### Statistical moments to describe sets of data – explained with invoices

Moments are terms statisticians use to describe the distribution of data points. Consider how a law department’s invoices can illustrate four moments (See my post of May 31, 2006 generally on statistics and references cited.)

The first central moment is zero (See my post of Nov. 30, 2005 on illustrations of means, modes, and medians.) and doesn’t do much for us.

The second central moment is the variance, the square root of which is the data set’s standard deviation. This figure describes how the amounts of invoices a law department receives are distributed around the average invoice value (See my post of Oct. 24, 2005 on standard deviations.).

The third central moment describes the lopsidedness of the distribution of bills, which statisticians call “skewness” (See my post of June 30, 2006.). If there are more bills to the left of the average, relatively more smaller than larger bills in other words, the distribution has “negative skewness”; if more bills to the right, it has “positive skewness.”

The fourth central moment describes whether the distribution of invoices is tall and skinny (on a column chart, mostly clustered around the average) or short and squat (columns stretching out on either side of the average). “Kurtosis” is a term for the degree of “peakedness” of the invoice data’s distribution. The fourth central moment of a normal distribution is three times the standard deviation raised to the fourth power. Kids, don’t try this without a parent present.

There are even higher-level statistics of this kind, but this is enough for the moment.