Likewise weighted arithmetic mean we can also calculate weighted geometric mean with the help of the following formula
G.M. 10 = A.L [log X1 x W1) + (log X2 x W2) +...+ (log Xn X Wn/W1 + W2 + W3 +...+ Wn]
= A.L. [Σ(log Xx W/ ΣW]
Symbolically G.M w = √(X1w1 x X2w2 x X3w3 x..x X2Wn
Proof:
If w1, W2, W3 .... Wn are weights assigned to different values of X1, X2, X3.....Xn
G.M. w √X1w1 X2W2; X3W3 ... Xn Wn
Taking logarithms of both sides
Log G.M . w log (X1W1, X2W2, X3W3 ... Xn Wn)
Where N = ΣW = W1, + W2, + W3 + ... + Xn
= log (X1w1 + log X2W2 + log X3W3 +... log XnWn / ΣW = ΣW log X/ΣW
G. M. w = antilog [Σ W log X / Σ W
Since it is difficult to fang nth root, the geometric mean can be calculated with the help of logarithms.
Illustration:
Find the weighted geometric mean from the following data:
| Group |
index number |
weights |
| Food |
260 |
46 |
| Fuel & lighting |
180 |
10 |
| Clothing |
220 |
8 |
House rent
|
230 |
20 |
| Education |
120 |
12 |
| Misc. |
200 |
4 |
Solution:
Calculation of weighted geometric mean
| Group |
Index No. X |
Weights W |
Log X |
W log X |
| Food |
260 |
46 |
2.4150 |
11.0900 |
| Fuel & lighting |
180 |
10 |
2.2553 |
22.5530 |
| Clothing |
220 |
8 |
2.3424 |
18.7392 |
| House rent |
230 |
20 |
2.3617 |
47.2340 |
| Education |
120 |
12 |
2.0792 |
24.9504 |
| Misc. |
200 |
4 |
2.3010 |
9.2040 |
| |
Σ W = 100 |
|
ΣW log X = |
233.7706 |
G.W. w = A.L [ΣW log X / Σw] = A.L [233.7706 / 100] = A.L 2.337 = 217.6
Illustration :
The weighted geometric mean of the four numbers 8, 25, 17 and 30 is 15.3 if the weights of the first three numbers are 5, 3 and 4 respectively find the weight of the fourth number.
Solution:
Let the weight of the fourth number be W1.
Calculation of geometric mean:
| X |
W |
Log X |
W.log X |
| 8 |
5 |
0.9031 |
4.5155 |
| 25 |
3 |
1.3979 |
4.1937 |
| 17 |
4 |
1.2304 |
4.9216 |
| 30 |
W1 |
1.4771 |
1.4771 + W1 |
| |
ΣW = 12 + W |
|
ΣW. Log X = 13.6308 + 1.4771 + W1 |
Log G.mw = [ΣW log X / ΣW] log 15.3 = 13.6308 + 1.4771 W1 / 12 + W1
1.1847 (12 + W1) = 13.6308 + 1.4771 W1
14.2164 + 1.1847 W1 = 13.6308 + 1.4771 W1
1.1847 W1 – 1.477 W1 = 13.6308 – 14.2164 = - 0.5856
W1 = 0.5856 / 0.2942 = 2.003 or 2 approx.
Thus the weight of the fourth number is 2.