# Analysis of a control chart

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### Introduction

If analysis of the control chart indicates that the process is currently under control (i.e. is stable, with variation only coming from sources common to the process) then data from the process can be used to predict the future performance of the process. If the chart indicates that the process being monitored is not in control, analysis of the chart can help determine the sources of variation, which can then be eliminated to bring the process back into control. A control chart is a specific kind of run chart that allows significant change to be differentiated from the natural variability of the process.

The control chart can be seen as part of an objective and disciplined approach that enables correct decisions regarding control of the process, including whether or not to change process control parameters. Process parameters should never be adjusted for a process that is in control, as this will result in degraded process performance.

The control chart is one of the seven basic tools of quality control.

### Chart details

A control chart consists of:

- Points representing a statistic (e.g., a mean, range, proportion) of measurements of a quality characteristic in samples taken from the process at different times [the data]
- The mean of this statistic using all the samples is calculated (e.g., the mean of the means, mean of the ranges, mean of the proportions)
- A center line is drawn at the value of the mean of the statistic
- The standard error (e.g., standard deviation/sqrt(n) for the mean) of the statistic is also calculated using all the samples
- Upper and lower control limits (sometimes called "natural process limits") that indicate the threshold at which the process output is considered statistically 'unlikely' are drawn typically at 3 standard errors from the center line

The chart may have other optional features, including:

- Upper and lower warning limits, drawn as separate lines, typically two standard errors above and below the center line
- Division into zones, with the addition of rules governing frequencies of observations in each zone
- Annotation with events of interest, as determined by the Quality Engineer in charge of the process's quality

### Chart usage

If the process is in control, all points will plot within the control limits. Any observations outside the limits, or systematic patterns within, suggest the introduction of a new (and likely unanticipated) source of variation, known as aspecial-causevariation. Since increased variation means increasedquality costs, a control chart "signaling" the presence of a special-cause requires immediate investigation.

This makes the control limits very important decision aids. The control limits tell you about process behavior and have no intrinsic relationship to any specification targets or engineering tolerance. In practice, the process mean (and hence the center line) may not coincide with the specified value (or target) of the quality characteristic because the process' design simply can't deliver the process characteristic at the desired level.

Control charts limit specification limits or targets because of the tendency of those involved with the process (e.g., machine operators) to focus on performing to specification when in fact the least-cost course of action is to keep process variation as low as possible. Attempting to make a process whose natural center is not the same as the target perform to target specification increases process variability and increases costs significantly and is the cause of much inefficiency in operations. Process capability studies do examine the relationship between the natural process limits (the control limits) and specifications, however.

The purpose of control charts is to allow simple detection of events that are indicative of actual process change. This simple decision can be difficult where the process characteristic is continuously varying; the control chart provides statistically objective criteria of change. When change is detected and considered good its cause should be identified and possibly become the new way of working, where the change is bad then its cause should be identified and eliminated.

The purpose in adding warning limits or subdividing the control chart into zones is to provide early notification if something is amiss. Instead of immediately launching a process improvement effort to determine whether special causes are present, the Quality Engineer may temporarily increase the rate at which samples are taken from the process output until it's clear that the process is truly in control. Note that with three sigma limits, one expects to be signaled approximately once out of every 370 points on average, just due to common-causes.

### Choice of limits

Shewhart set3-sigma(3-standard error) limits on the following basis.

- The coarse result ofChebyshev's inequalitythat, for anyprobability distribution, theprobabilityof an outcome greater thankstandard deviationsfrom themeanis at most 1/k2.
- The finer result of theVysochanskii-Petunin inequality, that for anyunimodalprobability distribution, theprobabilityof an outcome greater thankstandard deviationsfrom themeanis at most 4/(9k2).
- The empirical investigation of sundryprobability distributionsreveals that at least 99% of observations occurred within threestandard deviationsof themean.

Shewhart summarized the conclusions by saying:

... the fact that the criterion which we happen to use has a fine ancestry in highbrow statistical theorems does not justify its use. Such justification must come from empirical evidence that it works. As the practical engineer might say, the proof of the pudding is in the eating.

Though he initially experimented with limits based onprobability distributions, Shewhart ultimately wrote:

Some of the earliest attempts to characterize a state of statistical control were inspired by the belief that there existed a special form of frequency functionfand it was early argued that the normal law characterized such a state. When the normal law was found to be inadequate, then generalized functional forms were tried. Today, however, all hopes of finding a unique functional formfare blasted.

The control chart is intended as a heuristic.Deminginsisted that it is not ahypothesis testand is not motivated by theNeyman-Pearson lemma. He contended that the disjoint nature ofpopulationandsampling framein most industrial situations compromised the use of conventional statistical techniques.Deming's intention was to seek insights into thecause systemof a process...under a wide range of unknowable circumstances, future and past .... He claimed that, under such conditions,3-sigmalimits provided... a rational and economic guide to minimum economic loss...from the two errors:

- Ascribe a variation or a mistake to a special cause when in fact the cause belongs to the system (common cause).(Also known as aType I error)
- Ascribe a variation or a mistake to the system (common causes) when in fact the cause was special.(Also known as aType II error)

### Calculation of standard deviation

As for the calculation of control limits, thestandard deviation(error) required is that of thecommon-causevariation in the process. Hence, the usualestimator, in terms of sample variance, is not used as this estimates the total squared-error loss from bothcommon- and special-causesof variation.

An alternative method is to use the relationship between therangeof a sample and itsstandard deviationderived byLeonard H. C. Tippett, an estimator which tends to be less influenced by the extreme observations which typifyspecial-causes.

### Rules for detecting signals

The most common sets are:

- TheWestern Electric rules
- TheWheelerrules (equivalent to the Western Electric zone tests[5])
- TheNelson rules

There has been particular controversy as to how long a run of observations, all on the same side of the centre line, should count as a signal, with 6, 7, 8 and 9 all being advocated by various writers.

The most important principle for choosing a set of rules is that the choice be made before the data is inspected. Choosing rules once the data have been seen tends to increase theType I errorrate owing totesting effects suggested by the data.

### Performance of control charts

When a point falls outside of the limits established for a given control chart, those responsible for the underlying process are expected to determine whether a special cause has occurred. If one has, then that cause should be eliminated if possible. It is known that even when a process isin control(that is, no special causes are present in the system), there is approximately a 0.27% probability of a point exceeding3-sigmacontrol limits. Since the control limits are evaluated each time a point is added to the chart, it readily follows thateverycontrol chart will eventually signal the possible presence of a special cause, even though one may not have actually occurred. For a Shewhart control chart using3-sigmalimits, thisfalse alarmoccurs on average once every 1/0.0027 or 370.4 observations. Therefore, thein-control average run length(or in-control ARL) of a Shewhart chart is 370.4.

Meanwhile, if a special cause does occur, it may not be of sufficient magnitude for the chart to produce an immediatealarm condition. If a special cause occurs, one can describe that cause by measuring the change in the mean and/or variance of the process in question. When those changes are quantified, it is possible to determine the out-of-control ARL for the chart.

It turns out that Shewhart charts are quite good at detecting large changes in the process mean or variance, as their out-of-control ARLs are fairly short in these cases. However, for smaller changes (such as a1-or2-sigmachange in the mean), the Shewhart chart does not detect these changes efficiently. Other types of control charts have been developed, such as theEWMA chartand theCUSUMchart, which detect smaller changes more efficiently by making use of information from observations collected prior to the most recent data point.

### Criticisms

Several authors have criticised the control chart on the grounds that it violates thelikelihood principle.However, the principle is itself controversialand supporters of control charts further argue that, in general, it is impossible to specify alikelihood functionfor a process not in statistical control, especially where knowledge about thecause systemof the process is weak.

Some authors have criticised the use of average run lengths (ARLs) for comparing control chart performance, because that average usually follows ageometric distribution, which has high variability and difficulties.

### Reference

- en.wikipedia.org/wiki/Control_charts
- www.vertex42.com ›Excel Template
- www.statsoft.com/textbook/quality-control-chart
- www.asq.org/learn-about-quality/data...tools/.../control-chart.htm
- www.statgraphics.com/control_charts.ht
- www.skymark.com/resources/tools/control_charts.asp