Harmonic Mean

The harmonic mean is another way of expressing central tendency for a set of scores. The harmonic mean is obtained by dividing the number of observations by the sum of the reciprocal of the scores. That is, if we have n observations with scores x1,x2,…,xn, the harmonic mean is computed as follows:

There are certain situations in which the harmonic mean provides the most appropriate definition of the “average.” For example, the harmonic mean is useful when averaging rates of speed. To illustrate, suppose that a vehicle travels from city A to city B at an average speed of 40 miles per hour and returns to city A at an average speed of 60 miles per hour. The average speed is calculated as 48 miles per hour using the harmonic mean, as shown below:

In other words, the total amount of time for the round trip is equivalent to the time it would have taken to make the trip at a constant speed of 48 miles per hour. Had we used the arithmetic mean to compute the average velocity for this example, the result would have been 50 miles per hour.

The harmonic mean has applications in the behavioral sciences as well. An example can be found in the context of statistics. Suppose that two group means are compared using an independent t test. The denominator of the t statistic is the standard error,

which quantifies the sampling error associated with the mean difference in the numerator of the t formula. In this formula, s2pooled represents the pooled variance and is computed as

where ssk is the sums of squares for the kth group. After some algebraic manipulation, it is possible to[Page 426] rewrite the standard error formula using the harmonic mean h in place of n1 and n2 as follows:

To illustrate, suppose it was of interest to compute an independent t test with unequal group sizes, such as ss1 = 25, n1 = 12, ss2 = 35, and n2 = 20. Using standard formulae, the pooled variance is s2pooled = 2, and the standard error of the mean difference is sX¯1−X¯2 = .52. The harmonic mean sample size in this example is h = 15 (i.e., 2/[1/12][1/20]). Substituting the harmonic mean into the formula above yields a standard error of .52—the same result obtained using unequal ns. Thus, the standard error of the mean difference using two unequal groups is identical to the standard error that would have been obtained using two groups with a common n equal to h (i.e., n1 = n2 = h).

In sum, there are certain situations in which the harmonic mean provides the appropriate definition of the “average,” as when averaging rates and when computing the average sample size from a disparate collection of ns.