Statistical Process Control 5 Part 4 pptx

Variables and process variation 93number of individual readings would, therefore, be necessary before such achange was confirmed.The distribution of sample means reveals the change much

92 Variables and process variation

and when n = 4, SE = /2, i.e half the spread of the parent distribution of

individual items SE has the same characteristics as any standard deviation,and normal tables may be used to evaluate probabilities related to thedistribution of sample averages We call it by a different name to avoidconfusion with the population standard deviation

The smaller spread of the distribution of sample averages provides thebasis for a useful means of detecting changes in processes Any change inthe process mean, unless it is extremely large, will be difficult to detectfrom individual results alone The reason can be seen in Figure 5.5a, whichshows the parent distributions for two periods in a paint filling processbetween which the average has risen from 1000 ml to 1012 ml The shadedportion is common to both process distributions and, if a volume estimateoccurs in the shaded portion, say at 1010 ml, it could suggest either avolume above the average from the distribution centred at 1000 ml, or oneslightly below the average from the distribution centred at 1012 ml A large

Figure 5.5 Effect of a shift in average fill level on individuals and sample means Spread of sample means is much less than spread of individuals

Variables and process variation 93number of individual readings would, therefore, be necessary before such achange was confirmed.

The distribution of sample means reveals the change much more quickly,the overlap of the distributions for such a change being much smaller (Figure5.5b) A sample mean of 1010 ml would almost certainly not come from thedistribution centred at 1000 ml Therefore, on a chart for sample means,plotted against time, the change in level would be revealed almostimmediately For this reason sample means rather than individual values areused, where possible and appropriate, to control the centring of processes

The Central Limit Theorem

What happens when the measurements of the individual items are notdistributed normally? A very important piece of theory in statistical process

control is the central limit theorem This states that if we draw samples of size

n, from a population with a mean µ and a standard deviation , then as n

increases in size, the distribution of sample means approaches a normaldistribution with a mean µ and a standard error of the means of /n This

tells us that, even if the individual values are not normally distributed, thedistribution of the means will tend to have a normal distribution, and the largerthe sample size the greater will be this tendency It also tells us that the

Grand or Process Mean X will be a very good estimate of the true mean of

the population µ

Even if n is as small as 4 and the population is not normally distributed, the

distribution of sample means will be very close to normal This may beillustrated by sketching the distributions of averages of 1000 samples of sizefour taken from each of two boxes of strips of paper, one box containing arectangular distribution of lengths, and the other a triangular distribution(Figure 5.6) The mathematical proof of the Central Limit Theorem is beyondthe scope of this book The reader may perform the appropriate experimentalwork if (s)he requires further evidence The main point is that, when samples

of size n = 4 or more are taken from a process which is stable, we can assume that the distribution of the sample means X will be very nearly normal, even

if the parent population is not normally distributed This provides a soundbasis for the Mean Control Chart which, as mentioned in Chapter 4, has

decision ‘zones’ based on predetermined control limits The setting of these

will be explained in the next chapter

The Range Chart is very similar to the mean chart, the range of eachsample being plotted over time and compared to predetermined limits Thedevelopment of a more serious fault than incorrect or changed centring canlead to the situation illustrated in Figure 5.7, where the process collapsesfrom form A to form B, perhaps due to a change in the variation ofmaterial The ranges of the samples from B will have higher values than

94 Variables and process variation

Figure 5.6 The distribution of sample means from rectangular and triangular universes

Figure 5.7 Increase in spread of a process

Variables and process variation 95ranges in samples taken from A A range chart should be plotted inconjunction with the mean chart.

Rational subgrouping of data

We have seen that a subgroup or sample is a small set of observations on aprocess parameter or its output, taken together in time The two major problemswith regard to choosing a subgroup relate to its size and the frequency ofsampling The smaller the subgroup, the less opportunity there is for variationwithin it, but the larger the sample size the narrower the distribution of themeans, and the more sensitive they become to detecting change

A rational subgroup is a sample of items or measurements selected in a waythat minimizes variation among the items or results in the sample, andmaximizes the opportunity for detecting variation between the samples With

a rational subgroup, assignable or special causes of variation are not likely to

be present, but all of the effects of the random or common causes are likely

to be shown Generally, subgroups should be selected to keep the chance fordifferences within the group to a minimum, and yet maximize the chance forthe subgroups to differ from one another

The most common basis for subgrouping is the order of output orproduction When control charts are to be used, great care must be taken in theselection of the subgroups, their frequency and size It would not make sense,for example, to take as a subgroup the chronologically ordered output from anarbitrarily selected period of time, especially if this overlapped two or moreshifts, or a change over from one grade of product to another, or four differentmachines A difference in shifts, grades or machines may be an assignable

cause that may not be detected by the variation between samples, if irrational

subgrouping has been used

An important consideration in the selection of subgroups is the type ofprocess – one-off, short run, batch or continuous flow, and the type of dataavailable This will be considered further in Chapter 7, but at this stage it isclear that, in any type of process control charting system, nothing is moreimportant than the careful selection of subgroups

Chapter highlights

 There are three main measures of the central value of a distribution(accuracy) These are the mean µ (the average value), the median (themiddle value), the mode (the most common value) For symmetricaldistributions the values for mean, median and mode are identical Forasymmetric or skewed distributions, the approximate relationship is mean– mode = 3 (mean–median)

96 Variables and process variation

 There are two main measures of the spread of a distribution of values(precision) These are the range (the highest minus the lowest) and thestandard deviation  The range is limited in use but it is easy tounderstand The standard deviation gives a more accurate measure ofspread, but is less well understood

 Continuous variables usually form a normal or symmetrical distribution.The normal distribution is explained by using the scale of the standarddeviation around the mean Using the normal distribution, the proportionfalling in the ‘tail’ may be used to assess process capability or the amountout-of-specification, or to set targets

 A failure to understand and manage variation often leads to unjustifiedchanges to the centring of processes, which results in an unnecessaryincrease in the amount of variation

 Variation of the mean values of samples will show less scatter thanindividual results The Central Limit Theorem gives the relationshipbetween standard deviation (), sample size (n), and Standard Error ofMeans (SE) as SE = /n.

 The grouping of data results in an increased sensitivity to the detection ofchange, which is the basis of the mean chart

 The range chart may be used to check and control variation

 The choice of sample size is vital to the control chart system and depends

on the process under consideration

References

Besterfield, D (2000) Quality Control, 6th Edn, Prentice Hall, Englewood Cliffs NJ, USA Pyzdek, T (1990) Pyzdek’s Guide to SPC, Vol One: Fundamentals, ASQC Quality Press,

Milwaukee WI, USA.

Shewhart, W.A (1931 – 50th Anniversary Commemorative Reissue 1980) Economic Control of Quality of Manufactured Product, D Van Nostrand, New York, USA.

Wheeler, D.J and Chambers, D.S (1992) Understanding Statistical Process Control, 2nd Edn,

SPC Press, Knoxville TN, USA.

Variables and process variation 97

3 A bottle filling machine is being used to fill 150 ml bottles of a shampoo.The actual bottles will hold 156 ml The machine has been set to discharge

an average of 152 ml It is known that the actual amounts discharged follow

a normal distribution with a standard deviation of 2 ml

(a) What proportion of the bottles overflow?

(b) The overflow of bottles causes considerable problems and it hastherefore been suggested that the average discharge should be reduced

to 151 ml In order to meet the weights and measures regulations,however, not more than 1 in 40 bottles, on average, must contain lessthan 146 ml Will the weights and measures regulations be contravened

by the proposed changes?

You will need to consult Appendix A to answer these questions

4 State the Central Limit Theorem and explain how it is used in statisticalprocess control

5 To: International Chemicals Supplier

From: Senior Buyer, Perpexed Plastics Ltd

Subject: MFR Values of Polyglyptalene

As promised, I have now completed the examination of our delivery recordsand have verified that the values we discussed were not in fact inchronological order They were simply recorded from a bundle ofCertificates of Analysis held in our Quality Records File I have checked,however, that the bundle did represent all the daily deliveries made by ICSsince you started to supply in October last year

Using your own lot identification system I have put them into sequence asmanufactured:

98 Variables and process variation

37) 3.238) 3.439) 3.540) 3.041) 3.442) 3.543) 3.644) 3.045) 3.146) 3.447) 3.148) 3.6

49) 3.350) 3.351) 3.452) 3.453) 3.354) 3.255) 3.456) 3.357) 3.658) 3.159) 3.460) 3.5

61) 3.262) 3.763) 3.364) 3.1

I hope you can make use of this information

Analyse the above data and report on the meaning of this information

Worked examples using the normal distribution

1 Estimating proportion defective produced

In manufacturing it is frequently necessary to estimate the proportion ofproduct produced outside the tolerance limits, when a process is not capable

of meeting the requirements The method to be used is illustrated in thefollowing example: 100 units were taken from a margarine packaging unitwhich was ‘in statistical control’ or stable The packets of margarine were

weighed and the mean weight, X = 255 g, the standard deviation,  = 4.73 g.

If the product specification demanded a weight of 250 ± 10 g, how much

of the output of the packaging process would lie outside the tolerancezone?

Figure 5.8 Determination of proportion defective produced

Variables and process variation 99The situation is represented in Figure 5.8 Since the characteristics of thenormal distribution are measured in units of standard deviations, we must firstconvert the distance between the process mean and the Upper SpecificationLimit (USL) into  units This is done as follows:

Z = (USL – X)/,

where USL = Upper Specification Limit

X = Estimated Process Mean

 = Estimated Process Standard Deviation

Z = Number of standard deviations between USL and X

(termed the standardized normal variate)

Hence, Z = (260 – 255)/4.73 = 1.057 Using the Table of Proportion Under theNormal Curve in Appendix A, it is possible to determine that the proportion

of packages lying outside the USL was 0.145 or 14.5 per cent There are twocontributory causes for this high level of rejects:

(i) the setting of the process, which should be centred at 250 g and not 255 g,and

(ii) the spread of the process

If the process were centred at 250 g, and with the same spread, one maycalculate using the above method the proportion of product which wouldthen lie outside the tolerance band With a properly centred process, thedistance between both the specification limits and the process mean would

be 10 g So:

Z = (USL – X )/ = (X – LSL)/ = 10/4.73 = 2.11.

Using this value of Z and the table in Appendix A the proportion lying outsideeach specification limit would be 0.0175 Therefore, a total of 3.5 per cent ofproduct would be outside the tolerance band, even if the process mean wasadjusted to the correct target weight

2 Setting targets

(a) It is still common in some industries to specify an Acceptable QualityLevel (AQL) – this is the proportion or percentage of product that theproducer/customer is prepared to accept outside the tolerance band Thecharacteristics of the normal distribution may be used to determine thetarget maximum standard deviation, when the target mean and AQL are

100 Variables and process variation

specified For example, if the tolerance band for a filling process is 5 mland an AQL of 2.5 per cent is specified, then for a centred process:

Z = (USL – X )/  = (X – LSL)/ and (USL – X ) = (X – LSL) = 5/2 = 2.5 ml.

We now need to know at what value of Z we will find (2.5%/2) under thetail – this is a proportion of 0.0125, and from Appendix A this is theproportion when Z = 2.24 So rewriting the above equation we have:

max = (USL – X )/Z = 2.5/2.24 = 1.12 ml.

In order to meet the specified tolerance band of 5 ml and an AQL of 2.5per cent, we need a standard deviation, measured on the products, of atmost 1.12 ml

(b) Consider a paint manufacturer who is filling nominal one-litre cans withpaint The quantity of paint in the cans varies according to the normaldistribution with a standard deviation of 2 ml If the stated minimumquality in any can is 1000 ml, what quantity must be put into the cans onaverage in order to ensure that the risk of underfill is 1 in 40?

1 in 40 in this case is the same as an AQL of 2.5 per cent or aprobability of non-conforming output of 0.025 – the specification is one-sided The 1 in 40 line must be set at 1000 ml From Appendix A thisprobability occurs at a value for Z of 1.96 So 1000 ml must be 1.96below the average quantity The process mean must be set at:

(1000 + 1.96) ml = 1000 + (1.96  2) ml

= 1004 mlThis is illustrated in Figure 5.9

A special type of graph paper, normal probability paper, which is alsodescribed in Appendix A, can be of great assistance to the specialist inhandling normally distributed data

3 Setting targets

A bagging line fills plastic bags with polyethylene pellets which areautomatically heat-sealed and packed in layers on a pallet SPC charting of

Variables and process variation 101

the bag weights by packaging personnel has shown a standard deviation of

20 g Assume the weights vary according to a normal distribution If thestated minimum quantity in one bag is 25 kg what must the averagequantity of resin put in a bag be if the risk for underfilling is to be aboutone chance in 250?

The 1 in 250 (4 out of 1000 = 0.0040) line must be set at 25 000 g FromAppendix A, Average – 2.65 = 25 000 g Thus, the average target should be

25 000 + (2.65  20) g = 25 053 g = 25.053 kg (see Figure 5.10)

Figure 5.9 Setting target fill quantity in paint process

Figure 5.10 Target setting for the pellet bagging process

Part 3

Process Control

6 Process control using variables

Objectives

 To introduce the use of mean and range charts for the control of processaccuracy and precision for variables

 To provide the method by which process control limits may be calculated

 To set out the steps in assessing process stability and capability

 To examine the use of mean and range charts in the real-time control ofprocesses

 To look at alternative ways of calculating and using control charts limits

6.1 Means, ranges and charts

To control a process using variable data, it is necessary to keep a check on thecurrent state of the accuracy (central tendency) and precision (spread) of thedistribution of the data This may be achieved with the aid of control charts.All too often processes are adjusted on the basis of a single result or

measurement (n = 1), a practice which can increase the apparent variability.

As pointed out in Chapter 4, a control chart is like a traffic signal, theoperation of which is based on evidence from process samples taken atrandom intervals A green light is given when the process should be allowed

to run without adjustment, only random or common causes of variation beingpresent The equivalent of an amber light appears when trouble is possible.The red light shows that there is practically no doubt that assignable or specialcauses of variation have been introduced; the process has wandered.Clearly, such a scheme can be introduced only when the process is ‘instatistical control’, i.e is not changing its characteristics of average andspread When interpreting the behaviour of a whole population from a sample,often small and typically less than 10, there is a risk of error It is important

to know the size of such a risk

The American Shewhart was credited with the invention of control chartsfor variable and attribute data in the 1920s, at the Bell Telephone Laboratories,

106 Process control using variables

and the term ‘Shewhart charts’ is in common use The most frequently usedcharts for variables are Mean and Range Charts which are used together.There are, however, other control charts for special applications to variablesdata These are dealt with in Chapter 7 Control charts for attributes data are

to be found in Chapter 8

We have seen in Chapter 5 that with variable parameters, to distinguishbetween and control for accuracy and precision, it is advisable to group

results, and a sample size of n = 4 or more is preferred This provides an

increased sensitivity with which we can detect changes of the mean of theprocess and take suitable corrective action

Is the process in control?

The operation of control charts for sample mean and range to detect the state ofcontrol of a process proceeds as follows Periodically, samples of a given size(e.g four steel rods, five tins of paint, eight tablets, four delivery times) aretaken from the process at reasonable intervals, when it is believed to be stable orin-control and adjustments are not being made The variable (length, volume,weight, time, etc.) is measured for each item of the sample and the sample meanand range recorded on a chart, the layout of which resembles Figure 6.1 Thelayout of the chart makes sure the following information is presented:

 chart identification;

 any specification;

 statistical data;

 data collected or observed;

 sample means and ranges;

 plot of the sample mean values;

 plot of the sample range values

The grouped data on steel rod lengths from Table 5.1 have been plotted onmean and range charts, without any statistical calculations being performed, inFigure 6.2 Such a chart should be examined for any ‘fliers’, for which, at thisstage, only the data itself and the calculations should be checked The samplemeans and ranges are not constant; they vary a little about an average value

Is this amount of variation acceptable or not? Clearly we need an indication

of what is acceptable, against which to judge the sample results

Mean chart

We have seen in Chapter 5 that if the process is stable, we expect most of the

individual results to lie within the range X ± 3 Moreover, if we are sampling from a stable process most of the sample means will lie within the range X ±

3SE Figure 6.3 shows the principle of the mean control chart where we have

Figure 6.1 Layout of mean and range charts

108 Process control using variables

turned the distribution ‘bell’ onto its side and extrapolated the ± 2SE and

± 3SE lines as well as the Grand or Process Mean line We can use this toassess the degree of variation of the 25 estimates of the mean rod lengths,taken over a period of supposed stability This can be used as the ‘template’

to decide whether the means are varying by an expected or unexpectedamount, judged against the known degree of random variation We can alsoplan to use this in a control sense to estimate whether the means have moved

by an amount sufficient to require us to make a change to the process

If the process is running satisfactorily, we expect from our knowledge of thenormal distribution that more than 99 per cent of the means of successivesamples will lie between the lines marked Upper Action and Lower Action.These are set at a distance equal to 3SE on either side of the mean The chance

of a point falling outside either of these lines is approximately 1 in 1000,unless the process has altered during the sampling period

Figure 6.2 Mean and range chart