Question

How does r squared related to standard deviation?

Answer 1

R-squared measures how well the regression line fits the data. This is why higher R-squared values correlate with lower standard deviation.

The easiest way to see this is by playing with a data set in a spreadsheet software: make a dot plot, right click on a point to add a regression line, and tick the option to show the R-squared. Then, use the STDEV function to calculate the standard deviation.

Answer 2

I always think of this as measures of spread so the spread from the regression line and the spread from the distribution should be highly correlated.

so consider the regression line as a line that runs on the y axis at 0. Why 0 , well because that's what the mean is when we look at the distribution (theoretically for normal distribution). Now if we have say two points at `(1,0)` and `(-1,0)` then the distance from each of the points to the line would look like the standard deviation if we overlay a bell shape on top of the graph.