The normality rule for lm: what should be normal

Let give an answer to this question:

"In a lm, is it important that the variable are distributed Gaussianly or is it the residual?"

In a linear model (lm), it is the residuals that matter — not the distribution of the original variables.

Here is the key distinction:

✅  What must be (approximately) Gaussian?

•          The residuals (errors), conditional on the predictors, should be roughly normally distributed if you want valid confidence intervals, standard errors, and p-values.

❌  What does not need to be Gaussian?

•          The raw variables (predictor or response) do NOT need to follow a normal distribution.

•          Linear regression works fine with skewed variables, non-Gaussian predictors, etc.

Why residual normality matters

Normality of residuals ensures:

•          estimates of standard errors are valid

•          hypothesis tests (t-tests, F-tests) are reliable

•          confidence intervals behave correctly

The coefficients themselves remain unbiased even if residuals are not normal, as long as:

•          the model is linear

•          errors have mean zero and constant variance

•          errors are independent

Practical rule

•          Check residuals, not raw data.

•          If residuals are clearly non-normal and sample size is small, then consider transformations or robust methods.