Standard deviation of the residuals (s)

The standard deviation of the residuals, s, is roughly the typical size of a prediction error from the regression line, in the units of y.

The value ss answers a practical question: when you predict with this line, how far off are you usually? It is s=(yy^)2n2s = \sqrt{\dfrac{\sum (y - \hat{y})^2}{n - 2}}, so you square the residuals, average them using n2n - 2 because the line costs you two estimated coefficients, then take the square root, which returns you to the units of the response variable. For example, five residuals of 2, -1, 0, 1, and -2 give (yy^)2=4+1+0+1+4=10\sum (y - \hat{y})^2 = 4 + 1 + 0 + 1 + 4 = 10 and s=10/31.83s = \sqrt{10/3} \approx 1.83, so predictions typically miss by about 1.83 units of yy. Software prints it as s, and a smaller ss for the same response values means a tighter fit, the same tightness a residual plot shows visually.

Where this comes up

More regression and correlation terms, or browse the full statistics glossary.