Definition

Harmonic mean: $n$ over the sum of the reciprocals

Statistics • Mean • Harmonic Mean

Formula

H = \frac{n}{\frac{1}{x _{1}} + \frac{1}{x _{2}} + \dots + \frac{1}{x _{n}}}

The harmonic mean of

n

positive values

x_{1}, x_{2}, \dots, x_{n}

H = \frac{n}{\frac{1}{x _{1}} + \frac{1}{x _{2}} + \dots + \frac{1}{x _{n}}}

— the reciprocal of the average of the reciprocals, not the average of the reciprocals itself. For two values it reduces to

H = \frac{2 ab}{a + b}

, and it is defined only when every value is positive.

How to use it

1For two numbers, read the formula as $\frac{2 \times product}{sum}$ : that is the form almost every question wants.
2When the data are rates for a fixed amount of something — speed over a fixed distance, work per fixed job — the harmonic mean is the correct average, not the arithmetic mean.
3To check an answer, use $H \leq G \leq A$ : the harmonic mean never exceeds the geometric mean, which never exceeds the arithmetic mean.

Worked example

Find the harmonic mean of

3

and

6

Step-by-step

1
$H = \frac{2}{\frac{1}{3} + \frac{1}{6}} = \frac{2}{\frac{1}{2}}$
Why: Two values, so $n = 2$ over the sum of the two reciprocals.
2
$H = 4$

Answer

4

Watch out

Taking the average of the reciprocals and stopping gives

\frac{1}{H}

, not

H

— the final reciprocal is the step most often dropped. A zero among the values makes the harmonic mean undefined, and negative values make the ordering

H \leq G \leq A

fail.

Builds on

Negative exponent is a reciprocal

Also known as

harmonic mean, HM, H.M. of n numbers, reciprocal of the mean of reciprocals, 2ab/(a+b)

Harmonic mean: $n$ over the sum of the reciprocals

Formula

Worked example

Appears in past papers

2020 (II)

2020 (I)