Four moments of distribution: Mean, Variance, Skewness, and Kurtosis

The first moment of distribution is MEAN, the second moment is VARIANCE, the third is SKEWNESS, and the fourth one is KURTOSIS, and so on (Learning first four moments is enough). 

The mean and variance are raw moments, and the skewness and kurtosis are normalized/standardized moments (normalized with standard deviation). Unlike mean and variance, skewness and kurtosis are unit-free/dimensionless moments. For example, if our data is in inches (height), then mean and variance will be in inches, but skewness and kurtosis will be unit-free (not in inches). Therefore, skewness and kurtosis will not be affected by any linear change in the scale of the data (inches to centimeter).

The brief description of these four moments is shown in the Table below:

Mean and Variance are very popular metrics, but Skewness and Kurtosis are rarely discussed (but important attributes of a distribution). Therefore, we focus on these two measures.

Skewness

If the value of the skewness comes to be negative, it implies that the distribution is skewed negatively. That means that there are some extreme values in the data which drags the mean to the left side of the distribution. Similarly, the positive skewness indicates that there are some extreme values in the data which drags the mean to the right side of the distribution. Skewness is very useful measure when data has outliers. The positive and negatively skewed distributions are shown in the figure below.

Kurtosis

It is always positive because it is the fourth moment and as the power 4. It measures the peakness or flatness of the data under consideration. Some tools measure excess kurtosis which is measured as the kurtosis of the data minus three (the kurtosis of the normal distribution is 3). Excess kurtosis is termed as relative skewness (relative to a normal distribution). If excess kurtosis is positive, then the distribution is termed as Leptokurtic. But if the excess kurtosis is negative, then the distribution is called as Platykurtic. A distribution in between is termed as Mesokurtic.

The computation of these four moments is shown in the video given below:

Sources:

https://en.wikipedia.org/wiki/Moment_(mathematics)

http://researchbasics.education.uconn.edu/normal-distribution/

http://www.aip.de/groups/soe/local/numres/bookcpdf/c14-1.pdf