Harmonic Mean Calculator for grouped data

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 – 119 1
120 – 129 4
130 – 139 17
140 – 149 28
150 – 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 – 100 35
100 -120 55
120 -140 30
140 – 160 15
160 – 180 12
180 – 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:

Descriptive Statistics

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.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.

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