D3 Function

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Functions

D3 Function

computes the standard deviation of the range of n independent normal random variables.

Syntax

D3(n)

where n is the sample size, with $2\leq n\leq 25$ .

Description

The D3 function returns the standard deviation of the range of n independent, normally distributed random variables with the same mean and with unit standard deviation. The standard deviation returned is referred to as the control chart constant d₃. The values returned by the D3 function are accurate to ten decimal places.

The value d₃ can be expressed as

$d_3 = \sqrt{ 2 \int_{-\infty}^{\infty} \int_{-\infty}^y f(x,y) \,dx\,dy - d_2^2 }$

where

$f(x,y) = 1 - (\Phi(y))^n - (1-\Phi(x))^n + (\Phi(y) - \Phi(x))^n$

where $\Phi(\cdot)$ is the standard normal cumulative distribution function and d₂ is the expected range. Refer to Tippett (1925).

In other chapters d₃ is written as d₃(n) to emphasize the dependence on n.

In the SHEWHART procedure, d₃ is used to calculate control limits for r charts, and it is used in the estimation of the process standard deviation based on subgroup ranges.

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).

You can use the constant d₃ to calculate an unbiased estimate $(\hat{\sigma})$ of the standard deviation $\sigma_R$ of the range of a sample of n normally distributed observations from the sample range of n observations:

$\hat{\sigma}_R = ({sample range})(d_3/d_2)$

You can use the D2 function to calculate d₂.

Examples

The following statements result in a value of 0.8640819411:

   data;
      constant=d3(5);
      put constant;
   run;

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