Statistical Process Control 5 Part 6 pot

Other types of control charts for variables 163The advantage of using sample medians over sample means is that the formerare very easy to find, particularly for odd sample sizes where th

Figure 7.4 Median chart for herbicide batch moisture content

Other types of control charts for variables 163

The advantage of using sample medians over sample means is that the formerare very easy to find, particularly for odd sample sizes where the method ofcircling the individual item values on a chart is used No arithmetic isinvolved The main disadvantage, however, is that the median does not takeaccount of the extent of the extreme values – the highest and lowest Thus, themedians of the two samples below are identical, even though the spread ofresults is obviously different The sample means take account of thisdifference and provide a better measure of the central tendency

This failure of the median to give weight to the extreme values can be anadvantage in situations where ‘outliers’ – item measurements with unusuallyhigh or low values – are to be treated with suspicion

A technique similar to the median chart is the chart for mid-range The

middle of the range of a sample may be determined by calculating the average

of the highest and lowest values The mid-range (M) of the sample of five,~

of sample ranges and the control chart limits are calculated in a similar fashion

to those for the median chart

each rod Such multiple variation may be represented on the multi-vari chart.

In the multi-vari chart, the specification tolerances are used as controllimits Sample sizes of three to five are commonly used and the results areplotted in the form of vertical lines joining the highest and lowest values in thesample, thereby representing the sample range An example of such a chartused in the control of a heat treatment process is shown in Figure 7.5a The

Figure 7.5 Multi-vari charts

Other types of control charts for variables 165

longer the lines, the more variation exists within the sample The chart showsdramatically the effect of an adjustment, or elimination or reduction of onemajor cause of variation

The technique may be used to show within piece or batch, piece to piece,

or batch to batch variation Detection of trends or drift is also possible Figure7.5b illustrates all these applications in the measurement of piston diameters.The first part of the chart shows that the variation within each piston is verysimilar and relatively high The middle section shows piece to piece variation

to be high but a relatively small variation within each piston The last section

of the chart is clearly showing a trend of increasing diameter, with littlevariation within each piece

One application of the multi-vari chart in the mechanical engineering,automotive and process industries is for trouble-shooting of variation caused

by the position of equipment or tooling used in the production of similar parts,for example a multi-spindle automatic lathe, parts fitted to the same mandrel,multi-impression moulds or dies, parts held in string-milling fixtures Use ofmulti-vari charts for parts produced from particular, identifiable spindles orpositions can lead to the detection of the cause of faulty components and parts.Figure 7.5c shows how this can be applied to the control of ovality on aneight-spindle automatic lathe

7.4 Moving mean, moving range, and exponentially

weighted moving average (EWMA) charts

As we have seen in Chapter 6, assessing changes in the average value and thescatter of grouped results – reflections of the centring of the process and thespread – is often used to understand process variation due to common causesand detect special causes This applies to all processes, including batch,continuous and commercial

When only one result is available at the conclusion of a batch process orwhen an isolated estimate is obtained of an important measure on aninfrequent basis, however, one cannot simply ignore the result until more dataare available with which to form a group Equally it is impractical tocontemplate taking, say, four samples instead of one and repeating theanalysis several times in order to form a group – the costs of doing this would

be prohibitive in many cases, and statistically this would be different to thegrouping of less frequently available data

An important technique for handling data which are difficult or consuming to obtain and, therefore, not available in sufficient numbers toenable the use of conventional mean and range charts is the moving mean andmoving range chart In the chemical industry, for example, the nature ofcertain production processes and/or analytical methods entails long time

time-intervals between consecutive results We have already seen in this chapterthat plotting of individual results offers one method of control, but this may

be relatively insensitive to changes in process average and changes in thespread of the process can be difficult to detect On the other hand, waiting forseveral results in order to plot conventional mean and range charts may allowmany tonnes of material to be produced outside specification before one pointcan be plotted

In a polymerization process, one of the important process control measures

is the unreacted monomer Individual results are usually obtained once every

24 hours, often with a delay for analysis of the samples Typical data fromsuch a process appear in Table 7.1

If the individual or run chart of these data (Figure 7.6) was being used alonefor control during this period, the conclusions may include:

Other types of control charts for variables 167

April 16 – warning and perhaps a repeat sample

April 20 – action signal – do something

April 23 – action signal – do something

April 29 – warning and perhaps a repeat sample

From about 30 April a gradual decline in the values is being observed.When using the individuals chart in this way, there is a danger that decisionsmay be based on the last result obtained But it is not realistic to wait foranother three days, or to wait for a repeat of the analysis three times and thengroup data in order to make a valid decision, based on the examination of amean and range chart

The alternative of moving mean and moving range charts uses the data

differently and is generally preferred for the following reasons:

 By grouping data together, we will not be reacting to individual resultsand over-control is less likely

 In using the moving mean and range technique we shall be making moremeaningful use of the latest piece of data – two plots, one each on twodifferent charts telling us different things, will be made from eachindividual result

 There will be a calming effect on the process

The calculation of the moving means and moving ranges (n = 4) for the

polymerization data is shown in Table 7.2 For each successive group of four,the earliest result is discarded and replaced by the latest In this way it is

Figure 7.6 Daily values of unreacted monomer

Table 7.2 Moving means and moving ranges for data in unreacted monomer (Table 7.1)

value

4-day moving total

4-day moving mean

4-day moving range

Combination for conventional mean and range control charts

Other types of control charts for variables 169

possible to obtain and plot a ‘mean’ and ‘range’ every time an individualresult is obtained – in this case every 24 hours These have been plotted oncharts in Figure 7.7

The purist statistician would require that these points be plotted at the point, thus the moving mean for the first four results should be placed on thechart at 2 April In practice, however, the point is usually plotted at the lastresult time, in this case 4 April In this way the moving average and movingrange charts indicate the current situation, rather than being behind time

mid-An earlier stage in controlling the polymerization process would have been

to analyse the data available from an earlier period, say during February andMarch, to find the process mean and the mean range, and to establish the meanand range chart limits for the moving mean and range charts The process wasfound to be in statistical control during February and March and capable ofmeeting the requirements of producing a product with less than 0.35 per centmonomer impurity These observations had a process mean of 0.22 per cent

and, with groups of n = 4, a mean range of 0.079 per cent So the control chart

limits, which are the same for both conventional and moving mean and rangecharts, would have been calculated before starting to plot the moving meanand range data onto charts The calculations are shown below:

Moving mean and mean chart limits

be used as an effective forecasting method.

In the polymerization example one new piece of data becomes availableeach day and, if moving mean and moving range charts were being used,the result would be reviewed day by day An examination of Figure 7.7shows that:

Figure 7.8

 There was no abnormal behaviour of either the mean or the range on

16 April

 The abnormality on 18 April was not caused by a change in the mean of

the process, but an increase in the spread of the data, which shows as anaction signal on the moving range chart The result of zero for theunreacted monomer (18th) is unlikely because it implies almost totalpolymerization The resulting investigation revealed that the plant chemisthad picked up the bottle containing the previous day’s sample from whichthe unreacted monomers had already been extracted during analysis – sowhen he erroneously repeated the analysis the result was unusually low.This type of error is a human one – the process mean had not changed andthe charts showed this

 The plots for 19 April again show an action on the range chart This isbecause the new mean and range plots are not independent of the previousones In reality, once a special cause has been identified, the individual

‘outlier’ result could be eliminated from the series If this had been donethe plot corresponding to the result from the 19th would not show anaction on the moving range chart The warning signals on 20 and 21 Aprilare also due to the same isolated low result which is not removed from theseries until 22 April

Supplementary rules for moving mean and moving range charts

The fact that the points on a moving mean and moving range chart are notindependent affects the way in which the data are handled and the charts

interpreted Each value influences four (n) points on the four-point moving

mean chart

The rules for interpreting a four-point moving mean chart are that theprocess is assumed to have changed if:

1 ONE point plots outside the action lines

2 THREE (n – 1) consecutive points appear between the warning and action

lines

3 TEN (2.5n) consecutive points plot on the same side of the centreline.

If the same data had been grouped for conventional mean and range charts,

with a sample size of n = 4, the decision as to the date of starting the grouping

would have been entirely arbitrary The first sample group might have been 1,

2, 3, 4 April; the next 5, 6, 7, 8 April and so on; this is identified in Table 7.2

Other types of control charts for variables 173

as combination A Equally, 2, 3, 4, 5 April might have been combined; this iscombination B Similarly, 3, 4, 5, 6 April leads to combination C; and 4, 5, 6,

7 April will give combination D

A moving mean chart with n = 4 is as if the points from four conventional

mean charts were superimposed This is shown in Figure 7.9 The plottedpoints on this chart are exactly the same as those on the moving mean andrange plot previously examined They have now been joined up in theirindependent A, B, C and D series Note that in each of the series the problem

at 18 April is seen to be on the range and not on the mean chart As we arelooking at four separate mean and range charts superimposed on each other it

is not surprising that the limits for the mean and range and the moving meanand range charts are identical

The process overall

If the complete picture of Figure 7.7 is examined, rather than considering thevalues as they are plotted daily, it can be seen that the moving mean andmoving range charts may be split into three distinct periods:

Clearly, a dramatic change in the variability of the process took place in themiddle of April and continued until the end of the month This is shown by thegeneral rise in the level of the values in the range chart and the more erraticplotting on the mean chart.

An investigation to discover the cause(s) of such a change is required Inthis particular example, it was found to be due to a change in supplier offeedstock material, following a shut-down for maintenance work at the usualsupplier’s plant When that supplier came back on stream in early May, notonly did the variation in the impurity, unreacted monomer, return to normal,but its average level fell until on 13 May an action signal was given.Presumably this would have led to an investigation into the reasons for thelow result, in order that this desirable situation might be repeated andmaintained This type of ‘map-reading’ of control charts, integrated into agood management system, is an indispensable part of SPC

Moving mean and range charts are particularly suited to industrialprocesses in which results become available infrequently This is often aconsequence of either lengthy, difficult, costly or destructive analysis incontinuous processes or product analyses in batch manufacture The rules formoving mean and range charts are the same as for mean and range chartsexcept that there is a need to understand and allow for non-independentresults

Exponentially Weighted Moving Average

In mean and range control charts, the decision signal obtained depends largely

on the last point plotted In the use of moving mean charts some authors havequestioned the appropriateness of giving equal importance to the most recentobservation The exponentially weighted moving average (EWMA) chart is atype of moving mean chart in which an ‘exponentially weighted mean’ iscalculated each time a new result becomes available:

New weighted mean = (a  new result) + ((1 – a)  previous mean), where a is the ‘smoothing constant’ It has a value between 0 and 1; many people use a = 0.2 Hence, new weighted mean = (0.2  new result) + (0.8

Other types of control charts for variables 175

Figure 7.10 An EWMA chart

When viscosity of batch 1 becomes available,

New weighted mean (1) = (0.2  79.1) + (0.8  80.0)

= 79.82

When viscosity of batch 2 becomes available,

New weighted mean (2) = (0.2  80.5) + (0.8  79.82)

= 79.96

Previous data, from a period when the process appeared to be in control,

was grouped into 4 The mean range (R ) of the groups was 7.733 cSt.

 = R/d n = 7.733/2.059 = 3.756

SE = /[a/(2 – a)]

= 3.756[0.2/(2 – 0.2)] = 1.252LAL = 80.0 – (3  1.252) = 76.24

The EWMA has been used by some organizations, particularly in theprocess industries, as the basis of new ‘control/performance chart’ systems.Great care must be taken when using these systems since they do not showchanges in variability very well, and the basis for weighting data is ofteneither questionable or arbitrary

7.5 Control charts for standard deviation ( )

Range charts are commonly used to control the precision or spread ofprocesses Ideally, a chart for standard deviation () should be used but,because of the difficulties associated with calculating and understandingstandard deviation, sample range is often substituted

Significant advances in computing technology have led to the availability

of cheap computers/calculators with a standard deviation key Using suchtechnology, experiments in Japan have shown that the time required tocalculate sample range is greater than that for , and the number ofmiscalculations is greater when using the former statistic The conclusions ofthis work were that mean and standard deviation charts provide a simpler andbetter method of process control for variables than mean and range charts,when using modern computing technology

Other types of control charts for variables 177

The standard deviation chart is very similar to the range chart (see Chapter

6) The estimated standard deviation (s i) for each sample being calculated,plotted and compared to predetermined limits:

Statistical theory allows the calculation of a series of constants (Cn) whichenables the estimation of the process standard deviation () from the average

of the sample standard deviation (s) The latter is the simple arithmetic mean

of the sample standard deviations and provides the central-line on the standarddeviation control chart:

s = k

i = 1 si/k

where s = average of the sample standard deviations;

s i = estimated standard deviation of sample i;

k = number of samples.

The relationship between  and s is given by the simple ratio:

 = sC n

where  = estimated process standard deviation;

Cn = a constant, dependent on sample size Values for Cnappear inAppendix E

The control limits on the standard deviation chart, like those on the rangechart, are asymmetrical, in this case about the average of the sample standard

deviation (s) The table in Appendix E provides four constants B1

.001, B1 025,

B1

.975and B1

.999which may be used to calculate the control limits for a standard

deviation chart from s The table also gives the constants B.001, B.025, B.975and B.999 which are used to find the warning and action lines from theestimated process standard deviation,  The control chart limits for thecontrol chart are calculated as follows:

Upper Action Line at B1.001s or B.001

Upper Warning Line at B1

.025s or B.025Lower Warning Line at B1

.975s or B.975Lower Action Line at B1 999s or B.999

An example should help to clarify the design and use of the sigma chart Let

us re-examine the steel rod cutting process which we met in Chapter 5, and forwhich we designed mean and range charts in Chapter 6 The data has been

reproduced in Table 7.4 together with the standard deviation (s i) for each

Sample

number

Sample rod lengths

Sample mean (mm)

Sample range (mm)

Standard deviation (mm)