Geometric Mean: Overview, Questions, Preparation

A geometric mean is a mean that indicates the average trend of a number set.  The Geometric mean was invented by Pythagoras. This philosopher from Greece also developed harmonic mean and arithmetic mean.

The geometric mean includes roots and multiplication and does not have any addition and division. By multiplying numbers and then finding the n^th root of the result, you obtain the geometric mean. The n^th root refers to the number of numbers multiplied by you. 

The formula for the geometric mean

GM = _n√a1×a2×a3×…….×an
Where a1, a2...an are different numbers and n is the total count.

For example, the geometric mean of seven numbers can be calculated by multiplying seven numbers and then summing the result’s seventh root. The geometrical average of three numbers is the 3^rd root of the product of those three numbers. Suppose we said that using the 9_th root of a series, we find the geometric mean that informs you that nine numbers multiplied. What root will we take, to find the geometric mean of six numbers? Of course, the sixth root.

In many situations, a geometric mean is useful, particularly to calculate growth-related issues of the business. The geometric mean is also employed to calculate values, such as the human populace values or rates of interest on financial investment over time, which have to be compounded together or have an exponential existence.

Calculation of the Geometric Mean

Let us start with two numbers, 16 and 4. What will be the geometric mean of 16 and 4?

To start with,  we need to do 16 × 4 =64.

The next step will be to find out the square root of their product as there are only two numbers:

So the geometric mean of 16 and 4 is 8.

Geometric mean properties

• If you replace each object in the geometric mean data, the objects’ product will be the same.
• If you multiply the geometric mean of multiple series, it will be equal to the whole data elements’ geometric mean. 
• The geometric mean for a defined set of data is always less than the arithmetical mean for that particular data set.

Difference between Geometric Mean and Arithmetic Mean

Contrary to the geometric mean, arithmetic mean is the simple mathematical average for a defined set of numbers. You can easily calculate the arithmetic mean by adding the numbers and then calculating the average of those numbers by dividing them by the total count.

The average value of a group of numbers can be measured using geometric mean and arithmetic mean. The geometric mean works better when you need to display a pattern for a group of numbers. At the same time, arithmetic mean works better when you need to find a simple average.

Summary

This lesson has shown how to calculate the geometric mean by multiplying a group of n variables. We also studied how the geometric mean of any group of numbers can be calculated and how the geometric mean can be applied to situations where growth rates are based on multiplication, not addition, and how the formula can be written and used for the geometric mean.

It is a Part of Algebra syllabus which is having 30% weightage

1.How to calculate the GM of 7 and 28?

Solution.  Here we need to use the formula we learned in this lesson

Geometric Mean of 7, 28 = √(7×28) = √196

So, the geometric mean of 7 and 28 is 14.

2. How to calculate the GM of 3 and 4?

Solution.  Here we need to use the formula we learned earlier

GM = n√a1×a2×a3×…….×an

Geometric Mean of 3, 4 = √(3×4) = √12

So, the geometric mean of 3 and 4 is two√3=3.5.

3.How to calculate the GM of 4, 6, 8 and 9?

Solution.  Use the formula we learned earlier

GM = n√a1×a2×a3×…….×an

GM of 4, 6, 8 and 9=4√(4x6x8x9)=6.447

So, the geometric mean of 4, 6, 8 and 9 is 6.447

Q: How do you calculate the geometric mean?

Q. What are the various uses of geometric mean?

Q. Which is greater: Geometric mean or arithmetic mean?

Q. Can you use geometric mean to calculate growth rate for your savings?

Q. What is the most effective use of geometric mean?