Functions

D3 Function

Subsections:

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 $d_3$. The values returned by the D3 function are accurate to ten decimal places.

The value $d_3$ 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 $d_2$ 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 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 $d_3$ 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:

\[ \hat{\sigma }_ R = (\mbox{sample range})(d_3/d_2) \]

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;