The harmonic mean or HM is a numerical average. It is calculated by dividing the number of observations, or terms, in the series, by the sum of reciprocals of each term in the series. Therefore, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocal values.
In statistics, the measure of central tendency is used. A measure of central tendency is a single value that describes the way that a group of data aggregate around a central value. It represents the center of the data set.
There are three measures of central tendency. They are mean, median, and mode. In this page, you will learn about one of the important types of means called “harmonic mean” along with its definition, formula, and examples in detail.
Definition of HM
The harmonic mean (HM) is defined as the average of the reciprocals of the data points. It is based on all the observations. Harmonic mean gives less significance to the large values and more significance to the small values.
In general, the harmonic mean is applied when it is necessary to give more weight to smaller values. It is practiced in the case of times and average rates.
Since the harmonic mean is the average of reciprocals, the formula for defining the harmonic mean “HM” is as follows:
If a1, a2, a3,…, an are the individual terms up to n, then
Harmonic Mean, HM = n / [(1/a1)+(1/a2)+(1/a3)+…+(1/an)]
How to Calculate HM
If x1, x2, x3, x4, … are the given data points, then the algorithm to find the harmonic mean is as follows:
Step 1: Calculate the reciprocal of each value (1/x1, 1/x2, 1/x3, 1/x4, …)
Step 2: Determine the average of reciprocals from step 1.
Step 3: Finally, take the reciprocal of the average from the value in step 2.
Relationship Between AM, HM, and GM
The three means, such as the arithmetic mean, geometric mean, and harmonic mean, are known as Pythagorean means. The formulas for three different types of means are given below:
Arithmetic Mean = (y1 + y2 + y3 +…..+yn ) / n
Harmonic Mean = n / [(1/y1)+(1/y2)+(1/y3)+…+(1/yn)]
Geometric Mean = 
The relationship between G (geometric mean), H (harmonic mean), and A (arithmetic mean) is given by:
G=√AH
Or
G2 = A.H
Applications of HM
The following are the primary applications of the harmonic mean:
- The harmonic mean is applied in finance to average multiples like price-revenue ratios.
- It is also used by market technicians in order to determine patterns like Fibonacci Sequences and fractals.
- It can be used to calculate scalars such as speed. As we know, speed is expressed as a ratio of two calculative units, Km/hr.
- HM is used to estimate average rates in business firms as it assigns equal weight to all data points in a given sample.
Weighted Harmonic Mean
A weighted harmonic mean is used when we want to find the average of a set of observations, such as when equal weight is given to each data point. Let x1, x2, x3....xn be the observations and w1, w2, w3....wn be the corresponding weights. Then the formula for the weighted harmonic mean is given as follows:
Weighted HM = 
If we have normalized weights, then all weights sum to 1. i.e, w1 + w2 + w3 +....+ wn = 1
Suppose we have a frequency distribution with n items x1, x2, x3....xn having corresponding frequencies f1, f2, f3....fn then the weighted harmonic mean is give as:
Weighted HM = 
Examples
Example 1:
Find the harmonic mean for data 3, 4, 6, and 8.
Solution:
Given data: 3, 4, 6, 8
Step 1: Calculate the reciprocal of the values:
1/3 = 0.34
1/4 = 0.25
1/6 = 0.16
1/8 = 0.125
Step 2: Calculate the average of the reciprocal values obtained in step 1.
The total number of data points = 4.
Average = (0.34 + 0.25 + 0.16 + 0.125)/4
Average = 0.875/4
Step 3: Estimate the reciprocal of the average value obtained in step 2.
Harmonic Mean = 1/ Average
Harmonic Mean = 4/0.875
Harmonic Mean = 4.57
Hence, the harmonic mean for the data 3, 4, 6, 8 is 4.57.
Example 2:
Calculate the harmonic mean for the following given table:
| x |
2 |
5 |
3 |
6 |
8 |
7 |
| f |
1 |
3 |
7 |
3 |
9 |
4 |
Solution:
The harmonic mean is calculated as follows:
| x |
f |
1/x |
f/x |
| 2 |
1 |
0.5 |
0.5 |
| 5 |
3 |
0.2 |
0.6 |
| 3 |
7 |
0.34 |
2.34 |
| 6 |
3 |
0.16 |
0.5 |
| 8 |
9 |
0.125 |
1.125 |
| 7 |
4 |
0.142 |
0.57 |
|
N = 27 |
|
Σ f/x=5.635 |
The formula for weighted harmonic mean is
HMw = N / [ (f1/x1) + (f2/x2) + (f3/x3)+ ….(fn/xn) ]
HMw = 27 / 5.635
HMw = 4.791
Therefore, the harmonic mean, HMw, is 4.791.
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