Harmonic Mean

The harmonic mean $f_{\text{HM}}$ of a list of values $a_1, \ldots, a_k$ is the mean with the property that the reciprocal of the mean is equal to the average of the reciprocals.

Suppose one is given data that are to be weighted by their reciprocals. In that case, one might consider a mean with the property that $k$ times the reciprocal of the mean equals the sum of the reciprocals of the values:

$$\frac{k}{f_{\text{HM}}} = \frac 1{a_1} + \frac 1{a_2} + \frac 1{a_3} + \cdots + \frac 1{a_k}.$$

This leads to the harmonic mean defined as

$$f_{\text{HM}} = \frac{k}{\frac 1{a_1} + \frac 1{a_2} + \frac 1{a_3} + \cdots + \frac 1{a_k}}.$$

What is the harmonic mean of $1, \frac{3}{2}, 3?$

---

We have

$$\frac{3}{1 + \frac{2}{3} + \frac{1}{3}} = \frac{3}{2}.\ _\square$$

• The reciprocal of the harmonic mean is the arithmetic mean of the reciprocals.
• By the QM-AM-GM-HM inequality, the harmonic mean is smaller than either the arithmetic mean or geometric mean and is the smallest of the classical (Pythagorean) means.