P–P plot

A P–P plot plots two cumulative distribution functions (cdfs) against each other:[1] given two probability distributions, with cdfs "F" and "G", it plots ${\displaystyle (F(z),G(z))}$ as z ranges from ${\displaystyle -\infty }$ to ${\displaystyle \infty .}$ As a cdf has range [0,1], the domain of this parametric graph is ${\displaystyle (-\infty ,\infty )}$ and the range is the unit square ${\displaystyle [0,1]\times [0,1].}$

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.[2]

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. (need a graph for this paragraph)

As the above 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.[3] However, they are of general use, particularly where observations are not all modelled with the same distribution.

However, it has 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 stairstep (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.[3] 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,[4][5] 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,[4][5] 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.

• Probability plot