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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.
Geometric Mean Calculator for Grouped Data | |
---|---|
Type of Frequency Distribution | DiscreteContinuous |
Enter the Classes for X (Separated by comma,) | |
Enter the frequencies (f) (Separated by comma,) | |
Geometic Mean Results | |
Number of Observation (N): | |
Geometric Mean : | |
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:
Let me know in the comments if you have any questions on Geometric mean calculator for grouped data with examples and your thought on this article.