# Statistical Moments and R

In statistics, moments are quantities that are related to the shape of a set of numbers. “Shape of a set of numbers,” means “what a histogram based on the numbers looks like” — how spread out it is, how symmetric it is, and more.

A raw moment of order k is the average of all numbers in the set, with each number raised to the kth power before you average it. So the first raw moment is the arithmetic mean. The second raw moment is the average of the squared scores. The third raw moment is the average of the cubed scores, and so on.

A central moment is based on the average of deviations of numbers from their mean. (Beginning to sound vaguely familiar?) If you square the deviations before you average them, you have the second central moment. If you cube the deviations before you average them, that’s the third central moment. Raise each one to the fourth power before you average them, and you have the fourth central moment.

Two quick questions: 1. For any set of numbers, what’s the first central moment? 2. By what other name do you know the second central moment?

Two quick answers: 1. Zero. 2. Population variance.