The UNIVARIATE Procedure

Construction of Quantile-Quantile and Probability Plots

Figure 4.14 illustrates how a Q-Q plot is constructed for a specified theoretical distribution. First, the $n$ nonmissing values of the variable are ordered from smallest to largest:

\[  x_{(1)} \leq x_{(2)} \leq \cdots \leq x_{(n)}  \]

Then the $i$th ordered value $x_{(i)}$ is plotted as a point whose $y$-coordinate is $x_{(i)}$ and whose $x$-coordinate is $F^{-1}\left( \frac{i- 0.375 }{n + 0.25} \right)$, where $F(\cdot )$ is the specified distribution with zero location parameter and unit scale parameter.

You can modify the adjustment constants $-$0.375 and 0.25 with the RANKADJ= and NADJ= options. This default combination is recommended by Blom (1958). For additional information, see Chambers et al. (1983). Because $x_{(i)}$ is a quantile of the empirical cumulative distribution function (ecdf), a Q-Q plot compares quantiles of the ecdf with quantiles of a theoretical distribution. Probability plots (see the section PROBPLOT Statement) are constructed the same way, except that the $x$-axis is scaled nonlinearly in percentiles.

Figure 4.14: Construction of a Q-Q Plot

Construction of a Q-Q Plot