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# Geometric Mean

We already know that the mean of a set of numbers is often called the average, and is found by adding the numbers together and dividing by the number of numbers in the list. The geometric mean also tells us about the "middle" of a data set but we find it using a different method.
Let's define that in the proportion $\frac{a}{b}=\frac{c}{d}$ , b and c are called means, while a and d are called the extremes.
When the means of a proportion are the same number, that number is called the geometric mean of the extremes. So in the following proportion,
$\frac{p}{x}=\frac{x}{q}$
Cross multiplying gives you ${x}^{2}=pq$ . Taking the square root of both sides, we get $x=\sqrt{pq}$ as the geometric mean of p and q.

## Geometric mean definition

In math, the geometric mean, also called the GM, is the average value that identifies the central tendency of the set of numbers by finding the product of their values. Simply put, we multiply the numbers together and take the nth root of the multiplied numbers, where n equals the total number of data values given (their cardinality).
As an example, the geometric mean of a given set of numbers 3 and 2, the geometric mean is equal to $\sqrt{3}×2$ or $\sqrt{6}$ , which is approximately 2.449.
The geometric mean is not the same as the arithmetic mean, which for the numbers 3 and 2 would be $\frac{3+2}{2}=2.5$ .

## Properties of geometric mean

The important properties of the geometric mean are as follows:
• The GM for the provided data set is always less than the arithmetic mean for the same data set.
• If each object in the data set is substituted by the GM, the product of the subjects remains the same.

## Geometric mean examples

Example 1
Find the geometric mean of 25 and 9.
Since there are two numbers, the geometric mean of the two numbers is equal to the square root of their product.
$\mathrm{Geometric mean}=\sqrt{25×9}$
$=\sqrt{225}$
$=15$
The geometric mean of 25 and 9 is 15. (Note that the arithmetic mean of 25 and 9 is 17, which is more than the geometric mean, satisfying one of the important properties of geometric mean.)
Example 2
Find the geometric mean of 4, 10, and 25
Since there are three numbers, the geometric mean will be the cube root of their product.
$\mathrm{Geometric mean}=\sqrt[3]{4×10×25}$
$=\sqrt[3]{1000}$
$10$
The geometric mean of 4, 10, and 25 is 10.
Example 3
Find the geometric mean of 5, 10, 25, and 30
Since there are four numbers, the geometric mean will be the 4th root of their product.
$\mathrm{Geometric mean}=\sqrt[4]{5×10×25×30}$
$=\sqrt[4]{37500}$
$=13.915$
The geometric mean of 5, 10, 25, and 30 is approximately 13.915. The geometric mean is not always going to be a whole number.

## Get help learning about the geometric mean

The geometric mean has a lot of advantages and is used in a variety of fields. For example, it is used in stock indexes to calculate the annual return on the portfolio. It is also used in finance to find the average growth rates, which are also referred to as the compounded annual growth rate. In another field entirely, it is used in biology to study cell division and bacterial growth. The many uses of the geometric mean show that it's a useful concept worth knowing well.
If you are having a hard time understanding how to work out the geometric mean, you could use the help of a private tutor. A professional tutor can provide expert assistance as you learn, step by step, how to calculate the geometric mean and use it in certain applications. To learn more about how tutoring can help you understand the geometric mean and other related concepts, contact the Educational Directors at Varsity Tutors today.
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