Harmonic Mean

Theharmonic 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 theharmonic meandefined 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 $1, \frac{3}{2}, 3?$

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We have

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

• The reciprocal of the harmonic mean is thearithmetic meanof the reciprocals.
• By theQM-AM-GM-HMinequality, the harmonic mean is smaller than either thearithmetic meanorgeometric meanand is the smallest of the classical (Pythagorean) means.