Harmonic Mean Definition and Formula

The harmonic mean is a type of numerical average. It’s determined by multiplying the number of observations by the reciprocal of each series number. As a result, the harmonic mean is the reciprocal of the reciprocal arithmetic mean.

Harmonic Mean Definition

The reciprocal of the average of the reciprocals of the data values is the Harmonic Mean (HM). It is properly defined and is founded on all available evidence. To balance the values correctly, the harmonic mean provides less weight to high values and more weight to small values. In general, the harmonic mean is utilised when it is necessary to give smaller items more weight. It is employed in the computation of times and average rates.

Harmonic Mean Formula

Because the harmonic mean is the reciprocal of the average of reciprocals, the following formula is used to define the harmonic mean “HM”:

If x₁, x₂, x₃,…, x_n are the individual items up to n terms, then,

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

How to Find a Harmonic Mean?

If the given data values are a, b, c, d,…, then the steps to find the harmonic mean are as follows:

Step 1: Find the reciprocal of each value (1/a, 1/b, 1/c, 1/d, and so on).

Step 2: Calculate the average of the reciprocals found in step 1.

Step 3: Finally, multiply the average obtained in step 2 by its reciprocal.

Properties of Harmonic Mean

Harmonic means have a few characteristics that distinguish them from other forms of means.

1. If all of the observations are made at the same constant, say c, then the harmonic means determined for the observations are also c.

2. For any series with negative values, the harmonic mean can be calculated.

3. Because the reciprocal of 0 does not exist, the harmonic mean of a particular series cannot be determined if any of its values are 0.

4. If no two values in a particular series are equal, and no value is 0, the harmonic mean calculated will be less than the geometric and arithmetic means.

5. The harmonic mean, when compared to the geometric and arithmetic means, has the lowest value, i.e.AM > GM > HM.

Uses of Harmonic Mean

• In computing average pricing, average speed, and so on under particular conditions, the harmonic mean is utilised.

• In the financial sector, the harmonic mean is used to determine average multiples such as the price-earnings ratio.

• It’s also used to make Fibonacci sequences.

Harmonic Mean Example

Assume we have a series of 1, 3, 5, and 7. Each word has a difference of two. This creates a mathematical progression. We take the reciprocal of these terms to determine the harmonic mean. This is expressed as 1, 1/3, 1/5, and 1/7. (the sequence forms a harmonic progression). The total number of phrases (4) is then divided by the sum of the terms (1 + 1/3 + 1/5 + 1/7). As a result, the harmonic mean is calculated as 4 / (1 + 1/3 + 1/5 + 1/7) = 2.3864.

Solved Example 

1. What are the harmonic mean and harmonic arithmetic for the numbers 4, 5, and 10?

Solution:

The reciprocals of the numbers 4, 5, and 10 are as follows:

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

Upon adding them

0.25 + 0.20 + 0.10 = 0.55

Total values are 3 so divide it by 3

Average = 0.55/3

Our answer is the reciprocal of that average:

Harmonic Mean = 3/0.55

     = 5.454

CONCLUSION

The harmonic mean is a form of Pythagorean mean that measures central tendency. What it is is the reciprocal of the arithmetic mean of reciprocals of the observations. Larger values are given less weight, whereas lower values are given more weight. When calculating the average of ratios or rates of supplied numbers, it is commonly utilised. It is the most appropriate metric for ratios and rates since it equalises the weights of each data item.