Statistics - Geometric Mean of Discrete Series


When data is given alongwith their frequencies. Following is an example of discrete series:

Items510203040506070
Frequency251312057

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:

Items14364570105
Frequency251312

Solution:

Based on the given data, we have:

${x}$${f}$${logx}$${flogx}$
1411.14611.1461
3621.55633.1126
4511.65321.6532
7021.84503.6900
10512.02112.0211
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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