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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 .

### 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

where

where 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 (1996), and Wadsworth and others (1986).

You can use the constant d3 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 d2.

### Examples

The following statements result in a value of 0.8640819411:
   data;
constant=d3(5);
put constant;
run;


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