# Why do we say “Residual standard error”?

A standard error is the estimated standard deviation $\hat \sigma(\hat\theta)$ of an estimator $\hat\theta$ for a parameter $\theta$.

Why is the estimated standard deviation of the residuals called “residual standard error” (e.g., in the output of R’s summary.lm function) and not “residual standard deviation”? What parameter estimate do we equip with a standard error here?

Do we consider each residual as an estimator for “its” error term and estimate the “pooled” standard error of all these estimators?

I think that phrasing is specific to R’s summary.lm() output. Notice that the underlying value is actually called “sigma” (summary.lm()\$sigma). I don’t think other software necessarily uses that name for the standard deviation of the residuals. In addition, the phrasing ‘residual standard deviation’ is common in textbooks, for instance. I don’t know how that came to be the phrasing used in R’s summary.lm() output, but I always thought it was weird.