Harmonic Mean Formula
The harmonic mean of the data set is calculated using the formula, Let x1, x2, x3, x4, … xn is the n terms of the given data then the Harmonic Mean of the given data can be calculated by the formula,
Harmonic Mean (H.M) =
Proof of Harmonic Mean Formula
As harmonic mean is the inverse of the arithmetic mean of reciprocal data terms.
So the arithmetic mean for the data x1, x2, x3, …, xn is,
Arithmetic mean =
In harmonic mean, we consider the reciprocal of data values.
So the arithmetic mean of reciprocal data 1/x1, 1/x2, 1/x3, …, 1/xn is,
Arithmetic mean for reciprocal data = . . .(1)
It is known that the Harmonic mean is the inverse of the arithmetic mean of reciprocal data values from eq (1)
Harmonic Mean = Inverse of the Arithmetic Mean of Reciprocal Data
⇒ Harmonic Mean =
⇒ Harmonic Mean =
This is the harmonic mean formula of the given data set.
Harmonic Mean of Two Numbers
We can find the harmonic mean of the two numbers by using the formula discussed above, suppose the two numbers are, a and b
n = 2
Reciprocal of a and b is 1/a and 1/b
HM = 2/[1/a + 1/b]
HM = (2ab)/(a + b)
Weighted Harmonic Mean
It is similar to the Harmonic mean but in addition, to the normal value we take the weight value of the data set. If the weights of each data set is equal to 1 then it is the same as the Harmonic mean formula. Weighted Harmonic mean is calculated for the given set of weights of the data set, Suppose the weights of the data set are, w1,w2,w3,w4,…,wn and their values are x1, x2, x3, x4, …, xn is, then the weighted harmonic mean formula,
Weighted Harmonic Mean =
This formula is used when the weight of the given data set is given.
Harmonic Mean
Harmonic Mean is the type of mean that is used when we have to find the average rate of change, it is the mean calculated by taking the reciprocal values of the given value and then dividing the number of terms by the sum of the reciprocal values. The harmonic mean is one of the Pythagorean mean and the other two Pythagorean mean are,
- Arithmetic Mean
- Geometric Mean
These means tell us about various parameters of the data set.
Harmonic Mean also denoted as HM is the mean calculated by taking the reciprocal of the given set. In this article, we will learn about HM, its formula, examples, and others in detail.