In mathematics and statistics, the summary that describes the whole data set values can be easily described with the help of measures of central tendencies. The most important measures of central tendencies are mean, median, mode and the range. Among these, the mean of the data set will provide the overall idea of the data. The mean defines the average of numbers. The different types of mean are Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM). In this article, let us discuss the definition, formula, properties, applications, the relation between AM, GM, and HM with solved examples in detail.

Table of Contents:

• Definition
• Formula
• Difference Between AM and GM
• Relation Between AM, GM and HM
• Properties
• Applications
• Examples
• FAQs

Geometric Mean Definition

In Mathematics, the Geometric Mean (GM) is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values. Basically, we multiply the numbers altogether and take the nth root of the multiplied numbers, where n is the total number of data values. For example: for a given set of two numbers such as 3 and 1, the geometric mean is equal to √(3×1) = √3 = 1.732.

In other words, the geometric mean is defined as the nth root of the product of n numbers. It is noted that the geometric mean is different from the arithmetic mean. Because, in arithmetic mean, we add the data values and then divide it by the total number of values. But in geometric mean, we multiply the given data values and then take the root with the radical index for the total number of data values. For example, if we have two data, take the square root, or if we have three data, then take the cube root, or else if we have four data values, then take the 4th root, and so on. 

Geometric Mean Formula

The formula to calculate the geometric mean is given below:

The Geometric Mean (G.M) of a series containing n observations is the nth root of the product of the values.

Consider, if x₁, x₂ …. X_n are the observation, then the G.M is defined as:

This can also be written as;

Log GM = \(\frac{1}{n}\log (x_{1}\times x_{2}\times ….x_{n})\)

=\(\frac{1}{n}(\log x_{1}+\log x_{2}+….+\log x_{n})\)

=\(\frac{\sum \log x_{i}}{n}\)

Therefore, Geometric Mean, GM = \(Antilog\frac{\sum \log x_{i}}{n}\)

Where n = f₁ + f₂ +…..+ f_n

It is also represented as:

G.M. = \(\sqrt[n]{\prod_{i=1}^{n}x_{i}}\)

For any Grouped Data, G.M can be written as;

GM = \(Antilog\frac{\sum f \log x_{i}}{n}\)

Difference Between Arithmetic Mean and Geometric Mean

Relation Between AM, GM and HM

To learn the relation between the AM, GM and HM, first we need to know the formulas of all these three types of the mean.

Assume that “x” and “y” are the two number and the number of values = 2, then 

AM = (a+b)/2

⇒ 1/AM = 2/(a+b) ……. (1)

GM = √(ab)

⇒GM² = ab ……. (2)

HM= 2/[(1/a) + (1/b)]

⇒HM = 2/[(a+b)/ab

⇒ HM = 2ab/(a+b) ….. (3)

Now, substitute (1) and (2) in (3), we get

HM = GM² /AM

⇒GM² = AM × HM

Or else,

GM = √[ AM × HM]

Hence, the relation between AM, GM and HM is GM² = AM × HM

Geometric Mean Properties

Some of the important properties of the G.M are:

• The G.M for the given data set is always less than the arithmetic mean for the data set
• If each object in the data set is substituted by the G.M, then the product of the objects remains unchanged.
• The ratio of the corresponding observations of the G.M in two series is equal to the ratio of their geometric means
• The products of the corresponding items of the G.M in two series are equal to the product of their geometric mean.

Application of Geometric Mean

The greatest assumption of the G.M is that data can be really interpreted as a scaling factor. Before that, we have to know when to use the G.M. The answer to this is, it should be only applied to positive values and often used for the set of numbers whose values are exponential in nature and whose values are meant to be multiplied together. This means that there will be no zero value and negative value which we cannot really apply. Geometric mean has a lot of advantages and it is used in many fields. Some of the applications are as follows:

• It is used in stock indexes. Because many of the value line indexes which is used by financial departments use G.M.
• It is used to calculate the annual return on the portfolio.
• It is used in finance to find the average growth rates which are also referred to the compounded annual growth rate.
• It is also used in studies like cell division and bacterial growth etc.

Geometric Mean Examples

Here you are provided with geometric mean examples as follows

Question 1 : Find the G.M of the values 10, 25, 5, and 30

Solution : Given 10, 25, 5, 30

We know that,

GM = \(\sqrt[n]{\prod_{i=1}^{n}x_{i}}\)

= \(\sqrt[4]{10\times 25\times 5\times 30}\)

= \(\sqrt[4]{37500}\)

= 13.915

Therefore, the geometric mean = 13.915

Question 2 : Find the geometric mean of the following data.

Solution: Here n=5

GM =  \(Antilog\frac{\sum \log x_{i}}{n}\)

= Antilog 8.925/5

= Antilog 1.785

= 60.95

Therefore the G.M of the given data is 60.95

Question 3: Find the geometric mean of the following grouped data for the frequency distribution of weights.

Solution:

From the given data, n = 160

We know that the G.M for the grouped data is

GM = \(Antilog\frac{\sum f \log x_{i}}{n}\)

GM = Antilog ( 324.2 /160 )

GM = Antilog ( 2.02625 )

GM = 106.23

Therefore, the G.M = 106.23

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