Mean, Harmonic

The harmonic mean is a measure of central tendency, and like all measures of central tendency, it is used to identify a single numerical value that is most “typical” or representative of a data set. The harmonic mean is not to be confused with the much more prevalent arithmetic mean as they answer distinctly different questions about a data set. The arithmetic mean of a data set answers the question, “with what single number could each value in the data set be replaced without changing the sum?” However, the harmonic mean answers [Page 938]the question, “with what single number could each value in the data set be replaced without changing the sum of the reciprocals?” While it is of very limited use as a descriptive statistic, the harmonic mean is very useful as an inferential statistic. This is especially true for research in areas involving technology or digital communication where multiple varying-rate factors combine to produce a single overall effect. The harmonic mean is calculated by adding the reciprocals of all values in a data set, then dividing the number of values in that data set by that sum. It has a strong bias toward smaller values in the data set, meaning that effect of large outliers on the value of the harmonic mean is much smaller than the effect of small outliers. For example, the data set {3, 6, 6, 6, 7, 9, 9, 12, 14} has a harmonic mean of 6.65 because

$$\begin{array}{l}13+16+16+16+17+19+\\ 19+112+114=1.3532\\ \mathrm{and}\frac{9}{1.3532}=6.65.\end{array}$$

By comparison, that data set has an arithmetic mean of 8. Formally, the harmonic mean, H, of a set of n numerical data {x1, x2, . . . , xn} is defined as

$$1H=1n_i=1n1x_i\mathrm{or}H=n_i=1n1x_i.$$

This entry will examine harmonic mean in descriptive and inferential statistics, compare harmonic mean to other types of mean, and provide an example of how and when harmonic mean might be adopted.