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.

Formula:

          Formula to find H of

           

a1, a2,....an is =         n /(1/a1 + 1/a2 +.....+1/an)


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/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

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 = G
₂/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 





Harmonic mean to Arithmetic mean