Editing Control chart (section)

===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)