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).

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

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