Statistics - Geometric Mean of Continous Series

Quiz

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

Items	0-5	5-10	10-20	20-30	30-40
Frequency	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 −

Problem Statement:

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

Dividend declared (in %age)	0-10	10-20	20-30	30-40
No. of companies	5	8	3	4

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	${log x}$	${log x} \times m$
0-10	5	5	0.6990	3.4950
10-20	15	8	1.1761	9.4088
20-30	25	3	1.3979	4.1937
30-40	35	4	1.5441	6.1764
		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.

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