QQplot

A QQ plot, short for quantile-quantile plot, is a graphical method used to assess whether a dataset follows a specified theoretical distribution or to compare the distributions of two datasets. It works by plotting corresponding quantiles from the data and the reference distribution. If the data come from the reference distribution, the points lie approximately on a straight line, often the 45-degree line.

To construct a QQ plot, the data are first ordered, and sample quantiles are paired with the theoretical quantiles of the reference distribution at corresponding probabilities (for example i/(n+1) or (i-0.5)/n for the i-th order statistic). The plot uses the theoretical quantiles on one axis and the observed quantiles on the other. Variants exist for different reference distributions, such as the normal, t, exponential, or gamma distributions, and QQ plots can also compare two samples by plotting their respective quantiles against one another.

Interpretation centers on linearity. A roughly straight line suggests the sample distribution matches the reference distribution. Systematic deviations indicate departures: curvature can reflect skewness, while deviations at the ends can reveal heavier or lighter tails or the presence of outliers. QQ plots are commonly used to assess normality and to diagnose distributional assumptions in modeling; they can also highlight differences between two samples or validate fit in probability plots against non-normal references.

Limitations include subjectivity in visual interpretation, sensitivity to sample size and outliers, and the absence of a formal test. They are best used alongside quantitative normality tests and diagnostic tools.