Control chart
From Academic Kids

The control chart, also known as the 'Shewhart chart' or 'processbehaviour chart' is a statistical tool intended to assess the nature of variation in a process and to facilitate forecasting and management.
Contents 
History
The control chart was invented by Walter A. Shewhart while working for Western Electric. 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 processadjustment in reaction to nonconformance actually increased variation and degraded quality. Shewhart framed the problem in terms of Common and specialcauses 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 commoncause 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 ShewhartDeming tradition has been championed by Donald J. Wheeler. Control charts play a central role in the Six Sigma management strategy.
Details
A control chart is a run chart of a sequence of quantitative data with three horizontal lines drawn on the chart:
 An upper controllimit (also called an upper natural processlimit drawn three standard deviations above the centre line; and
 A lower controllimit (also called a lower natural processlimit 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 specialcause (see Rules below). The run chart provides a context in which to interpret signals and can be beneficially annotated with events in the business.
picture to follow
Choice of limits
Shewhart set 3sigma 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/k^{2}.
 The finer result of the VysochanskiiPetunin inequality , that for any unimodal probability distribution, the probability of an outcome greater than k standard deviations from the mean is at most 5/9k^{2}.
 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.
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 NeymanPearson 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, 3sigma 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 commoncause variation in the process. Hence, the usual estimator, in terms of sample variance, is not used as this estimates the total squarederror loss from both common and specialcauses 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 specialcauses .
Rules for detecting signals
The two most common sets are:
 Western Electric rules; and
 Donald J. Wheeler's rules.
See also 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 3sigma 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 ShewhartDeming tradition.
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.
Types of chart
 Individuals/ movingrange chart (ImR chart or XmR chart)
 XbarR chart (Shewhart chart)
 pchart
 npchart
 cchart
 uchart
 Averages as individuals chart
 Threeway chart
 zchart
 EWMA chart (ExponentiallyWeighted Moving Average chart)
Monitoring and Control (http://www.itl.nist.gov/div898/handbook/pmc/pmc.htm)
See also
 Statistical process control
 Total Quality Management
 Six Sigma
 Create xBar, R, and S control charts using an online calculator (http://www.sqconline.com/control_chart_calculator.html)
Bibliography
 Deming, W E (1975) On probability as a basis for action, The American Statistician, 29(4), pp146152
 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 ProcessBehaviour Chart ISBN 0945320566
 Wheeler, D J & Chambers, D S (1992) Understanding Statistical Process Control ISBN 0945320132
NIST/SEMATECH eHandbook of Statistical Methods (http://www.itl.nist.gov/div898/handbook/index.htm)