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

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