QQplots

QQ plots, or quantile-quantile plots, are graphical tools for assessing whether a dataset follows a given theoretical distribution or to compare two empirical distributions. They plot the ordered sample quantiles against the corresponding quantiles of a reference distribution (or against the quantiles of another sample).

Construction: sort the data x(1) ≤ x(2) ≤ ... ≤ x(n). For i = 1,...,n, compute the theoretical quantile q(i) of the reference distribution at probability p_i = (i-0.5)/n (or i/n). Plot the points (q(i), x(i)). If comparing two samples, plot the paired quantiles (x(i), y(i)) from each sample. A reference line, often drawn as y = x or a fitted line, aids interpretation.

Interpretation: If the data come from the reference distribution, the points lie close to the reference line. Deviations reveal departures from the reference, such as skewness, heavier or lighter tails, or extreme values. For a normal reference distribution, an approximately linear pattern suggests normality; systematic S-shaped curves indicate skewness or kurtosis differences; pronounced tails produce bending away from the line and potential outliers.

Applications and variations: QQ plots are widely used to assess normality in statistical modeling, to compare distributions, and to evaluate the fit of a theoretical distribution. Variants include detrended QQ plots and tail-focused versions. Software implementations include R's qqnorm and qqplot, Python's scipy.stats.probplot, and MATLAB functions.

Limitations: interpretation is subjective and relies on visual inspection. Sensitivity depends on sample size and the chosen reference distribution; small samples yield noisy plots, large samples may reveal trivial departures. QQ plots do not provide formal hypothesis tests by themselves and are often used alongside tests such as Shapiro-Wilk or Anderson-Darling.