quantilequantile

Quantile-quantile plots, commonly called Q-Q plots, are graphical diagnostics used to compare probability distributions by plotting their quantiles against each other. They can compare a sample with a theoretical distribution (for example, normal, exponential) or compare two empirical samples. A typical workflow orders the data x(1) ≤ ... ≤ x(n). For a reference distribution with cumulative distribution function F, plotting positions p_i are chosen (such as p_i = (i − 0.5)/n), and the corresponding theoretical quantiles q_i = F^{-1}(p_i) are computed. The points (q_i, x(i)) form the Q-Q plot (some conventions plot (x(i), q_i)). If the two distributions match, the points lie approximately along the 45-degree line y = x.

Interpretation centers on deviations from the reference line. Systematic departures from linearity suggest differences in location or scale, while curvature indicates skewness. Tails that lie above or below the line signal heavier or lighter tails than the reference distribution. Outliers appear as isolated points far from the line. QQ plots are especially common for assessing normality, but they can compare any two distributions or assess how well residuals from a model follow a specified reference distribution.

Limitations and variations: QQ plots are diagnostic rather than formal tests and can be sensitive to sample size and the choice of plotting positions. They emphasize tail behavior and may be less informative about features elsewhere in the distribution. Other variants include P-P plots and Q-Q plots with different reference frames, but all serve the same core purpose: visual comparison of distributions.