Harmonic Mean

What is Harmonic Mean?

We are familiar with calculating the arithmetic mean, in which the sum of values is divided by the number of values. Now in this article let us study what is harmonic mean in statistics, properties of harmonic mean(HM), harmonic mean examples..A simple way to define harmonic mean is: It is the reciprocal of the arithmetic mean of the reciprocals of the observations. Harmonic mean is used to calculate the average of a group of numbers. The number of elements will be averaged and divided by the sum of the reciprocals of the elements.The most common examples of ratios are that of speed and time,work and time etc.

Harmonic Mean Definition

The Harmonic Mean (HM) is defined as the reciprocal of the arithmetic mean of the reciprocals of the observations. Harmonic mean gives less weightage to the larger values and more weightage to the smaller values to balance the values properly. The harmonic mean is generally used when there is a necessity to give greater weight to the smaller items.The harmonic mean is often used to calculate the average of the ratios or rates of the given values. It is the most appropriate measure for ratios and rates because it equalizes the weights of each data point.

HM Formula

By the harmonic mean definition, harmonic mean is the reciprocal of the arithmetic mean, the formula to define the harmonic mean “H” is given as follows:

Harmonic Mean(H) = n / [(1/x1)+(1/x2)+(1/x3)+…+(1/xn)]


Where,

n is total number of terms

 x1, x2, x3,…, xn are the individual values up to nth terms.

Solved Example

Example 1 : Find the harmonic mean of the following data {8, 9, 6, 11, 10, 5} ?

Solution:

Given data: {8, 9, 6, 11, 10, 5}

We have harmonic mean formula as:

Harmonic Mean(H) = n / [(1/x1)+(1/x2)+(1/x3)+…+(1/xn)]

n = 6

So Harmonic mean = 6/ [ (⅛) + (1/9) + (⅙ ) + ( 1/11 ) + ( 1/10 ) + ( ⅕)]

= 6/0.7936

= 7.560

Harmonic mean(H) = 7.560

Weighted Harmonic Mean

Weighted harmonic mean is a special case of harmonic mean where all the weights are equal to 1. It is similar to the simple harmonic mean. If the set of weights such as w1, w2, w3, …, wn connected with the sample space x1, x2, x3,…., xn, then the weighted harmonic mean is defined by

If the frequencies “f” is supposed to be the weights “w”, then the harmonic mean is calculated as follows:

If x1, x2, x3,…., xn are n items with corresponding frequencies f1, f2, f3, …., fn, then the weighted harmonic mean is

HMw = N / [ (f1/x1) + (f2/x2) + (f3/x3) + (f4/x4)….+ (fn/xn) ]


Also written as 

HM = \[\frac{W}{Σ\frac{w}{x}}\]


Where w = weight and x is the variable

Properties of Harmonic Mean

  • If all the observations taken are constants, say c, then the harmonic mean of the observations is also c.

  • The harmonic mean has the least value as compared to the geometric mean and the arithmetic mean i,e AM > GM > HM

Uses of Harmonic Mean

Harmonic mean is basically used in 

  • In calculating average prices, average speed etc under certain conditions.

  • The harmonic mean is useful in the finance sector to calculate the average multiples like the price-earnings ratio

  • It is also used in computing Fibonacci Sequences.

Merits and Demerits of Harmonic Mean

The merits and demerits of harmonic mean are as follows:

Merits:

1. It is rigidly defined

2. It is based on all observation 

3. It is least affected by fluctuation in sampling

4. It gives to greater importance to small items

5. It is capable of further algebraic treatment.

Demerits:

1. It is difficult to calculate

2. It does not give equal weight to every item

3. It may not be represented in the actual data

4. It is not defined for negative value.

Steps to Calculate Harmonic Mean

Step 1: Calculate the reciprocal (1/value) for every value.

Step 2: Find the average of those reciprocals, by just adding them and divide by number of total values

Step 3: Then do the reciprocal of that average.

Solve Example 

What is the harmonic mean of 4, 5 and 10?

Solution:

The reciprocals of 4, 5 and 10 are:

1/4 = 0.25 ;  1/5 = 0.20 ;   1/10 = 0.10

Now add them

 0.25 + 0.20 + 0.10 = 0.55

Total values are 3 so divide it by 3

Average = 0.55/3

The reciprocal of that average is our answer:

Harmonic Mean = 3/0.55

     = 5.454 (to 3 places)

FAQ (Frequently Asked Questions)

1. State the Relationship Between Arithmetic Mean, Geometric Mean and Harmonic Mean?

In statistics, three means such as arithmetic mean, geometric mean, harmonic means are known as Pythagorean means. 

The formulas for three different types of means are:

Arithmetic Mean = (a1 + a2 + a3 +…..+an ) / n

Harmonic Mean = n / [(1/a1)+(1/a2)+(1/a3)+…+(1/a3)]

Geometric Mean = n√a1 x a2 x a3  x…….. xan  

If G is the geometric mean, H is the harmonic mean, and A is the arithmetic mean, then the relationship between them is given by-

G = √AH


2. What is Arithmetic Mean?

Answer: Arithmetic mean is simply defined as a value that is computed by dividing the sum of a set of terms by the number of total terms. Arithmetic mean is the same as the average of data. Suppose we are given a ‘ n ‘ number of data and we need to compute the arithmetic mean, all that we need to do is add up all the numbers and divide the result by the total numbers. The Arithmetic mean is written as AM.

The formula for Arithmetic Mean is given by

AM = (a1 + a2 + a3 +…..+an ) / n

Where a1, a2 , ….an are individual values.