Using Mean, Standard Deviation, Skewness, and Kurtosis

Previous blogs have discussed the properties of the first four moments which can be computed from a data set. The next step is to use these easily computed statistics in everyday applications. When presented with a set of data, it is important to understand what information may be hidden in the “sea of numbers”. We know that the Mean gives us the central tendency of the data, the Standard Deviation explains the dispersion about the Mean, the Skewness represents the symmetry/asymmetry of the data, and the Kurtosis is related to the shape or peakedness characteristics. In essence, we are using these numerical quantities to explain the properties of the underlying distribution or probability density function (PDF). These statistics can be used to qualitatively perform distribution fitting for you data.

Since a set of data can have any Mean and Standard Deviation, we can use these statistics to determine the location and relative dispersion. Using the Skewness and Kurtosis, we can learn much more as shown in the table below;

Skewness	Kurtosis*	Classical Distribution
0	1.8	Uniform PDF
Any negative number	2.4	Left-skewed Triangular PDF
0	2.4	Symmetric Triangular PDF
Any positive number	2.4	Right-skewed Triangular PDF
0	3	Normal PDF
0.63	3.26	Raleigh PDF
2	9	Exponential PDF

Using this information, you can make as-like comparisons of your data to the properties of some of the known classical distribution. While you may not be able to conclude that the data set is from a population with a particular distribution, you will be able to infer that “based on the data, the uncertainty is representative of that of a _______ distribution”.

*Note that these quantities are for Kurtosis where 3 = Normal PDF. If quantities for Excess of Kurtosis, where 0 = Normal PDF, are desired, then subtract 3 from the values shown.