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BOXCHART Statement: SHEWHART Procedure

Percentile Definitions

You can use the PCTLDEF= option to specify one of five definitions for computing quantile statistics (percentiles). Let n equal the number of nonmissing values for a variable, and let $x_{1}, x_{2},\ldots ,x_{n}$ represent the ordered values of the process variable. For the tth percentile, set $p =t/100$ , and express $np$ as

$np = j + g$

where j is the integer part of $np$ , and g is the fractional part of $np$ .

The tth percentile (call it y) can be defined in five ways, as described in the next five sections.

PCTLDEF=1

This uses the weighted average at $x_{np}$

$y = (1- g)x_{j} + gx_{j+1}$

where $x_{0}$ is taken to be $x_{1}$ .

PCTLDEF=2

This uses the observation numbered closest to $np$

$y=x_{i}$

where i is the integer part of $np + 1/2$ .

PCTLDEF=3

This uses the empirical distribution function

$\begin{array}{ll} y=x_{j} & \mbox{if } \, g=0 \\ y=x_{j+1} & \mbox{if } \, g > 0 \\ \end{array}$

PCTLDEF=4

This uses the weighted average aimed at $x_{p(n+1)}$

$y=(1-g)x_{j} + gx_{j+1}$

where $(n+1)p=j+g$ , and where $x_{n+1}$ is taken to be $x_{n}$ .

PCTLDEF=5

This uses the empirical distribution function with averaging

$\begin{array}{ll} y=(x_{j} + x_{j+1})/2 & \mbox{if } \, g = 0 \\ y=x_{j+1} & \mbox{if } \, g > 0 \\ \end{array}$

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