In statistical process control, the control chart, also known as the 'Shewhart chart' or 'process-behaviour chart' is a tool to determine whether a manufacturing or business process is in a state of statistical control or not. If the chart indicates that the process being monitored is not in control, the pattern it reveals can help determine the source of variation to be eliminated to bring the process back into control. A control chart is a specific kind of run chart. Statistical process control (SPC) is a method for achieving quality control in manufacturing processes. ... Process (lat. ... Statistical Process Control, or SPC is a method for achieving quality control in manufacturing processes. ... A simple run chart showing data collected over time. ...

The control chart is one of the seven basic tools of quality control (along with the histogram, Pareto chart, check sheet, cause-and-effect diagram, flowchart, and scatter diagram). See quality management glossary. For the Jurassic 5 album, see Quality Control (album) In engineering and manufacturing, quality control and quality engineering are involved in developing systems to ensure products or services are designed and produced to meet or exceed customer requirements. ... For the histogram used in digital image processing, see Color histogram. ... Pareto Chart A Pareto Chart is a special type of Histogram where the values being plotted are arranged in descending order. ... The check sheet is a simple document that is used for collecting data in real-time and at the location where the data is generated. ... The Ishikawa diagram is a graphical method for finding the most likely causes for an undesired effect. ... A flowchart that a member of the Wikipedia community could use for guidance when dealing with a difficult editor. ... A scatterplot or scatter graph is a graph used in statistics to visually display and compare two or more sets of related quantitative, or numerical, data by displaying only finitely many points, each having a coordinate on a horizontal and a vertical axis. ...

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 realized the importance of reducing variation in a manufacturing process. Moreover, they had realized 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." ^[1] 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. Walter Andrew Shewhart (March 18, 1891 - March 11, 1967) was a physicist, engineer and statistician, sometimes known as the father of statistical quality control. ... Bell Laboratories (also known as Bell Labs and formerly known as AT&T Bell Laboratories and Bell Telephone Laboratories) was the main research and development arm of the United States Bell System. ... In telecommunication, Telephony encompasses the general use of equipment to provide voice communication over distances. ... For the British rock band of the same name, see Amplifier (band). ... 1920 (MCMXX) was a leap year starting on Thursday. ... Special cause Common- and special-causes are the two distinct origins of variation, in a process that features in the statistical thinking and methods of Walter A. Shewhart and W. Edwards Deming. ... is the 136th day of the year (137th in leap years) in the Gregorian calendar. ... For the rap album, see 1924 (album). ... Statistical Process Control, or SPC is a method for achieving quality control in manufacturing processes. ... Special cause Common- and special-causes are the two distinct origins of variation, in a process that features in the statistical thinking and methods of Walter A. Shewhart and W. Edwards Deming. ...

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.^[2] The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... Probability density function of Gaussian distribution (bell curve). ... The graph of the probability density function of the normal distribution is sometimes called the bell curve or the bell-shaped curve; see normal distribution. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ...

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. William Edwards Deming (October 14, 1900–December 20, 1993) was an American statistician, college professor, author, lecturer, and consultant. ... “USDA” redirects here. ... The United States Census Bureau (officially Bureau of the Census as defined in Title ) is a part of the United States Department of Commerce. ... William Edwards Deming (October 14, 1900–December 20, 1993) was an American statistician, college professor, author, lecturer, and consultant. ... Combatants Allied powers: China France Great Britain Soviet Union United States and others Axis powers: Germany Italy Japan and others Commanders Chiang Kai-shek Charles de Gaulle Winston Churchill Joseph Stalin Franklin Roosevelt Adolf Hitler Benito Mussolini Hideki Tōjō Casualties Military dead: 17,000,000 Civilian dead: 33,000... William Edwards Deming (October 14, 1900–December 20, 1993) was an American statistician, college professor, author, lecturer, and consultant. ... Look up scap in Wiktionary, the free dictionary. ... the first thing that was invented was the automatic DILDO. Education grew explosively because of a very strong demand for high school and college education. ... The 1960s decade refers to the years from 1960 to 1969. ...

More recent use and development of control charts in the Shewhart-Deming tradition has been championed by Donald J. Wheeler. Donald J. Wheeler is an internationally recognized expert on SPC and data analysis. ...

Details

A control chart consists of the following:

• Points representing averages of measurements of a quality characteristic in samples taken from the process versus time
• A centre line, drawn at the process mean
• Upper and lower control limits ("natural process limits") that indicate the threshold at which the process output is considered statistically unlikely

The chart may contain other optional features, including: In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ...

• Upper and lower warning limits, drawn as separate lines, typically two standard deviations above and below the centre 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

Image File history File links ControlChart. ...

If the process is in control, most 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 a special-cause. Since increased variation means increased costs, the control chart "signaling" the presence of special requires immediate investigation. Special cause Common- and special-causes are the two distinct origins of variation, in a process that features in the statistical thinking and methods of Walter A. Shewhart and W. Edwards Deming. ... Cost of poor quality, or COPQ, can be defined as costs that would disappear if systems, processes, and products were perfect. ...

Note that in practice, the long-term process mean (and hence the centre line) may not coincide exactly with the ideal value (or target) of the quality characteristic because equipment simply can't control the process to the desired precision or because it's too costly to put the process on target.

Control charts omit specification limits 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. It is generally much easier to put a process on target than it is to keep variability from creeping into the process and omitting the specification limits reinforces this thinking. Process capability studies do examine the relationship between the natural process limits (that drive the control limits) and specification limits, however. “Specification” redirects here. ... The Process Capability Study answers the question, is my process good enough? This is quite different from the question answered by a Control chart, which is, has my process changed? Properly, use of a Control Chart to establish that a process is stable and predictable precedes the use of a...

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. Special cause Common- and special-causes are the two distinct origins of variation, in a process that features in the statistical thinking and methods of Walter A. Shewhart and W. Edwards Deming. ...

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: In probability theory, Chebyshevs inequality (also known as Tchebysheffs inequality, Chebyshevs theorem, or the Bienaymé-Chebyshev inequality), named after Pafnuty Chebyshev, who first proved it, states that in any data sample or probability distribution, nearly all the values are close to the mean value, and provides a... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... Probability is the likelihood that something is the case or will happen. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... In probability theory, the Vysochanskiï-Petunin inequality gives a lower bound for the probability that a random variable with finite variance lies within a certain number of standard deviations of the variables mean. ... A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... Probability is the likelihood that something is the case or will happen. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ...

... 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: In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ...

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: William Edwards Deming (October 14, 1900–December 20, 1993) was an American statistician, college professor, author, lecturer, and consultant. ... One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ... In statistics, the Neyman-Pearson lemma states that when doing a hypothesis test between two point hypotheses H0: θ=θ0 and H1: θ=θ1, then the likelihood-ratio test which rejects H0 in favour of H1 when is the most powerful test of size α. ... Sampling Frame: The source from which a sample is drawn. ... William Edwards Deming (October 14, 1900–December 20, 1993) was an American statistician, college professor, author, lecturer, and consultant. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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

Type I errors (or α error, or false positive) and type II errors (β error, or a false negative) are two terms used to describe statistical errors. ... Type I errors (or α error, or false positive) and type II errors (β error, or a false negative) are two terms used to describe statistical errors. ...

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. In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... Special cause Common- and special-causes are the two distinct origins of variation, in a process that features in the statistical thinking and methods of Walter A. Shewhart and W. Edwards Deming. ... In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ... Special cause Common- and special-causes are the two distinct origins of variation, in a process that features in the statistical thinking and methods of Walter A. Shewhart and W. Edwards Deming. ...

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 . In descriptive statistics, the range is the length of the smallest interval which contains all the data. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... Special cause Common- and special-causes are the two distinct origins of variation, in a process that features in the statistical thinking and methods of Walter A. Shewhart and W. Edwards Deming. ...

Rules for detecting signals

The most common sets are:

• The Western Electric rules
• The Wheeler 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. This article or section does not cite its references or sources. ... Donald J. Wheeler is an internationally recognized expert on SPC and data analysis. ... Nelson rules are a method in process control of determining if some measured variable is out of control. ...

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 statistics, hypotheses suggested by the data must be tested differently from hypotheses formed independently of 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 percentiles 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. 1935 (MCMXXXV) was a common year starting on Tuesday (link will display full calendar). ... British Standards is the new name of the British Standards Institute and is part of BSI Group which also includes a testing organisation. ... Egon Sharpe Pearson (11 August 1895 — 12 June 1980) a son of Karl Pearson, was like his father, a British statistician, and succeeded him as professor of statistics at University College London and as editor of the journal Biometrika. ... // A percentile is the value of a variable below which a certain percent of observations fall. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...

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. In statistics, the likelihood principle is a controversial principle of statistical inference which asserts that all of the information in a sample is contained in the likelihood function. ... Look up likelihood in Wiktionary, the free dictionary. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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. In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before...

Types of charts

An XbarR chart is a specific member of a family of control charts. ... In statistical process control, the individual/moving-range chart is a control chart for variables data that uses individual measurements of a quality characteristic. ... P-chart The P chart is very similar to the X-bar chart except that the statistic being plotted is the sample proportion rather than the sample mean. ... In industrial statistics, the np-chart is a type of control chart that is very similar to the p-chart except that the statistic being plotted is a number count rather than a sample proportion of items. ... Also see moving average (disambiguation). ... In statistics, signal processing, and econometrics, a time series is a sequence of data points, measured typically at successive times, spaced at (often uniform) time intervals. ...

See also

• Walter A. Shewhart
• Common cause and special cause
• Analytic and enumerative statistical studies
• W. Edwards Deming
• Statistical process control
• Total Quality Management
• Six Sigma
• Process capability

Walter Andrew Shewhart (March 18, 1891 - March 11, 1967) was a physicist, engineer and statistician, sometimes known as the father of statistical quality control. ... Special cause Common- and special-causes are the two distinct origins of variation, in a process that features in the statistical thinking and methods of Walter A. Shewhart and W. Edwards Deming. ... In Some Theory of Sampling (1950, Chapter 7), W. Edwards Deming introduced concepts he labeled Analytic and enumerative statistical studies. ... William Edwards Deming (October 14, 1900–December 20, 1993) was an American statistician, college professor, author, lecturer, and consultant. ... Statistical process control (SPC) is a method for achieving quality control in manufacturing processes. ... Total Quality Management (TQM) is a management strategy aimed at embedding awareness of quality in all organizational processes. ... The often-used six sigma symbol. ... The Process Capability Study answers the question, is my process good enough? This is quite different from the question answered by a Control chart, which is, has my process changed? Properly, use of a Control Chart to establish that a process is stable and predictable precedes the use of a...

Notes

Bibliography

• 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 0-521-30553-5.
• Oakland, J (2002) Statistical Process Control ISBN 0-7506-5766-9.
• Shewhart, W A (1931) Economic Control of Quality of Manufactured Product ISBN 0-87389-076-0.
• Shewhart, W A (1939) Statistical Method from the Viewpoint of Quality Control ISBN 0-486-65232-7.
• Wheeler, D J (2000) Normality and the Process-Behaviour Chart ISBN 0-945320-56-6.
• Wheeler, D J & Chambers, D S (1992) Understanding Statistical Process Control ISBN 0-945320-13-2.
• Wheeler, Donald J. (1999). Understanding Variation: The Key to Managing Chaos - 2nd Edition. SPC Press, Inc. ISBN 0-945320-53-1.

External links

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• NIST/SEMATECH e-Handbook of Statistical Methods
• Monitoring and Control with Control Charts