Statistics How To

Moment in Statistics: Defintion, Examples

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 sth moment = (x1s + x2s + x3s + . . . + xns)/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 1st moment around zero for discrete distributions = (x11 + x21 + x31 + . . . + xn1)/n
= (x1 + x2 + x3 + . . . + xn)/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):
integral moments

Second (s=2).

The 2nd moment around the mean = Σ(xi – μx)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 3rd moment = (x13 + x23 + x33 + . . . + xn3)/n

The third is skewness.
what is a moment

Fourth (s=4).

The 4th moment = (x14 + x24 + x34 + . . . + xn4)/n

The fourth is kurtosis.
what is a fourth 2

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.


If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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Moment in Statistics: Defintion, Examples was last modified: December 9th, 2017 by Stephanie

21 thoughts on “Moment in Statistics: Defintion, Examples

  1. Muntoo Meddler

    Shouldn’t the headings be:

    First (s=1).
    Second (s=2).
    Third (s=3).
    Fourth (s=4).

  2. hbogert

    The 1st moment = (x11 + x21 + x31 + . . . + xn1)/n
    = (x1 + x2 + x3 + . . . + xn).

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

  3. John

    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!

  4. Andale Post author

    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 :)

  5. collins pchumba loktari

    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.