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.

**Questions**? Post a comment and I’ll do my best to help!

Thank u it’s really good explanation and very helpfull

Shouldn’t the headings be:

First (s=1).

Second (s=2).

Third (s=3).

Fourth (s=4).

Thanks for spotting that. It’s fixed.

Shouldn’t there be a denominator after the equals sign?

Yes! Thanks for catching that…it’s now fixed.

should be like that

i dont understand at all..

You might want to note the difference between moments about zero (shown here) and central moments (i.e. moments about the mean) because the 2nd moment about zero as you have shown here is *not* the same thing as variance (which is the moment about the mean or [(x1-u)^2 + (x2-u)^2 + … + (xn-u)^2]/n).

Otherwise, I like your explanation!

Thanks for your suggestion. I decided to change the formula to the one for variance (as that’s the one most used in statistics). I also clarified that it’s the moment about the mean. I hope that’s clearer :)

Please how do you obtain the kth moment of Transmuted Lindley Distribution?

Go to this article: Lindley distribution, then scroll down to

moments. You’ll find the kth moment of Transmuted Lindley Distribution there.Regards,

Stephanie

Thank you so much for the help and clarification.

how can one get moments about an arbitrary point in a grouped data?.And what does [r] stand for in the formula for calculating moments.Thank you.

Which formula has r?

What is the moment about the origin n (e.g n=1,2,4,….) for a set of numbers, thank you.

Does this help? http://www.statisticshowto.com/method-moments/

May I know the final expression of the 5th and the 6th raw moment of binomial distribution

Manju, there isn’t a one-size fits all answer for that, i.e. there’s no simple expression. Anything beyond the 2nd moment is very rarely used in statistics. You may want to start here: http://www.maa.org/sites/default/files/benyi01200522430.pdf

thank for it

it really helpful thanks