Table of Contents
Definition of Harmonic Mean (H.M.)
Harmonic Mean of a set of values of a variable gives a measure of their central tendency and is defined as the reciprocal of the arithmetic mean of the reciprocals of the variable values. Clearly, harmonic mean becomes undefined when anyone variable value is zero.
The formula of Harmonic Mean:
For unweighted data :
if H be the simple harmonic mean of n values x1, x2, …, xn of a variable x, then by definition,
For weighted data :
if variable x posesses values x1, x2, …, xn with frequencies(weight) f1, f2, …, fn respectivly then H is given by
![H=\frac{1}{\frac{1}{N}\left[\frac{f_{1}}{x_{1}}+\frac{f_{2}}{x_{2}}+\cdots+\frac{f_{n}}{x_{n}}\right]}=\frac{N}{\sum \frac{f}{x}} \text { where } N=\Sigma f](https://raynoc.com/wp-content/ql-cache/quicklatex.com-e7e78e9c2d415d66a65c3d1304524bd1_l3.png "Rendered by QuickLaTeX.com")
Example 1 : Calculate the harmonic mean 46.1, 121, 127.3, 202
Solution :
Calculation of H.M
x | 1/x |
46.1 | 0.0217 |
121 | 0.0083 |
127.3 | 0.0079 |
202 | 0.0050 |
Total | 0.0429 |
Example 2. Calculate the H.M.
Class | 10-14 | 15-19 | 20-24 | 25-29 | 30-34 |
f | 4 | 6 | 8 | 2 | 2 |
Solution:
Calculation of H.M.
Class | f | Mid- value (x) | 1/x | f/x |
10-14 | 4 | 12 | 0.0833 | 0.3332 |
15-19 | 6 | 17 | 0.0588 | 0.3528 |
20-24 | 8 | 22 | 0.0455 | 0.3640 |
25-29 | 2 2 | 27 | 0.0370 | 0.0740 |
30-34 | 2 | 32 | 0.0312 | 0.0624 |
Total | 22 | - | - | 1.1864 |
Use of Harmonic Mean
It is especially useful in an averaging speed ( when equal or different distances are covered with different speeds) and prices ( when certain fixed amounts or different amounts are spent in purchasing a commodity at different prices).
Advantages and Disadvantages of Harmonic Mean
Advantages:
- In the definition of H.M., the reciprocal of the values of the variable are involved; hence, it gives the largest weight to the smallest observation and the smallest weight to the largest observation. Therefore, it is not very much indulgenced by one or two large observations.
- It is capable of further algebraic treatment
- Like A.M. and G.M. it is based on all observations.
- It is capable of further algebraic treatment.
- it is suitable for some particular type of averages.
Disadvantages:
- It is not readily understood, nor, can it be calculated at ease.
- H.M. can’t be calculated if any of the observations is zero.
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