QQ Plot
Study guide: Exam MAS-I
A QQ Plot (Quantile-Quantile Plot) is a graphical diagnostic that compares the quantiles of a dataset against the quantiles of a theoretical distribution (or another dataset) to assess goodness of fit.
Construction: Plot the
i-th order statisticx_{(i)}on the vertical axis against the theoretical quantileF^{-1}(p_i)on the horizontal axis, with plotting positionp_i = \frac{i}{n+1}(or\frac{i-0.5}{n}).
Interpretation (sample quantiles on the vertical axis):
| Pattern | Implication |
|---|---|
| Points fall on a straight line | Data fits the theoretical distribution well |
| Low end above the line, high end below it | Data has lighter (shorter) tails than the theoretical distribution |
| Low end below the line, high end above it | Data has heavier (longer) tails — fat-tailed data |
| Points above the line in the upper tail | Right tail is heavier than theoretical |
| Systematic curvature | Distribution family is wrong |
- A Normal QQ plot plots sample quantiles against Normal quantiles; a straight line indicates normality
- Used to assess model assumptions in Generalized Linear Models and linear regression (Residual Plots on a QQ scale check normality of residuals)
- Both tail cases can bend the middle of the plot into an S-like pattern, so read the direction of the departure at the two ends, not the shape of the middle
Interpreting a Normal QQ Plot of Residuals After fitting a linear regression to claim severity data, a Normal QQ plot of the residuals shows points that closely follow the reference line in the center but curve sharply upward in the upper-right tail. What does this indicate?
Answer The upward curve in the upper tail indicates that the residuals have a heavier right tail than the Normal distribution. This suggests the normality assumption for residuals may be violated — perhaps a log transformation of severity or a GLM with a Gamma distribution would be more appropriate.