Geometric Mean Formula

We have used the arithmetic mean in many data-related problems. Here we will see another such term frequently used with data analysis. A geometric mean formula is used to calculate the geometric mean of a set of numbers. It is a type of mean that indicates the central tendency of a set of numbers by using the product of their values. It is also defined as the nth root of the product of n numbers. The geometric mean is properly defined only for a positive set of real numbers. In this article, we will discuss the geometric mean formula with examples. Let us begin learning!

Geometric Mean Formula

What is the Geometric Mean?

The geometric mean is the mean value of a set of products. Its calculation is commonly used to determine the performance results of an investment or portfolio. It can be stated as “the nth root value of the product of n numbers.”

The geometric mean should be used when working with percentages, which are derived from values. The geometric mean is a very useful tool for calculating portfolio performance. It is because it takes into account the effects of compounding.

The formula for Geometric Mean

The geometric mean is used as a proportion in geometry and therefore it is sometimes called the “mean proportional”. The mean proportional of two positive numbers a and b, will e the positive number x, so that:

\(\frac{a}{x}=\frac{x}{b}\)

i.e. after doing cross multiplication we get

x=\(\sqrt{a \times  b}\)

In general for n multiple numbers as a_1,a_2, a_3,…..,a_n then geometric mean GM will be the nth root of the product of the numbers. In terms of formula it is:

GM = \(\sqrt[n]{a_1 \times  a_2 \times a_3 \times …….\times a_n}\)

Some real-life uses of geometric mean:

1. Aspect Ratios:

The geometric mean has been used in film and video also to find the appropriate aspect ratios i.e. the proportion of the width to the height of a screen or image. It is used to find an appropriate balancing between the two aspect ratios as well as for distorting or cropping both ratios equally.

2. Computer Science:

Computers use mind-boggling amounts of large data which generally requires the summarization for many applications using various statistical measurements.

3. Medicine:

The Geometric Mean has many applications in the medical industry also. It is known as the “gold standard” for some measurements, including for the calculation of gastric emptying time.

4. Proportional Growth:

It is very useful in finding the growth rate. The geometric mean is used for calculating the proportional growth as well as demand growth.

Solved Example

1: Find the geometric mean of 4 and 3?

Solution: Using the formula for G.M.,

a=4  and b=3

Geometric Mean will be:

x= √(4×3)

= 2√3

So, GM will be 3.46

Example-2: Find the geometric mean of 5 numbers as 4, 8, 3, 9 and 17?

Solution:

Here multiple numbers are taken.

n = 5

Find geometric mean using the formula:

GM = \(\sqrt[n]{a_1 \times  a_2 \times a_3 \times …….\times a_n}\)

Putting values of numbers,

We get

GM = \(\sqrt[5]{4 \times 8 \times 3 \times 9 \times 17}\)

i.e. GM = \(\sqrt[5]{4 \times 8 \times 3 \times 9 \times 17}\)

i.e. GM= \(\sqrt[5]{14688}\)

So, geometric mean = 6.81