Syntax

D3

where is the sample size, with .

Description

The D3 function returns the standard deviation of the range of 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 . The values returned by the D3 function are accurate to ten decimal places.

The value can be expressed as

where

where is the standard normal cumulative distribution function and is the expected range. Refer to Tippett (1925).

In other chapters is written as to emphasize the dependence on .

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

For more information, refer to the American Society for Quality Control (1983), the American Society for Testing and Materials (1976), Montgomery (1996), and Wadsworth, Stephens, and Godfrey (1986).

You can use the constant to calculate an unbiased estimate of the standard deviation of the range of a sample of normally distributed observations from the sample range of observations:

You can use the D2 function to calculate .

Examples

The following statements result in a value of 0.8640819411:

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