Geometric Mean: Definition, Formulas, Function

Geometric Mean of two positive numbers is the positive square root of the multiplication of the two positive numbers. Similarly, the geometric mean of three, four\(,…….,n\) positive numbers are the \({{\rm{3}}^{{\rm{rd}}}}{\rm{,}}{{\rm{4}}^{{\rm{th}}}}{\rm{, \ldots \ldots }}{n^{{\rm{th}}}}\) root of the multiplication of the positive numbers, respectively. The geometric mean represents the central value of a group of numbers. Geometric mean is not defined for negative numbers. In this article, we shall discuss more about geometric mean like its definition, function, properties, applications, formulas etc.

Definition of Geometric Mean

Geometric mean is the average value, also known as the mean, which denotes the central value of a group of positive numbers by calculating the product of their values with proper power raised for the product. We multiply all the numbers together and then obtain the \({n^{{\rm{th}}}}\) root of the multiplied numbers, where \(n\) is the total number of data points. For instance, the geometric mean of a pair of numbers such as \(3\) and \(1\) is equal \(\sqrt {3 \times 1} = \sqrt 3 .\)

The geometric mean, in other words, is defined as the \({{\rm{n}}^{{\rm{th}}}}\) root of the product of n numbers. The geometric mean differs from the arithmetic mean. In the case of arithmetic mean, we add the data values and then divide them by the total number of values. In geometric mean, however, we multiply the given data values before taking root with the radical index for the total number of data variables.

For the geometric mean, we must take the square root if we have two data points, the cube root if we have three data points, the \({{\rm{4}}^{{\rm{th}}}}\) root if we have four data points, and so on.

The Use of Geometric Mean

The  geometric mean can be used to find the average of a set of values contributing to a product. It takes the product of a set of values of size \(n\) and returns the \({n^{{\rm{th}}}}\) root of the result (also known as geometric mean).

Example: What is the geometric mean of \(2\) and \(8\)?

To find the geometric mean of \(2\) and \(8,\) let us take  \(a = 2,b = 8\)

Formula to find the geometric mean \( = \sqrt {ab} \)

Hence, the geometric mean of \(2\) and \(8\) is \(\sqrt {2 \times 8} = \sqrt {16} = 4.\)

Formula to Find Geometric Mean

The formula to find the geometric mean is given below:

In case there are two numbers \(a\) and \(b,\) the geometric mean is given by \( = \sqrt {a \times b} .\)

If there are many numbers \({x_1},{x_2},{x_3} \ldots {x_n},\) then, the geometric mean is \(\sqrt[n]{{{x_1} \times {x_2} \times {x_3} \ldots {x_n}}}\)

\( \Rightarrow \) Geometric Mean \( = \sqrt[n]{{{x_1} \times {x_2} \times {x_3} \ldots {x_n}}}\)

\( \Rightarrow \) Geometric Mean \( = {\left( {{x_1} \times {x_2} \times {x_3} \ldots {x_n}} \right)^{\frac{1}{n}}}\)

Learn Geometric Progression

Properties of Geometric Mean

Some properties of the geometric mean are given below.

1. For every given set of data, the geometric mean is always less than the arithmetic mean.
2. The product of the data remains the same when the geometric mean is substituted for each data in the data set.
3. If \(a\) and \(b\) are two positive numbers and \(G\) is the geometric mean of these two numbers then \(a, G, b\) will be in geometric progression, where \(G = \sqrt {ab} .\)

Application of Geometric Mean

The geometric mean can be interpreted as a scaling factor. The geometric mean should only be used with positive numbers, and it is frequently applied to a group of numbers whose values are exponential, and these values are known to be multiplied together. It means that we cannot apply geometric mean on zero and negative numbers. The geometric mean has various advantages and is utilized in a variety of fields. The following are some of the applications are given below:

1. It is a component of stock indexes. Geometric mean is employed in many of the value line indexes used by financial departments.
2. Geometric mean is used to calculate the portfolio’s annual return.
3. Geometric mean is used in finance to calculate average growth rates, commonly known as compounded annual growth rates.
4. Geometric mean is also utilized in research on cell division and bacterial development, among other things.

Difference Between Geometric Mean and Arithmetic Mean

Solved Examples – Geometric Mean

Q.1. What is the geometric mean of \(4\) and \(3\)?
Ans: To find the geometric mean of \(4\) and \(3.\)
Let us take \(a=4, b=3\)
Formula to find the geometric mean \( = \sqrt {ab} \)
\( = \sqrt {4 \times 3} = \sqrt {12} \)
\( = \sqrt {12} = 2\sqrt 3 \)
Therefore, the geometric mean of \(4\) and \(3\) is \(2\sqrt 3 .\)

Q.2: Find the geometric mean of \(2, 3,\) and \(6.\)
Ans: To find the geometric mean of \(2, 3\) and \(6.\)
Let us take \(a=2, b=3, c=6\)
Formula to find the geometric mean \( = \sqrt[3]{{abc}}\)
\( = \sqrt[3]{{2 \times 3 \times 6}} = \sqrt[3]{{36}}\)
\( = \sqrt[3]{{36}}\)
Therefore, the geometric mean of \(2, 3\) and \(6\) is \(\sqrt[3]{{36}}.\)

Q.3. Find the geometric mean of \(3, 5,\) and \(12.\)
Ans: To find the geometric mean of \(3, 5\) and \(12.\)
Let us take \(a=3, b=5, c=12\)
Formula to find the geometric mean \( = \sqrt[3]{{abc}}\)
\( = \sqrt[3]{{3 \times 5 \times 12}} = \sqrt[3]{{180}}\)
\( = \sqrt[3]{{180}}\)
Therefore, the geometric mean of \(3, 5\) and \(12\) is \(\sqrt[3]{{180}}.\)

Q.4. Find the geometric mean of \(4, 8, 3,\) and \(9.\)
Ans: To find the geometric mean of \(4, 8, 3,\) and \(9.\)
Let us take \(a=4, b=8, c=3, d=9\)
Formula to find the geometric mean \( = \sqrt[4]{{a \times b \times c \times d}}\)
\( = \sqrt[4]{{4 \times 8 \times 3 \times 9}}\)
\( = \sqrt[4]{{864}}\)
Therefore, the geometric mean of \(4, 8, 3\) and \(9\) is \(\sqrt[4]{{864}}.\)

Q.5. Find the geometric mean of \(1, 2, 4, 8\) and \(16.\)
Ans: To find the geometric mean of \(1, 2, 4, 8\) and \(16.\)
Let us take \(a=1, b=2, c=4, d=8, e=16\)
Formula to find the geometric mean \( = \sqrt[5]{{a \times b \times c \times d \times e}}\)
\( = \sqrt[5]{{1 \times 2 \times 4 \times 8 \times 16}}\)
\( = \sqrt[5]{{1024}} = 4\)
Therefore, the obtained geometric mean is \(4.\)

Summary

In this article, we have discussed the definition of geometric mean between two numbers, as well as more than two numbers. We also discussed that geometric mean can be calculated only for positive numbers. Later we discussed properties of geometric mean, application of geometric mean, a difference of geometric mean with arithmetic mean etc. In the end, we have discussed some examples along with frequently asked questions which will help students to understand geometric mean in better way.

Frequently Asked Questions (FAQs) – Geometric Mean

Q.1. What is the geometric mean of \(4\) and \(9\)?
Ans: To find the geometric mean of \(4\) and \(9.\)
Let us take \(a=4, b=9\)
Formula to find the geometric mean\( = \sqrt {ab} \)
\( = \sqrt {4 \times 9} = \sqrt {36} \)
\( = \sqrt {36} = 6\)
Therefore, the geometric mean of \(4\) and \(9\) is \(6.\)

Q.2. What is the geometric mean of \(4\) and \(10\)?
Ans: To find the geometric mean of \(4\) and \(10.\)
Let us take \(a=4, b=10\)
Formula to find the geometric mean\( = \sqrt {ab} \)
\( = \sqrt {4 \times 10} = 2\sqrt {10} \)
Therefore, the geometric mean of \(4\) and \(10\) is \(2\sqrt {10} .\)

Q.3. What does geometric mean measure?
Ans: The geometric mean is a mean or average that uses the product of the numbers with proper power raised to the product, to show the central tendency or typical value of a set of numbers.

Q.4. Why is geometric mean used?
Ans: The geometric mean has various advantages and is utilized in a variety of fields. The following are some of the applications are given below:
1. It is a component of stock indexes.
2. Geometric mean is used to calculate the portfolio’s annual return.
3. Geometric mean is used in finance to calculate average growth rates, commonly known as compounded annual growth rates.
4. Geometric mean is also utilized in research on cell division and bacterial development, among other things.

Q.5. What is geometric mean?
Ans: The Geometric Mean is a sort of average in which we multiply the numbers together and then take the square root of two, the cube root of three, and so on, depending on the number of data used.

Q.6. What are the three measures of central tendency?
Ans: Three measures of central tendency are mean, mode and median.

Q.7. Write the applications of the geometric mean.
Ans: Some of the applications of the geometric mean are:
1. It is a component of stock indexes. Geometric mean is employed in many of the value line indexes used by financial departments.
2. It is used to calculate the portfolio’s annual return.
3. It is used in finance to calculate average growth rates, commonly known as compounded annual growth rates.
4. It is also utilized in research on cell division and bacterial development, among other things.

Q.8. Write one of the differences between geometric mean and arithmetic mean.
Ans: One of the differences between geometric mean and arithmetic mean is:
The geometric mean can be calculated by multiplying all the numbers in the data set and then taking the nth root of the result, where n is the total number of data.
The arithmetic mean, known as the mean, is calculated by adding all the values in a data collection and divided by the number of data points in the set.

PRACTICE QUESTIONS ON GEOMETRIC MEAN

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