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Control charts, also known as Shewhart charts or process-behaviour charts, in statistical process control are tools used to determine whether or not a manufacturing or business process is in a state of statistical control. Contents [hide]
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1 Overview 2 History 3 Chart details o 3.1 Chart usage o 3.2 Choice of limits o 3.3 Calculation of standard deviation 4 Rules for detecting signals 5 Alternative bases 6 Performance of control charts 7 Criticisms 8 Types of charts 9 Se

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Control charts
, also known as
Shewhart charts
or
process-behaviour charts
, in statistical process controlare tools used to determine whether or not a manufacturing or business process is
in a state of statistical control.
Contents
[hide]
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1 Overview
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2 History
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3 Chart details
o
3.1 Chart usage
o
3.2 Choice of limits
o
3.3 Calculation of standard deviation
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4 Rules for detecting signals
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5 Alternative bases
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6 Performance of control charts
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7 Criticisms
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8 Types of charts
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9 See also
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10 Notes
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11 Bibliography
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12 External links
[edit] Overview
If analysis of the control chart indicates that the process is currently under control (i.e. is stable,with variation only coming from sources common to the process) then no corrections or changesto process control parameters are needed or desirable. In addition, data from the process can beused to predict the future performance of the process. If the chart indicates that the process beingmonitored is not in control, analysis of the chart can help determine the sources of variation,which can then be eliminated to bring the process back into control. A control chart is a specifickind of run chartthat allows significant change to be differentiated from the natural variability of the process.The control chart can be seen as part of an objective and disciplined approach that enables correctdecisions regarding control of the process, including whether or not to change process control parameters. Process parameters should never be adjusted for a process that is in control, as thiswill result in degraded process performance.
[1]
A process that is stable but operating outside of desired limits (e.g. scrap rates may be in statistical control but above desired limits) needs to beimproved through a deliberate effort to understand the causes of current performance andfundamentally improve the process.
[2]
The control chart is one of the seven basic tools of quality control.
[3]
[edit] History
The control chart was invented byWalter A. Shewhart while working for Bell Labsin the 1920s.
The company's engineers had been seeking to improve the reliability of their telephonytransmission systems. Becauseamplifiersand other equipment had to be buried underground,there was a business need to reduce the frequency of failures and repairs. By 1920 the engineershad already realized the importance of reducing variation in a manufacturing process. Moreover,they had realized that continual process-adjustment in reaction to non-conformance actuallyincreased variation and degraded quality. Shewhart framed the problem in terms of Common- andspecial-causesof variation and, on May 16, 1924, wrote an internal memo introducing the controlchart 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 thirdof that page was given over to a simple diagram which we would all recognize today as aschematic control chart. That diagram, and the short text which preceded and followed it, setforth all of the essential principles and considerations which are involved in what we know todayas process quality control.
[4]
Shewhart stressed that bringing a production process into a state of statistical control,where there is onlycommon-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 statisticaltheories, he understood data from physical processes typically produce a normal distribution curve (aGaussian 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 motionof particles). Dr. Shewhart concluded that while every process displaysvariation, some processes display controlled variation that is natural to the process, while othersdisplay uncontrolled variation that is not present in the process causal system at all times.
[5]
In 1924 or 1925, Shewhart's innovation came to the attention of W. Edwards Deming, thenworking at the Hawthorne facility. Deming later worked at theUnited States Department of Agricultureand then became the mathematical advisor to theUnited States Census Bureau. Over
the next half a century,Deming became the foremost champion and proponent of Shewhart's
work. After the defeat of Japanat the close of World War II,Demingserved 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, andthe use of the control chart, widely in Japanese manufacturing industry throughout the 1950s and1960s.
[edit] Chart details
A control chart consists of:
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Points representing a statistic (e.g., a mean, range, proportion) of measurements of aquality characteristic in samples taken from the process at different times [the data]
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The mean of this statistic using all the samples is calculated (e.g., the mean of the means,mean of the ranges, mean of the proportions)
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A center line is drawn at the value of the mean of the statistic
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The standard error (e.g., standard deviation/sqrt(n) for the mean) of the statistic is alsocalculated using all the samples
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Upper and lower control limits (sometimes called natural process limits ) that indicatethe threshold at which the process output is considered statistically 'unlikely' are drawntypically at 3 standard errors from the center line
The chart may have other optional features, including:
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Upper and lower warning limits, drawn as separate lines, typically two standard errorsabove and below the center line
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Division into zones, with the addition of rules governing frequencies of observations ineach zone
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Annotation with events of interest, as determined by the Quality Engineer in charge of the process's quality
[edit] Chart usage
If the process is in control (and the process statistic is normal), 99.7300% of all the points willfall between 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 aspecial-causevariation. Since increased variation means increased quality costs, a control chart
signaling the presence of a special-cause requires immediate investigation.This makes the control limits very important decision aids. The control limits tell you about process behavior and have no intrinsic relationship to any specificationtargets or engineering
tolerance.In practice, the process mean (and hence the center line) may not coincide with thespecified value (or target) of the quality characteristic because the process' design simply cannotdeliver the process characteristic at the desired level.Control charts limitspecification limitsor targets because of the tendency of those involved withthe process (e.g., machine operators) to focus on performing to specification when in fact theleast-cost course of action is to keep process variation as low as possible. Attempting to make a process whose natural center is not the same as the target perform to target specification increases process variability and increases costs significantly and is the cause of much inefficiency inoperations. Process capabilitystudies 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 simple decision can be difficult where the process characteristic iscontinuously varying; the control chart provides statistically objective criteria of change. When
change is detected and considered good its cause should be identified and possibly become thenew 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 provideearly notification if something is amiss. Instead of immediately launching a process improvementeffort to determine whether special causes are present, the Quality Engineer may temporarilyincrease 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, common-causevariations result in
signals less than once out of every twenty-two points for skewed processes and about once out of every three hundred seventy (1/370.4) points for normally-distributed processes.
[6]
The two-sigmawarning levels will be reached about once for every twenty-two (1/21.98) plotted points innormally-distributed data. (For example, the means of sufficiently large samples drawn from practically any underlying distribution whose variance exists are normally distributed, accordingto the Central Limit Theorem.)
[edit] Choice of limits
Shewhart set
3-sigma
(3-standard error) limits on the following basis.
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The coarse result of Chebyshev's inequalitythat, for any probability distribution,the
probability of an outcome greater than
k
standard deviationsfrom themeanis at most
1/
k
2
.
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The finer result of theVysochanskii-Petunin inequality, that for any unimodal probability
distribution, the probabilityof an outcome greater than
k
standard deviationsfrom themeanis at most 4/(9
k
2
).
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The empirical investigation of sundry probability distributions reveals that at least 99%
of observations occurred within threestandard deviations of themean.
Shewhart summarized 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.
[
citation needed
]
Though he initially experimented with limits based on probability distributions,Shewhart
ultimately wrote:
Some of the earliest attempts to characterize a state of statistical control were inspired by thebelief that there existed a special form of frequency function
f
and it was early argued that thenormal law characterized such a state. When the normal law was found to be inadequate, then generalized functional forms were tried. Today, however, all hopes of finding a unique functional form
f
are blasted.
[
citation needed
]
The control chart is intended as a heuristic.Deminginsisted that it is not a hypothesis testand is
not motivated by the Neyman-Pearson lemma. He contended that the disjoint nature of population
andsampling frame in most industrial situations compromised the use of conventional statistical
techniques. Deming's intention was to seek insights into the cause systemof a process
...under awide range of unknowable circumstances, future and past....
[
citation needed
]
He claimed that, under such conditions,
3-sigma
limits provided
... a rational and economic guide to minimum economicloss...
from the two errors:
[
citation needed
]

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