# Statistics - Geometric Mean of Discrete Series

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

 Items Frequency 5 10 20 30 40 50 60 70 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 Frequency 14 36 45 70 105 2 5 1 3 12

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.