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