Skewed and leptokurtic distributions of law department size

Normal bell-curve distributions of data produce a skewness statistic – the left or right spread of the data – of about zero. The more skewness is positive the more the data bunches toward the low end of the range, the more negative if it bunches at the higher end. Values of two standard errors of skewness (ses) or more are skewed significantly.

The standard error can be estimated roughly using the square root of 6 divided by the number of data points. I tested this by the distribution of law departments in the General Counsel Metrics benchmark survey by number of lawyers. The standard error of skewness is therefore the square root of six divided by 813 – the number of law departments. Since two times the standard error of skewness (0.085907356 as calculated by Excel) is 0.17, the survey’s distribution of lawyers is significantly skewed. Being positive at 5.95, it means lots of departments are at the small end, which is true.

Now shift visually from left-or-right spread to kurtosis, which describes up-and-down, squashed or stretched bell shapes. Normal distributions produce a kurtosis statistic of about zero. A positive value indicates a leptokurtic distribution (taller than normal), the opposite of a platykurtic distribution (flatter). Values of two standard errors of kurtosis (sek) or more differ from mesokurtic – normal appearance – to a significant degree.

The sek can be estimated roughly using the formula of the square root of 24 divided by the number of data points. Since two times the standard error of kurtosis (0.171814712) is 0.34 and the kurtosis statistic was 44.6, my distribution has significant kurtosis.