Harmonic Mean

In Statistics, to represent the data or value in series, the measure of central tendency is used. A measure of central tendency is a single value which describes the way that the group of data clusters around a central value. It defines the centre of the dataset. There are three measures of central tendency. They are:

  • Mean
  • Median
  • Mode

In this article, let us discuss one of the important types of mean called “Harmonic Mean” with the definition, formula, and examples in detail.

Harmonic Mean Definition

The Harmonic Mean (HM) is defined as the reciprocal of the arithmetic mean of the given data values. It is based on all the observations, and it is rigidly defined. Harmonic mean gives less weightage to the large values and large weightage to the small values to balance the values properly. In general, the harmonic mean is used when there is a necessity to give greater weight to the smaller items. It is applied in the case of times and average rates.

Harmonic Mean Formula

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

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

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

Relationship Between Arithmetic Mean, Geometric Mean and Harmonic Mean

The 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/an)]

Geometric Mean = \(\sqrt[n]{a_{1}.a_{2}.a_{3}…a_{n}}\)

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

\(G = \sqrt{AH}\)

Weighted Harmonic Mean

Calculating weighted harmonic mean is similar to the simple harmonic mean. It is a special case of harmonic mean where all the weights are equal to 1. 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

\(HM_{w}=\frac{\sum_{i=1}^{n}w_{i}}{\sum_{i=1}^{n}\frac{w_{i}}{x_{i}}}\)

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)+ ….(fn/xn) ]

Note:

  1. f values are considered as weights
  2. For continuous series, mid-value = (Lower limit + Upper limit)/2 is taken as x

Harmonic Mean Example

Question:

Calculate the harmonic mean for the following data:

x

1

3

5

7

9

11

f

2

4

6

8

10

12

Solution:

The calculation for the harmonic mean is shown in the below table:

x

f

1/x

f/x

1

2

1

2

3

4

0.333

1.333

5

6

0.2

1.2

7

8

0.143

1.143

9

10

0.111

1.111

11

12

0.091

1.091

N =42

Σ f/x = 7.878

The formula for weighted harmonic mean is

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

HMw = 42 / 7.878

HMw = 5.331

Therefore, the harmonic mean, HMw is 5.331

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