Morrison on Metrics: 5 kinds of averages

Is your knowledge above- or below-average when it comes to averages?

Everyone in a law department ought to understand plain-vanilla averages—and be above-average if they know four more variations.

 

1. Mean: The mean of a group of numbers—a data set—is its arithmetic average. Add up the lawyers in each of the 10 law firms you pay the most and divide that sum by 10, that’s the arithmetic average of their lawyers—the mean of that data set. With small numbers of observations, means give a distorted result if there is an abnormally high or low value. So, consider a trimmed mean.

2. Trimmed Mean: A trimmed mean shaves off some number of extreme values. A common choice is to omit the 2 ½ percent of the numbers at the low end and the 2 ½ percent at the high end of a sorted list. Thus, if you have revenue for 50 companies sorted high to low, you would drop the two smallest and the two largest. Then calculate the mean. You might not be sure the trim is sufficiently deep. So, consider a middle average.

3. Middle Average: Another technique to avoid anomalous outliers might be called the average of the middle. After ranking a data set, drop the top and bottom quartile (25 percent) and calculate the average of the middle 50 percent. If you want to know your average discount granted, look at all your invoices from last year, rank them from high to low and drop the top and bottom 25 percent. Then average the middle half. You could think of this as a huge trimmed mean. But neither the trimmed mean nor the middle average reflect some characteristic of the numbers being calculation. So, consider a weighted average.

4. Weighted Average: Each figure in a data set is assigned a weight. For example, count the number of contracts reviewed for six business units during a year and weight those counts by the annual revenue of the unit. You multiply the number of contracts for a unit by the unit’s revenue and divide the total of those six products by the total revenue of the units. Weightings are the equivalent of having that many like items with the same value involved in the average. If you have no basis for a weighting, you might aggregate your two numbers into an all-encompassing average. So, use an overall average.

5. Overall Average: This is my term for another measure of the center of a data set. An overall average gives more influence to larger figures in a data set. It does so because it adds up all of the figures for the metric that is in the numerator (the top number of a fraction) and divides it by the sum of all of the numbers in the denominator (the bottom number). The weighted average totals both pieces of the ratio and divides them. If you do this for legal staff per billion dollars of revenue, very large law departments influence the overall average more than do the smaller ones.

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