Harmonic Mean Formula
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,
- Mean
- Median
- 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 x_{1}, x_{2}, x_{3}, x_{4}, … x_{n} is the given data then the Harmonic Mean of the given data can be calculated by the formula:
Harmonic Mean (H.M) =
Proof
Harmonic mean is the inverse of arithmetic mean of reciprocal data terms.
So the arithmetic mean for the data x_{1}, x_{2}, x_{3}, …, x_{n} is,
Arithmetic mean =
In harmonic mean we consider reciprocal of data values.
So the arithmetic mean of reciprocal data 1/x_{1}, 1/x_{2}, 1/x_{3}, …, 1/x_{n }is,
Arithmetic mean for reciprocal data = ->(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}
=
=
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 w_{1},w_{2},w_{3},w_{4},…,w_{n} and the data x_{1}, x_{2}, x_{3}, x_{4}, …, x_{n} is,
Weighted harmonic mean =
Let’s look into the relationship between types of means,
Relation between Arithmetic Mean, Harmonic Mean, and Geometric Mean
For the data x_{1}, x_{2}, x_{3}, x_{4}, …, x_{n},
Arithmetic Mean = (x_{1 }+ x_{2 }+ x_{3 }+ x_{4 }+….+ x_{n})/n
Harmonic Mean = n/((1/x_{1}) + (1/x_{2}) + (1/x_{3}) + (1/x_{4}) + … + (1/x_{n}))
Geometric mean = n√(x_{1 }+ x_{2 }+ x_{3 }+ x_{4 }+ … + x_{n})
From these formulas we can write it as,
Geometric mean = √(Arithmetic Mean × Harmonic Mean)
Geometric mean^{2 }= Arithmetic Mean × Harmonic Mean
From the Above formula,
Harmonic Mean = Geometric mean^{2}/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=
=
= 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=
=
= 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=
=
= 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.M^{2}/A.M
= 7^{2}/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., ∑w_{i}, 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 Mean_{w }= ∑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., ∑w_{i}, 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 Mean_{w }= ∑w/∑w/x
= 15/0.831
= 18.05
Weighted Harmonic mean for the given data is 18.05
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