PPPlots

PPPlots, short for probability-probability plots, are graphical diagnostics used in statistics to compare the distribution of a dataset with a reference distribution or to contrast two empirical distributions. A PP plot conveys how well cumulative probabilities match between the compared distributions, rather than comparing quantiles as in Q-Q plots. In the common form, for each observation x, the empirical CDF F_hat(x) is plotted against the reference CDF F0(x). If the data come from the reference distribution, the points tend to fall near the 45-degree diagonal line y = x. Variants may plot F0(x) on the x-axis with F_hat(x) on the y-axis or plot the CDF values of two samples against each other.

PPPlots are used for goodness-of-fit assessment and model validation across fields such as statistics, finance, environmental science, and quality control. They help detect deviations in location, scale, skewness, or tail behavior that might not be obvious from other diagnostics. They are particularly useful for validating assumed theoretical distributions or for comparing two observed distributions directly.

Limitations include sensitivity to sample size, especially in the tails, and potential distortions from ties or discrete data. Interpretation can be subjective, and a PPPlot by itself is not a formal statistical test; it is typically used in conjunction with formal goodness-of-fit tests or other diagnostics to draw conclusions about distributional similarity.

Software implementations exist in major statistical environments, with PPPlots provided as part of distribution-fitting or diagnostic plotting toolkits in R, Python, MATLAB, and other platforms.