For the calculation of harmonic mean in continuous series the procedure is the similar as applied to the discrete series. The most important difference is that here we take the reciprocal of the mid -points.
Illustrations:
From the following data compute the value of harmonic mean
| Class interval |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
| Frequency |
4 |
6 |
10 |
7 |
3 |
Solution: Calculation of harmonic mean
| Class interval |
Mid-points |
Frequency |
f/m |
| 10-20 |
15 |
4 |
0.267 |
| 20-30 |
25 |
6 |
0.240 |
| 30-40 |
35 |
10 |
0.286 |
| 40-50 |
45 |
7 |
0.156 |
| 50-60 |
55 |
3 |
0.055 |
| |
|
N= 30 |
Σ(f/m) = 1,004 |
H.M N/ Σ (f/m) = 30 / 1.004 = 29.88.
Illustration:
An automobile driver travels from Spain to hill station (100 km). The distance travel at an average speed of 30 km. per hour. The driver then makes the return trip at an average speed of 20 km. per hour what is his average speed over the whole distance (200 km)?
Solution:
If the problem is given to a layman, it is most likely to compute the arithmetic mean of two speeds
X = (30 km. + 20 km.)/ 2 = 25 km p.h
But this is not the right average harmonic mean. This would be more suitable in the situation harmonic mean of 30 and 20 i.e.
H.M = 1 / 1/20 + 1/20 = 2 / 10 / 120 = 2 x 120 / 10 = 24 km.p.h.
It can be proved that the harmonic mean is the appropriate average in this case by tabulating the time and distance for each trip separately as follows:
| |
Distance (km) |
Average speed km p.h |
Time taken |
| Going |
100 |
30 |
3 hours 20 minutes |
| Returning |
100 |
|
5 hours |
| Total |
200 |
|
8 hours 20 minutes |
Thus the total time required for covering a distance of 200 km is 8 hours 20 minutes which gives an average speed of 24 km.h instead of 25 km. p.h
The above problem can be changed in such a manner that the arithmetic mean is the appropriate average. suppose the driver makes the similar trip but it is given that he travels at 30 km. per hour for half of the distance and at 20 km. per hour for another half . Now the correct answer about the average speed would be given by the arithmetic mean average speed = (30 + 20)/2 = 25 km. per hour. To verify the result we again make a table of time & distance at each speed.
| Speed km. p.h. |
Distance |
Time required |
| 30 |
120 km |
120/30 = 4 hours |
| 20 |
80 km |
80/20 = 4 hours |
| Total |
200km. |
8 hours |
Thus the driver has covered 200km. in 8 hours. And hence the average speed is 25 km. per hour the above example clearly describes that when distances are the same of the two speeds harmonic mean gives the right answer but when times are same the arithmetic mean of the rates of speed gives the right answer.