Solved Examples on Harmonic Mean
Example 1: Find the Harmonic Mean for the data 10, 20, 5, 15, 10.
Solution:
Given,
10, 20, 5, 15, 10
n = 5
Harmonic Mean=
⇒ Harmonic Mean =
⇒ Harmonic Mean = 5/(0.1 + 0.05 + 0.2 + 0.06 + 0.1)
⇒ Harmonic Mean = 5/0.51
⇒ Harmonic Mean = 9.8Hence, the Harmonic mean for the given data is 9.8 .
Example 2: Find Harmonic Mean if the Arithmetic Mean of the given data is 10 and the Geometric mean is 7.
Solution:
Given,
Arithmetic Mean (AM) = 10
Geometric Mean (GM) = 7We know that,
Harmonic Mean(HM) = (G.M)2/A.M
⇒ HM = 72/10
⇒ HM = 49/10
⇒ HM = 4.9Hence, the Harmonic mean from the given Arithmetic and geometric mean is 4.9
Example 3: Find 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) = 15Geometric Mean(GM) = √(Arithmetic Mean × Harmonic Mean)
⇒ GM = √(20 × 15)
⇒ GM = √300
⇒ GM = 17.32Hence, the Geometric mean from the given Arithmetic and Harmonic mean is 17.32
Example 4: Find the weighted harmonic mean for the given data.
Weight(w) | Data(x) |
---|---|
1 | 20 |
2 | 30 |
3 | 10 |
2 | 15 |
Solution:
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
Weight Harmonic Mean = ∑w / ∑(w/x)
⇒ Weight Harmonic Mean = 8/0.53
⇒ Weight Harmonic Mean = 15.09The weighted Harmonic mean for the given data is 15.09 .
Example 5: Find the weighted harmonic mean for the given data.
x | 10 | 15 | 20 | 25 | 30 |
---|---|---|---|---|---|
w | 2 | 3 | 4 | 5 | 1 |
Solution:
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
Weighted Harmonic Mean = ∑w / ∑(w/x)
⇒ Weight Harmonic Mean = 15/0.831
⇒ Weight Harmonic Mean = 18.05The weighted Harmonic mean for the given data is 18.05 .
Harmonic Mean
Harmonic Mean is the type of mean that is used when we have to find the average rate of change, it is the mean calculated by taking the reciprocal values of the given value and then dividing the number of terms by the sum of the reciprocal values. The harmonic mean is one of the Pythagorean mean and the other two Pythagorean mean are,
- Arithmetic Mean
- Geometric Mean
These means tell us about various parameters of the data set.
Harmonic Mean also denoted as HM is the mean calculated by taking the reciprocal of the given set. In this article, we will learn about HM, its formula, examples, and others in detail.