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# Morrison on Metrics: Inter-Quartile Range, Mode and Range

Earlier this year, this column explored two ways to describe data dispersion: trimmed means and weighted averages. Those two are siblings in a large family. Others members of the family include the inter-quartile range, the mode and the range. Those three I explain here and in a later column I will turn to standard deviations. The family of mathematical descriptors makes a difference to general counsel who want to master business-speak and who want to extract the most regarding their department's data.

Start with inter-quartile calculations. When you order a set of numbers from largest to smallest, the middle number in that ranking is the median. Midway between the median and the lowest number stands the first quartile figure; midway between the median and the highest figure stands the third quartile figure. The difference between the first quartile and the third quartile becomes the inter-quartile difference. Average the first and third quartile figures and you have the inter-quartile mean. Inter-quartiles avoid the unusual and misleading data points at either end of a ranked list.

Consider some from the 701 law departments in the fourth release of General Counsel Metrics (GCM). The inter-quartile difference for lawyers was 20, for paralegals 6, and other legal staff 10. If you calculate the average of the two quartile figures they are for lawyers 13, for paralegals 4, and for all other legal staff 6. These numbers tell, for example, that this set of law departments has much more variability on the lawyer side than on the non-lawyer side. Similar calculations could be done to compare industries or countries on the spread of their benchmark metrics.

Next, divide the difference between each of those quartile pairs by the median. Thus, the first quartile number of lawyers was 3 while the third quartile was 23, so the difference was 20. The median being 7 for lawyers, 7 divided into 20 leaves 2.85. The calculation for paralegals resulted in 3.0 and for other staff 3.33. Thus, in terms of spread in relation to median, the three sets of staff numbers were quite similar. In fact, lawyers were even more tightly clustered.

The mode of a set of numbers is simply the number that is most common. Not surprisingly, in the GCM group, the mode for lawyers is one. More law departments had a single lawyer than any other number of lawyers. For paralegals the mode is also one and for other staff, zero. The mode often does not tell very much about a set of numbers but here it tells us once again that many law departments are small. It also says that many small law departments are interested in benchmark metrics.

The range states the difference between the largest number in a set and the smallest. For those law departments that provided revenue figures, the GCM range was more than \$285 billion. In any benchmark set from law departments the range tends to be the largest revenue, number of lawyers or legal staff, or whatever in the set.

With these math tools, it is possible to describe many characteristics of a set of data regarding variability. The descriptions let you understand the data and compare it to other sets as well as to your own figures.

### Rees Morrison

Rees Morrison, Esq. is a partner at Altman Weil, Inc. with countless interests in legal data analytics. He is also the founder of General Counsel Metrics, LLC....

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