What problem or game are variance and standard deviation optimal solutions for?

For a given random variable (or a population, or a stochastic process), mathematical expectation is the answer to a question What point forecast minimizes the expected square loss?. Also, it is the optimal solution to a game Guess the next realization of a random variable (or a new draw from a population), and I will punish you by the squared distance between the value and your guess if you have linear disutility in terms of the punishment. Median is the answer to a corresponding question under absolute loss and mode is the answer under "all or nothing" loss.

Questions: Does variance and standard deviation answer any similar questions? What are they?

The motivation for this question stems from teaching basic measures of central tendency and spread. Whereas the measures of central tendency can be motivated by decision-theoretic problems above, I wonder how one could motivate the measures of spread.

• Very interesting question. My initial approach would be that the "game" is qualitatively the same as what you already describe, except that the question expects (no pun intended) the answer to be about a range of values instead of one point, since spread without a point of reference is rather incomplete (if not meaningless) information. – Emil Sep 3 '18 at 13:08
• Note that variance is itself an expectation - if $Y=(X-\mu)^2$ then $\text{Var}(X)=E(Y)$. – Glen_b Sep 3 '18 at 22:54
• @Glen_b, you are right, and I got that (I should have included that in the question text). "Guess the difference between the next value and the expectation and I will punish you quadratically" would be the game. Is that the best there is? Does not sound very practical or very fun a game, IMHO. – Richard Hardy Sep 4 '18 at 6:12