# Geometrical Mean for Ungrouped and Grouped Data

## Geometrical Mean for Ungrouped and Grouped Data

Geometric Mean (G.M): The nth root of the product of the values is called Geometric Mean.

Geometric Mean for Ungrouped Data: If x₁, x₂, …, xn be n observations, then geometric mean is given by $$G\text{ }=\text{ }{{\left( x.\ x.\ldots .\ {{x}_{n}} \right)}^{\frac{1}{n}}}$$,

$$\log G=\frac{1}{n}\left( \log {{x}_{1}}+\log {{x}_{2}}+….+\log {{x}_{n}} \right)$$,

$$\log G=\frac{1}{n}\sum\limits_{i=1}^{n}{\log {{x}_{i}}}$$,

$$G=anti\log \left[ \frac{1}{n}\sum\limits_{i=1}^{n}{\log {{x}_{i}}} \right]$$.

Geometric Mean for Grouped Data: If x₁, x₂, …, xn be n observations, whose corresponding frequencies are f₁, f₂, …, fn then geometrical mean is given by $$G.M={{\left( {{x}_{1}}^{{{f}_{1}}}.{{x}_{2}}^{{{f}_{2}}}……..{{x}_{n}}^{{{f}_{n}}} \right)}^{\frac{1}{N}}}=anti\log \left[ \frac{1}{N}\sum\limits_{i=1}^{n}{{{f}_{i}}\log {{x}_{i}}} \right]$$.

Examples:

1) Find the geometrical mean of the number 3, 3², …, 3ⁿ.

Solution: Given,

3, 3², …, 3ⁿ

$$G.M={{\left( {{3.3}^{2}}{{…..3}^{n}} \right)}^{\frac{1}{n}}}$$,

$$={{3}^{\left( \frac{1+2+…+n}{n} \right)}}$$ (∵ First n terms 1, 2, 3, …, n = n (n + 1)/ 2)

$$={{3}^{\left( \frac{\frac{n(n+1)}{2}}{n} \right)}}$$,

$$={{3}^{\left( \frac{n(n+1)}{2n} \right)}}$$,

$$={{3}^{\left( \frac{(n+1)}{2} \right)}}$$.

2) Find the geometrical mean of the number 2, 2², …, 2ⁿ

Solution: $$G.M={{\left( {{2.2}^{2}}{{…..2}^{n}} \right)}^{\frac{1}{n}}}$$ (∵ First n terms 1, 2, 3, …, n = n (n + 1)/ 2)

$$={{2}^{\left( \frac{1+2+…+n}{n} \right)}}$$,

$$={{2}^{\left( \frac{\frac{n(n+1)}{2}}{n} \right)}}$$,

$$={{2}^{\left( \frac{n(n+1)}{2n} \right)}}$$,

$$={{2}^{\left( \frac{(n+1)}{2} \right)}}$$.