## 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 .

*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 *d*_{3}. The values returned
by the D3 function are accurate to ten decimal places.
The value *d*_{3} can be expressed as

where

where is the standard normal cumulative distribution
function and *d*_{2} is the expected range. Refer to Tippett (1925).

In other chapters *d*_{3} is written as *d*_{3}(*n*) to emphasize the
dependence on *n*.

In the SHEWHART procedure, *d*_{3} 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 (1996), and Wadsworth and others (1986).

You can use the constant *d*_{3} to calculate an unbiased estimate
of the standard deviation of the range of
a sample of *n* normally distributed observations from the sample
range of *n* observations:

You can use the D2 function to calculate *d*_{2}.

*Examples*

The following statements result in a value of 0.8640819411:

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

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.