Karl Pearson showed that if we know the first four moments of a distribution, we can construct a density function that is consistent with those moments. This can provide a neat way to build density functions that approximate a given set of data. For instance, for a given data set, let us suppose that:

denoting estimates of the mean, and of the second, third and fourth central moments. The Pearson family consists of 7 main *Types*, so our first task is to find out which type this data is consistent with. We do this with **mathStatica**'s PearsonPlot function:

**Fig. 1: ** The chart for the Pearson system

The big black dot in Fig. 1 is in the *Type I* zone. Then, the fitted Pearson density and its domain are immediately given by:

The actual data used to create this example is grouped data depicting the number of sick people (freq) at different ages (X):

We can easily compare the histogram of the empirical data with our fitted Pearson pdf:

**Fig. 2: ** The data histogram and the fitted Pearson pdf