Harmonic Mean for grouped data
Harmonic mean is an important measure of central tendency of the data. Harmonic mean is used for calculating average of ratios. Most commonly used ratios are speed and time, work and time, dividend per share of companies, cost and units materials, etc.
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 $N=\sum f$ is the total no. of observations.
Harmonic Mean Calculator for grouped data
Use this calculator to find the Harmonic Mean for grouped data (frequency distribution).
Harmonic Mean for Grouped Data Calculator | |
---|---|
Type of Frequency Distribution | DiscreteContinuous |
Enter the Classes for X (Separated by comma,) | |
Enter the frequencies (f) (Separated by comma,) | |
Harmonic Mean Results | |
Number of Observation. (N): | |
Harmonic Mean : | |
frequency distribution : | |
How to find harmonic mean of grouped data?
Step 1 - Select type of frequency distribution (Discrete or continuous)
Step 2 - Enter the Range or classes (X) seperated by comma (,)
Step 3 - Enter the Frequencies (f) seperated by comma
Step 4 - Click on "Calculate" for harmonic mean calculation
Step 5 - Gives output as number of observation (n)
Step 6 - Calculate harmonic mean
Step 7 - Display frequency distribution
Harmonic Mean for Grouped Data 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 | |
tot | Total | 55 | 2.9167 |
The harmonic mean of $X$ is
$$ \begin{aligned} HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}}\\ &= \frac{55}{\frac{3}{10}+\frac{12}{15}+\frac{25}{20}+\frac{10}{25}+\frac{5}{30}}\\ &= \frac{55}{2.9167}\\ &= 18.8571 \end{aligned} $$
Harmonic Mean for Grouped Data 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 | |
tot | Total | 55 | 2.4991 |
The harmonic mean of $X$ is
$$ \begin{aligned} HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}}\\ &= \frac{55}{\frac{2}{12.5}+\frac{13}{17.5}+\frac{21}{22.5}+\frac{14}{27.5}+\frac{5}{32.5}}\\ &= \frac{55}{2.4991}\\ &= 22.0077 \end{aligned} $$
Harmonic Mean for Grouped Data Example 3
Find the Harmonic Mean of distribution of weights of 75 students at virtual University in the table:
Weight | Frequency |
---|---|
110 â 119 | 1 |
120 â 129 | 4 |
130 â 139 | 17 |
140 â 149 | 28 |
150 â 159 | 25 |
Solution
Class | mid-value ($x$) | Freq ($f$) | $f/x$ | |
---|---|---|---|---|
110-119 | 114.5 | 1 | 0.0087 | |
120-129 | 124.5 | 4 | 0.0321 | |
130-139 | 134.5 | 17 | 0.1264 | |
140-149 | 144.5 | 28 | 0.1938 | |
150-159 | 154.5 | 25 | 0.1618 | |
tot | Total | 75 | 0.5228 |
The harmonic mean of $X$ is
$$ \begin{aligned} HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}}\\ &= \frac{75}{0.5228}\\ &= 143.4473\;\; minutes \end{aligned} $$
Harmonic Mean for Grouped Data Example 4
Following is the frequency distribution about the weight of earheads in grams:
Weight of earheads in gms | No. of earhead |
---|---|
40 - 60 | 6 |
60 - 80 | 8 |
80 â 100 | 35 |
100 -120 | 55 |
120 -140 | 30 |
140 â 160 | 15 |
160 â 180 | 12 |
180 â 200 | 9 |
Calculate harmonic mean for the given frequency distribution.
Solution
Class | mid-value ($x$) | Freq ($f$) | $f/x$ | |
---|---|---|---|---|
40-60 | 50 | 6 | 0.12 | |
60-80 | 70 | 8 | 0.1143 | |
80-100 | 90 | 35 | 0.3889 | |
100-120 | 110 | 55 | 0.5 | |
120-140 | 130 | 30 | 0.2308 | |
140-160 | 150 | 15 | 0.1 | |
160-180 | 170 | 12 | 0.0706 | |
180-200 | 190 | 9 | 0.0474 | |
tot | Total | 170 | 1.5719 |
The harmonic mean of $X$ is
$$ \begin{aligned} HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}}\\ &= \frac{170}{1.5719}\\ &= 108.1493\;\; grams \end{aligned} $$
Harmonic Mean for Grouped Data Example 5
Following is the data about the dividend yield (in percent) for the number of companies:
Dividend Yield (in percent) | No. of Companies |
---|---|
2 - 6 | 10 |
6 - 10 | 12 |
10-14 | 18 |
14-18 | 8 |
Find harmonic mean for the dividend yield.
Solution
Class | mid-value ($x$) | Freq ($f$) | $f/x$ | |
---|---|---|---|---|
2-6 | 4 | 10 | 2.5 | |
6-10 | 8 | 12 | 1.5 | |
10-14 | 12 | 18 | 1.5 | |
14-18 | 16 | 8 | 0.5 | |
tot | Total | 48 | 6 |
The harmonic mean of $X$ is
$$ \begin{aligned} HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}}\\ &= \frac{48}{6}\\ &= 8\;\; percent \end{aligned} $$
Conclusion
In this tutorial, you learned about formula for harmonic mean for grouped data and how to calculate harmonic mean for grouped data. You also learned about how to solve numerical problems based on harmonic mean for grouped data.
To learn more about other descriptive statistics measures, please refer to the following tutorials:
Let me know in the comments if you have any questions on Harmonic mean calculator for grouped data with examples and your thought on this article.