**Geometric mean :**

Geometric mean is one of the measures of central tendency which can be defined as follows.

For a given set of n positive observations, the geometric mean is defined as the n-th root of the product of the observations.

Let the variable "x" assume "n" values as given below

All the above values are being positive, then the GM of x is given by

For a grouped frequency distribution, the GM is given by

Where, N = ∑f

1) Logarithm of G for a set of observations is the AM of the logarithm of the observations; i.e. logG = (1/r)∑logx.

2) If all the observations assumed by a variable are constants, say K > 0, then the GM of the observations is also K.

3) GM of the product of two variables is the product of their GM‘s i.e. if z = xy, then

GM of "z" = (GM of "x") x (GM of "y")

4) GM of the ratio of two variables is the ratio of the GM’s of the two variables i.e. if z = x / y, then

GM of "z" = (GM of "x") / (GM of "y")

5) Like arithmetic mean, GM also possess some mathematical properties.

6) It is rigidly defined.

7) It is based on all the observations.

8) It is difficult to comprehend.

9) It is difficult to compute.

10) It has limited applications for the computation of average rates and ratios and such like things.

**Problem 1 : **

Find the GM of 2, 4 and 8.

**Solution :**

For the given data, the formula to find geometric-mean is given by

Fitting the given data in to the above formula, we get

GM = (2 x 4 x 8)⅓

GM = (2⁶)⅓

GM = 2²

**GM = 4**

**Problem 2 : **

Find the GM of 3, 6 and 12.

**Solution :**

For the given data, the formula to find geometric-mean is given by

Fitting the given data in to the above formula, we get

GM = (3 x 6 x 12)⅓

GM = (6³)⅓

**GM = 6**

**Problem 3 :**

Find the GM for the following distribution:

**Solution :**

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