Is there such a thing as 'percentage of standard deviation explained'? I have been reading about percentage of variance explained.
As nicely put in this blog post variance is hard to visualize and not directly comparable to the mean.
https://assessingpsyche.wordpress.com/2014/07/10/two-visualizations-for-explaining-variance-explained/
So I wonder, since from variance we can calculate standard deviation, can we calculate percentage of standard deviation explained since we can do so for variance? or is this nonsense?
 A: Percentage of variance can be meaningful because it the regression is it's additive. The percentage of the standard deviation is not going to be additive or linear, which will make this measure difficult to use and understand.
In a typical regression $y=X\beta+\varepsilon$ you have $\sigma^2_y=\sigma^2_{X\beta}+\sigma^2\varepsilon$, so you can re-write it as:
$$1=\frac {\sigma^2_{X\beta}}{ \sigma^2_y}+\frac{\sigma^2_\varepsilon}{\sigma^2_y},$$
where the first term on the right hand side is the percentage of variance explained. This nicely allocated the variance into explained and unexplained components.
You can see that this doesn't work out nicely for the standard deviation:
$$1\ne \frac {\sigma_{X\beta}}{ \sigma_y}+\frac{\sigma_\varepsilon}{\sigma_y},$$
actually you can show that:
$$\frac {\sigma_{X\beta}}{ \sigma_y}+\frac{\sigma_\varepsilon}{\sigma_y}=
\sqrt{1+2\frac {\sigma_{X\beta}}{ \sigma_y}\frac{\sigma_\varepsilon}{\sigma_y}},$$
i.e. the sum of percentages of explained and unexplained standard deviations is greater than 100%. For instance, if only a half of variance is explained, in terms of the standard deviation it would look like 71% is explained, but it's misleading, because in this case 71% is unexplained too! Here's the math:
$$\frac {\sigma^2_{X\beta}}{ \sigma^2_y}=\frac{\sigma^2_\varepsilon}{\sigma^2_y}=\frac 1 2$$
$$\frac {\sigma_{X\beta}}{ \sigma_y}=\frac{\sigma_\varepsilon}{\sigma_y}=\frac{1}{\sqrt 2}\approx 0.71$$
