# Statistics - Geometric Mean of Continous Series

When data is given based on ranges alongwith their frequencies. Following is an example of continous series:

 Items Frequency 0-5 5-10 10-20 20-30 30-40 2 5 1 3 12

In case of continous series, a mid point is computed as $\frac{lower-limit + upper-limit}{2}$ and Geometric Mean is computed using following formula.

## Formula

$G.M. = Antilog\ \frac{\sum f \times \log x}{N} \\[7pt] \, = Antilog\ \frac{f_{1} \log x_{1} + f_{2} \log x_{2} + .... + f_{n} \log x_{n}}{N}$

Where −

• ${G.M.}$ = Geometric Mean

• ${x_1,x_2,x_3,...,x_n}$ = Different values of mid points in ranges.

• ${f_1,f_2,f_3,...,f_n}$ = Corresponding frequencies.

• ${N = \sum f}$

### Example

Problem Statement:

A record of dividend declared (in percentage) in the past four years by 20 companies is as follows:

Dividend declared
(in %age)
0-1010-2020-3030-40
No. of companies5834

What is the average percentage of dividend declared by the companies?

Solution:

The average percentage of dividend declared will he calculated by using Geometric Mean.

Dividend declared
(in %age)
Mid-pt
m
Frequency
f
${log x}$${log x} \times m$
0-10550.69903.4950
10-201581.17619.4088
20-302531.39794.1937
30-403541.54416.1764
20 23.2739

Based on the above mentioned formula, Geometric Mean $G.M.$ will be:

$G.M. = Antilog\ \frac{\sum f \times \log x}{N} \\[7pt] \, = Antilog\ of\ \frac{23.2739}{20} \\[7pt] \, = Antilog\ of\ 1.1637 \\[7pt] \, = 14.58$

The average percentage of Geometric Mean declared by the companies is 14.58.