RMS error of the SD line Freedman's Statistics (p. 147 of the hardcover edition) says that if $Y$ is estimated using the SD line (rather than the regression line) then the RMS error of the residuals is $\sqrt{2(1-|r|)}\sigma_Y$, where $r$ is the correlation. What's the derivation for this result (and under what assumptions is it true)?
 A: This textbook emphasizes two fundamental ideas about regression:


*

*Standardize the variables.  That is, choose units of measurement in which their means are zero and they have unit variances.

*Model a "football shaped" scatterplot as an iid realization of an approximately binormal variable.
Everything follows from these ideas.  To make it even simpler, let's negate $Y$ if necessary so that the correlation, $\rho$, is non-negative.  The regression line is
$$y = \rho x,$$
whence variation (of $Y$) around that line is in the form
$$Y = \rho X + \varepsilon$$
with $\varepsilon$ uncorrelated with $X$: this will be crucial in subsequent calculations.
Computing variances (remembering that we have standardized both $X$ and $Y$) gives
$$1 = \text{Var}(Y) = \text{Var}(\rho X + \varepsilon) = \rho^2 \text{Var}(X) + \text{Var}(\varepsilon) = \rho^2 + \text{Var}(\varepsilon),$$
whence $\text{Var}(\varepsilon) = 1 - \rho^2$.
Consider now the variation of $Y$ around the "standard deviation" line which, in standardized coordinates, has equation $y = f(x) = x$.  (This is where the simplification to $\rho \ge 0$ is used; for negative $\rho$, $f(x)=-x$.)  Exploiting what we know about the distributions of $Y$, $X$, and $\varepsilon$, as well as bilinearity of covariance, we split the variation of $Y$ around the SD line into the variation around the regression line plus the variation of the SD line around the regression line, thereby deducing that the expected squared error of residuals must be
$$\mathbb{E}[(Y-f(X))^2] = \mathbb{E}[(Y - \rho X - (1-\rho)X)^2] = \mathbb{E}[(\varepsilon - (1-\rho)X)^2]$$
$$=(1-\rho^2) + (1-\rho)^2 = 2(1-\rho).$$
Taking square roots gives the stated result.
Looking back at this derivation, it appears that binormality is not essential.  The crucial assumptions concern covariances; in particular, we only need the regression residuals $\varepsilon$ to be uncorrelated with $X$.
