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Harmonic Mean is one of the measures of central tendency and dispersion. Generally known as mathematical averages, a Mean can be subdivided into Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM). Harmonic Mean is used to calculate the average of speed; for example, if we want to find the average speed of a vehicle moving from one direction to the other, we use Harmonic Mean. Harmonic Mean is given by the reciprocal of the Arithmetic Mean of the reciprocals of the observations.
Harmonic Mean Formulas
- Harmonic Mean Formula for Harmonic Mean of numbers:
(A) The formula to calculate Harmonic Mean of Two Numbers:
The 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, the number of observations will be equal to 2.
The 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 = Number of observations / the sum of reciprocal of the given two numbers:
HM of x1, x2 = 2 / (1/x1+ 1/x2)
Taking LCM, we will get the final result, as follows:
HM of x1, x2 = 2 x1 x2 / x1 + x2
(B) The formula to calculate Harmonic Mean of Three Numbers:
Similarly, the Harmonic Mean of three numbers x1, x2, and x3 will be:
HM of x1, x2, x3 = 3 / (1/x1+ 1/x2 + 1/x3)
Taking LCM, we will get the result as follows:
HM of x1, x2, x3 = 3 x1 x2x3 / x1 x2+ x2 x3+ x1 x3
The same pattern can be followed further to get the Harmonic Mean of n numbers as well.
(C) The formula to calculate Harmonic Mean of ‘n’ Numbers:
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
The Sum of Reciprocal of the observations will be:
1/x1 + 1/x2 + 1/x3+…………………………..+ 1/xn …………………….(i)
Now, calculate the Arithmetic Mean of reciprocal of observations 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.
- Harmonic Mean Formula for Harmonic Mean between two numbers ‘a’ and ‘b’:
H1 is called one Harmonic Mean between two numbers ‘a’ and ‘b’ if:
a, H1, b form a Harmonic Progression (HP)
Similarly, H1 and H2 are called two Harmonic Means between two numbers ‘a’ and ‘b’ if,
a, H1, H2, b form a Harmonic Progression (HP)
Relationship with Other Mean Formulas
For the given positive numbers, the Arithmetic Mean is always greater than or equal to the Geometric Mean of given numbers; similarly, the Geometric Mean for the given numbers is always greater than or equal to the Harmonic Mean of the given numbers.
AM ≥ GM ≥ HM
Conclusion
In this article, all the formulas that could be used in Harmonic Mean were discussed. The formula for Harmonic Mean of two numbers, three numbers, and subsequently ‘n’ numbers were explained with examples. The derivation of formulas and their relationship with each other was also discussed. How the formula for Harmonic Mean between two numbers is different from Harmonic Mean of numbers was described. The mathematical interpretation and relation of Harmonic Mean with other Mean formulas such as Arithmetic Mean and Geometric Mean were also clarified.
