Geometric Mean
The geometric mean is another measure of central tendency based on mathematical footing, like arithmetic mean. The geometric mean can be defined as:
“The geometric mean is the nth positive root of the product of 'n' positive given values.”
Hence, the geometric mean for a value containing values such as is denoted by of and given as:
(for ungrouped data)
If we have a series of positive values with repeated values such as which are repeated times respectively, then the geometric mean will become:
(For Grouped Data)
Where
Example:
Find the geometric mean of the values 10, 5, 15, 8, 12.
Solution:
Here, and
Example:
Find the geometric mean of the following data:












Solution:
We may write it as below:
Here ,
, , , ,
Using the formula of geometric mean for grouped data, the geometric mean in this case will become:
The method explained above to calculate the geometric mean is useful when the values in the given data are small in number and an electronic calculator is available. When a set of data contains a large number of values then we need an alternate way to compute the geometric mean. The modified or alternative way of computing the geometric mean is given as:
For Ungrouped Data

For Grouped Data



Example:
Find the geometric mean of the values 10, 5, 15, 8, 12












Total


Example:
Find the geometric mean for the following distribution of students’ marks:
Marks





No. of Students





Solution:
Marks

No. of Students

Mid Points


















Total



