# Geometric Mean - Definition, Formula, and Examples

*Publish Date:*In mathematics and statistics, central tendencies are used to give an overall idea of the whole set of data. The measures for central tendencies are:

- Mean (Average of the whole data)
- Median (middle point of the whole data)
- Mode (most occurring reading)

Where “Mean” has further two types: Arithmetic mean and Geometric mean. Both are different in definition, accuracy, and calculation.

## Geometric mean definition:

According to Statisticshowto, the definition of the geometric mean is:

*“The geometric mean is a type of average, usually used for growth rates, like population growth or interest rates.”*

It is also defined as “nth root of the product of n values”. In simple words, you have to multiply all the given values, suppose n number of values. Then take its root by the total number of values i.e nth root.

## Geometric mean formula:

For n number of values, the formula of the geometric mean is:

Another form of this formula is

This formula states that: product of n values under nth root. Some other forms of this formula are:

And

## How to find the geometric mean?

Well, the answer to that is: by using formula ;). On a serious note, it can be calculated both manually and online. Let’s first learn to find it manually by examples.

Example # 1:

Find the geometric mean of 1, 3, 7, and 11.

Solution:

**Step 1:** Count the values:

Here we have 4 values. It means we will take the 4th root.

**Step 2:** Multiply all the values:

= 1 x 3 x 7 x 11

= 231

**Step 3:** Take 4th root:

= \sqrt[4]{231}

= 3.8985

Example # 2:

Henry started a business with rupees $50,000. In the first year, he had a profit of 10%. In the second year, he earned a profit of 25%. What amount does Henry have at the end of the second year?

Solution:

**Step 1:** Understand the statement.

- We know that the initial amount is = $50,000
- In the first year, this amount was = 110 % or 1.1
- In the second year, 1st year amount was = 120% or 1.2

**Step 2:** Find the geometric mean.

G.M = (1.1 1.2)^{12}

G.M = (1.32)^{0.5}

G.M = 1.1489

This is the average profit per year.

**Step 3:** Multiply this with the initial amount.

For 2 years.

= $50,000 * 1.1489 * 1.1489

= $65,999

## Online calculators:

Now, you can find geometric mean through online calculators. These calculators usually only require values.

## Properties of geometric mean:

Below are some properties of geometric mean:

- For the same set of data, the geometric mean is always less than the arithmetic mean i.e G.M < A.M. Unless all the values are the same in a set. In this case, both measures would be the same.
- The product of all the values in a data set would be equal to the product of all the values replaced by the geometric mean.
- For two series, the product of two values is equal to the product of their geometric mean.

## Applications of Geometric mean:

Many have this misconception that mathematics has no use in daily life. But this is wrong. Even measures like geometric mean are widely used in different fields.

**Films and videos:**

You would be surprised to learn that geometric mean is used in film making and photography. It is used to find the aspect ratio to find a compromise between the width and height of a screen.

**Finance:**

The most common use of geometric mean is in the field of finance. It is used to calculate compounded annual growth rate, portfolio returns, and in constructing stock indexes.

**Medicine:**

Yes, You read it right! It is also used in the field of medicine. According to an article, it is the “golden standard” for certain medical measures.

**HDI:**

HDI stands for human development index. It is an index presented by UNO for the development of different countries and solving different worldwide problems. According to UNDP:

*“The HDI is the geometric mean of normalized indices for each of the three dimensions.” *

## FAQs:

**What is the difference between arithmetic mean and geometric mean?**

Arithmetic mean is “the sum of values divided by the total number of values” while geometric means “nth root of the product of n values”

A.M = X1 + X2 + … Xn /n while

**What is the geometric mean of 4 and 9?**

G.M = (4 9)^{12}

G.M = (36)^{0.5}

G.M = 6

**What is more accurate; geometric mean or arithmetic mean?**

For volatile data, the geometric mean is more accurate and for the independent data set, the arithmetic mean is more accurate.