For convenience and context, I'm looking at the formula for standard error of the slope coefficient from here: https://www.statology.org/standard-error-of-regression-slope/

enter image description here

If X is not normally distributed but the residuals are, is the estimated slope coefficient's distribution characterized by the T-distribution?

  • $\begingroup$ Residuals are never independently Normally distributed, so you likely are trying to ask about the errors in the model. $\endgroup$
    – whuber
    May 31 at 21:00
  • $\begingroup$ @whuber, yes your comment makes sense. $\endgroup$
    – Jitty
    Jun 1 at 1:55
  • $\begingroup$ In Kutner's Applied Linear Statistical Models Ch2 Page 42, it explains that the sampling distribution of $b_1$ is normal because $b_1$ is a linear combination of $Y_i$ and if we assume $Y_i$ is normally distributed, then a linear combination of $Y_i$ is also normally distributed. I think this is the answer I was looking for, if I'm not misunderstanding the passage. Would love feedback on this.. $\endgroup$
    – Jitty
    Jun 19 at 0:58
  • $\begingroup$ It contradicts what you write. Because the estimated slope is Normal, it is not characterized by a Student t distribution. $\endgroup$
    – whuber
    Jun 19 at 17:01

1 Answer 1


Yes! The usual linear regression model does not assume that X is normal. The assumptions are independence, homoscedasticity, linearity and normality of residuals.

  • 2
    $\begingroup$ Your comment about residuals is not quite correct, as my linked answer discusses. $\endgroup$
    – Dave
    May 31 at 19:47
  • $\begingroup$ The model makes no assumptions whatsoever about residuals: you seem to use "residual" (observed difference between response and fitted value) instead of "error" (a random variable in the model). $\endgroup$
    – whuber
    May 31 at 21:01

Not the answer you're looking for? Browse other questions tagged or ask your own question.