The Harmonic mean is based on the reciprocal of the averaged number, it is defined as the reciprocal of the arithmetic mean of the reciprocal of the individual observations. Thus,
H.M. = N / (1/X1 + 1/X2 + 1/X3 +........+ 1/Xn
When the number of items is very large, the computation of harmonic mean in the above manner becomes very tedious. To simplify the calculations we obtain the reciprocal of the various items from the table and apply the following
In the individual observations, H.M. = N/∑ (/X)
In the discrete series, H.M. = N/∑(f x /X)
In the continuous series, H.M. = N/∑(fx/X) = N / ∑(f/X)
Calculation of harmonic mean-individual observations:
In individual series, the harmonic mean is computed by applying the following formula as shown:
H.M. = N/(1/X1 + 1/X2 + 1/X3 +........1/Xn
X1, X2, X3, etc, refer to the different items of the variable.
Illustration: - To find the harmonic mean of the following :
2574 475 75 5 0.8 0.08 0.005 0.0009
Solution:- Calculation of harmonic mean
| X |
(1/X) |
X |
(1/X) |
| 2574 |
0.0004 |
0.8 |
1.2500 |
| 475 |
0.0021 |
0.08 |
12.5000 |
| 75 |
0.0133 |
0.05 |
200.0000 |
| 5 |
0.2000 |
0.0009 |
1111.1111 |
| |
|
|
∑(1/X) = 1325.0769 |
H.M. = N/∑ (1/X) = 8 / 1325.0769 = 0.006.