A Harmonic Progression is a sequence if the reciprocals of its terms are in Arithmetic Progression, and harmonic mean (or shortly written as HM) can be calculated by dividing the number of terms by reciprocals of its terms.
For example, Terms t1, t2, t3 is HP if and only if
Then, Harmonic Mean =
HM gives less weightage to large values and more weightage to small values and thus does the balancing act properly. The Harmonic mean has an application in many fields like physics, finance, geometry, trigonometry etc. Harmonic Mean is used when we need to give greater weights to smaller items.
Table of Content:
- What is Harmonic Mean?
- Formula for Harmonic Mean
- Harmonic Mean of Two Numbers
- Weighted Harmonic Mean
- Steps to Calculate
- Solved Problems
What is Harmonic Mean?
Harmonic Mean in statistics is the reciprocal of the arithmetic mean of the values. It is based on all observations and is rigidly defined. It is applied in the case of times and average rates.
Relationship between Arithmetic mean, Geometric Mean and Harmonic Mean
For n terms, a1, a2, a3, ……., an.
Arithmetic mean =
Geometric mean =
Harmonic mean =
In statistics, Arithmetic Mean, Geometric Mean and Harmonic Mean are called Pythagorean Means.
Where Arithmetic mean is denoted as A, Geometric Mean as G and Harmonic Mean as H.
Harmonic Mean Formula
If x1,x2,….,xn are the n individual items, the Harmonic mean is given by
Harmonic Mean =
Harmonic Mean of Two Numbers
Harmonic Mean of two numbers is an average of two numbers.
In particular, Let a and b be two given numbers and H be the HM between them a, H, b are in HP.
Again, if three terms are in HP, then the middle term is called the Harmonic Mean between the other two, so if a, b, c are in HP, then b is the HM of a and c.
Single HM of n Positive Numbers:
Let n positive numbers be a1, a2, a3, …, an and H be the HM of these numbers, then
Points to Remember :
1. HM of a, b, c is
2. The AM between two numbers a and b is
It does not follow that HM between the same numbers is
Insert n-Harmonic Mean Between Two numbers
Let a and b be two given numbers and H1, H2, H3,….., Hn are n HM’s between them.
Then, a, H1, H2, H3, …., Hn, b will be in HP, if D be the common difference of the corresponding AP.
b = (n+2)th term of HP.
This implies, D = (1/b – 1/a)/(n+1)
Harmonic Mean For Grouped Data
Consider x1, x2, x3, …. ,xn are n individual values and f1, f2, f3, …..,fn are the frequencies, then
Weighted Harmonic Mean
The Harmonic Mean as defined is the special case,
When all of the weights are equal to 1, and
Equivalent to any weighted HM considering all weights are equal.
If a set of weights w1, w2,……………,wn is associated with the sample space x1, x2,……….…, xn, the Weighted Harmonic Mean is defined by
How to Calculate the Harmonic Mean
Below are Steps to find the harmonic mean of any data:
Step 1: Understand the given data and arrange it.
Step 2: Set up the harmonic mean formula (Given above)
Step 3: Plug the value of n and sum of reciprocal of all the entries into the formula.
Step 4: Solve and get your result.
Harmonic Mean Problems
Example 1: If H be the harmonic mean between x and y, then show that
Solution: We have,
By componendo and dividendo, we have
Which is true as, x, H, y are in HP. Hence, the required result.
Example 2: Find the harmonic mean for integers from 15 to 24.
Solution: There are 10 integers between 15 to 24 i.e. n = 10
Let x1 = 15, x2 = 16, …………..…, x10 = 24
Sum of reciprocal of all the terms = (1/15)+(1/16)+(1/17)+(1/18)+(1/19)+(1/20)+(1/21)+(1/22)+(1/23)+(1/24)+(1/25) = 1/1.906
Therefore the harmonic mean is:
HM = (number of terms) / (Sum of reciprocal of all the terms)