A Q-Q plot reveals at a glance whether two distributions share a shape
Histograms can hide the difference between a bell curve and something with dangerously heavy tails. Sort two sets of numbers, plot them against each other, and see whether the dots form a straight line. That is a Q-Q plot, one of the most revealing and least known tools in statistics.
The name stands for quantile-quantile. Each point pairs a quantile of one distribution, say its tenth percentile, with the same quantile of another. If the two distributions are identical, the points fall along the 45-degree line where y equals x. If one is simply a shifted or stretched version of the other, the points still form a straight line, just a different one. Because it compares whole distributions rather than matched pairs, the two groups need not be paired or even the same size.
Building one is easiest with two datasets of equal length: sort both from smallest to largest and plot them against each other in order. With unequal sizes, you interpolate so that each point compares values at the same underlying probability. Most often, though, the plot pits real data against a theoretical model, such as a normal probability plot that checks whether measurements look bell-shaped.
Reading the shape is the skill. Points always climb from left to right. A trend flatter than the diagonal means the horizontal distribution is more spread out; a steeper one means the vertical distribution is. An arc or S-curve signals that one distribution is more skewed or has heavier tails than the other. Fitting a line through the points gives rough measures of relative location and scale, and the correlation between paired quantiles offers a numerical score of how well the shapes match.
One subtlety is choosing which theoretical quantiles to plot against n data points. Statisticians have proposed many formulas, with James Filliben suggesting one in 1975, though for large samples the differences hardly matter. The R language includes ready-made functions for these plots, and they are widely used in genetics to scan genome-wide association results.
Source: Q–Q plot
