### More power to power-law distributions

Cases cited by the US Supreme Court, punitive damages awarded by federal courts, damages sought by class action plaintiffs, revenues of law firms in a given year, lawyers at law firms, relative renown of law firms, and many other aspects of the legal industry exhibit what are called power-law distributions (See my post of May. 27, 2007: Zipf’s law is a power law.).

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 some index. For example, a power-law formula with an index will predict how commonly a \$50,000 regulatory fine will occur, if you have discovered the power-law index based on a number of such fines (See my post of July 25, 2005: also example of regulatory fines.). Stated differently, a variable (for example, the size of the largest law firm) is a function of another variable (for example, rank by size) with an index exponent (rank raised to a power),” as explained in the McKinsey Quarterly, 2009, No. 1 at 11.

As compared to the familiar bell curve distribution, a power-law distribution has only one tail at the high mark (the largest fine, payment to a firm, settlement amount, etc.) and then drops off quickly to a long “tail” of significantly smaller events. IEEE Spectrum, May 2008, Vol. 45 at 18, offers more. The curve of frequency versus rank shows a steep decline at first, followed by a long tail that looks rather flat when plotted on a linear scale. On a log-log plot, the tail becomes a straight line.