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Harmonic Mean Formula

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  • Last Updated : 01 Feb, 2022
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Statistics deals with data analysis and presents the data/values in an organized manner. To represent data in a series statistics use the concept of central tendency. The measure of central tendency is a value that describes the group of clusters of data around a central value. It is the center of a set of data. 

There are three measures of central tendencies,

  1. Mean
  2. Median
  3. Mode

Now let’s know an important type of mean called Harmonic Mean.

Harmonic Mean

Harmonic Mean is defined as reciprocal of average of reciprocal of all the data values. It gives less weightage to the larger values and large weightage to smaller values. This harmonic mean is mainly used in the finance sector.

Let x1, x2, x3, x4, … xn is the given data then the Harmonic Mean of the given data can be calculated by the formula:

Harmonic Mean (H.M) = \frac{n}{(1/x_1)+(1/x_2)+(1/x_3)+(1/x_4)+...+(1/x_n)}

Proof

Harmonic mean is the inverse of arithmetic mean of reciprocal data terms.

So the arithmetic mean for the data x1, x2, x3, …, xn is,

Arithmetic mean = \frac{x_1+x_2+x_3+...+x_n}{n}

In harmonic mean we consider reciprocal of data values.

So the arithmetic mean of reciprocal data 1/x1, 1/x2, 1/x3, …, 1/xn  is,

Arithmetic mean for reciprocal data = \frac{(1/x_1)+(1/x_2)+(1/x_3)+...+(1/x_n)}{n} ->(1)

It is known that Harmonic mean is inverse of arithmetic mean of reciprocal data values from-(1)

Harmonic Mean = Arithmetic Mean of Reciprocal data-1

= (\frac{(1/x_1)+(1/x_2)+(1/x_3)+...+(1/x_n)}{n})^{-1}

\frac{n}{(1/x_1)+(1/x_2)+(1/x_3)+...+(1/x_n)}

Weighted Harmonic Mean

It is Similar to Harmonic mean but in addition, is to this is weights. If weights are equal to 1 then it is the same as Harmonic mean. Weighted Harmonic mean is calculated for the given set of weights w1,w2,w3,w4,…,wn and the data x1, x2, x3, x4, …, xn is,

Weighted harmonic mean = \frac{\sum_{i=1}^{i=n}W_i}{\sum_{i=1}^{i=n}\frac{w_i}{x_i}}

Let’s look into the relationship between types of means,

Relation between Arithmetic Mean, Harmonic Mean, and Geometric Mean

For the data x1, x2, x3, x4, …, xn,

Arithmetic Mean = (x1 + x2 + x3 + x4 +….+ xn)/n

Harmonic Mean = n/((1/x1) + (1/x2) + (1/x3) + (1/x4) + … + (1/xn))

Geometric mean = n√(x1 + x2 + x3 + x4 + … + xn)

From these formulas we can write it as,

Geometric mean = √(Arithmetic Mean × Harmonic Mean)

Geometric mean2 = Arithmetic Mean × Harmonic Mean

From the Above formula,

Harmonic Mean = Geometric mean2/Arithmetic Mean

Let’s look into a few examples of finding Harmonic Mean,

Sample Problems

Question 1: What is the Harmonic Mean for the data 10, 20, 5, 15, 10.

Solution:

Given data

10, 20, 5, 15, 10

n = 5 (As total size is 5)

Harmonic mean= \frac{n}{(1/x_1)+(1/x_2)+(1/x_3)+(1/x_4)+...+(1/x_n)}

\frac{5}{(1/10)+(1/20)+(1/5)+(1/15)+(1/10)}

= 5/(0.1 + 0.05 + 0.2 + 0.06 + 0.1)

= 5/0.51

= 9.8

Hence Harmonic mean for the given data is 9.8

Question 2: What is the Harmonic Mean for the data 100, 50, 75, 25.

Solution:

Given data

100, 50, 75, 25

n = 4 (As total size is 4)

Harmonic mean=  /frac{n}/{(1/x_1)+(1/x_2)+(1/x_3)+(1/x_4)+...+(1/x_n)}

\frac{4}{(1/100)+(1/50)+(1/75)+(1/25)}

= 4/(0.01+0.02+0.013+0.04)

= 4/0.083

= 48.2

Hence Harmonic mean for the given data is 48.2

Question 3: What is the Harmonic Mean for the data 1, 5, 3.

Solution:

Given data

1, 5, 3

n = 3 (As total size is 3)

Harmonic mean= \frac{n}{(1/x_1)+(1/x_2)+(1/x_3)+(1/x_4)+...+(1/x_n)}

\frac{3}{(1/1)+(1/5)+(1/3)}

= 3/(1 + 0.2 + 0.33)

= 3/1.53

= 1.96

Hence Harmonic mean for the given data is 1.96

Question 4: What is the Harmonic Mean if the Arithmetic mean is 10 and the Geometric mean is 7.

Solution:

Given, 

Arithmetic Mean (A.M) = 10

Geometric Mean (G.M) = 7

Harmonic Mean = G.M2/A.M

= 72/10

= 49/10

= 4.9

Hence Harmonic mean from given Arithmetic and geometric mean is 4.9

Question 5: What is the Geometric Mean if the Arithmetic mean is 20 and the Harmonic mean is 15.

Solution:

Given

Arithmetic Mean (A.M) = 20

Harmonic Mean (H.M) = 15

Geometric Mean = √(Arithmetic Mean × Harmonic Mean)

= √(20 × 15)

= √300

= 17.32

Hence Geometric mean from given Arithmetic and Harmonic mean is 17.32

Question 6: What is the weighted harmonic mean for the given data.

Weight(w)

Data(x)

1

20

2

30

3

10

2

15

Solution:

Find Summation of Weights i.e., ∑wi, 1/x and w/x and ∑w/x

Weights(w)

x

1/x

w/x

1

20

0.05

0.05

2

30

0.03

0.06

3

10

0.1

0.3

2

15

0.06

0.12

∑w=8

 

 

∑w/x=0.53

Harmonic Meanw = ∑w/∑w/x

= 8/0.53

= 15.09

Weighted Harmonic mean for the given data is 15.09

Question 7: What is the weighted harmonic mean for the given data.

x

10

15

20

25

30

w

2

3

4

5

1

Solution:

Find Summation of Weights i.e., ∑wi, 1/x and w/x and ∑w/x

x

w

1/x

w/x

10

2

0.1

0.2

15

3

0.066

0.198

20

4

0.05

0.2

25

5

0.04

0.2

30

1

0.033

0.033

 

∑w = 15

 

∑w/x = 0.831

Harmonic Meanw = ∑w/∑w/x

= 15/0.831

= 18.05

Weighted Harmonic mean for the given data is 18.05


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