Control chart

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. A control chart is a more specific kind of a run chart.

The control chart is one of the seven basic tools of quality control, which include the histogram, Pareto chart, check sheet, control chart, cause-and-effect diagram, flowchart, and scatter diagram. See Quality Management Glossary.

History
The control chart was invented by Walter A. Shewhart while working for Bell Labs in the 1920s. 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. Dr. Shewhart's boss, George Edwards, recalled: "Dr. Shewhart prepared a little memorandum only about a page in length. About a third of that page was given over to a simple diagram which we would all recognize today as a schematic control chart. That diagram, and the short text which preceded and followed it, set forth all of the essential principles and considerations which are involved in what we know today as process quality control." 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.

Dr. Shewhart created the basis for the control chart and the concept of a state of statistical control by carefully designed experiments. While Dr. Shewhart drew from pure mathematical statistical theories, he understood data from physical processes never produce a "normal distribution curve" (a Gaussian distribution, also commonly referred to as a "bell curve"). He discovered that observed variation in manufacturing data did not always behave the same way as data in nature (Brownian motion of particles). Dr. Shewhart concluded that while every process displays variation, some processes display controlled variation that is natural to the process, while others display uncontrolled variation that is not present in the process causal system at all times.

In 1924 or 1925, Shewhart's innovation came to the attention of W. Edwards Deming, then working at the Hawthorne facility. Deming later worked at the United States Department of Agriculture and then became the 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.

Details
A control chart is a run chart of a sequence of quantitative data with five horizontal lines drawn on the chart:
 * A centre line, drawn at the process mean;
 * An upper warning limit drawn two standard deviations above the centre line;
 * An upper control-limit (also called an upper natural process-limit drawn three standard deviations above the centre line;
 * A lower warning limit drawn two standard deviations below the centre line;
 * 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.

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 4/(9k2).
 * The empirical investigation of sundry probability distributions reveals 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.''

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

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:


 * 1) Ascribe a variation or a mistake to a special cause when in fact the cause belongs to the system (common cause). In statistics this is a Type I error
 * 2) Ascribe a variation or a mistake to the system (common causes) when in fact the cause was special. In statistics this is a Type II error

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 most common sets are:


 * The Western Electric rules;
 * The Donald J. Wheeler's rules;
 * The Nelson 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.

Alternative bases
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.

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 is in control (that is, no special causes are present in the system), there is approximately a 0.27% probability of a point exceeding 3-sigma control limits. Since the control limits are evaluated each time a point is added to the chart, it readily follows that every control chart will eventually signal the possible presence of a special cause, even though one may not have actually occurred. For a Shewhart control chart using 3-sigma limits, this false alarm occurs on average once every 1/0.0027 or 370.4 observations. Therefore, the in-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 immediate alarm 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 a 1- or 2-sigma change in the mean), the Shewhart chart does not detect these changes efficiently. Other types of control charts have been developed, such as the EWMA chart and the CUSUM chart, 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 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.

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

Types of charts

 * Individuals/ moving-range chart (ImR chart or XmR chart)
 * XbarR chart (Shewhart chart)
 * p-chart
 * np-chart
 * c-chart
 * u-chart
 * Averages as individuals chart
 * Three-way chart
 * z-chart
 * EWMA chart (Exponentially-Weighted Moving Average chart)
 * CUSUM chart (Cumulative Sum chart)