Statistics Definitions > Moment

If you do a casual Google search for “What is a Moment?”, you’ll probably come across something that states the first moment **is** the mean or that the second measures how wide a distribution is (the variance). Loosely, these definitions are right. Technically, a moment is defined by a mathematical formula that just so happens to equal formulas for some measures in statistics.

## The formula.

The *s*th moment = (x_{1}^{s} + x_{2}^{s} + x_{3}^{s} + . . . + x_{n}^{s})/n.

This type of calculation is called a geometric series. You should have covered geometric series in your college algebra class. If you didn’t (or don’t remember how to work one), don’t fret too much; In most cases, you won’t have to actually perform the calculations. You just have to have a general grasp of the meaning.

## Moment List.

### First (s=1).

The *1*st moment around zero for discrete distributions = (x_{1}^{1} + x_{2}^{1} + x_{3}^{1} + . . . + x_{n}^{1})/n

= (x_{1} + x_{2} + x_{3} + . . . + x_{n})/n.

This formula is identical to the formula to find the sample mean. You just add up all of the values and divide by the number of items in your data set. For continuous distributions, the formula is similar but involves an integral (from calculus):

### Second (s=2).

The *2*nd moment around the mean = Σ(x_{i} – μ_{x})^{2}.

The second is the variance.

**In practice, only the first two moments are ever used in statistics. However, more moments exist (they are usually used in physics):**

### Third (s=3).

The *3*rd moment = (x_{1}^{3} + x_{2}^{3} + x_{3}^{3} + . . . + x_{n}^{3})/n

The third is skewness.

### Fourth (s=4).

The *4*th moment = (x_{1}^{4} + x_{2}^{4} + x_{3}^{4} + . . . + x_{n}^{4})/n

The fourth is kurtosis.

## Higher Orders.

Higher-order terms(above the 4th) are difficult to estimate and equally difficult to describe in layman’s terms. You’re unlikely to come across any of them in elementary stats. For example, the 5th order is a measure of the relative importance of tails versus center (mode, shoulders) in causing skew. For example, a high 5th means there is a heavy tail with little mode movement and a low 5th means there is more change in the shoulders.

**Next**: Sheppard’s correction for moments calculated from grouped data.

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