Geometric mean formula

Geometric mean formula
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If a bacteria grows 10% in the first hour, 30% in the second hour, and 25% in the third hour. What will be the mean growth rate of the bacteria?

We cannot simply get the answer to this by working out the arithmetic mean here. We would have to use a special type of mean called geometric mean. 

This is because whenever we have a situation with growth during some period of time, it requires the use of a geometric mean to get the correct mean.

geometric mean application

In this lesson, we would learn about the geometric mean and its formula. Let's get started!

Lesson Plan

What Is the Geometric Mean? 

A mean indicates the central tendency of a set of numbers. The geometric mean (or GM) is a very interesting type of special mean that indicates the central tendency of a set of numbers by using the product of their values.

geometric mean

The geometric mean is an average but it is calculated in a special way.

For calculating the geometric mean of \(\text n\) numbers, we multiply the numbers together and then take an \(\text n\)th root of the result.

It is defined as the \(n^{th}\) root of the product of \(n\) numbers.

Let's understand using two simple examples.

geometric mean formula


What Is the Geometric Mean Formula?

How do you define geometric mean?

The geometric mean of \(\text n\) numbers is calculated as the \(\text {n^{th}}\) root of the product of those\(\text n\) numbers. 

How to find the geometric mean?

The general formula for geometric mean is given as:

Geometric mean


How do You Calculate the Geometric Mean?

For calculating the geometric mean of \(n\) numbers: multiply them all together and then take the \(n^{th}\) root.

Example: 

Find the geometric mean of 2, 3, and 6?

First, we will multiply the numbers together and then we will take the cube root (since there are three numbers)

\[=  \sqrt[3] {2 \times 3 \times 6} = 3.30\]


Geometric Mean Altitude Theorem

The geometric mean formula in a triangle finds its application through the geometric mean altitude theorem.

Theorem statement:

In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of this altitude is the geometric mean of the lengths of those two segments. 

geometric mean theorem


Use of Geometric Mean in Statistics

The geometric mean formula is used widely in statistics.

Geometric mean finds its application in statistics to study economic growth rates of a country, interest growth, population growth, medicinal research data evaluation, etc.


Geometric Mean Calculator

How to use the following calculator:
Enter the numbers separated by a comma. Press the submit button to get the result. 


Dealing with Negative Numbers

It is impossible to find the geometric mean with a negative number. Often, the collected data has negative numbers, especially the data involving population growth rate or financial returns.

Mathematicians work around this problem by converting the negative number into a suitable positive equivalent.

Example:

A city recorded a population growth rate in three consecutive years as +12%, -8%, and +2% respectively. What will be the net rate of population growth?

To calculate the geometric mean with a negative number, we can convert all the three numbers into positive equivalents as:

\(12%= \dfrac {12}{100} + 1 = 1.12\)

\(-8% = \dfrac {-8}{100} + 1 =  -0.08 + 1 = 0.92\)

\(12% = \dfrac {12}{100} + 1 = 1.12\)

The new equivalent numbers will be 1.12, 0.92, and 1.02 respectively. The GM of the three numbers is 1.0167

On subtracting 1 from this value, +1.67% is the net rate of population growth

What will be the geometric mean for an investment that shows a negative growth?

Real-life example: 

Julia is an investor and her annual returns are as shown in the table for five consecutive years. She wants to know her average return for this 5-year period.

Stock trader application geometric mean

Year Return %
1 5%
2 10%
3 20%
4 -50%
5 20%

 

The arithmetic mean in this case would overstate the return and will not reflect the true returns.

Using the arithmetic mean, Julia's total return is \(\dfrac {5\% + 10\% + 20\% - 50\% + 20\%}{5} = 1\%\)

The resultant answer of 1% return is misleading.

A more accurate return could be calculated using the following geometric mean formula for the return.

geometric mean

\[\sqrt[5] {(1 + 0.05)(1 + 0.1)(1 + 0.2)(1 – 0.5)(1 + 0.2)} – 1 = -0.03621\]

It means she had a negative return of -3.62%.

Geometric mean vs Arithmetic mean:

The geometric mean formula better helps here because when the return is compounded, the investor needs to use the geometric mean to calculate the final value of the investment rather than the arithmetic mean.

As we have a situation with percentage growth during some period of time, we must remember that it requires the use of geometric mean.

 The geometric mean is always less than the arithmetic mean for any two positive unequal numbers.


Antilog of a Geometric Mean

The geometric mean of \(\text n\) numbers is calculated as the \(\text {n^{th}}\) root of the product of those\(\text n\) numbers. Finding the \(n^{th}\) root of a number is often difficult. The calculation becomes easier with the formula given below in terms of antilog-

geometric mean and log

Example:

A company recorded its annual growth rate of profits for five consecutive years (in percentage) as 50, 72, 54, 82, 93. What will be the geometric mean of the annual growth rate of profits?

Using the formula to calculate the geometric mean:

\(\text { G.M. }=\text { Antilog } \frac{\sum_{i=1}^{n} \log x_{i}}{n}\)

\[\begin{align*}\text { G.M. }&=\text { Antilog } \frac{\sum_{i=1}^{n} \log x_{i}}{n}\\ &= \text { Antilog } \dfrac{9.1710}{5} \\ &= \text { Antilog } 1.8342 \\ &= 68.26 \end{align*}\]

The company's annual growth rate of profits is \(68.26\%\)

 
important notes to remember
Important Notes
  1. The geometric mean (or GM) is a type of mean that indicates the central tendency of a set of numbers by using the product of their values.
  2. The geometric mean of \(\text n\) numbers is calculated as the \(\text {n^{th}}\) root of the product of those\(\text n\) numbers.
  3. The geometric mean uses the product of numbers to find the central tendency as opposed to the arithmetic mean which uses their sum.
  4. For any two positive unequal numbers, the geometric mean is always less than the arithmetic mean.

Solved Examples

Example 1

 

 

Help Natalie calculate the geometric mean of 2 and 32. She also wants to know what is the geometric mean of both numbers compares to their arithmetic mean.

Solution

For geometric mean, first, we will multiply the numbers together, and then we will take the square root (because there are two numbers) = \( \sqrt{2 \times 32} = 8\)

Arithmetic mean of 2 and 32 = \(\dfrac {2 + 32}{2} = 17\)

The geometric mean of two positive numbers is always lesser than the arithmetic mean. Here the arithmetic mean is 17, while the geometric mean is 8

\(\therefore\)The geometric mean of 2 and 32 is 8
Example 2

 

 

Jason is finding it difficult to calculate the mean of the first five positive integers. Can you explain to him the steps for it?

Solution

The first five positive integers are 1, 2, 3, 4, and 5

For finding the GM of the above 5 numbers  we will multiply the numbers together and then we will take the 5th root (since there are five numbers.)

\[\begin {align*} &= \sqrt[5] {1 \times 2 \times 3 \times 4 \times 5} \\ &= \sqrt[5] {120}\\& \approx 2.605\end{align*}\]

\(\therefore\) The mean of the first five positive integers is approx. 2.605
Example 3

 

 

Anna is puzzled when her teacher asked "12 is the geometric mean of 36 and what another number?" Can you help her with finding the other number?

Solution

Let the unknown number be 'x'.

As per the above statement:

\[\begin {align*}\sqrt{36 \times x} &= 12\\ \sqrt{36 \times x} &= \sqrt{144}\\ x &= \dfrac{144}{36}\\ x &= 4 \end {align*}\]

\(\therefore\) 12 is the geometric mean of 36 and 4
Example 4

 

 

Four tire models were tested on their durability and stiffness using an index. The record of this testing is given below. 

geometric mean

Model Durability Stiffness
A 5 3
B 10 4
C 20 1
D 10 3

The higher the scores, the better is the performance of the model as per this index. Which tire model should be considered the best performer?

Solution

The higher is the geometric mean, the better will be the performance. 

Calculating the geometric mean of the given four models:

Model A: Geometric mean = \(\sqrt {5\times3} = 3.87\)

Model B: Geometric mean = \(\sqrt {10\times4} = 6.32\)

Model C: Geometric mean = \(\sqrt {20\times1} = 4.47\)

Model D: Geometric mean = \(\sqrt {10\times3} = 5.48\)

\(\therefore\) Model B is the best performer.
Example 5

 

 

If a bacteria population increases 10% in the first hour, by 30% in the second hour, and 25% in its third hour. What will be the mean growth rate?

geometric mean application

Solution

Let's consider only 100 bacteria to understand this.

Bacteria population in the first hour = 110 (after a 10% growth).

Thus, the growth rate in the first hour \(= 100 + 100 \times 0.1 = 1.1\)

The bacteria growth rate in the second hour = 1.3 (after a 30% growth)

Total bacteria at the end of the second hour \(= 110 + 110 \times 0.3 = 1.43 = 143\) bacteria

The bacteria growth rate in the third hour = 1.25 (after a 25% growth)

Total bacteria at the end of the second hour \(= 143 + 143 \times 0.25 = 1.78 = 178.75\) bacteria

On finding the geometric mean of the above three growth rates we would get:

\[\begin {align*}&= \sqrt[3]{1.1 \times 1.3 \times 1.25}\\ &=\sqrt[3]{1.7875}\\ &\approx 1.21\end {align*}\]

\(\therefore\)The mean growth rate of the bacteria over a period of 3-hours is 12.1%
 
Challenge your math skills
Challenging Question
  1. The population in a country increased at the rate of 25% in the first year and 15% in the next year. In the third year, it decreased at a rate of 5%. What is the average rate of growth?

  2. The geometric mean of the following set of numbers is 9. Can you find the value of  \(x\)?

          1,    3,    \(x\),    27,    81

Interactive Questions on Geometric Mean Formula

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 
 
 



Let's Summarize

The mini-lesson targeted the fascinating concept of geometric mean and its formula. We also learned geometric mean formula in a triangle,  geometric mean formula with negative numbers,  geometric mean formula examples,  geometric mean problems,  geometric mean vs arithmetic mean, and geometric mean formula in statistics. 

The math journey around geometric mean formula starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds, done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions

1. What are equal ratios?

For calculating the geometric mean of 4 and 9. The calculation would be \(\sqrt{4 \times9} = \sqrt{36} = 6\)

The ratio of the first number (4) and the geometric mean (9) is 2/3
The ratio of the second number (9) and the geometric mean (6) is 9/6, which on reciprocation simplifies to 2/3

Ratios thus obtained are 'equal'.
The geometric mean averages all of the multipliers which we put into the equation. 

2. Explain logarithms values and the geometric mean?

Geometric mean is defined as the \(n^{th}\) root of the product of \(n\) numbers.

The geometric mean is related to logarithms as the geometric mean can be expressed as the exponential of the arithmetic mean of logarithms.

geometric mean and log

3. What is arithmetic mean?

The arithmetic mean is the sum of the data items divided by the number of items in the set:

\[\dfrac{2 + 3 + 4}{ 3} = 3\]
Hence 3 is the arithmetic mean of the given numbers.

  
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