Geometric mean is defined as the Nth root of the product of N items. For example, if we have two items then we will take square root, if we take three items then we take cube root and so on.
It can be calculated from following formula:
Where X1 X2 …. Xn = various items in a series
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n = number of items.
Properties of Geometric Mean:
1. Geometric mean is defined as the nth root of the n items therefore the n the power of the GM is always equal to the product of all the n items.
Symbolically:
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GM = (X1 X2 X3…. Xn)1/n
Gn = X1 X2 X3…. Xn
2. If the information of more than one group is given that is GM1+ GM2 and N1, N2 then we can calculate the combined geometric mean through following formula:
GM12 = Antilog (N1 log GM1 + N2log GM2/N1+N2)
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3. If given number of items in more than two series are equal, then GM of the product of the corresponding items in the series is same as the product of GM of items.
4. The total of the deviations of the logs of the values from the log of the GM is always zero.
5. The geometric mean of the ratios of the corresponding items of two series with equal number of observation is equal to the ratio of geometric means.
In case of Individual Series:
Log GM = ∑ Log X/N
GM = Antilog (∑ Log X)/N
Where GM = geometric mean.
X1, X2, X3…Xn = various items of the series.
Example 1:
Calculate the GM from the following data:
X= 9 112 87 0.6 1.8 0.09 24.9
Solution:
Calculation of GM
In case of Discrete Series:
Log GM = ∑ log x/N
GM= Antilog [∑f log x/N]
Where, GM = geometric mean.
Example 2:
Find the GM from the following data:
Solution:
Calculation of GM
In case of Continuous Series:
The following formula will be used in continuous series for calculation:
GM = ∑f log x/N
Or, GM = antilog ∑f log m/N
Where, GM = geometric mean.
m = mid-value of various class intervals
∑f log m = sum of the product of the log m with their respective frequency.
N = No. of frequencies.
Example 3:
Calculate the value of GM:
Combined Geometric Mean:
If we have given the information about more than two groups, that is, GM1 GM2, N1, and N2 then there is need to calculate their combined geometric mean.
If can be calculated from the following formula:
GM12 = Antilog (n1log GM1+ log GM2/n1+ n2)
GM12 = Combined geometric mean
GM1 = GM of one group
GM2 = GM of second group
n1 = No. of terms in fast group
n2 = No. of terms in second group.
Example 4:
Initially, the population of a town was 5,000. At first, it rose at the rate of 2% per annum for two years. What will be the population of the town after 5 years?
Solution:
Taking log on both sides
log x = log 5000+ 5 log 1.02
Log x = 3.6990 + 50 (0.086)
x = A.L. [3.7420]
x = 5521.0
Example 5:
The population of a town was 10,000. At first, it increased at the rate of 3% per annum for the first population of the town after 5 years?
Solution:
We are given-
P0 = 10000, r = 0.03
r = 0.02, n1 = 3, n2 = 2, Pn = ?
Pn= P0 (1 + r1)3 – (1 – r2)2
Pn = 10000 (1 + 0.03)3 (1 – 0.02)2
Pn = 10000 (1.03)3 (0.98)2
P = 10000 × 1.0496 = 10496
Weighted Geometric Mean:
Weighted geometric mean defined as the computation of geometric mean by assigning the weights to different items according to their importance. It is used where the values of variables are not of equal importance.
It can be calculated from the following formula:
Weighted GM = Antilog [∑W log X/∑W]
Where W1, W2 W3 ………… , Wn weighted assigned to the variable values.
∑W log X = sum of the product log X with the assigned weights.
∑W = sum of weights.
Example 6:
Find the weighted GM from the following data:
Solution:
Calculation of weighted GM:
Merits and Demerits of Geometric Mean:
Merits:
1. It includes all the items given in a series.
2. It has rigidity.
3. It is useful for further mathematical applications.
4. It assigns the weights to items according to their importance.
5. It has sampling stability.
6. It is least affected by the extreme items.
7. It is most suitable for the measurement of ratios and percentages.
Demerits:
1. It is difficult as compared to other measures of central tendency because it can be solved by taking logs and antilog of values to calculate GM.
2. The value of GM is undefined in case of zero and negative values.
3. It exists at those points where the very few actual values lie.