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.