Control chart 435754 220974142 2008-06-22T13:50:23Z Lightbot 7178666 Units/dates/other In [[statistical process control]], the '''control chart''', also known as the ''''Shewhart chart'''' or ''''process-behaviour chart'''' is a tool used to determine whether a manufacturing or business [[Process (general)|process]] is in a state of [[statistical control]] or not. If the chart indicates that the process is currently under control then it can be used with confidence to predict the future performance of the process. 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]] that allows significant change to be differentiated from the natural variability of the process. This is key to effective process control and improvement. 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]]). ==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 [[amplifier]]s 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." <ref> [http://www.porticus.org/bell/doc/western_electric.doc ''Western Electric - A Brief History'']</ref> Shewhart stressed that bringing a production process into a state of [[statistical control]], where there is only [[Common- and special-causes|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.<ref>"Why SPC?" British Deming Association SPC Press, Inc. 1992</ref> 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, [[W. Edwards Deming|Deming]] became the foremost champion and exponent of Shewhart's work. After the defeat of [[Japan]] at the close of [[World War II]], [[W. Edwards Deming|Deming]] served as statistical consultant to the [[SCAP|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 consists of the following: *Points representing measurements of a quality characteristic in samples taken from the process at different times [the data] *A centre line, drawn at the process characteristic [[mean]] which is calculated from the data *Upper and lower control limits (sometimes called "natural process limits") that indicate the threshold at which the process output is considered statistically 'unlikely' The chart may contain other optional features, including: *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 However in the early stages of use the inclusion of these items may confuse inexperienced chart interpreters. [[Image:ControlChart.svg]] 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 a [[Common- and special-causes|special-cause]] variation. Since increased variation means increased [[quality costs]], a control chart "signaling" the presence of a special-cause requires immediate investigation. The control limits tell you about process behaviour and have no intrinsic relationship to any [[specification]] targets or [[engineering tolerance]]. 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 deliver the process characteristic at the desired level or because it's too costly to put the process on target. Control charts omit [[Specification (technical standard)|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 centre is not the same as the target increases process variability and 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 can be difficult where the process characteristic is continuously varying, the control chart provides statistically objective criteria of change. When change is detected then if the change is 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- and special-causes|common-cause]]s. ===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 deviation]]s from the [[mean]] is at most 1/''k''<sup>2</sup>. *The finer result of the [[Vysochanskii-Petunin inequality]], that for any [[monotonic function|unimodal]] [[probability distribution]], the [[probability]] of an outcome greater than ''k'' [[standard deviation]]s from the [[mean]] is at most 4/(9''k''<sup>2</sup>). *The empirical investigation of sundry [[probability distribution]]s reveals that at least 99% of observations occurred within three [[standard deviation]]s of the [[mean]]. Shewhart summarised the conclusions by saying: <BLOCKQUOTE> ''... 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.'' </BLOCKQUOTE> Though he initially experimented with limits based on [[probability distribution]]s, Shewhart ultimately wrote: <BLOCKQUOTE> ''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.'' </BLOCKQUOTE> The control chart is intended as a heuristic. [[W. Edwards Deming|Deming]] insisted that it is not a [[hypothesis test]] and is not motivated by the [[Neyman-Pearson lemma]]. He contended that the disjoint nature of [[population (statistics)|population]] and [[sampling frame]] in most industrial situations compromised the use of conventional statistical techniques. [[W. Edwards Deming|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).'' (Also known as a [[Type I and type II errors|Type I error]]) #''Ascribe a variation or a mistake to the system (common causes) when in fact the cause was special.'' (Also known as a [[Type I and type II errors|Type II error]]) ===Calculation of standard deviation=== As for the calculation of control limits, the [[standard deviation]] required is that of the [[Common- and special-causes|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 (statistics)|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 [[Common- and special-causes|special-cause]]s . ==Rules for detecting signals== The most common sets are: *The [[Western Electric rules]] *The [[Donald J. Wheeler|Wheeler]] rules (equivalent to the Western Electric zone tests<REF>{{Citation | last = Wheeler | first = Donald J. | last2 = Chambers | first2 = David S. | author-link = Donald J. Wheeler | publication-date = 1992 | title = Understanding [[statistical process control]] | edition = 2 | publication-place = [[Knoxville, Tennessee]] | publisher = SPC Press | isbn = 9780945320135 | oclc = 27187772 | page = 96 }}</REF>) *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 [[Type I error]] rate owing to [[testing hypotheses suggested by the data|testing effects suggested by the data]]. ==Alternative bases== In 1935, the [[British Standard|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 [[percentile]]s 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. <!-- example here? --> 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 and difficulties ==Types of charts== {| class="wikitable" ! Chart ! Process observation ! Process observations relationships ! Process observations type ! Size of shift to detect |- | [[XbarR chart]] | Quality characteristic measurement within one subgroup | Independent | Variables | Large (≥ 1.5σ) |- | [[XbarS chart]] | Quality characteristic measurement within one subgroup | Independent | Variables | Large (≥ 1.5σ) |- | [[Shewhart individuals control chart]] (ImR chart or XmR chart) | Quality characteristic measurement for one observation | Independent | Variables | Large (≥ 1.5σ) |- | [[Three-way chart]] | Quality characteristic measurement within one subgroup | Independent | Variables | Large (≥ 1.5σ) |- | [[p-chart]] | Fraction nonconforming within one subgroup | Independent | Attributes | Large (≥ 1.5σ) |- | [[np-chart]] | Number nonconforming within one subgroup | Independent | Attributes | Large (≥ 1.5σ) |- | [[c-chart]] | Number of nonconformances within one subgroup | Independent | Attributes | Large (≥ 1.5σ) |- | [[u-chart]] | Nonconformances per unit within one subgroup | Independent | Attributes | Large (≥ 1.5σ) |- | EWMA chart | [[Exponentially weighted moving average]] of quality characteristic measurement within one subgroup | Independent | Attributes or variables | Small (< 1.5σ) |- | [[CUSUM]] chart | Cumulative sum of quality characteristic measurement within one subgroup | Independent | Attributes or variables | Small (< 1.5σ) |- | [[Time series]] model | Quality characteristic measurement within one subgroup | Autocorrelated | Attributes or variables | N/A |} ==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]] == Notes == {{reflist}} ==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== :''Note: Before adding your company's link, please read [[WP:Spam#External_link_spamming]] and [[WP:External_links#Links_normally_to_be_avoided]].'' *[http://www.itl.nist.gov/div898/handbook/index.htm NIST/SEMATECH e-Handbook of Statistical Methods] *[http://www.itl.nist.gov/div898/handbook/pmc/pmc.htm Monitoring and Control with Control Charts] [[Category:Product management]] [[Category:Quality]] [[Category:Quality control tools]] [[Category:Statistical charts and diagrams]] [[de:Qualitätsregelkarte]] [[fr:Carte de contrôle]] [[ja:管理図]] [[pt:Carta de controle]] [[su:Grafik kadali]]