Single HM of n Positive Numbers

Harmonic Mean is one of the measures of average calculation. For two numbers, we can derive the Harmonic Mean of the given numbers and we can also calculate the Harmonic Mean between those given numbers. Calculating a single Harmonic Mean of ‘n’ positive numbers is similar to the Harmonic Mean of two or more numbers. Just like Arithmetic Mean, calculating Harmonic Mean is very simple. The harmonic mean of given numbers will be equal to the number of observations divided by the sum of reciprocal of the given numbers. We will discuss a stepwise approach to calculate the Single Harmonic Mean of ‘n’ positive real numbers. Let’s discuss in detail the concepts revolving around the topic and derivation formulas.

Single Harmonic Mean of n ‘positive’ numbers:

Single Harmonic Mean of n ‘positive’ numbers can be defined as the reciprocal of the Arithmetic Mean of the reciprocal of the observations. Now, let us break the definition of Harmonic Mean stated here to get a deep insight of the meaning of the terms used therein:

Let us assume that we have n number of observations and the observations are x1, x2, x3,…………………xn

Sum of Reciprocal of the observations will be:

1/x1 + 1/x2 + 1/x3+…………………………..+ 1/xn         …………………….(i)

Now, the Arithmetic Mean of reciprocal of observations can be calculated as follows:

( 1/x1 + 1/x2 + 1/x3+…………………………..+ 1/xn ) / n        ………………………….(ii)

Now, the reciprocal of Arithmetic Mean of the reciprocal of the observations will be simply the reciprocal of what we just calculated above in step (ii):

n / (1/x1 + 1/x2 + 1/x3+…………………………..+ 1/xn ) ………………………(iii)

Resulting equation (iii) is called the Harmonic Mean.

Let us understand the Harmonic Mean n ‘positive’ numbers with the help of a practical example below:

Example 1:

Let us assume that we have been given 4 observations and those are 2, 4, 6 and 10. Calculate Harmonic Mean for the given observations.

Solution:

Given Observations are:  2,4,6, 10

No. of Observations (n) = 4

Step 1:

Sum of Reciprocal of the observations will be:

1/x1 + 1/x2 + 1/x3+…………………………..+ 1/xn         …………………….(i)

Putting the values in the equation (i) above, we will get

= 1/2 + 1/4 + 1/6 + 1/10

Step 2:

The Arithmetic Mean of reciprocal of observations can be calculated as follows:

(1/x1 + 1/x2 + 1/x3+…………………………..+ 1/xn ) / n        ………………………….(ii)

Putting the values in the equation (ii) above, we will get

= (1/2 + 1/4 + 1/6 + 1/10) / 4

Step 3:

Now, the reciprocal of Arithmetic Mean of the reciprocal of the observations will be simply the reciprocal of what we just calculated above in step (ii):

n / (1/x1 + 1/x2 + 1/x3+…………………………..+ 1/xn ) ………………………(iii)

Putting the values in the equation (iii) above, we will get

= 4 / (1/2 + 1/4 + 1/6 + 1/10)

= 3.92

Hence, Harmonic Mean for the given observations = 3.92

Example 2:

 Insert three Harmonic Means between two numbers 15 and 15/61.

Solution:

Given Harmonic Progression (H.P.) will be as below:

15, H1, H2, H3, 15/61

a1 = 15/1 (first term)

a5 = 15/61 (fifth term)

We know that:  A1, A2,  A3,………………………….An is an A.P.

Here

A1 = 1/15 (Reciprocal of a1)

A5 = 61/15 (Reciprocal of a6)

Hence, we can calculate d from here:

a+4d = 61/15

1/15 + 4d = 61/15

d = 1

Now, as we have value of d, we can calculate all the missing values as follows:

A2 = 1/15+ 1 = 16/15

Hence H1 = 15/16

A3 = 1/15+2*1 = 31/15

H2 = 15/31

A4 = 1/15 + 3*1 = 46/15

H3 = 15/46

Properties of Mean:

• If all the observations of the series are constant ‘K’, the mean will also be ‘K’. The same property applies for all the means be it Arithmetic Mean, Geometric Mean, or Harmonic Mean.
• If the deviation in the series is taken from the mean, the sum of deviations from the mean will be zero.
• If we change the origin, the same change will consequently be seen in the Mean as well.
• Similarly, if we change the scale, the same scale change will be seen eventually in the Mean.

Conclusion:

In this article, we discussed how to calculate the single Harmonic Mean of ‘n’ positive numbers. First, we learned the derivation of the formula for Harmonic Mean of ‘n’ positive numbers. We saw practical examples for a better understanding of the formulas derived above and the application of the same in the problems. Then we discussed how the formula for Harmonic Mean between two numbers is different from the Harmonic Mean of numbers. We also discussed mathematical interpretation and relation of Harmonic Mean with other Mean formulas such as Arithmetic Mean and Geometric Mean. We hope this study material will be helpful for you and will give you a deep understanding of the topic.