This choice of "plotting position" (choice of quantile of the theoretical distribution) has occasioned less controversy than the choice for Q–Q plots. 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. 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. However, they are of general use, particularly where observations are not all modelled with the same distribution. P–P plots are sometimes limited to comparisons between two samples, rather than comparison of a sample to a theoretical model distribution. Notably, it will pass through the point (1/2, 1/2) if and only if the two distributions have the same median. (need a graph for this paragraph)Īs 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. Example Īs 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. The degree of deviation makes it easy to visually identify how different the distributions are, but because of sampling error, even samples drawn from identical distributions will not appear identical. The comparison line is the 45° line from (0,0) to (1,1), and the distributions are equal if and only if the plot falls on this line. 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. Given two probability distributions, with cdfs " F" and " G", it plots ( F ( z ), G ( z ) ) This behavior is similar to that of the more widely used Q–Q plot, with which it is often confused.Ī P–P plot plots two cumulative distribution functions (cdfs) against each other: It works by plotting the two cumulative distribution functions against each other if they are similar, the data will appear to be nearly a straight line. What do you think the unknown distribution looks like? Draw a rough sketch of a possible PDF for the unknown distribution.In statistics, a P–P plot ( probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. Generate some samples from the unknown distribution.How do the data points compare to the normal CDF? Is it clearer with a linear or probability scale? Generate some samples from one of the skewed distributions.How do the data points compare to the normal CDF? Generate some samples from a normal distribution.Drag the dashed green line up and down to see how the two vertical axes are related. ![]() Click ‘Show normal curve’ to see the normal distribution that the probability scale is based on. Click ‘Probability scale’ to transform the vertical axis to a probability scale. ![]() Click ‘Show normal CDF’ to show the CDF of a normal distribution with the same mean and standard deviation as the sample.Īt first, the vertical axis shows the quantiles on a linear scale. Click ‘New sample’ to generate new data, or choose between a normal, left skewed, or right skewed distribution for sampling, or an unknown distribution.Ĭlick ‘Show estimated CDF’ to show an estimate of the empirical CDF based on the data. The applet initially shows data from a sample of size 19, sorted and plotted against the corresponding quantile on the vertical axis. This applet shows the relationship between a plot of an estimated empirical CDF and a normal probability plot.
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