## Comparison of P-P Plots and Q-Q Plots

A P-P plot compares the empirical cumulative distribution function
of a data set with a specified theoretical cumulative distribution
function *F*(·). A Q-Q plot compares the quantiles of a data
distribution with the quantiles of a standardized theoretical
distribution from a specified family of distributions. There are
three important differences in the way P-P plots and Q-Q plots
are constructed and interpreted:
- The construction of a Q-Q plot does not require that the
location or scale parameters of
*F*(·) be specified.
The theoretical quantiles are computed from a standard
distribution within the specified family. A linear point
pattern indicates that the specified family reasonably
describes the data distribution, and the location
and scale parameters can be estimated visually as the
intercept and slope of the linear pattern. In contrast,
the construction of a P-P plot requires the location and
scale parameters of *F*(·) to evaluate the cdf at the
ordered data values.
- The linearity of the point pattern on a Q-Q plot is
unaffected by changes in location or scale. On a P-P
plot, changes in location or scale do not necessarily
preserve linearity.
- On a Q-Q plot, the reference line representing a particular
theoretical distribution depends on the location and scale
parameters of that distribution, having intercept and slope
equal to the location and scale parameters. On a P-P plot,
the reference line for any distribution is always the
diagonal line
*y*=*x*.

Consequently, you should use a Q-Q plot if your objective is to
compare the data distribution with a family of distributions
that vary only in location and scale, particularly if you want
to estimate the location and scale parameters from the plot.

An advantage of P-P plots is that they are discriminating in
regions of high probability density, since in these regions the
empirical and theoretical cumulative distributions change more
rapidly than in regions of low probability density. For example,
if you compare a data distribution with a particular normal
distribution, differences in the middle of the two distributions
are more apparent on a P-P plot than on a Q-Q plot.

For further details on P-P plots, refer to Gnanadesikan
(1997) and Wilk and Gnanadesikan (1968).

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.