Harmonic mean is a central value of a data set which can be computed using arithmetical operations. It is generally used in averaging rates, ratios etc. It is well defined and suitable for algebraic treatment. It takes all the observations into account. In the case of outliers even though it mitigates the effect of very large values, but affected by very small values. Harmonic mean is the smallest when compared to the Arithmetic and Geometric Means.

## Harmonic Mean Definition

Harmonic Mean can be defined as the ratio of the number of elements in the data set to the sum of the reciprocals of each value in the set. Harmonic mean is used to calculate the average of a set of numbers. It can be calculated by dividing the number of observations by the reciprocal of each number in the series.

## Harmonic Mean Formula

The Harmonic Mean HM of n observations x1, x2,......xn is given by the formula

HM = $\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+........+\frac{1}{x_{n}}}$

When f1, f2,......f3 are the corresponding frequencies of the above observation then

HM = $\frac{N}{\frac{f_{1}}{x_{1}}+\frac{f_{2}}{x_{2}}+.......\frac{f_{n}}{x_{n}}}$

where N = f1 + f2+.......+ fn.

## Harmonic Mean Example

### Solved Examples

Question 1: Find the AM, GM and HM of the four values 3, 6, 24 and 48. Verify HM < GM < AM.
Solution:

Arithmetic Mean = $\frac{3+6+24+48}{4}$ = $\frac{81}{4}$ = 20.25

Geometric Mean = $\sqrt[4]{3\times 6\times 24\times 48}$ = 12

Harmonic Mean = $\frac{4}{\frac{1}{3}+\frac{1}{6}+\frac{1}{24}+\frac{1}{48}}$

= $\frac{64}{9}$ ≈ 7.11

The computed values verify HM < GM < AM.

Question 2: James, Janet, Jenny, Jack and Jerome work together and produce wooden toys. If they can correspondingly complete making one toy in 4.5, 6, 6, 5.5 and 5 hrs, find the average number of toys completed in one hour. If they work 8 hrs per day for 5 days in a week find the average number of toys produced by a person in one week?
Solution:

Average rate of work is $\frac{1}{HM}$ of the five times given.

HM of 4.5, 6, 6, 5.5 and 5 = $\frac{5}{\frac{1}{4.5}+\frac{1}{6}+\frac{1}{6}+\frac{1}{5.5}+\frac{1}{5}}$

= $\frac{5}{\frac{2}{9}+\frac{1}{6}+\frac{1}{6}+\frac{2}{11}+\frac{1}{5}}$

= $\frac{5\times 990}{2\times 110+165+165+2\times 90+198}$

= $\frac{4950}{928}$ = $\frac{2475}{464}$

Average Number of units produce in one hr = $\frac{1}{HM}$ = $\frac{464}{2475}$ ≈ 0.1875

Average number of toys produced by a person in one week = 5 x 8 x 0.1875 = 7.5

Question 3: An airplane covers a distance of 1000 Km in two phases. During the first phase it averaged a speed of 400 Km/hr and during the second phase the average speed was 600 Km/hr.  Find the average speed for the entire trip.
Solution:

Average speed of the trip is the Harmonic Mean of the average speeds of the two phases.

Harmonic Mean of the two average speeds = $\frac{2}{\frac{1}{400}+\frac{1}{600}}$

= $\frac{2\times 1200}{3+2}$ = $\frac{2400}{5}$ = 480 Km/hr

The average speed of the plane for the entire trip = 480 Km/hr.