Mathematics derivation

Harmonic Mean Definition:
     Harmonic mean is used to calculate the average of a set of numbers. Here the number of elements will be averaged and divided by the sum of the reciprocals of the elements. The Harmonic mean is always the lowest mean.

Harmonic Mean Formula :
Harmonic Mean = N/(1/a₁+1/a₂+1/a₃+1/a₄+.......+1/a_N)
where
              X = Individual score
              N = Sample size (Number of scores)



Harmonic Mean Example: To find the Harmonic Mean of 1,2,3,4,5.

  Step 1: Calculate the total number of values.
            N = 5

  Step 2: Now find Harmonic Mean using the above formula.
            N/(1/a₁+1/a₂+1/a₃+1/a₄+.......+1/a_N)
            = 5/(1/1+1/2+1/3+1/4+1/5)
            = 5/(1+0.5+0.33+0.25+0.2)
            = 5/2.28
            So, Harmonic Mean = 2.19

This example will guide you to calculate the harmonic mean manually.
 

If a, b and c are in Arithmetic Series, then 1/a, 1/b, 1/c are in Harmonic series.

So, if b is the Arithemetic mean of a and c. 1/b is the Harmonic Mean of 1/a and 1/c

Mathematical Derivation:

For easiness, lets say x, y, z are in HM i.e. x = 1/a, y = 1/b, and z=1/c

Now, using the formula of AM, we get b = (a+c)/2.

Now, Harmonic Mean = 1/b = 2/(a+c)

Converting Values in x, y, and Z. We get. HM = y = 2/(1/x+1/z)

                                                                                     = 2/((x+z)/xz)

                                                                                     = 2*xz/(x+z)

                                                                                     = 2xz/(x+z)

Above is the derivation for harmonic Mean of two numbers x and z.

USAGE:

Some things exhibit harmonic properties like velocity.

Example. If we travel a certain distance of 100 km from A to B, at 20 Km/hr and then B to A at 50 Km/hr, then what is the average velocity.

Solution: Since velocity exhibit Harmonic Properties, Average Velocity is HM between them,

Therefore, Average Velocity = 2*50*20/(50+20)

                                                   = 200/7 km/hr

If we calculate direct average i.e. AM, it will be wrong. Here AM of 50 & 20 will be 35 km/hr.

Cross Check:

Total Time taken from A to B = 100/20 = 5 Hr

Total time taken from B to A = 100/5 = 2 Hr

Total time taken for return journey = 2 Hr+5 Hr = 7 Hr

Total Distance for Return Journey = 100+100 = 200 km

So, Average Velocity = 200/7 km/hr