# How to estimate the standard deviation of residuals on a residual plot?

So I understand that on the normal histogram and scatter plot, we estimate the standard deviation by looking at the portion which contains $\frac23$ of the data.

In terms of a residual plot from a regression, however, it is a randomly sized region around (x or y)= 0. How do we estimate the standard deviation of the residuals?

• Please explain what you mean by a "residual plot" (there are many ways to plot residuals and many possible models to produce residuals), and what you meany by "(x or y)= 0." Perhaps you could post an image of such a plot?
– whuber
Oct 4, 2016 at 17:59
• @whuber Actually, I think we can reasonably assume we're dealing with ordinary linear regression. It doesn't matter what the residuals are plotted against as long as the plotted residuals are not standardized or otherwise transformed in some way (well, they could reasonably be divided by $\sqrt{1-h_{ii}}\,$ I suppose, to adjust for the effect of leverage on variance, but outside that sort of adjustment, untransformed). SInce we can state such restrictions in an answer I think this can be reopened still have an answer that would be very useful for beginners who don't quite see how to do it. Oct 5, 2016 at 1:30
• @CoolKid if I have somehow misinterpreted your question, you can post a new question with the necessary clarifications (as whuber describes) to make it explicit what you need Oct 5, 2016 at 5:21

You can do this on a plot of residuals - plain, raw residuals, not standardized or studentized - vs anything at all - residuals vs fitted, residuals vs x (indeed any predictor in a multiple regression), residuals vs index number, residuals vs a variable you didn't use in the regression, whatever you like. It's only the fact that the residuals ($y-\hat{y}$) are plotted on one of the axes that's important.

The residuals are most typically plotted on the y-axis so in that case it's the y-axis you pay attention to. It doesn't make a difference what's on the x-axis when doing this, you don't pay any attention to it. [If your residuals have been plotted on the x-axis for some reason then that is the direction you pay attention to.]

Here's a plot of residuals vs fitted values for regression on a particular data set. There are 50 observations (though one of the residuals near 0 is not visible because it lays exactly over another point). Since we want an interval containing 2/3 of the points, we'll have 1/6 of the points above the top of the interval and 1/6 of the points below it.

Since 50/6 is 8-and-a-bit, let's leave 8 points outside either end, so we'll mark our lines about halfway between the 8th and 9th points from the top and bottom of the plot. [You could mount an argument for drawing the lines right at the 9th point and saying that's 8.5 points outside each of the lines and 50-8.5-8.5 = 33 between them, but this is pretty approximate anyway so I won't worry too much. For example, 2/3 is slightly too small so I'll stick with 8 points outside each end of the interval.]

Measuring the distance between those two horizontal lines gives a distance of about 2.2 in this case. Since that interval containing the middle 2/3 of the data should be about 2 standard deviations wide, we estimate the standard deviation to be $s\approx 2.2/2 = 1.1$.

The actual standard deviation of residuals from the regression output ($\sqrt{SSE/(n-2)}\,$) was $s = 1.102$.

Not too shabby!