Geometric Mean Calculator for Grouped Data with Examples

Geometric mean for grouped data

Geometric mean is another measure of central tendency. Geometric mean is used when the data is in terms of rates and ratios.

Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. The geometric mean of $X$ is denoted by $GM$ and is given by

$$ \begin{eqnarray*} GM& =\bigg(\prod_{i=1}^n x_i^{f_i}\bigg)^{1/N}\\ \therefore \log (GM) &=\frac{1}{N}\sum_{i=1}^n f_i\log(x_i) \end{eqnarray*} $$

where,$N=\sum_i f_i$ total number of observations.

Geometric Mean Calculator

Use geometric mean calculator to calculate the Geometric Mean for grouped data and frequency distribution.

How to find geometric mean of grouped data?

Step 1 - Select type of frequency distribution (Discrete or continuous)

Step 2 - Enter the Range or classes (X) seperated by comma (,)

Step 3 - Enter the Frequencies (f) seperated by comma

Step 4 - Click on "Calculate" for geometric mean calculation

Step 5 - Gives output as number of observation (n)

Step 6 - Calculate geometric mean

Step 7 - Display frequency distribution

Geometric Mean for Grouped Data Example 1

Compute geometric mean for the following frequency distribution.

x	5-8	9-12	13-16	17-20	21-24
f	2	13	21	14	5

Solution

	Class Boundries	mid-value ($x_i$)	Freq ($f_i$)	$log(x_i)$	$f_i*log(x_i)$
	4.5-8.5	6.5	2	1.8718	3.7436
	8.5-12.5	10.5	13	2.3514	30.5682
	12.5-16.5	14.5	21	2.6741	56.1561
	16.5-20.5	18.5	14	2.9178	40.8492
	20.5-24.5	22.5	5	3.1135	15.5675
tot		Total	55		146.8846

The log of geometric mean is

$$ \begin{aligned} \log (GM) &=\frac{1}{N}\sum_{i=1}^n f_i\log(x_i)\\ &=\frac{146.8846}{55}\\ &=2.6706 \end{aligned} $$

The geometric mean is

$$ \begin{aligned} GM & =\exp(\log (GM))\\ &=\exp(2.6706)\\ &=14.4486 \end{aligned} $$

Geometric Mean for Grouped Data Example 2

Find the geometric mean of distribution of weights of 75 students at virtual University in the table:

Weight	Frequency
110-119	1
120-129	4
130-139	17
140-149	28
150-159	25

Solution

	Class Boundries	mid-value ($x_i$)	Freq ($f_i$)	$log(x_i)$	$f_i*log(x_i)$
	109.5-119.5	114.5	1	4.7406	4.7406
	119.5-129.5	124.5	4	4.8243	19.2972
	129.5-139.5	134.5	17	4.9016	83.3272
	139.5-149.5	144.5	28	4.9733	139.2524
	149.5-159.5	154.5	25	5.0402	126.005
tot		Total	75		372.6224

The log of geometric mean is

$$ \begin{aligned} \log (GM) &=\frac{1}{N}\sum_{i=1}^n f_i\log(x_i)\\ &=\frac{372.6224}{75}\\ &=4.9683 \end{aligned} $$

The geometric mean is

$$ \begin{aligned} GM & =\exp(\log (GM))\\ &=\exp(4.9683)\\ &=143.7822 \;\; minutes \end{aligned} $$

Geometric Mean for Grouped Data Example 3

Following is the frequency distribution about the weight of earheads in grams:

Weight of earheads in gms	No. of earhead
60-80	22
80-100	38
100-120	45
120-140	35
140-160	20

Calculate geometric mean for the given frequency distribution.

Solution

	Class Boundries	mid-value ($x_i$)	Freq ($f_i$)	$log(x_i)$	$f_i*log(x_i)$
	109.5-119.5	70	22	4.2485	93.467
	119.5-129.5	90	38	4.4998	170.9924
	129.5-139.5	110	45	4.7005	211.5225
	139.5-149.5	130	35	4.8675	170.3625
	149.5-159.5	150	20	5.0106	100.212
tot		Total	160		746.5564

The log of geometric mean is

$$ \begin{aligned} \log (GM) &=\frac{1}{N}\sum_{i=1}^n f_i\log(x_i)\\ &=\frac{746.5564}{160}\\ &=4.666 \end{aligned} $$

The geometric mean is

$$ \begin{aligned} GM & =\exp(\log (GM))\\ &=\exp(4.666)\\ &=106.2718 \;\; grams \end{aligned} $$

Geometric Mean for Grouped Data Example 4

For the following data about the diameter (in cm) of defective screw, calculate geometric mean.

Diameter (in cm)	no. of defective screw
5	5
15	10
25	12
35	8
45	3

Solution

	$x_i$	Freq ($f_i$)	$log(x_i)$	$f_i*log(x_i)$
	5	5	1.6094	8.047
	15	10	2.7081	27.081
	25	12	3.2189	38.6268
	35	8	3.5553	28.4424
	45	3	3.8067	11.4201
tot	Total	38		113.6173

The log of geometric mean is

$$ \begin{aligned} \log (GM) &=\frac{1}{N}\sum_{i=1}^n f_i\log(x_i)\\ &=\frac{113.6173}{38}\\ &=2.9899 \end{aligned} $$

The geometric mean is

$$ \begin{aligned} GM & =\exp(\log (GM))\\ &=\exp(2.9899)\\ &=19.8837 \;\; cm \end{aligned} $$

Conclusion

In this tutorial, you learned about formula for geometric mean for grouped data and how to calculate geometric mean for grouped data. You also learned about how to solve numerical problems based on geometric mean for grouped data.

To learn more about other descriptive statistics measures, please refer to the following tutorials:

Descriptive Statistics

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