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Geometric Mean - Definition, Formula, and Examples

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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:

G.M = \sqrt[ n ]{ { a }^{ 1 } × { a }^{ 2 } × { a }^{ 3 } ..... { a }^{ n }}

Another form of this formula is 

G.M = ( { a }^{ 1 } × { a }^{ 2 } × { a }^{ 3 } ..... { a }^{ n } )^{\tiny\dfrac{1}{n}}

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

G.M = AntiLog \space \dfrac{\sum{log \space x_i}}{n}

And 

G.M = \sqrt[n]{ \prod_{i=1}^n} \space x_i \enspace \enspace OR \enspace \enspace \Big( \prod_{i=1}^n \space x_i \Big)^{\tiny \dfrac{1}{n}}

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

G.M = \sqrt[ n ]{ { a }^{ 1 } × { a }^{ 2 } × { a }^{ 3 } ..... { a }^{ n }}
  • 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.