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The control chart, also known as the 'Shewhart chart' or 'process-behaviour chart' is a statistical tool intended to assess the nature of variation in a process and to facilitate forecasting and management.
The control chart was invented by Walter A. Shewhart while working for the Western Electric Company . The company's engineers had been seeking to improve the reliability of their telephony transmission systems. Because amplifiers and other equipment had to be buried underground, there was a business need to reduce the frequency of failures and repairs. By 1920 they had already realised the importance of reducing variation in a manufacturing process. Moreover, they had realised that continual process-adjustment in reaction to non-conformance actually increased variation and degraded quality. Shewhart framed the problem in terms of Common- and special-causes of variation and, on May 16 1924, wrote an internal memo introducing the control chart as a tool for distinguishing between the two. Shewhart stressed that bringing a production process into a state of statistical control, where there is only common-cause variation, and keeping it in control, is necessary to predict future output and to manage a process economically.
In 1938, Shewhart's innovation came to the attention of W. Edwards Deming, then working at the United States Department of Agriculture but about to become mathematical advisor to the United States Census Bureau. Over the next half a century, Deming became the foremost champion and exponent of Shewhart's work. After the defeat of Japan at the close of World War II, Deming served as statistical consultant to the Supreme Commander of the Allied Powers. His ensuing involvement in Japanese life, and long career as an industrial consultant there, spread Shewhart's thinking, and the use of the control chart, widely in Japanese manufacturing industry throughout the 1950s and 1960s.
More recent use and development of control charts in the Shewhart-Deming tradition has been championed by Donald J. Wheeler . Control charts play a central role in the Six Sigma management strategy.
- An upper control-limit (also called an upper natural process-limit drawn three standard deviations above the centre line; and
- A lower control-limit (also called a lower natural process-limit drawn three standard deviations below the centre line.
Common cause variation plots as an irregular pattern, mostly within the control limits. Any observations outside the limits, or patterns within, suggest (signal) a special-cause (see Rules below). The run chart provides a context in which to interpret signals and can be beneficially annotated with events in the business.
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Choice of limits
Shewhart set 3-sigma limits on the following basis.
- The coarse result of Chebyshev's inequality that, for any probability distribution, the probability of an outcome greater than k standard deviations from the mean is at most 1/k2.
- The finer result of the Vysochanskii-Petunin inequality , that for any unimodal probability distribution, the probability of an outcome greater than k standard deviations from the mean is at most 5/9k2.
- The empirical investigation of sundry probability distributions that at least 99% of observations occurred within three standard deviations of the mean.
Shewhart summarised 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.
Some of the earliest attempts to characterise a state of statistical control were inspired by the belief that there existed a special form of frequency function f and it was early argued that the normal law characterised such a state. When the normal law was found to be inadequate, then generalised functional forms were tried. Today, however, all hopes of finding a unique functional form f are blasted.
The control chart is intended as a heuristic. Deming insisted that it is not an hypothesis test and is not motivated by the Neyman-Pearson lemma. He contended that the disjoint nature of population and sampling frame in most industrial situations compromised the use of conventional statistical techniques. Deming's intention was to seek insights into the cause system of a process ...under a wide range of unknowable circumstances, future and past .... He claimed that, under such conditions, 3-sigma limits 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).
- Ascribe a variation or a mistake to the system (common causes) when in fact the cause was special.
Calculation of standard deviation
As for the calculation of control limits, the standard deviation required is that of the common-cause variation in the process. Hence, the usual estimator, in terms of sample variance, is not used as this estimates the total squared-error loss from both common- and special-causes of variation.
An alternative method is to use the relationship between the range of a sample and its standard deviation derived by Leonard H. C. Tippett , an estimator which tends to be less influenced by the extreme observations which typify special-causes .
Rules for detecting signals
The two most common sets are:
- Western Electric rules ; and
- Donald J. Wheeler 's 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 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 the economic losses arising from error 1 owing to testing effects suggested by the data.
In 1935, the British Standards Institution, under the influence of Egon Pearson and against Shewhart's spirit, adopted control charts, replacing 3-sigma limits with limits based on percentage points of the normal distribution. This move continues to be represented by John Oakland and others but has been widely deprecated by writers in the Shewhart-Deming tradition.
Several authors have criticised the control chart on the grounds that it violates the likelihood principle. However, the principle is itself controversial and supporters of control charts further argue that, in general, it is impossible to specify a likelihood function for a process not in statistical control, especially where knowledge about the cause system of the process is weak.
Types of chart
- Individuals/ moving-range chart (ImR chart or XmR chart)
- XbarR chart (Shewhart chart)
- Averages as individuals chart
- Three-way chart
- EWMA chart (Exponentially-Weighted Moving Average chart)
- Deming, W E (1975) On probability as a basis for action, The American Statistician, 29(4), pp146-152
- Deming, W E (1982) Out of the Crisis: Quality, Productivity and Competitive Position ISBN 0521305535
- Oakland, J (2002) Statistical Process Control ISBN 0750657669
- Shewhart, W A (1931) Economic Control of Quality of Manufactured Product ISBN 73890760
- Shewhart, W A (1939) Statistical Method from the Viewpoint of Quality Control ISBN 0486652327
- Wheeler, D J (2000) Normality and the Process-Behaviour Chart ISBN 0945320566
- Wheeler, D J & Chambers, D S (1992) Understanding Statistical Process Control ISBN 0945320132
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