## PROBLEMS ON HARMONIC MEAN

Harmonic mean(H) is one of the measures of central tendency and also based on arithmetic mean and geometric mean. It is useful for quantitative data. It is also a kind of average.

Definition :

This  is nothing but the reciprocal of the arithmetic mean of the reciprocals of the given numbers.

In other words to find the H of the given numbers, we have to divide n (total number of given numbers) by the sum of the reciprocals of the given numbers.

Example 1 :

Find the H of 3, 4,  5,  6, 7 and 8

Solution :

The total number of values  =  6

Let us find H using the formula

H  =  n /(1/a₁ + 1/a₂ +.....+1/aₓ)

=  6/(1/3+1/4+1/5+1/6+1/7+1/8)

=  6/(0.333+0.25+0.20+0.166+0.142+0.125)

=  6/1.216

=  4.93

Example 2 :

Find the H of 1, 2, 5, 7, 9

Solution :

The total number of values = 5

Let us find H using the formula

H  =  n /(1/a + 1/a +.....+1/aₓ)

=  6/(1/1+1/2+1/5+1/7+1/9)

=  6/(1 + 0.5 + 0.2 + 0. 14 + 0.11)

=  6/1.95

=  3.07

## Harmonic mean of two numbers

Example 1 :

Find the H of two numbers 50 and 30.

Solution :

Instead of x1 and x2 we are having 50 and 30.

=  2(50) (30) / (50 + 30 )

=  (100 x 30)/80

=  3000/80

=  37.5

Example 2 :

Find the H of two numbers 12 and 15.

Solution :

Instead of x1 and x2 we are having 12 and 15.

=  2(12) (15) / (12 + 15 )

=  (24 x 15)/27

=  360/27

=  13.3

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