In a normal Q-Q plot of regression residuals, which pattern signals heavy tails rather than right skew?

## How a Q-Q plot is built Sort the `n` residuals from smallest to largest. The `i`-th sorted residual is the empirical quantile at roughly the fraction `(i - 0.5) / n` of the data. Look up the quantile of a standard normal distribution at that same fraction. Plot the pair. Do this for every observation and you have a normal Q-Q plot: sample quantiles on the vertical axis, theoretical normal quantiles on the horizontal. If the residuals come from a normal distribution, sample and theoretical quantiles move together and the points lie on a straight line. The line's slope reflects the residual standard deviation and its intercept the mean, so the *straightness* is what matters, not whether the line happens to be the 45-degree diagonal. **State the convention before you interpret.** Some tools put theoretical quantiles on the vertical axis; every up/down reading below then flips. Saying which axis is which is part of a correct answer. ## Heavy tails A heavy-tailed distribution is symmetric-ish but produces extreme values more often than a normal with the same spread. In quantile terms, its low quantiles are more negative and its high quantiles are more positive than the normal's. On the plot: the points at the far left lie **below** the line, the points at the far right lie **above** it. The middle of the plot tracks the line reasonably well. The result is a shallow S rotated so both ends flare away from the line in opposite vertical directions. Practical reading: a handful of observations are much further from the fit than the bulk. Least squares is sensitive to those, so a small number of rows carry disproportionate weight in the estimate, and any interval that depends on the shape of the error distribution rather than on an average is affected. ## Right skew A right-skewed distribution has a long upper tail and a short, bunched lower tail. Its high quantiles are more extreme than the normal's, but its low quantiles are **less** extreme — closer to the centre. On the plot: the far-right points lie above the line, and the far-left points also lie above the line, because they are not as negative as a normal would predict. The whole curve is convex — it bends upward — and the biggest departure is at the top. It looks like one bend, not two flares. That single distinguishing test is worth memorising: **opposite directions at the two ends means tail weight; the same direction at both ends means asymmetry.** ## The other two shapes - **Light tails** (a distribution more compact than normal, sometimes from a bounded outcome): the left end sits above the line and the right end below it — the S in the opposite orientation to heavy tails. This is the least worrying pattern for inference; intervals built assuming normality are conservative. - **Left skew**: the mirror of right skew, concave, both ends below the line, biggest departure at the bottom. A discrete or heavily rounded outcome produces yet another signature: horizontal steps, because many residuals share exactly the same value. ## How much departure is real The extreme order statistics are the noisiest points on the plot. Even with genuinely normal residuals, the largest of 30 observations can sit visibly off the line; with `n = 20` you should expect the ends to wander. This is why so many people over-diagnose non-normality: they read the two or three most eye-catching points instead of the systematic shape. Two disciplines help. First, judge the *shape*, not individual points: is the deviation systematic and in a consistent direction over many points, or is it one point at the very end? Second, calibrate by simulation: generate several samples of the same size from a normal distribution, draw their Q-Q plots, and see how much wobble is normal at that `n`. Some tools draw this directly as a band around the reference line. At the other end, with tens of thousands of observations the plot will show real but tiny departures with great clarity. Then the question stops being 'is it exactly normal' — nothing is — and becomes 'is the departure big enough to matter for what I am using the model for'. ## What to do with the finding The Q-Q plot is a description, not a verdict. Right skew in the residuals often means the response should be modelled on a different scale, or that a predictor explaining the long tail is missing. Heavy tails often mean a mixture of populations that the model treats as one, or genuine rare large deviations that ordinary least squares handles poorly because it squares them. One caution that catches people out: the plot is about the **residuals**, not the response. A strongly skewed response variable is perfectly compatible with well-behaved residuals, because the predictors may explain exactly that skew. Checking the marginal distribution of the outcome instead of the residuals is a classic mistake.