## Harmonic mean for grouped data

Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. Then the harmonic mean of $X$ is denoted by $HM$ and is given by

\begin{aligned} HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}} \end{aligned}

where

• $N=\sum f$ total no. of observations

## Example 1

Compute harmonic mean for the following frequency distribution.

x 10 15 20 25 30
f 3 12 25 10 5

### Solution

$x$ Freq ($f$) $f/x$
10 3 0.3
15 12 0.8
20 25 1.25
25 10 0.4
30 5 0.1667
Total 55 2.9167

The harmonic mean of $X$ is

\begin{aligned} HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}}\\ &= \frac{55}{\frac{3}{10}+\frac{12}{15}+\frac{25}{20}+\frac{10}{25}+\frac{5}{30}}\\ &= \frac{55}{2.9167}\\ &= 18.8571 \end{aligned}

## Example 2

Compute harmonic mean for the following frequency distribution.

x 10-15 15-20 20-25 25-30 30-35
f 2 13 21 14 5

### Solution

Class mid-value ($x$) Freq ($f$) $f/x$
10-15 12.5 2 0.16
15-20 17.5 13 0.7429
20-25 22.5 21 0.9333
25-30 27.5 14 0.5091
30-35 32.5 5 0.1538
Total 55 2.4991

The harmonic mean of $X$ is

\begin{aligned} HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}}\\ &= \frac{55}{\frac{2}{12.5}+\frac{13}{17.5}+\frac{21}{22.5}+\frac{14}{27.5}+\frac{5}{32.5}}\\ &= \frac{55}{2.4991}\\ &= 22.0077 \end{aligned}