Continuous Geometric Mean:
G.M = anti log (Σ ƒ log m /N)
The steps are shown below:
(i) At first Find out the mid-points of the classes and take their logarithms.
(ii) Then Multiply these logarithms with the respective frequencies of each class and obtain the total Σ ƒ log m.
(iii) Divide the total obtained in step (ii) by the total frequency and take the anti log of the value so obtained.
Illustration:
| Marks |
Frequency |
Marks |
Frequency |
| 4-8 |
6 |
24-28 |
12 |
| 8-12 |
10 |
28-32 |
10 |
| 12-16 |
18 |
32-36 |
6 |
| 16-20 |
30 |
36-40 |
2 |
| 20-24 |
15 |
|
|
Solution:
Calculation of Geometric Mean:
| Marks |
m.p.m |
F |
Log m |
Fx log m |
| 4-8 |
6 |
6 |
0.7782 |
4.6692 |
| 8-12 |
10 |
10 |
1.0000 |
10.0000 |
| 12-16 |
14 |
18 |
1.1461 |
20.6298 |
| 16-20 |
18 |
30 |
1.2553 |
37.6590 |
| 20-24 |
22 |
15 |
1.3424 |
20.1360 |
| 24-28 |
26 |
12 |
1.4150 |
16.9800 |
| 28-32 |
30 |
10 |
1.4771 |
14.7710 |
| 32-36 |
34 |
6 |
1.5315 |
|
| 36-40 |
38 |
2 |
1.5798 |
3.1596 |
| |
|
N= 109 |
Σ ƒ x log m = |
137.1936 |
G.M. A.L. (Σ ƒ log X / N) = AL(137.1936 / 109) = A.L 1.2587 = 18.14.
Illustration:
The price of a commodity is increased by 5% from 2006 to 2007, 8% from 2007 to 2008 & 7% from 2008 to 2009. The average increase from 2007 to 2009 is quoted as 26% and not 30%. Explain it and verify the result.
Solution:
The appropriate average is the geometric mean and not the arithmetic mean. The arithmetic mean of 5,8,77 is 30 but this is not the right answer. The right answer should be obtained if we calculate the geometric mean.
| % rise |
X price at the end of the year taking preceding year as 100 |
Log X |
| 5 |
105 |
2.0212 |
| 8 |
108 |
2.0334 |
| 77 |
177 |
2.2480 |
| |
|
Σ log x = 6.3026 |
G.M = A. L (Σ log X N) = AL. (2.1009) = 126.2
The average increase from 2007 to 2009 = 126.2 – 100 = 26.2 % or approx 26% verification when the average rise is 30%
| Year |
Rate to change |
Total change |
Price at the end of each year |
| I year |
30% on 100 |
30 |
13.0169.0 |
| II year |
30% on 130 |
39 |
219.7 |
| III year |
30% on 169 |
50.7 |
|
When the average rise is 26%
| I year |
26% on 100.00 |
26.00 |
126.00 |
| II year |
26 % on 126.00 |
32.76 |
158.76 |
| III year |
26 % on 158.76 |
41.28 |
200.04 |
When the rise is of 5, 8 and 77% the changed price at the end of each year:
| I year |
5% on 100.0 |
5.00 |
105.00 |
| II year |
8% on 105.0 |
8.40 |
113.40 |
| III year |
77% on 113.4 |
87.318 |
200.00 |
The above calculations make it clear that in the second and third case the price at the end of the third year is almost similar as the slight difference being due to approximation of 26.2 to 26. And hence, the average increase is 26%.