[Maths Class Notes] on Geometric Mean Formula Pdf for Exam

Mathematics and statistics use the measures of central tendency to express the summary of all the values in a data collection.

The mean, median, mode, and range are the most essential measurements of central tendency. Among these, the data set’s mean provides an overall picture of the data. It’s an average of the data set’s numbers.

There are different types of mean

• Arithmetic Mean (AM)

• Geometric Mean (GM)

• Harmonic Mean (HM)

The geometric mean is the average value or mean that, by applying the root of the product of the values, displays the central tendency of a set of numbers or data.

It is a metric unit used to calculate the performance of a single investment or an investment portfolio. For example, for a set of 2 numbers such as 24 and 1. The geometric mean for the given set of two numbers is equal to [sqrt {(24 + 1)} = sqrt {25} = 5]

The geometric mean is also written as G.M.

Fundamentally, 

• Total ‘n’ values are multiplied together

• The nth root is being taken out of the numbers, where n is the total number of values.

In this article, we will discuss the geometric mean, geometric mean definitions, and formula, the geometric mean formula for grouped data, properties of geometric mean, etc. is.

Formula:

Below is the formula for Geometric mean calculation: 

If X₁, X₂….X_n is the observation, then the Geometric mean is defined as:

GM = [sqrt[n]{x_1 times x_2 times x_3…..x_n}]

GM = [{(x_1 times x_2 times x_3…..x_n)^{1/n}}]

OR

Log GM = [frac {1} {n}log{(x_1 times x_2 times x_3…..x_n)}]

= [frac {1} {n}{(logx_1 + logx_2 + …..+logx_n)}]

= [frac {varepsilon log x_i} {n}]

GM = Anti log [frac {varepsilon log x_i} {n}]

Where, n = f₁ + f₂ + …….+f_n

It can also be written as:

GM = [sqrt[n]{prod^n_{i=n}x_i}]

GM can be written as follows for any grouped data:

GM = Antilog  [frac {varepsilon flog x_i} {n}]

The geometric mean, to put it another way, is the nth root of the product of n values.

Multiply all of your values together to get the geometric mean, then take a root of it. The number of values in your dataset determines the root. Take the square root if you have two values. Take the cube root with three values. Take the 4th root using four values, and so on.

Positive values must be present in all of your data. The geometric mean is usually always less than the arithmetic mean for any given dataset. When your dataset contains identical integers, an exception arises (e.g., all 5s). The geometric mean equals the arithmetic mean in this example.

 

Geometric Mean Formula for Grouped Data

If we have a set of n positive values with some repeated values such as x₁,x₂,x₃…..x_n, and the values are repeating s₁,s₂,s₃…….s_k times, then the geometric mean formula for grouped data is defined as.

 

[G.M. of X = overline{X} = sqrt[n]{x_{1}^{s_{1}}, x_{2}^{s_{2}}, x_{3}^{s_{3}} . . . . x_{k}s_{k}}]

 

Geometric Mean Formula for Ungrouped Data

The geometric mean formula for ungrouped data for value X consisting n observation such as x₁,x₂, x₃…x_n is represented by G, M of X and is derived as

 

[G.M. of X = overline{X} = sqrt[n]{x_{1}, x_{2}, x_{3} . . . . x_{n}}]

 

Properties

The following are the properties of Geometric mean:

1. The geometric mean for a given data is always less than the arithmetic means for a given data set.

2. The ratio of the associated observation of the geometric mean in two series is equivalent to the ratio of their geometric means.

3. The product of the associated observation of the geometric mean in two series is equivalent to the product of their geometric means.

4. If the geometric mean replaces each observation in the given data set, then the product of observations does nor change.

Algebraic Properties

• The geometric mean is less than the arithmetic mean for any set of positive numbers but when all of a series’ values are equal, however, G.M. equals the arithmetic mean.

• If any value in a series is 0, the geometric mean is infinity, which is unsuitable.

• If the number of negative values is odd, it cannot be calculated. This is due to the fact that the product of the values will turn negative, and we will be unable to determine the root of a negative product.

• The product of the values equals the geometric mean raised to the nth power.

• The geometric mean of any set of numbers with the same N and product is the same.

• The product of the geometric mean’s each side ratio will be equal to both sides.

• Even when each number in a series is replaced by its geometric mean, the series’ products remain the same.

• The sum of the deviations of the original values’ logarithms above and below the G.M.’s logarithm is equal.

Solved Examples:

1. Find the geometric mean of the numbers 2, 3, and 6?

i) Take the cubed root of the numbers multiplied together (because there are three numbers) = [frac {(2 times 3 times 6 times)1}{3}] = 3.30 

2. Find the geometric mean of 4,8.3,9 and 17?

i) Multiply the numbers together

ii) Take the 5th root (because there are 5 numbers) = [{(4 times 8 times 3 times 9 times 17)}^{1/5}] = 6.81

3. Geometric mean of [frac {1}{2}] , [frac {1}{4}] , [frac {1}{5}] , [frac {9}{72}] and [frac {7}{4}] is ?

i) Multiply the numbers together 

ii) 5th root is to be taken: ([frac {1}{2} times frac {1}{4} times frac {1}{5} frac {9}{72} frac {7}{4}^(frac {1}{5})] = 0.35.

4. Over ten years, the average monthly wage in a particular municipality increased from $2,500 to $5,000. What is the average annual increase using the geometric mean?

i) Geometric mean.

 [{(2500 times 5000)}^{1/2}] = 3535.53390593. 

ii) Divide by 10 (to get the ten-year average increase).

[frac {3535.53390593}{10}] = 353.53. 

As per GM, the average increase is 353.53.

5. Find the geometric mean for the following data.

 

 

Solution:

 

Geometric mean of X = Antilog [frac {sum f log x}{sum f}]

= Antilog [frac {(119.1074)}{48}]

= Antilog (2.4814)

= 11.958

 

6. Calculate the geometric mean of 10,5,15,8,12.

 

Solution:

Given, x₁ = 10 ,x₂= 5,x₃ = 15,x₄ = 8,x₅= 12

N= 5

Using the geometric mean formula,

[sqrt[n]{x_{1}, x_{2}, x_{3}. . . . x_{n}}]

[G.M. of X = overline{X} = sqrt[5]{10 times 5 times 15 times 8 times 12}]

[sqrt[5]{72000}]

= [(72000)^{frac{1}{5}}]

= 9.36.

7. Find the geometric mean for the following data.

 

Solution: Here n = 5

Geometric Mean = [frac {sum log x_i}{n}]

= Antilog [frac {8.925}{5}]

= Antilog  1.785

 

Quiz Time

1. The geometric mean is calculated by

1. (a + b)²

2. (a – b)/2

3. ± √ab

4. ± ab

 

2. The three geometric means between 2 and 32 are

1. 6,10,14

2. 10,12,14

3. 6,8,10

4. 4,8,16

 

3. If a, G and b are in geometric progression the ‘G’ is considered as

1. Arithmetic mean

1. Geometric mean

2. Standard deviation

3. None of the above