Moment in Statistics

Concise Notes for Skewness and Kurtosis

What is a Moment in Statistics?

We generally use moments in statistics, machine learning, mathematics, and other fields to describe the characteristics of a distribution.

Let’s say the variable of our interest is X then, moments are X’s expected values. For example, E(X), E(X²), E(X³), E(X⁴),…, etc.

Figure 1: Moments in Statistics.

Moments in statistics:

1) First Moment: Measure of the central location. (MEAN)

2) Second Moment: Measure of dispersion/spread.(VARIANCE)

3) Third Moment: Measure of asymmetry.

4) Fourth Moment: Measure of outliers/tailedness.

Now we are very familiar with the first moment(mean) and the second moment(variance).

The third moment is called skewness, and the fourth moment is known as kurtosis.

The third moment measures the asymmetry of distribution while the fourth moment measures how heavy the tail values are. Physicists generally use the higher-order moments in applications of physics. Let’s have a look at the visualization of the third and fourth moments.

Third Moment(Skewness):

1) No Skew:

Figure 2: Graph data with no skew.

2) Positive Skew:

Figure 3: Graph of data with a positive skew.

3) Negative Skew:

Figure 4: Graph of data with a negative skew.

Fourth Moment(Kurtosis):

Figure 5: Graph representing the types of kurtosis.