What is Harmonic Mean: Uses, Advantages and Limitations

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". NCERT solutions on statistics are helpful for those who want to learn more about the harmonic mean in detail.

The Harmonic Mean (HM) is the reciprocal of the arithmetic mean for the given data values. It gives 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. 

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 the formula of Harmonic Mean will be:

$\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}}$

The formula of harmonic mean is 

$\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}}$

Here:

• $x_i$ is the ith data point in the sample
• 1/ $x_i$ is the reciprocal of every data point
• n is the total number of observations
• $\underset{i=1}{\overset{n}{\sum }}\frac{1}{x_i}$ is the sum of all reciprocals

Let us start the derivation

Formula for Harmonic mean is derived from the meaning arithmetic mean definition.

Arithmetic mean A of reciprocals of numbers $\frac{\frac{x_{\mathrm{1, }}}{}}{}$ $\frac{\frac{x_{\mathrm{2....., }}}{}}{}$ $\frac{\frac{x_{\mathrm{n\; is\; given\; by}}}{}}{}$

$A=\frac{\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}}{n}$

Now, let us take the reciprocal of arithmetic mean A which is:

$\frac{1}{A}=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}}$

The reciprocal of arithmetic mean of reciprocals is defined as harmonic mean H

H= $\frac{1}{A}=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}}$

The following are the uses of the harmonic mean. Those who understand these can perform well in exams like JEE Main and NEET:

1. Average Speeds: Whenever someone travels equal distances at different speeds, harmonic mean gives average speed for the entire trip.
2. Finance: The harmonic mean can calculate average multiples such as the price-earnings ratio. This is useful to compare the performance of different investments.
3. Electronics: While dealing with parallel circuits, harmonic mean can calculate equivalent resistance of resistors connected in parallel.
4. Hydrology: The harmonic mean can calculate average permeability rates when dealing with layers of soil or rock.
5. Biology and Ecology: It is used for calculating average growth rates or densities in populations while dealing with different segments that have varying rates.
6. Data Analysis: The harmonic mean is used in situations where you require average ratios or rates, such as in performance metrics, where different samples or groups have different sizes.

Let us take a look at the advantages of the harmonic mean in detail:

• Appropriate for Rates and Ratios: The harmonic mean provides accurate averages when dealing quantities such as speed, density, or any ratio of two different units of measure.
• Less Influence from Large Values: It is less affected by large values in dataset compared to arithmetic mean. This is useful where large values could skew the average.
• Useful for Time and Distance Problems: It is ideal for scenarios involving constant distances travelled at different speeds, providing a more accurate average speed.
• Suitable for Certain Types of Data: In fields such as finance, it is used to average the multiples, such as price-earnings ratios that provides a more accurate reflection of performance.

Exams like IISER and IIT JAM often ask questions that involve the use of harmonic mean. Therefore, knowing the importance of the harmonic mean is significant for students of such entrance exams.

The weighted harmonic mean is a special case, where all the weights are 1. Say, if the set of weights $W_1,W_2,W_3,\cdots ,W_n$ are connected with sample space $X_1,X_2,X_3,\cdots ,X_n$ then it will be

${\mathrm{HM}}_w=\frac{\underset{i=1}{\overset{n}{\sum }}{w}_i}{\underset{i=1}{\overset{n}{\sum }}\frac{w_i}{x_i}}$

Calculating the weighted harmonic mean is almost the same as calculating the simple harmonic mean.

In case, instead of weights 'w', frequencies 'f' are used then weighted harmonic mean will be as mentioned below.

If $X_1,X_2,X_3,\cdots ,X_n$ are n number of items with corresponding frequencies  $f_1,f_2,f_3,\cdots ,f_n$ . In this case, the weighted harmonic mean will be:

${\mathrm{HM}}_w=\frac{N}{\left[\frac{f_1}{x_1}+\frac{f_2}{x_2}+\frac{f_3}{x_3}+\cdots +\frac{f_n}{x_n}\right]}$

 

Let us understand the difference between geometric mean and harmonic mean:

Parameter	Harmonic Mean	Geometric Mean
Definition	Reciprocal of average of the reciprocals of the data.	The geometric mean meaning is the nth root of the product of the data.
Use Cases	Finding average rates and ratios, including average speed, density, or other rates of change.	For determining products and growth rates, including average growth over time or geometric properties.
Sensitivity to Extremes	Less sensitive to larger values, whereas it is more sensitive to smaller values.	Less sensitive to extreme values as compared to arithmetic mean, but it is more balanced than harmonic mean.
Example	Average speed when travelling equal distances at different speeds.	Average growth rates over multiple periods.
Applicability	Suitable for datasets where values are rates or ratios.	Suitable for datasets that have exponential growth or multiplicative relationships.

 

Let us take a look at some of the illustrated examples to understand harmonic mean in more detail:

1. Find the harmonic mean between 2 and 4. 
Solution:

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

Here, a1 and a2 are 2 and 4 respectively.

H.M=  (2x4)/3

=8/3

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

Solution:

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.
Solution:

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

= 4.17