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.
Formula:
Formula to find H of
a_{1}, a_{2},....a_{n} is = n /(1/a_{1} + 1/a_{2} +.....+1/a_{n}) |
Example 1:
Find the H of 3,4,5,6,7 and 8
Solution:
I Step; The total number of values = 6
II Step; 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:
I Step;
The total number of values = 5
II Step; Let us find H using the formula
H = n /(1/a₁ + 1/a₂ +.....+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
Find the H of two numbers
x₁ and x₂.
H of x₁ and x₂ is
= 2x₁x₂ ÷ ( x₁ + x₂)
Relation between arithmetic mean(A), geometric mean(G) and harmonic mean(H)
H = G₂/A
Example 1:
Find the "H" of two numbers 50 and 30.
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.
Instead of x₁ and x₂ we are having 12 and 15.
= 2(12) (15) / (12 + 15 )
= (24 x 15)/27
= 360/27
= 13.3