Harmonic Mean (HM) is defined as the reciprocal of the arithmetic average of the reciprocal of the value of various items.
It can be solved through the following formula:
In Case of Individual Series:
Example 1:
Calculate the HM from the following data:
X: 18 12 16 21 7 9
Solution:
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Calculation of HM:
HM = N/∑(1/x)= 6/1.0358 =5.7926
In case of Discrete Series:
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HM = N/or∑f/∑(f/x)
Example 2:
Find the value of HM
HM N/∑f/x =50/4.213 = 11.868
In case of continuous series:
HM = N or ∑f/∑(f/m)
where, HM = harmonic mean
m = mid-value of various class intervals.
N = number of items in a series.
Example 3:
Calculate HM from the following data:
Solution:
H.M. = ∑f/∑f/x = 95/3.9 = 24.36
Merits and Demerits:
Merits:
1. It is mostly used to compute the average speed.
2. It is simple to understand.
3. It has rigidity.
4. It include all the items in a given series.
5. It gives the weight of all the items according to their importance in a given series.
6. It is time and rate based, because of this reason, it gives the best result than other.
Demerits:
1. It is difficult to calculate.
2. Its calculation is not possible if the zero or negative items are given in a series.
3. Sometimes, it assigns of more weights to the small items.
4. It is effected by the extreme values.
Relation between Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM):
There is a relationship between AM, GM and HM. This relation can be defined through following points:
1. The geometric mean of the two positive numbers is same as the GM of their arithmetic mean and harmonic mean.
Symbolically,
GM = VAM × √HM
Where GM = Geometric mean
AM = Arithmetic mean
HM = Harmonic mean
2. When the original values differ in size in any distribution, that is, AM is greater than GM and GM is greater than HM.
Symbolically,
AM > GM > HM
3. When in the series, the values are equal then AM is equal to GM and GM is equal to HM.
Symbolically,
AM = GM = HM