QQavg

QQavg is a statistical measure used to summarize the overall deviation between observed data and a specified theoretical distribution in a quantile-quantile (QQ) plot. It is commonly used to assess how closely a dataset follows a reference distribution, such as the normal distribution, though it can be adapted to other references.

Calculation begins by ordering the data x(1) ≤ … ≤ x(n). For each i, compute the theoretical quantile qi from the reference distribution at probability pi = (i − 0.5)/n. A basic formulation is QQavg = (1/n) ∑i=1..n |x(i) − qi|. A standardized variant can use residuals (x(i) − qi)/s, where s is a scale estimate, to reduce the influence of the data’s magnitude and to compare datasets on a common scale.

Interpretation and use: A smaller QQavg indicates closer agreement with the reference distribution, while larger values point to deviations such as skewness or heavy tails. QQavg is typically used alongside a visual examination of the QQ plot and other normality or distributional tests. Its absolute-value form makes it more robust to outliers than squared-error alternatives, but its value is still sensitive to sample size and the chosen reference distribution.

Limitations: QQavg provides a single summary and may mask localized departures in parts of the distribution. It does not constitute a formal hypothesis test by itself, and results depend on sample size, data quality, and the appropriateness of the chosen reference distribution.