Editing Control chart (section)

==Chart details==

A control chart consists of:
* Points representing a statistic (e.g., a [[mean]], range, proportion) of measurements of a quality characteristic in samples taken from the process at different times (i.e., the data)
* 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) - or for a reference period against which change can be assessed.  Similarly a median can be used instead.
* A centre line is drawn at the value of the mean or median of the statistic
* The [[standard deviation]] (e.g., sqrt(variance) of the mean) of the statistic is calculated using all the samples - or again for a reference period against which change can be assessed. in the case of XmR charts, strictly it is an approximation of standard deviation, the {{what|date=November 2021}} does not make the assumption of homogeneity of process over time that the standard deviation makes.
* Upper and lower [[control limits]] (sometimes called "natural process limits") that indicate the threshold at which the process output is considered statistically 'unlikely' and are drawn typically at 3 standard deviations from the center line

The chart may have other optional features, including:
* More restrictive upper and lower warning or control limits, drawn as separate lines, typically two standard deviations above and below the center line. This is regularly used when a process needs tighter controls on variability. 
* 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' quality
* Action on special causes
(n.b., there are several rule sets for detection of signal; this is just one set. The rule set should be clearly stated.)

# Any point outside the control limits
# A Run of 7 Points all above or all below the central line - Stop the production 
#* Quarantine and 100% check 
#* Adjust Process.
#* Check 5 Consecutive samples
#* Continue The Process.
# A Run of 7 Point Up or Down - Instruction as above

===Chart usage===

If the process is in control (and the process statistic is normal), 99.7300% of all the points will fall 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 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.

This makes the control limits very important decision aids. The control limits provide information about the process behavior and have no intrinsic relationship to any [[specification]] targets or [[engineering tolerance]]. In practice, the process mean (and hence the centre line) may not coincide with the specified value (or target) of the quality characteristic because the process design simply cannot deliver the process characteristic at the desired level.

Control charts limit [[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 perform to target specification increases process variability and increases 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 an increase in process variability.<ref> Statistical Process Controls for Variable Data. Lean Six sigma. (n.d.). Retrieved from https://theengineeringarchive.com/sigma/page-variable-control-charts.html. </ref> This simple decision can be difficult where the process characteristic is continuously 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 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 is clear that the process is truly in control. Note that with three-sigma limits, [[Common- and special-causes|common-cause]] variations 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.<ref name="Wheeler112010">{{cite web|last=Wheeler|first=Donald J.|title=Are You Sure We Don't Need Normally Distributed Data?|url=http://www.qualitydigest.com/inside/quality-insider-column/are-you-sure-we-don-t-need-normally-distributed-data.html|publisher=Quality Digest|access-date=7 December 2010|date=1 November 2010}}</ref> The two-sigma warning levels will be reached about once for every twenty-two (1/21.98) plotted points in normally distributed data. (For example, the means of sufficiently large samples drawn from practically any underlying distribution whose variance exists are normally distributed, according to the Central Limit Theorem.)

===Choice of limits===
Shewhart set ''3-sigma'' (3-standard deviation) 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 [[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>).
*In the [[Normal distribution]], a very common [[probability distribution]], 99.7% of the observations occur within three [[standard deviation]]s of the [[mean]] (see [[Normal distribution#Standard deviation and coverage|Normal distribution]]).

Shewhart summarized the conclusions by saying:

<blockquote>
''...&nbsp;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.''<ref>{{cite book |last=Shewhart |first=W A |title=Economic Control of Quality of Manufactured Product |publisher=Van Nordstrom |year=1931 |page=18 }}
</ref>
</blockquote>

Although he initially experimented with limits based on [[probability distribution]]s, Shewhart ultimately wrote:

<blockquote>
''Some of the earliest attempts to characterize 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 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.''<ref>{{cite book |last1=Shewart |first1=Walter Andrew |last2=Deming |first2=William Edwards |title=Statistical Method from the Viewpoint of Quality Control |date=1939 |publisher=Graduate School, The Department of Agriculture |location=University of California |isbn=9780877710325 |page=12 |url=https://books.google.com/books?id=GF9IAQAAIAAJ}}</ref>
</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....''{{Citation needed|date=December 2010}} He claimed that, under such conditions, ''3-sigma'' limits provided ''...&nbsp;a rational and economic guide to minimum economic loss...'' from the two errors:{{Citation needed|date=December 2010}}

#''Ascribe a variation or a mistake to a special cause (assignable 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]] or False Positive)
#''Ascribe a variation or a mistake to the system (common causes) when in fact the cause was a special cause (assignable cause).'' (Also known as a [[Type I and type II errors|Type II error]] or False Negative)

===Calculation of standard deviation===

As for the calculation of control limits, the [[standard deviation]] (error) 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 Henry Caleb Tippett|Leonard H. C. Tippett]], as an estimator which tends to be less influenced by the extreme observations which typify [[Common- and special-causes|special-cause]]s.{{Citation needed|reason=What is the relationship? Ratio? How is it used? There's no way to look it up.|date=March 2019}}