Varentropy ($V_H(X)$)

The varentropy of a discrete random variable $X$ is the variance of its information content:It measures variation in surprise across outcomes, while Shannon entropy is the expected value of that surprise. Assuming finite Shannon entropy, the variance may still be infinite; finite varentropy requires an integrable squared logarithm.

For a Bernoulli random variable with success probability $0<p<1$,Indeed the information content takes values $-{\log }_{2}p$ and $-{\log }_2(1-p)$, and the variance of a two-valued random variable is $p(1-p)$ times the squared difference. At $p=1/2$, every outcome supplies one bit, so Shannon entropy is maximal but varentropy is zero. For independent and identically distributed random variables, the information content of a length-$n$ word is a sum, giving variance $n{V}_H(X)$ when finite.