Harmonic Mean Merits and Demerits

The measure of central tendency is used in statistics to represent data or values in series. A single value that reflects how a bunch of data clusters around a core value is called a measure of central tendency. It designates the data set’s centre. There are three approaches to access central tendency. The three are the mean, median, and mode. This article will teach you about one of the most essential types of mean, the “Harmonic Mean,” including its definition, formula, and examples.

To correctly balance the data, the harmonic mean provides less weight to the larger values and more weight to the smaller values. When it’s necessary to give tiny elements more weight, the harmonic mean is typically used. When calculating the average of ratios or rates of given variables, the harmonic mean is frequently utilised. Because it equalises the weights of each data point, it is the most suited metric for ratios and rates.

A harmonic mean is the Pythagorean mean. The harmonic mean is obtained by dividing the number of terms in a data series by the sum of all reciprocal terms. When compared to the geometric and arithmetic means, the harmonic mean will always be the lowest.

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)]

Weighted Harmonic Mean

The weighted harmonic mean is calculated in the same way as the simple harmonic mean. It’s a type of harmonic mean in which all of the weights are equal to one. The weighted harmonic mean is defined by the set of weights w₁, w₂, w₃, …, w_n associated to the sample space x₁, x₂, x₃,…, x_n.

The harmonic mean is determined as follows if the frequencies “f” are meant to represent the weights “w”:

The weighted harmonic mean is if x₁, x₂, x₃,…, x_n are n elements with corresponding frequencies f₁, f₂, f₃, …., f_n.

HM_w = N / [ (f₁/x₁) + (f₂/x₂) + (f₃/x₃)+ ….(f_n/x_n) ]

Merits and Demerits of Harmonic Mean

The benefits of the harmonic mean are as follows:

1. It is very clearly defined.

2. It is calculated using all of a series’ observations; it cannot be calculated without considering any of the series’ items.

3. It can be further algebraically treated.

4. It produces greater results when the goals to be reached are the same for all methods used.

5. It gives the smallest component in a series the most weight.

6. It can be calculated for any negative value in a series.

7. It transforms a skewed distribution into a normal distribution.

8. It produces a curve that is straighter than the arithmetic and geometric mean.

The harmonic mean has the following drawbacks:

1. It is difficult to comprehend for a prudent person.

2. Its calculation is time-consuming because it requires obtaining the reciprocals of the numbers.

3. When the means used for diverse ends are the same, it does not produce better or more accurate results.

4. Its algebraic treatment is substantially more constrained than the arithmetic mean’s.

5. The values of the extreme items have a significant impact.

6. It is impossible to determine whether any of the things are zero.

Harmonic Mean Uses

The following are the most common applications of harmonic means:

• In finance, the harmonic mean is used to calculate average multiples such as the price-earnings ratio.

• It is also utilised by market technicians to discover patterns such as Fibonacci Sequences.

CONCLUSION

A harmonic mean is the Pythagorean mean. We calculate it by multiplying the number of terms in a data series by the sum of all reciprocal terms. When compared to the geometric and arithmetic means, it will always be the lowest.

The reciprocal of the average of the reciprocals of the data values is the Harmonic Mean (HM). It is precisely defined and is based on all observations. 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 used in the case of times and average values.