Geometric Mean and Useful Results: Formula, Properties & Examples

Geometric Mean and Useful Result: In mathematics and statistics, measures of central tendencies describe the summary of whole data set values. The most important measures of central tendencies are mean, median, mode, and range. Among these, the mean is defined as average numbers in the data set. The different types of mean are arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM).

In the following article, let us learn more about the definition, formula, properties, applications of geometric mean, advantages and disadvantages, and the relation between AM, GM, and HM, along with some solved examples.

What is Arithmetic Mean?

The arithmetic mean or average is the ratio of the sum of all observations to the total number of observations. If a data set consists of the observations \({b_1},{b_2},{b_3}, \ldots ,{b_n},\) then the arithmetic mean \(B\) is given by

\(B = \frac{1}{n}\sum\limits_{i = 1}^n {{b_i}} \)

Example: The arithmetic mean of \(2, 8, 5, 9, 6,\) and \(12\) is \(\frac{{2 + 8 + 5 + 9 + 6 + 12}}{6} = \frac{{42}}{6} = 7\)

What is Harmonic Mean?

The harmonic mean is the ratio of the number of observations to the sum of the reciprocal of each observation. In other words, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the observations.

Example: The harmonic mean of \(1, 4,\) and \(4\) is \(\frac{3}{{\left( {1 + \frac{1}{4} + \frac{1}{4}} \right)}} = \frac{3}{{1.5}} = 2.\)

What is Geometric Mean?

The geometric mean (GM) is the average value or the mean that signifies the central tendency of the set of numbers by taking the root of the product of their values. In other words, we multiply the \(n\) values together and take out the \({n^{{\text{th}}}}\) root of all the numbers

Example: For three numbers \(9,4\) and \(2,\) \(GM = \sqrt[3]{{9 \times 4 \times 2}}\)

\( = \sqrt[3]{{72}}\)

\(GM = \sqrt[3]{{72}}\)

Geometric mean is also defined as the \({n^{{\rm{th}}}}\) root of the product of \(n\) values. Suppose, if you have two values, take the square root; if you have three, take the cube root; if you have four, then take the \({4^{{\rm{th}}}}\) root, and so on.

Formula

If \(G\) is the geometric mean between \(a\) and \(b,\) then the geometric mean between two numbers \(a\) and \(b\) is 

\(G = \sqrt {ab} \)

The geometric mean of a data with \(n\) observations is the \({n^{{\rm{th}}}}\) root of the product of the values. 

Consider, \({x_1},{x_2}….{x_n}\) are \(n\) observations, for which our target is to determine the geometric mean. The formula is to calculate the geometric mean is given below

\(GM = \sqrt[n]{{{x_1} \times {x_2} \times \ldots {x_n}}}\)

\( \Rightarrow GM = {\left( {{x_1} \times {x_2} \times \ldots {x_n}} \right)^{\frac{1}{n}}}\)

It is also written as

\(GM = \sqrt[n]{{\prod\limits_{i = 1}^n {{x_i}} }}\)

Now, taking logarithm on both sides, then we get,

\(\log (GM) = \log {\left( {{x_1} \times {x_2} \times \ldots {x_n}} \right)^{\frac{1}{n}}}\)

\( \Rightarrow \log (GM) = \frac{1}{n}\log \left( {{x_1} \times {x_2} \times \ldots {x_n}} \right)\)

\( \Rightarrow \log (GM) = \frac{1}{n}\left( {\log {x_1} + \log {x_2} + \cdots + \log {x_n}} \right)\)

\( \Rightarrow \log (GM) = \frac{{\left( {\sum\nolimits_{i = 1}^{i = n} {\log {x_i}} } \right)}}{n}\)

Hence, the geometric mean is 

\(GM = {\rm{Antilog}}\frac{{\left( {\sum\nolimits_{i = 1}^{i = n} {\log {x_i}} } \right)}}{n}\)

Note: For grouped data, the geometric mean can be calculated as

\(GM = {\rm{Antilog}}\frac{{\left( {\sum f \log {x_i}} \right)}}{n}\)

where \(n = {f_1} + {f_2} + \cdots + {f_n}\)

Geometric Mean Vs Arithmetic Mean

The comparison of geometric mean and arithmetic mean is given below.

Relation Between AM, GM, and HM

Let \(a\) and \(b\) are the two numbers. 

Arithmetic mean between two numbers \(a\) and \(b\) is \(AM = \frac{{(a + b)}}{2}\)

\( \Rightarrow \frac{1}{{AM}} = \frac{2}{{(a + b)}} \ldots (i)\)

Geometric mean between two numbers \(a\) and \(b\) is \(GM = \sqrt {ab} \)

\( \Rightarrow G{M^2} = ab \ldots (ii)\)

Similarly, the harmonic mean between two numbers \(a\) and \(b\) is \(HM = \frac{2}{{\frac{1}{a} + \frac{1}{b}}}\)

\( \Rightarrow HM = \frac{2}{{\frac{{a + b}}{{ab}}}}\)

\( \Rightarrow HM = \frac{{2ab}}{{a + b}} \ldots (iii)\)

Now, substitute equation \((i)\) and equation \((ii)\) in equation \((iii),\) then we get

\(HM = \frac{{G{M^2}}}{{AM}}\)

\( \Rightarrow G{M^2} = AM \times HM\)

\( \Rightarrow GM = \sqrt {AM \times HM} \)

Properties of Geometric Mean

Some of the properties of geometric mean are:

• Geometric mean for the given set is always less than the arithmetic mean of that data.
• If each value in the data set is substituted by the geometric mean, then product of the values remains the same.
• The ratio of corresponding observations of the geometric mean in two series is equal to the ratio of their geometric means.
• The product of corresponding observations of the geometric mean in two series is equal to the product of their geometric means.

Applications of Geometric Mean

The geometric mean is applied to data that has only positive values, but is sometimes used for the set of numbers whose values are exponential and whose values are meant to be multiplied together. Hence, geometric mean has many advantages and is used in many fields. Some of the applications of GM are given below:

• Geometric mean is used in stock indexes because many of values line indexes which are used by financial departments.
• It is used to find the annual return on investment portfolios.
• The geometric mean is also used in finance to find the average growth rates known as the compounded annual growth rate.
• Geometric mean is used in biological studies like cell division and bacterial growth rate.

Advantages and Disadvantages of Geometric Mean

Advantages:

• The geometric mean is rigidly defined; its value is fixed.
• The geometric mean is based on all the items of the series.
• The geometric mean can determine the correct average while dealing with percentages and ratios.
• The geometric mean is less affected by sampling fluctuations.
• It is helpful for algebraic calculation and other mathematical treatments.

Disadvantages:

• One of the significant drawbacks of the geometric mean is that if any one of the observations is negative, then the geometric mean value will be imaginary despite the quantity of the other observations.
• It requires a different mathematical knowledge (logarithm, ratio, roots, etc) to determine the geometric mean, and it is complex to calculate as it involves \({n^{{\rm{th}}}}\) root and difficult to understand.

Solved Examples

Q.1. Find the geometric mean of \(20\) and \(30.\)
Ans: Given: \(20\) and \(30\)
Geometric mean, \(GM = \sqrt {ab} \)
Here,
\(a=20\)
\(b=30\)
\( \Rightarrow GM = \sqrt {20 \times 30} \)
\( = \sqrt {600} \)
\(\therefore GM = 10\sqrt 6 \)

Q.2. Find the geometric mean of \(3,5,7,9.\)
Ans: Given: \(3,5,7,9\)
Geometric mean is given by \({\left( {{x_1} \times {x_2} \times {x_3} \ldots \times {x_n}} \right)^{\frac{1}{n}}}\)
\( \Rightarrow GM = {(3 \times 5 \times 7 \times 9)^{\frac{1}{4}}}\)
\( = {(945)^{\frac{1}{4}}}\)
\(\therefore GM = 5.54444337\)

Q.3. Find the geometric mean of \(10,25,5,\) and \(30.\)
Ans: Given: \(10,25,5,\) and \(30\)
Geometric mean is \({\left( {{x_1} \times {x_2} \times {x_3} \ldots \times {x_n}} \right)^{\frac{1}{n}}}\)
\( \Rightarrow GM = {(10 \times 25 \times 5 \times 30)^{\frac{1}{4}}}\)
\( \Rightarrow GM = {(37500)^{\frac{1}{4}}}\)
\(\therefore GM = 13.915\)
Hence, the geometric mean of \(10, 25, 5,\) and \(30\) is \(13.915.\)

Q.4. Find the geometric mean of \(5,15,25,\) and \(30.\)
Ans: Given: \(5,15,25,\) and \(30\)
Geometric mean is \({\left( {{x_1} \times {x_2} \times {x_3} \ldots \times {x_n}} \right)^{\frac{1}{n}}}\)
\( \Rightarrow GM = {(5 \times 15 \times 25 \times 30)^{\frac{1}{4}}}\)
\( = {(56250)^{\frac{1}{4}}}\)
\(\therefore GM = 15.4003514\)
Hence, the geometric mean of \(5,15,25,\) and \(30\) is \(15.4003514.\)

Q.5. Calculate the geometric mean of the following data:

Ans: As we know, for grouped data, the geometric mean can be calculated as
\(GM = {\mathop{\rm Antilog}\nolimits} \frac{{\left( {\sum f \log {x_i}} \right)}}{n}\)
where \(n = {f_1} + {f_2} + \cdots + {f_n}\)
Here, \(n=5\)
\( \Rightarrow GM = {\rm{Antilog}}\left( {\frac{{8.925}}{5}} \right)\)
\( = {\rm{Antilog}}(1.785)\)
\(\therefore GM = 60.95\)
Hence, the geometric mean of the given data is \(60.95.\)

Q.6. Find the geometric mean for the following grouped data for the frequency distribution of weights.

Ans:

From the given data, we have \(n=160\)
As we know, the geometric mean for the grouped data is \(GM = {\rm{Antilog}}\frac{{\left( {\sum f \log {x_i}} \right)}}{n}\)
\(GM = {\rm{Antilog}}\left( {\frac{{324.2}}{{160}}} \right)\)
\(GM = {\rm{Antilog}}(2.02625)\)
\(GM=106.23\)
Hence, the required \(GM\) is \(106.23\)

Summary

Geometric mean is the value that denotes the central tendency of a set of numbers by calculating the \({n^{{\rm{th}}}}\) root of the product of their values, where \(n\) is the number of observations. So, the geometric mean of \(n\) observations is given by \(GM = \sqrt[n]{{{x_1} \times {x_2} \times \ldots {x_n}}}.\) While the arithmetic mean is calculated by adding the observations, the geometric mean is achieved by multiplication. Hence, geometric mean cannot be calculated if there are odd number of negative values in the data. The harmonic mean is the ratio of the number of observations to the sum of the reciprocal of each observation. The relation between AM, GM and HM is given by \(G{M^2} = AM \times HM.\)

Frequently Asked Questions (FAQs)

Q.1. What is a geometric mean, and how is it solved?
Ans: The geometric mean can be determined by multiplying all the values of the given data set and taking the \({{n^{{\rm{th}}}}}\) root for the final result.
The geometric mean of \(n\) data values can be determined by using the formula
\({\left( {{x_1} \times {x_2} \times {x_3} \ldots \times {x_n}} \right)^{\frac{1}{n}}}\)

Q.2. What is geometric mean and examples?
Ans: Geometric mean takes several values and multiplies them together and sets them to the \(\frac{1}{{{n^{{\rm{th}}}}}}\) power, where \(n\) is the number of observations.
Example: For the given set of two numbers such as \(4\) and \(9,\) the geometric mean is given by,
\(GM = \sqrt {(4 \times 9)} \)
\( = \sqrt {36} \)
\(\therefore GM = 6\)

Q.3. When is geometric mean useful?
Ans: The geometric mean is useful when numbers in the series are not independent of each other or if numbers tend to make large fluctuations.

Q.4. How is geometric mean used in real life?
Ans: The geometric mean can be used in real life to find growth rates, portfolio returns and stock indexes.

Q.5. What is the difference between arithmetic and geometric means?
Ans: While the arithmetic mean is the ratio of the sum of given values to the total number of values, geometric mean is the \({n^{{\rm{th}}}}\) root of the product of the given \(n\) values.

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