### Power-law statistics and law department metrics

What statisticians call power-law relationships describe the frequency of events occurring according to their size or severity, such as how often earthquakes of different Richter scale magnitudes happen. Power-law relationships are characterized by a number called an index. For each ten-fold increase in the amount paid in settlement of employment discrimination cases, for example, the probability of there being a settlement of a given size decreases by a factor of ten raised to the power of this index. For example, the index will predict how commonly a $50,000 regulatory fine will occur, if you have developed the power-law index based on a number of such fines.

As compared to the familiar bell curve (normal) distribution, a power-law distribution differs because it has only one tail (the smallest fine, payment to a firm, settlement amount) and no peak. Payments by a large law department to its law firms over a period of years, treating each year’s payment as a single data point, would likely follow a power-law distribution. To go from $10 to $100, and from $100 to $1000 – a logarithmic increase – would show a pattern of decreasing frequency described by the power-law index.

I find this interesting, because power-law relationships often crop up in complex and highly interacting systems (which fairly describes outside counsel spending). Like logarithmic charts (or, to show off, log-log charts), the power index helps identify patterns in data that law departments would otherwise overlook.