Percentile Definitions

The BOXPLOT Procedure

Percentile Definitions

You can use the PCTLDEF= option to specify one of five definitions for computing quantile statistics (percentiles). Suppose that n equals the number of nonmissing values for a variable and that x₁, x₂, ... ,x_n represents the ordered values of the analysis variable. For the tth percentile, set p =t/100.

For the following definitions numbered 1, 2, 3, and 5, express np as

np = j + g

where j is the integer part of np, and g is the fractional part of np. For definition 4, let

(n+1)p=j+g

The tth percentile (call it y) can be defined as follows:

PCTLDEF=1

weighted average at x_np

y = (1 - g)x_j + gx_j+1

where x₀ is taken to be x₁

PCTLDEF=2

observation numbered closest to np

y = x_i

where i is the integer part of np + 1/2 if $g \neq 1/2$ . If g=1/2, then y=x_j if j is even, or y=x_j+1 if j is odd.

PCTLDEF=3

empirical distribution function

y = x_j if g = 0

y = x_j+1 if g > 0

PCTLDEF=4

weighted average aimed at x_p(n+1)

y=(1 - g)x_j + gx_j+1

where x_n+1 is taken to be x_n

PCTLDEF=5

empirical distribution function with averaging

y = (x_j + x_j+1)/2 if g = 0

y = x_j+1 if g > 0

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