Geometric Mean Formula with Explanation and Solved Examples

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The geometric mean is the average mean that determines the central tendency of a finite set of real numbers by using the product of their values. It is commonly used to calculate growth rates such as population or interest rates.

Geometric mean, arithmetic mean (AM), and harmonic mean (HM) are the three types of means.
It is often used for a set of numbers whose values are exponential in nature, like a set of growth figures.
The square root of the product of two positive numbers is the geometric mean of the two positive values.
The measurement of central tendencies includes mean, median, mode, and range.
Interest rates attached to any financial investments are the most common example of geometric mean.
The geometric mean formula for n real numbers is as follows:

G.M. = √∏ᵢ₌₁ⁿ xᵢ

where n: number of samples
√∏ᵢ₌₁ⁿ xᵢ: n^th square root of the product of the numbers.

Table of Content

Geometric Mean Definition
Formula for Geometric Mean
Things to Remember
Sample Questions

Key Terms: Geometric Mean Formula, Geometric Mean, Central Tendency, Mean, Arithmetic Mean, Real Numbers, Harmonic Mean

Geometric Mean Definition

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Geometric mean refers to the nth root value of the product of n numbers. It signifies the central value of a set of positive numbers by calculating the product of their values with suitable power raised for the product.

The geometric mean of two numbers, x and y, is equal to the length of side of a square whose area is equal to the area of a rectangle.
It cannot be calculated from the arithmetic mean.
In this mean, first data values are multiplied, and then the root is taken for the total number of data values for required radical index.
The nth root of the multiplied numbers is obtained by multiplying all of the numbers together, where n is number of data points.
This mean is used for positive set of real numbers.
When it comes to calculate the performance of a portfolio, the geometric mean is a fantastic instrument.

Example of Geometric Mean

Example: Find the geometric mean of 5 and 5.

Solution: Using the formula for G.M., the geometric mean of 5 and 5 will be:

Geometric Mean will be √(5 x 5)

= 5

So, GM = 5

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Relevant Concepts
Sequence and Series	Difference between Sequence and Series	Geometric Distribution
Infinite Series	Geometric Probability	Geometric Series Formula

Formula to find Geometric mean

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The geometric mean is a calculation that involves multiplying the numbers together, then computing the square root if there are two numbers, cube root if there are three numbers, and so on.

The geometric mean is frequently referred to as the "mean proportional" since it is utilised as a proportion in geometry.
The positive number a will be the mean proportional of two positive values x and y.
The geometric mean of two positive numbers, x and y, is given by the formula:

x ⁄ a = a ⁄ y

a² = xy

a= √xy

Example of Formula to find Geometric mean

Example: Find the geometric mean of 1,2,7,10?

Ans: Since Geometric Mean is given as (x₁ × x₂ × x₃...× x_n)^1/n

= (1 × 2 × 7 × 10)^1/4

= 11.8

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Related Topics
Difference between Mean and Median	Series Formula	Differentiation and Integration Formula
Harmonic Mean Formula	Difference between Average and Mean	Bayes Theorem Formul

Things to Remember

The geometric mean is a mean or average in mathematics that employs the product of two integers to illustrate the centre tendency or usual value of a set of numbers.
A set of numbers whose values are supposed to be multiplied together or are exponential are used in these numbers.
If all data set members are equal, in that case, the geometric and arithmetic means are equal.
The G. M. for a given set of values is always less than the arithmetic mean of the given data.
The geometric means of the two series are proportional to the ratio of their associated G.M observations.

Sample Questions

Ques: Describe geometric mean in brief? (3 marks)

Ans: The Geometric Mean (GM) is an average value or mean representing the central tendency of a group of numbers by calculating the product of their values.

Measures of central tendency are used in mathematics and statistics to express the summary of all the values in a data collection.
The mean, median, mode, and range are the essential measurements of central tendency.
It gives you a broad perspective of the data.
It is the average of all the numbers in the data collection.
The three types of means are geometric (GM), arithmetic (AM), and harmonic (HM).

Ques: Tell one point of difference between geometric mean and arithmetic mean? (2 marks)

Ans: By summing the given data values and dividing them by the total number of values, the arithmetic mean can be computed. By multiplying all of the values in the given data values and then determining the nth root of the product, the geometric mean can be calculated.

Ques: Write any four applications of Geometric mean? (3 marks)

Ans: The following are some of the applications:

The geometric mean is employed in stock indexes since it is used in many value-line indexes used by financial departments.
It is used to calculate the investment portfolio's annual return.
In finance, the geometric mean is used to calculate average growth rates, also known as compounded annual growth rates (CAGR).
The geometric mean is also used in biological investigations, such as cell division and bacterial growth rate.

Ques: What does the term "Geometric Mean" in statistics mean? (1 mark)

Ans: The value or mean of a set of data points computed by raising the product of the points to the reciprocal of the number of data points is known as the geometric mean.

Ques: Why is the geometric mean considered a more accurate indicator of returns than the arithmetic mean? (2 marks)

Ans: It considers the compounding that occurs from period to period, the geometric mean differs from the arithmetic average, or arithmetic mean, in how it is calculated. As a result, investors prefer the geometric mean to the arithmetic mean as a more accurate indicator of returns.

Ques: Only positive numbers should be utilised with the geometric mean. How would you describe it? (2 marks)

Ans: Only positive integers should be used with the geometric mean, and it is commonly applied to a collection of numbers whose values are exponential and can be multiplied together. This indicates that we can't use the geometric mean on negative or zero integers.

Ques: Describe the Geometric Mean theorem? (2 marks)

Ans: The height of a right triangle is the length of a line drawn perpendicularly from the hypotenuse to the 90° vertex. The length of the altitude of the right triangle is the geometric mean of both segment lengths if this line divides the hypotenuse into two segments. This phenomenon is known as the geometric mean theorem.

Ques: Find the geometric mean of the given data values 5, 15, and 30? (3 marks)

Ans: Given data values are 5 25, 15, and 30.

Using the geometric mean formula,

G.M=n√x1×x2×…xn

GM = 3√5 × 15 × 30

GM = 13.10

Ques: If AM and HM of the data sets are 16 and 100 respectively, then find the GM? (3 marks)

Ans: Given that,

Arithmetic Mean (AM) = 16
Harmionic Mean (HM) = 100

We know that the relation between AM, GM, and HM is given as

GM = √ [ AM × HM]

Substitute the values in the above equation,

GM = √[16 × 100]

GM = 40

Ques: Find the geometric mean of 3 and 3? (2 marks)

Ans: Using the formula for G.M., the geometric mean of 3 and 3 will be:

Geometric Mean will be √(3×3)

= 3

So, GM = 3.46

Ques: Calculate the geometric mean of 2 and 8? (3 marks)

Ans: Let us assume that

a = 2
b = 9

Here, the number of the data or terms, n = 2

If n = 2, then the formula for the geometric mean is √(ab).

Therefore, GM = √(2×9)

GM = 3√2

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Geometric Mean Formula

Geometric Mean Definition

Example of Geometric Mean

Formula to find Geometric mean

Example of Formula to find Geometric mean

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Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

4.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

5.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

6.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

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Geometric Mean Formula

Geometric Mean Definition

Example of Geometric Mean

Formula to find Geometric mean

Example of Formula to find Geometric mean

Check More:

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

4. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

5.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

6. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

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SUBSCRIBE TO OUR NEWS LETTER

1.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

4.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

5.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

6.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).