C4 Function

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Functions

C4 Function

computes the expected value of the standard deviation of n independent normal random variables.

Syntax

C4(n)

where n is the sample size, with $n\geq2$ .

Description

The C4 function returns the expected value of the standard deviation of n independent, normally distributed random variables with the same mean and with standard deviation of 1. This expected value is referred to as the control chart constant c₄.

The value c₄ is calculated as

$c_4 = \frac{\Gamma(\frac{n}2) \sqrt{2/(n-1) } } {\Gamma(\frac{n-1}2) }$

where $\Gamma(\cdot)$ is the gamma function. As n grows, c₄ is asymptotically equal to (4n-4)/(4n-3).

For more information, refer to the ASQC Glossary and Tables for Statistical Quality Control, the ASTM Manual on Presentation of Data and Control Chart Analysis, Montgomery (1991), and Wadsworth and others (1986).

In other chapters, c₄ is written as c₄(n) to emphasize the dependence on n.

You can use the constant c₄ to calculate an unbiased estimate $(\hat{\sigma})$ of the standard deviation $\sigma$ of a normal distribution from the sample standard deviation of n observations:

$\hat{\sigma} = ({sample standard deviation})/c_4$

where the sample standard deviation is calculated using n-1 in the denominator. In the SHEWHART procedure, c₄ is used to calculate control limits for s charts, and it is used in the estimation of the process standard deviation based on subgroup standard deviations.

Examples

The following statements result in a value of 0.939985603:

   data;
      constant=c4(5);
      put constant;
   run;

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