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The expected value of the sample standard deviation is

$$E(s) = c_4(n)\sigma$$

where

$$ c_4(n) = \sqrt{2\over n-1}{\Gamma({n\over2})\over\Gamma({n-1\over2})} $$

The page on Wikipedia led me to believe that was related to the order of the approximating polynomial, although this doesn't appear to be the case.


Wikipedia cites Ben W. Bolch, "More on unbiased estimation of the standard deviation", The American Statistician, 22(3), p.27 (1968) for the formula

This includes a small table of $a_1$ and $a_2$ values, where $a_2(n) = c_4(n)^{-1}$ (the multiplicative inverse) and $a_1(n) = a_2(n)\sqrt{n-1\over k}$ (where $k$ is a value depending on $n$).


Bolch's paper cites W. H. Holtzmann, "The unbiased estimate of the population variance and standard deviation", American Journal of Psychology, 63, 615-617 (1950).

This includes a table of $C_N$, which is equal to $a_2(n)$ or $c_4(n)^{-1}$.


Neither paper makes use of the $c_4$ notation, which leads me to believe that this notation may have been invented later. Searching for "control chart constants" seems to consistently use the same naming convention, but none seem to reference the origin.

So where does the original table using the $c_4$ and other constants first appear? And why was the subscript $4$ chosen?

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    $\begingroup$ They originate in tables--I haven't been able to find the ultimate origin. See r-bar.net/control-chart-constants-tables-explanations for instance. The "4" in $c_4$ hasn't any intrinsic mathematical meaning AFAIK. BTW, $c_4$ is not the approximation: it is the exact value. $\endgroup$ – whuber May 12 at 15:48
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Note: This is a partial answer.


Wikipedia mentions Duncan, A. J., Quality Control and Industrial Statistics 4th Ed., Irwin (1974) ISBN 0-256-01558-9, p.139.

An older edition of this book contains a control chart constant table with $c_1$, $c_2$, $A_1$, $A_2$ and a few other constants (but not $c_3$ or $c_4$), which it cites as being reproduced from Table B2 of A.S.T.M. Manual on Quality Control of Materials, p.115.

It mentions $c_1$ and $c_2$ also being given in Table 29 of W.A. Shewhart, Economic Control of Quality of Manufactured product, (New York: D. Van Nostrand & Co., 1931), p.185.

The table is labeld Correction Factors $c_1$ and $c_2$. The $c_1$ and $c_2$ constant are defined as

$$\begin{align} c_1 &= \sqrt{n-2 \over n} \\ c_2 &= \sqrt{2\over n}{\Gamma({n\over 2})\over\Gamma({n-1\over2})} \end{align}$$

in equation (65) and (66), and the table lists values for $c_1(n)$ and $c_2(n)$ for $n$ some selected values in the range $3..100$.

We note that $c_4(n) = \sqrt{n\over n-1}c_2$, which is just a bessel correction, as $c_2$ is the mean of the scaled chi distribution. ($c_1$ being the mode).

$$\begin{align} \breve\sigma &= c_1\sigma \\ \bar\sigma &= c_2\sigma \end{align}$$

where the scaled chi distribution is

$$ \chi = \sqrt{{1\over n}\sum_{i=1}^{n-k} X_i^2} = \sqrt{{1\over n}\sum_{i=1}^{n-k} \sigma^2Z_i^2} = \sigma\sqrt{{1\over n}\sum_{i=1}^{n-k} Z_i^2} $$

where $Z_i$ are i.i.d. drawn from the standard normal distribution, with $n-k$ degrees of freedom. (?)


This doesn't show where $c_4$ first appear, but it does give a problable origin of $c_1$ and $c_2$, which appear in an early control chart table.

This points to $c_4$ being added to a later control chart table later, with a different subscript to avoid confusion. It is possible that the letter $c$ was used as it is the first letter in correction factor, given the table's name.


I am unfortunately not able to find a copy of A.S.T.M. Manual on Quality Control of Materials, and the library is currently closed due to the lockdown, so I am unable to investigate this any further.

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