With 5,000 rows, a regression's residuals are clearly non-normal in the Q-Q plot. Are the coefficient p-values still usable?

## Rank the assumptions by what they damage A linear model leans on four things, and they are not equally important: 1. **The mean function is right.** If it is not, the coefficients are biased and predictions are systematically wrong by region. Nothing about sample size repairs this. This is the assumption to worry about first. 2. **The errors are independent.** Violating it usually leaves the estimate roughly centred but makes the reported standard errors badly wrong, often far too small, which is worse than it sounds because you get confident wrong answers. 3. **The error variance is constant.** Violating it leaves the estimate unbiased but makes the usual standard-error formula the wrong formula. 4. **The errors are normal.** This one buys exactness in small samples and correct prediction intervals; it is not needed for the estimate to be unbiased, and at large n it is not needed for approximately correct coefficient inference either. The classical optimality result for least squares — that among linear unbiased estimators it has the smallest variance — requires the first three and says nothing about normality. That is the fact to reach for when someone insists all regression inference needs normal errors. ## Why n = 5,000 protects the coefficients A least-squares coefficient is a weighted sum of the outcomes: a linear combination of many independent contributions. Large-sample theory says such a combination has an approximately normal sampling distribution as the sample grows, as long as the errors have finite variance, are independent, and no small handful of observations carries most of the weight in the design. So the object that has to be normal for a p-value to be right is not the error distribution but the **sampling distribution of the estimator**, and at n = 5,000 the second is close to normal even when the first is visibly not. The t-distribution used for the test converges to the normal at those degrees of freedom too, so the arithmetic barely changes. The p-value is approximate rather than exact, but the approximation error is typically negligible next to everything else that is uncertain about the model. The conditions matter, though. 'No small handful of observations carries most of the weight' is a real requirement: with a very lopsided predictor distribution, a few rows can dominate, and the effective sample size for that coefficient is much smaller than 5,000. Extremely heavy tails, where variance is barely finite, also slow the approximation down. ## Why prediction intervals are not protected There are two intervals people conflate. A **confidence interval for the mean response** at a given set of predictor values expresses uncertainty about where the regression line sits. Its width is driven by the uncertainty in the estimated coefficients, which shrinks roughly like `1 / sqrt(n)`. Large n makes it narrow and approximately correct. A **prediction interval for a single new observation** expresses uncertainty about where one future point will land. Its width has two parts: the same shrinking estimation uncertainty, plus the irreducible spread of a single error, of size about the error standard deviation `sigma`. That second part does not shrink at all. As n grows, the interval converges to roughly `yhat` plus or minus a multiple of `sigma` — and which multiple gives 95 percent coverage depends entirely on the **shape** of the error distribution. If the errors are right-skewed, the symmetric normal-based interval is wrong in a specific way: it is too wide on the low side and too short on the high side, so the actual coverage is not 95 percent and, worse, the misses are asymmetric. This is exactly the situation in which practitioners are surprised, because they know 'large n fixes normality' as a slogan and apply it to the wrong quantity. ## The small-sample case At n = 15 the picture reverses. The estimator's sampling distribution is not guaranteed to be close to normal, so the t-based p-value and confidence interval are only exact if the errors really are normal. With visible skew or heavy tails at that sample size, both the coefficient inference and the prediction interval are suspect. Sensible responses are to model the response on a scale where the errors behave, to use an interval procedure that does not assume a normal error shape, or to be honest that the model supports a direction rather than a calibrated number. ## Non-normality as a symptom Even when the p-values survive, a clearly non-normal residual picture at large n is information worth using. - **Strong right skew** often means the response is multiplicative or bounded below, and the model would be better on a different scale. - **Heavy tails** often mean a mixture of subpopulations treated as one, or a rare regime the model has no term for. Least squares squares the errors, so those rows quietly buy a lot of influence over the fit. - **Bimodality** in the residuals almost always means a missing categorical distinction. So the professional answer is not just 'it is fine at this sample size'. It is: the coefficient inference is approximately valid, the prediction intervals are not, and the picture is telling me something about the model I should investigate before I ship it.