Calculate Harmonic Mean

Harmonic Mean is one of the measures of Central Tendency and Dispersion. Means, generally known as mathematical averages, can be subdivided into Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM). Harmonic Mean is generally used to calculate the average speed. For example, if we want to determine the average speed of a vehicle moving from one direction to the other, it is important to calculate the Harmonic Mean. Harmonic Mean is given by the reciprocal of Arithmetic Mean of the reciprocals of the observations. Moving forward, we will learn in detail how we can calculate Harmonic Mean using different steps, learn deriving formulas, and see some practical examples for knowing the implications of the formulas derived.

How to Calculate Harmonic Mean?

Harmonic mean of numbers:

Harmonic mean of two given numbers,  x1 and  x2 will be equal to the number of observations divided by the sum of reciprocals of the given two numbers.

Mathematically, we can write it as follows:

For two numbers, no. of observations will be equal to 2.

Sum of Reciprocal of given observations/numbers will be = 1/x1+ 1/x2

Putting the values derived above in the formula, we will get:

HM of x1, x2 = No. of observations/ the sum of reciprocal of the given two numbers

HM of x1, x2 = 2 / (1/x1+ 1/x2)

Taking L.C.M., we will get final result as below:

= 2 x1 x2 / x1 + x2

Similarly, Harmonic mean of three numbers  x1 , x2 and x3 will be

HM of x1, x2, x3 = 3 / (1/x1+ 1/x2 + 1/x3)

Taking L.C.M., we will get as follows:

= 3 x1 x2x3 / x1 x2+ x2 x3+ x1 x3

Follow this pattern to solve the Harmonic Mean problems of n numbers as well.

Single Harmonic Mean of n ‘positive’ numbers

Single Harmonic Mean of n ‘positive’ numbers can be defined as the reciprocal of the Arithmetic Mean of the reciprocal of the observations. Now, let us break the definition of Harmonic Mean stated here to get a deep insight of the meaning of the terms used therein:

Let us assume that we have n number of observations and the observations are x1, x2, x3,…………………xn

Sum of Reciprocal of the observations:

1/x1 + 1/x2 + 1/x3+…………………………..+ 1/xn         …………………….(i)

Now, the Arithmetic Mean of the reciprocal of observations can be calculated as follows:

( 1/x1 + 1/x2 + 1/x3+…………………………..+ 1/xn ) / n        ………………………….(ii)

Now, the reciprocal of Arithmetic Mean of the reciprocal of the observations will be simply the reciprocal of what we just calculated above in step (ii):n / (1/x1 + 1/x2 + 1/x3+…………………………..+ 1/xn ) ………………………(iii)

The resulting equation (iii) is called the Harmonic Mean.

‘n’ Harmonic Mean between two numbers a and b:

H1is called one Harmonic Mean between two numbers ‘a’ and ‘b’ if a, H1, b forms a Harmonic Progression (H.P.).

Similarly, H1and H2are called two Harmonic Means between two numbers ‘a’ and ‘b’ if a, H1 , H2, b forms a Harmonic Progression (H.P.).

Conclusion

In this article, we discussed how to calculate Harmonic Mean and all the formulas used in Harmonic Mean. First, we learnt Formula for Harmonic Mean of two numbers, three numbers and subsequently ‘n’ numbers. We learnt the derivation of formulas and their relationship with each other. Then we discussed how the formula for Harmonic Mean between two numbers is different from Harmonic Mean of numbers. We also discussed mathematical interpretation and relation of Harmonic Mean with other Mean formulas such as Arithmetic Mean and Geometric Mean. We hope this study material will be helpful for you and will give you a deep understanding of the topic.