A P-P plot (also Probability-Probability plot or Percent-Percent plot) is a probability plot for assessing how closely two data sets agree, which plots the two cumulative distribution functions against each other.
The Q-Q plot is more widely used, but they are both referred to as "the" probability plot, and are potentially confused.
A P-P plot plots two cumulative distribution functions (cdfs) against each other: given two probability distributions, with cdfs "F" and "G", it plots (F(z),G(z)) as z ranges from to As a cdf has range [0,1], the domain of this parametric graph is and the range is the unit square
Thus for input z the output is the pair of numbers giving what percentage of f and what percentage of g fall at or below z.
The comparison line is the 45° line from (0,0) to (1,1) – the distributions are equal if and only if the plot falls on this line – any deviation indicates a difference between the distributions.
As an example, if the two distributions do not overlap, say F is below G, then the P-P plot will move from left to right along the bottom of the square – as z moves through the support of F, the cdf of F goes from 0 to 1, while the cdf of G stays at 0 – and then moves up the right side of the square – the cdf of F is now 1, as all points of F lie below all points of G, and now the cdf of G moves from 0 to 1 as z moves through the support of G.
As this example illustrates, if two distributions are separated in space, the P-P plot will give very little data – it is only useful for comparing probability distributions that have nearby or equal location. Notably, it will pass through the point (1/2, 1/2) if and only if the two distributions have the same median.
P-P plots are sometimes limited to comparisons between two samples, rather than comparison of a sample to a theoretical model distribution. However, they are of general use, particularly where observations are not all modelled with the same distribution.
However, they have found some use in comparing a sample distribution from a known theoretical distribution: given n samples, plotting the continuous theoretical cdf against the empirical cdf would yield a stair-step (a step as z hits a sample), and would hit the top of the square when the last data point was hit. Instead one only plots points, plotting the observed kth observed points (in order: formally the observed kth order statistic) against the k/(n+1) quantile of the theoretical distribution. This choice of "plotting position" (choice of quantile of the theoretical distribution) has occasioned less controversy than the choice for Q-Q plots. The resulting goodness of fit of the 45° line gives a measure of the difference between a sample set and the theoretical distribution.
A P-P plot can be used as a graphical adjunct to a tests of the fit of probability distributions, with additional lines being included on the plot to indicate either specific acceptance regions or the range of expected departure from the 1:1 line. An improved version of the P-P plot, called the SP or S-P plot, is available, which makes use of a variance-stabilizing transformation to create a plot on which the variations about the 1:1 line should be the same at all locations.