Calculus How To

Moment: Definition, Examples

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What is a Moment?

The most common definitions you’ll come across for moments include:

  • The first is the mean(average),
  • The second is a measure of how wide a distribution is (the variance).

For most basic purposes in calculus and physics, these loose definitions are all you’ll need. However, it’s important to know that there are two different kinds of “moment”: raw moments (moments about zero) and central moments. They are defined differently.

  • The rth moment about the origin of a random variable X = μ′r = E(Xr). The mean (μ) is the first moment about the origin.
  • The rth moment about the mean of a random variable X is μr = E [(X – μ)r ]. The second moment about the mean of a random variable is the variance (σ2).

The formula.

The rth moment = (x1r + x2r + x3r + … + xnr)/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. The formula might look a little daunting, but all you have to do is replace the exponent “r” with the number of the moment you’re trying to find. For example, if you want to find the first moment, replace r with 1. For the second moment, replace r with 2.

First Moment (r = 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 in statistics. 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:
integral moments

Second Moment (r = 2)

The 2nd moment around the mean = Σ(xi – μx)2.
This is equal to the variance.
The Σ symbol means to “add up”. See: What is sigma notation?

In practice, only the first two moments are ever used in statistics. Several more moments are common in physics:

Third (s = 3).

The 3rd moment = (x13 + x23 + x33 + … + xn3)/n


The 3rd moment (skewness) = (x13 + x23 + x33 +… + xn3)/n

Skewness gives you information about a distribution’s “shift”, or lack of symmetry. Distributions with a left skew have long left tails; Distributions with a right skew have long right tails.

A right-skewed distribution.


The fourth is kurtosis. Kurtosis tells you how a data distribution compares in shape to a normal (Gaussian) distribution (which has a kurtosis of 3).

Positive kurtosis (left) and negative kurtosis (right)

  • Positive kurtosis = a lot of data in the tails.
  • Negative kurtosis = not much data in your tails.

Higher R.

Higher-order terms(above the 4th) are difficult to estimate and equally difficult to describe in layman’s terms.

For example, the 5th r measures the relative importance of tails versus center (mode, shoulders) in the cause of a distribution’s skew.

  • A high 5th = heavy tail, little mode movement
  • Low 5th = more change in the shoulders.

In calculus based statistics, the first two moments of distribution are most important. The mean tells us what the average values look like, and the variance tells us about the spread.

In physics, all moments are used, including higher-order n. Sometimes they are calculated from the definition; other times they are calculated using a moment generating function.

Rth Moment of a Distribution: Notation

  • When r = 1, we are looking at the first moment of a distribution X. We’d write this simply as μ, and we can write μ = E(X). This is just the mean of the distribution.
  • For r = 2, we have the second moment. This happens to be the variance of our distribution. We can write this as μ2‘= EX2, but usually we just write it as σ2.


Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 145-149, 1984.

Stephanie Glen. "Moment: Definition, Examples" From Calculus for the rest of us!

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