TITLE: Is Standard Deviation the same as Entropy?
QUESTION [24 upvotes]: We know that standard deviation (SD) represents the level of dispersion of a distribution. Thus a distribution with only one value (e.g., 1,1,1,1) has SD equals to zero. Similarly, such a distribution requires little information to be defined. On the other hand, a distribution with high SD requires many bits of information to be defined, therefore we can say its entropy level is high.

So my question: is SD the same as entropy? 
If not, which relationship exist between these two measurements?

REPLY [13 votes]: More counterexamples: 

Let X take be a discrete random variable taking two values $(-a,a)$ with equal probability. Then the variance $\sigma_X^2=a^2$ increases with $a$, but the entropy is constant $H(X)=1$ bit.
Let $X$ be a discrete rv taking values on $1 \cdots N$ with some arbitrary non-uniform distribution $p(X)$. If we permute the values of $p(X)$, the variance will change (decrease if we move the larger values towards the center), but the entropy is constant.
Let $X$ be a continuous rv with uniform distribution on the interval $[-1,1]$ $p(X)=1/2$. Let modify it so that its probability (on the same support) is bigger towards the extremes : say, $p(Y)=|Y|$. Then $\sigma^2_Y > \sigma_X^2$ but $H(Y)< H(X)$ (the uniform distribution maximes the entropy for a fixed compact support).