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Statistics - Geometric Mean of Discrete Series
When data is given alongwith their frequencies. Following is an example of discrete series:
| Items | 5 | 10 | 20 | 30 | 40 | 50 | 60 | 70 |
|---|---|---|---|---|---|---|---|---|
| Frequency | 2 | 5 | 1 | 3 | 12 | 0 | 5 | 7 |
In case of discrete series, 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 variable x.
${f_1,f_2,f_3,...,f_n}$ = Corresponding frequencies of variable x.
${N = \sum f}$
Example
Problem Statement:
Calculate Geometric Mean for the following discrete data:
| Items | 14 | 36 | 45 | 70 | 105 |
|---|---|---|---|---|---|
| Frequency | 2 | 5 | 1 | 3 | 12 |
Solution:
Based on the given data, we have:
| ${x}$ | ${f}$ | ${logx}$ | ${flogx}$ |
|---|---|---|---|
| 14 | 1 | 1.1461 | 1.1461 |
| 36 | 2 | 1.5563 | 3.1126 |
| 45 | 1 | 1.6532 | 1.6532 |
| 70 | 2 | 1.8450 | 3.6900 |
| 105 | 1 | 2.0211 | 2.0211 |
| Total | 11.623 |
Based on the above mentioned formula, Geometric Mean $G.M.$ will be:
$G.M. = Antilog\ \frac{\sum f \times \log x}{N} \\[7pt]
\, = Antilog\ of\ \frac{11.623}{5} \\[7pt]
\, = Antilog\ of\ 2.3246 \\[7pt]
\, = 211.15$
The Geometric Mean of the given numbers is 211.15.
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