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- Statistics - Discussion
Statistics - Geometric Mean of Continous Series
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 −
${G.M.}$ = 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) | 0-10 | 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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