# The ties that bind: A moment-ratio diagram for probability distributions

A moment-ratio diagram allows a modeller to see several probability distributions at one time. The figure below shows the skewness versus the kurtosis for several probability distributions, where the skewness and kurtosis are defined by

$\gamma&space;_{3}=E\left&space;[&space;\left&space;(&space;\frac{X-\mu_{X}}{\sigma_{X}}&space;\right&space;)&space;^{3}\right&space;];\;&space;\;&space;\gamma&space;_{4}=E\left&space;[&space;\left&space;(&space;\frac{X-\mu_{X}}{\sigma_{X}}&space;\right&space;)&space;^{4}\right&space;]$

and μX and σX are the mean and the standard deviation of some implied univariate random variable X.

These moment-ratio diagrams are useful for:

1. quantifying the proximity between various univariate distributions based on their third and fourth moments;
2. illustrating the versatility of a particular distribution based on the range of values that the various moments can assume (the two-parameter beta distribution which covers significant areas in the figure below is an example of this versatility);
3. creating a short list of potential probability models based on the proximity of the point associated with a data set to a particular probability model; and
4. clarifying the limiting relationships between various well-known distribution families (the Student’s t distribution approaching the standard normal distribution as the degrees of freedom increases is apparent in the figure).

A moment-ratio diagram can also indicate which distributions play a central role in probability and statistics. The normal distribution at the point (0, 3) and the exponential distribution at the point (2, 9) clearly play a central role in the figure.

Further details concerning moment-ratio diagrams is given in Vargo, Pasupathy, and Leemis.1

FIGURE A moment-ratio diagram of skewness versus kurtosis for several probability distributions.

Lawrence M. Leemis is a professor in the Department of Mathematics at the College of William & Mary in Williamsburg, Virginia. Raghu Pasupathy is an associate professor in the Department of Statistics at Purdue University, Indiana.