### Skewness describes oddly-shaped distributions of numbers, such as invoice amounts

We all understand averages, and many of us understand medians, but few of us understand another descriptive statistic for numbers: skewness (See my post of June 30, 2006: skewness; and March 23, 2007: third central moment.).

Imagine a bell curve distribution of the bills your legal department received during the past five years, depicted from low on the left to high on the right by the amount of the bill. If the average of the bills is to the left of the bell’s top, the median, relatively more smaller than larger bills in other words, the distribution has “negative skewness”; if the average is to the right of the median, it has “positive skewness.” Skewness describes the degree to which a distribution of numbers tips to the right or the left of the central point.

In a recent survey of 139 large legal departments, the average size of the departments – expressed as numbers of lawyers – was 69, more than double the median of 35. The skewness score was very high at a positive 6.5. A different group of 43 law departments interviewed for the same study was more normally distributed with a skewness of 1.4. This data comes from a paper discussed by Prof. Michele Beardslee at the Georgetown Conference on the Future of Law Firms at 19-20.

When you have some free time, here is how to calculate skewness:

- Calculate the mean (average) and standard deviation of your set of figures
- Subtract the mean from each figure (such as the amount of the invoices in my example) and cube that figure (i.e., raise it to the third power)
- Sum the cubed deviations
- Multiply the number of scores minus 1 times the cubed standard deviation of the figures (i.e., raised to the third power)
- Skewness = step 3 divided by step 4.