Harmonic Mean: Overview, Questions, Preparation

Statistics 2021 ( Statistics )

Rachit Kumar Saxena

Rachit Kumar SaxenaManager-Editorial

Updated on Aug 5, 2021 06:47 IST
Table of Contents
  1. What is Harmonic Mean?
  2. Weightage of Harmonic Mean in Class 11
  3. Illustrated Examples on Harmonic Mean
  4. FAQs on Harmonic Mean

What is Harmonic Mean?

In statistics, the metric of central propensity is used to describe data or meaning in a sequence. A central propensity metric is a single value that defines how a data community clusters around a central value. It determines the data set's core. Three tests of central inclination occur. They are the mean, median, and mode. With the description, formula, and in-depth examples given in this post, you will learn an important form of the mean called the "Harmonic Mean". 

Harmonic Mean

The Concept of Harmonic Mean 

The Harmonic Mean (HM) is the reciprocal of the arithmetic mean for the given data values. The harmonic mean gives the big values a lower weight and the small values a higher weight to match them accurately. It is commonly used where smaller things need to be assigned greater weight. In the case of time and mean rates, it is added. 

The Formula of Harmonic Mean 

Since the inverse of the arithmetic mean is the harmonic mean, the formula for the harmonic mean 'HM' is: 

If x1, x2, x3,..., xn, up to n quantities, are the individual products, then, 

Harmonic Mean, HM = n / [(1/x1)+(1/x2)+(1/x3)+...+(1/xn)]

Uses of the Harmonic Mean 

The major applications of harmonic means are as follows: 

  • In economics, the harmonic mean is related to average multiples such as the price-earnings ratio. 
  • Industry technicians often use it to assess trends such as Fibonacci Sequences. 

Harmonic Mean Merits and Demerits 

The merits of the harmonic mean are as follows: 

  1. It is constrained rigidly. 
  2. It focuses on all the views of a set, i.e., no object of a series can be computed without it. 
  3. It supports the algebraic approach. 
  4. It produces a more consistent outcome if the outcomes in question are the same using the different methods to compute the mean. 
  5. It gives the smallest object in a sequence the greater weight. 
  6. It can help locate a negative value in a sequence. 
  7. It derives a distorted distribution from a regular distribution. 
  8. It creates a straighter curve than those created by the arithmetic mean (A.M) and geometric mean (G.M). 

The demerits of the harmonic series are: 

  1. The harmonic mean is substantially influenced by severe entity values. 
  2. It is difficult to calculate if all of the objects are empty. 
  3. The measurement of the harmonic mean is tedious, as it uses the reciprocals of the sum.

Weightage of Harmonic Mean in Class 11

This topic is taught in the chapter Statistics. You will learn about the harmonic mean and its application in statistics. The weightage of this chapter is ten marks.

Illustrated Examples on Harmonic Mean

1. Find the harmonic mean between 2 and 4. 

H.M.= 2/(1/a1)+(1/a2)

Here, a1 and a2 are 2 and 4 respectively.

H.M=  (2x4)/3


2. Determine the harmonic mean of the first five odd numbers.


H.M = 5/(1+1/3+1/5+1/7+1/9)

= 2.79

3. Find the harmonic mean of the values 2, 4, 8, 12.

H.M = 4/{(12+6+3+2)/24}

= 4.17

FAQs on Harmonic Mean

Q: What is the harmonic meaning? 

A:  A harmonic mean is a form of average commonly used for statistics that reflects a rate or ratio such as accuracy and data retrieval recall. 

Q: Why is it called the harmonic mean? 

A: The association between a harmonic average and a harmonic progression gives the name harmonic mean. 

Q: What is the average harmonic of any two numbers? 

A: A sum of two numbers is the harmonic mean of two numbers.

Q: Does the harmonic mean influence extreme values? 

A: It does not influence extreme values. 

Q: How does the harmonic series relate to the arithmetic sequence? 

A: The progression created by the reciprocals of an arithmetic progression is called a harmonic sequence (or harmonic progression).


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