- 0.2 99 87 99 5 99 98

95

90

80 70 60 50 40 30

20

I I I

I I I I ~I"'

f.-" 1- 1-

I

--

1--

1005

10

--

1o13 I I I _I_

-.

k--"':'"

--

0 003

I' '

.....-

_ I_

-:::.+- - 134 I

USL

I

I I

- -· ----- - - - - - i-I ·

I

- 54

I

I I

LSL I

0.13

0.5 1.0

10

20 30 40 50 60 70

x

80

90

95

98 99 99.5 99.87

99.997

Average of the two high est classes (see section 12.11 Lf% ; (100 t 99.2)/2; 99.6 Xu ;

(100 + 90)/2; 95

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81

IW!I.!.M 12.1 Probability Plots Estimates of capability need data whose 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 few 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 section 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 13). A data set 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-fit 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 probabil 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. The normal distribution is explai ned on page 65.

82

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Figure 12.2: Distribution Information from Probabilit y 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 C Truncated distribution A diagonal line that bends to the vertical, from data with missing high values such as from a sma ll grade batch

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 E Skewed distribution A smooth curve from data where mean, mode and median are different. See pages 86 to 89

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83

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 lar distribution (section 10.1) non normal means other distributions, it does not mean abnormal.

12.3 Snap-Shot Capability Estimations Apart from giving a sim ple 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 time. 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 respectively 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 lower case letter sigma equivalent to s in English, here it

signifies a standard deviation, process spread is six standa rd 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.

84

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12.4 Capability Estimations for non normal Distributions Figure 12.3:111ustration of a distribution truncated at zero Mode

Mean

- - - - --

3sta ndard deviations - - - - - -

~-------- P rocess

spre a d - - - - - - - - - +

If prel imina ry work ind icates t hat 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 t he pola rity of measurements, fo r example, the so called 'one-sided' distributions described on page 86. not reporting results above or below particular values.

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85

iffi!I.!.M

reporting results beyond th e precision of the measurement method . having differing standards of measurement, for example, from sh ift to shift. reporting combined results off 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, Process Spread is three standard deviations plus the width zero to the mode. When the mode of a distribution is at zero its Process 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 in t he approximately normal t ail are used t o calculate t he standard deviation.

86

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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 technique s are described in various academic texts (see Betteley et al and others referenced in the Bibliography).

Amongst them is the use of probability papers 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 lin es/best-fit 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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87

fMii.UM Figure 12.4: Illustration of a paper used for extreme skew distributions

(also see AppendixJ, page 124)

100 - 99 - -

99.99 99.87 99

70

50

-

-- -

20

30

10

I

97 96

90

I

98

:

------

95

- +-

-- -

-

·-

-

- - - - - - r-- - - - - - - - - -

I I

94 93

I I

92

_.........

91

I

90

:

89

88

:I

87 86

85 84

I

--- -- - ~

83

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13. Bibliography The terms and symbols in this guide are widely accepted in man ufacturing industry. However, readers should note 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 0- 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, ISB N 3- 446- 19572- 6 Explains how to manage the acceptance of measurement systems and production facilities as well as 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 of process capability and the main control charts. Walpole,RF and Myers,RH (1993) Probability and Statistics for Engineers and Scientists, 5th edition, New York: Macmillan A brief account of the main types of control chart. Grant, EL and Leavenworth, RS (1988) Statistical Quality Control, 6th edition, NewYork: McGraw Hi ll Technical details of the main types of control chart. Montgomery, DC (1985) Introduction to Statistical Quality Control, New York: Wiley Detailed treatment of process capability and the main control charts. Mitra,A (1993) Fundamenta ls of Quality Control and Improvement, NewYork: M acMillan Detailed treatment of process capability

90

www.smmt.co.uk

International Organisation for Standardisation publications available through British Standards Institution, 389 Chiswick High Road, London, W4 4AL ISO 7870 Control charts - General guide and int roduction ISO 8258 Shewhart control charts Related publications available from SMMT - see inside back cover

www.smmt.co.uk

91

14. Appendices A Constants for variables control charts

93

Control chart forms reduced from A3 size. A worked example is shown on facing pages for each form.

B Mean and range process control chart C Mean and standard deviation process control chart D

Median and range process control chart E p chart for proportion of detectives F np chart for number of 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

92

www.smmt.co.uk

94 99

102 106 110 114

118 122 124

Appendix A - Constants for Variables Control Charts

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A2 1.880 1.023 0.729 0.577 0.483 0.419 0.373 0.337 0.308 0.285 0.266 0.249 0.235 0.223 0.212 0.203 0.194 0.187 0.180 0.173 0.167 0.162 0.157 0.153

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1.880 1.187 0.796 0.691 0.548 0.508 0.433 0.412 0.362

2.659 1.954 1.628 1.427 1.287 1.182 1.099 1.032 0.975 0.927 0.886 0.850 0.817 0.789 0.763 0.739 0.718 0.698 0.680 0.663 0.647 0.633 0.619 0.606

0 0.030 0.11 8 0.185 0.239 0.284 0.321 0.354 0.382 0.406 0.428 0.448 0.466 0.482 0.497 0.510 0.523 0.534 0.545 0.555 0.565

3.267 2.568 2.266 2.089 1.970 1.882 1.815 1.761 1.716 1.679 1.646 1.618 1.594 1.572 1.552 1.534 1.518 1.503 1.490 1.477 1.466 1.455 1.445 1.435

0.798 0.886 0.921 0.940 0.952 0.959 0.965 0.969 0.973 0.975 0.978 0.979 0.981 0.982 0.984 0.985 0.985 0.986 0.987 0.988 0.988 0.989 0.989 0.990

1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078 3.173 3.258 3.336 3.407 3.472 3.532 3.588 3.640 3.689 3.735 3.778 3.819 3.858 3.895 3.931

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4§!1.!.1 Appendix C - Mean and Standard Deviation Process Control Charts

The appearance of a mean and standard deviation process control chart is similar to the mean and range chart shown in Appendix B. lt can be hand drawn, using the form on page 100 when sample sizes are smal l, but it is best used when sample sizes are 25 or more units and computer programmes are available for calculation and drawing. To calculate s (the sample standard deviation)

either

enter the sample un it values into a pocket calculator and press the u(n-1) or equivalent key.

or

use the formula

S=

lx,- R)2 + lx2 - R)2 + lx 3 - )()2 + etc up to lxn -R) n-1

where x1, x2,x3, etc are the values of units in the sample

Do not confuse s (sample standard deviation) with u (population standard deviation)!

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Appendix H - c Chart for Number of Defects I

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/'.r M! '

I

Draw LCL at zero when th1s calculatiOn giVes a negat1ve result

>-

'-'

z

UJ

:::> d

UJ

a:

f---1---

....

0

a:

1-

z

0

(.)

en en

UJ

(.)

V)

~

w

0

a: 0..

Q> c:

E

.,"

~

o;

"C

:;;.

Q;

E ..cE

c: E ., );; E "'Q; .:;:, = ..::. .!'!? .s c., 1c c 'E

... Q;

a:

c( :1:

-"' Q.

~ g 0

·~ 0

:c.,

Q.

.,"'"'

" " " .... et

(.)

.

.0

0

0

a. E 0

(.)

..

c

E .5! en
E :.;;

s., .," :!::.," .... .::: ena. (.)

....,. E

en

(afdWeS JO Sj/Ufl 1/e U/ S~ajap JO Jaqwnu)

120

www.smmt.co.uk

~

SHIFT DATE

TIME

BY c (faults)

Mean of c values= c

I

I

Upper control line = c+ 3

F c-

lower control line = c- 3\/ c

~

3

a<> 0

~

~

~'---------'

I .

I

Draw LCL at zero when th1s calculatiOn . g1ves a negative result

Appendix I - Normal Probability Paper

~

CAPABLITV ASSESSMENT

~

f---f----

for feature with normal distribution

f-----xu is the class upper boundary

Tally chart

Class

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

rr-

f------

-

Lf%

f-

I I

I I

-

f---f----

~I

I

ff-

I

f---- t--

f------ I -

f---- t-f---- I -

f------ I -

f---- t-f---- I L _ _-

I- 5rr

MEASURED VALUES

122

1

6

11

16

21

26

31

36

41

46

51

56

61

66

71

76

81

86

91

96

101

106

111

116

121

2

7

12

17

22

27

32

37

42

47

52

57

62

67

72

JJ

82

87

92

97

102

107

112

117

122

3

8

13

18

23

28

33

38

43

48

53

58

63

68

73

78

83

BB

93

98

103

108

113

118

123

4

9

14

19

24

29

34

39

44

49

54

59

64

69

74

79

84

89

94

99

104

109

114

119

124

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

105

110

115

120

125

www.smmt.co.uk

4Mii.UM

REPORT

Date

-

Delete as appropriate CAPABLE

I

NOTCAPABLE

I

SffiiNG ON NOMINAL

I

SETIING OFF NOMINAL

9.997

99.87 99.5 99 98

95

90

80 70 60 50 40 30

20

10

5

80

90

95

2 1.0 0.5

0.003 -

0.13

t5rr

-

-

-

I I

I

I .003

0.13

0.5 1.0 2

10

20 30 40 50 60 70

98 99 99.5 99.87

99.997

l:l% INFORMATION SUMMARY Upper spe cification limit

u

Nomin al

N L A B

Lower specifica tion limit Xu at

line/ +50' intersection

Xu at line/-5a intersection

Difference I= A - B) rr estimate (=C/10) Tolerance band(= U- L) Process sprea d(= 6rr) Caoabilitv index(= Ti P) Process mean Pro cess settin Q(= x- NI % above specification

c rr T

p

c X

% below specification

www.smmt.co.uk

123

Appendix J - Probability Paper for Extreme Skew Distribution

CAPABLITY ASSESSMENT

----.;;-

far feawre with normal distribution

_-

xu is the class upper boundary Lf%

l:l

I

Ta lly chart

Cl ass

--=--- ff__ - f_ - f_ - f- f-

r-

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---

r--r--r--r--MEASURED VALUES I

124

6

11

16

21

26

31

36

41

46

51

56

61

66

71

76

81

86

91

96

101

106

111

116

121

2

7

12

17

22

27

32

37

42

47

52

57

62

67

72

77

82

87

92

97

102

107

112

117

12i

3

8

13

18

23

28

33

38

43

48

53

58

63

68

73

78

83

88

93

98

103

108

113

118

12:

4

9

14

19

24

29

34

39

44

49

54

59

64

69

74

79

84

89

94

99

104

109

114

119

12•

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

105

110

115

120

m

www.smmt.co.uk

4ff4!1.!.1

REPORT

Location (geography) Process (or operation) Equipment (or machine)

Produ ct (or component) Feature Performed by

Date

Delete as appropriate

CAPABLE

NOT CAPABLE

SETTI NG OFF NOMINAL

SETTING ON NO MINAL

-t--t--t-

-r

I

_L 99

9987

99

90

70

50

30

20

05

10

or-olr

0 13

0 05

0 01

99.95

99.99

I I I

I

I I

I

I

I !0- 1

----L

-l-1-

10

30

50

70

80

90

95

97 98

99

99.5 99.7 99.8

99.87

l:f%

-+-

-t-----J-

--r0.13

INFORMATION SUMMARY

Up pe r specification limit

u

Nominal

N

Lower spec ifi catio n limit

L

Xu at

A

line/99.87 pe rcentile Xu at line/0.13 pe rcentil e Process spread I= A- B) Tolerance band I= U- L) Capability index I= T/PI Proc ess mode

B p

T C, X

Process setting I= x- NI % above specifi ca tion % below spec ifi cation

www.smmt.co.uk

125

15. Subject Index Topics and Terms

AMM assignable causes attributes charts average movement of the mean c chart capability estimation (snap-shot) capa bility indexes capability index interpretation centring chance causes chart design chart for moving mean chart for sample size of one chart for small batch runs chart pattern interpretation chart pattern chance occurrence chart scales charting purpose charting strategy charting summary Cm, Cp and Pp Cmk, Cpk and Ppk co-ordinators common disturbances control lines cusum chart

126

www.smmt.co.uk

Page

71 11 28, 42 71 43, 118 84 85 60 53 11 28 71

72 74 44 52 29 25 21 23 58 59 20 11 30 78

Topics and Terms

Page

customers 9 detectives charts 40 defects charts 40 distribution 65 disturbances 11 disturbance elimination 52 executive role 20 expectation 12 facilitators 20 fact -holders 20 frequency table (illustrated) 66 histogram (illustrated) 66, 67 individuals and moving range chart 72 lim its 12, 60 management ro le 20 mean and range chart 35, 94 mean and standard deviation chart 37, 99 mean 14 median and range chart 38, 102 mode 14 nominal 12 non normal distribution 85 non normal process spread 87 normal 84 normal distribution (check for) 67 np chart 43, 110

Topics and Terms

one-sided distribution optimum p chart performance limits probability paper probability plot interpretation probability plots process capa bility process control process elements process spread R R (R bar) range s (s bar) a (lower case Greek sigma) (sigma circumflex) sample size for attributes sample size for variables sample size of one charts sampl ing of attributes sample of variables setting setting indexes short production run chart

s

a

Page

86 12 42, 106 12 82 84 82 55 22 22 84 35 36 35 37 37 57 57 40 32 72

40 32 53 59 74

Page

Topics and Terms

sigma limits skewed distrib ution special disturbances specification limits standard deviation of process standard deviation of sa mple standard tolerances standardised chart standardised deviate statistical control suppliers tally chart (illustrated) targets tolerance truncated distribution u chart variables charts variables charts constants work-teams (x bar) ~ (x double bar) (x wavy bar or x tilde) (x bar wavy bar) Z values Z values interpretation

x

x x

61 83 11 12 57 99 13 76 76 11

9 66 12 12 85 43, 114 28, 32 93 20 35 36 38 39 59 62

www.smmt.co.uk

127

Other publications available from the SMMT Continual Improvement Tools & Techniques A Guide For Business Improvement Process Management 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 Motor Manufacturers & Traders Ltd, Forbes House, Halkin Street, London SW1 X 70S

Tel +44 (0)20 7344 1612/16 11 Fax +44 (0)20 7344 1603 e-mail: [email protected]

Society of Motor Manufacturers and Traders Limited Forbes House, Ha lkin Street, London SWlX 70S Telephone 020 7235 7000 Fax: 020 7235 7112