Geometric Mean Formula
Geometric Mean is the measure of the central tendency used to find the central value of the data set in statistics. There are various types of mean that are used in mathematics including Arithmetic Mean(AM), Geometric Mean(GM), and Harmonic Mean(HM). In geometric mean, we first multiply the given number altogether and then take the nth root of the given product.
In this article, we will learn about Geometric Mean Definition, Geometric Mean Formula, Examples, and others in detail.
Table of Content
What is Geometric Mean?
Geometric Mean is the nth root of the product of the given dataset. It gives the central measure of the data set. To find the geometric mean of various numbers we first multiply the given numbers and then take the nth root of the given number. Suppose we are given 3 numbers 3, 9, and 27 then the geometric mean of the given values is calculated by taking the third root of the product of the three given data. The calculation of Geometric Mean is shown below:
âˆ›(3Ã—9Ã—27) = âˆ›(729) = 9
Thus, geometric mean is the measure of the central tendency that is used to find the central value of the data set.
Geometric Mean Definition
Geometric Mean is defined as the nth root of the product of “n” number of given dataset
Geometric Mean Formula
The formula used to calculate the geometric mean of the given values is added below. Suppose we are given ‘n’ numbers x_{1}, x_{2}, x_{3}, …, x_{n} then its geometric mean is calculated using the formula,
The other formula used to find the geometric mean is,
GM = Antilog (âˆ‘ log x_{k})/n
where,
- âˆ‘log x_{k} is Logarithm Value of sum of all Values in a Sequence
- n is the Number of values in the Sequence
Geometric Mean Formula Derivation
Suppose x_{1}, x_{2}, x_{3}, x_{4}, ……, x_{n} are the values of a sequence whose geometric mean has to be evaluated.
So, the geometric mean of the given sequence can be written as,
GM = âˆš(x_{1} Ã— x_{2} Ã— x_{3} Ã— … Ã— x_{n})
GM = (x_{1} Ã— x_{2} Ã— x_{3} Ã— … Ã— x_{n})^{1/n}
Taking log on both sides of the equation we get,
log GM = log (x_{1} Ã— x_{2 }Ã— x_{3} Ã— … Ã— x_{n})^{1/n}
Using, log formula log a^{b} = b log a,
log GM = (1/n) log (x_{1} Ã— x_{2} Ã— x_{3} Ã— … Ã— x_{n})
Using property, log (ab) = log a + log b,
log GM = (1/n) (log x_{1} + log x_{2} + log x_{3} + … + log x_{n})
log GM = (âˆ‘ log x_{k})/n
Taking antilog on both sides we get,
GM = Antilog (âˆ‘ log x_{k})/n
This derives the formula for geometric mean of a series.
Geometric Mean of Two Numbers
Suppose we are given two numbers ‘a’ and ‘b’ then the geometric mean of the two numbers is calculated as,
GM of (a, b) = âˆš(ab)
This is explained by the example added below,
Example: Find the geometric mean of 4 and 16.
Solution:
Given Numbers = 4 and 16
GM of 4 and 16 = âˆš(4Ã—16) = âˆš(64) = 8
Thus, the GM of 4 and 16 is 8
Arithmetic Mean Vs Geometric Mean
The difference between Arithmetic Mean and the Geometric Mean is explained in the table below,
Arithmetic Mean |
Geometric Mean |
---|---|
Arithmetic mean is the measure of the central tendency it is found by taking sum of all the values and then dividing it by the numbers of values. |
Geometric mean is also the measure of the central tendency. It is calculating by first taking the product of all n value and then taking the n the roots of the values. |
Arithmetic Mean Formula, AM = (Sum of Value)/(Number of Values) AM = (x_{1} + x_{2} + … + x_{n})/n |
Geometric Mean Formula, GM = (x_{1} Ã— x_{2} Ã— … Ã— x_{n})^{1/n} |
Example: Find the arithmetic mean of 4, 6, 10, 8 Given values,
Number of Values = 4 Sum of Value = 4+6+10+8 = 28 AM = 28/4 = 7 |
Example: Find the geometric mean of 4, 6, 10, 8 Given values,
Number of Values = 4 Product of Value = 4Ã—6Ã—10Ã—8 = 1920 GM = (1920)^{1/4} = 6.2 |
How to Find the Geometric Mean
The geometric mean of two numbers is found using the formula, GM = âˆš(ab), where a and b are the two numbers.
Example: What is the geometric mean of 36 and 4?
Solution:
Let the geometric mean of 36 and 4 is g,
g = âˆš(36.4) = âˆš(144)
g = 12
Thus, the geometric mean of 36 and 4 is 12.
Relation Between AM, GM and HM
There is relation between (arithmetic mean) AM, (geometric mean) GM and (harmonic mean) HM that is used to find any one value if other two values are given. Suppose we are given two numbers ‘a’ and ‘b’ then AM, GM and HM is calculated as,
AM = (a+b)/2…(i)
GM = âˆš(ab)…(ii)
HM = 2ab/(a+b)…(iii)
from (i), (ii) and (iii)
HM = GM^{2}/AM
GM^{2} = AM Ã— HM
This is the required AM, GM and HM inequalities.
Geometric Mean Properties
Various properties of the geometric mean are,
- GM of the given data set is always smaller than or equal to AM, i.e. AM â‰¥ GM
- Products of corresponding items of G.M in two series are equal to product of their geometric mean, etc.
Geometric Mean Theorem
Geometric mean theorem states that, if h is the altitude of the right angle triangle and a and b are the two segments of the hypotenuse then,
h = âˆš(a.b)
It is also stated as,
h^{2} = a.b
Geometric Mean Application
Geometric mean is the measure of the central tendency that is highly used in mathematics and related fields. Some applications of the Geometric mean are,
- It is used to study various types of graphs.
- It is used in the study of Stock markets.
- It is used in the explaining and identifying various patterns in big data.
- It is used in explained various biological process, such as bactericla growth, DNA synthesising,etc
Read More,
Geometric Mean Examples
Example 1: Calculate the geometric mean of the sequence, 2, 4, 6, 8, 10, 12.
Solution:
Given,
- Sequence, 2, 4, 6, 8, 10, 12
Product of terms (P) = 2 Ã— 4 Ã— 6 Ã— 8 Ã— 10 Ã— 12 = 46080
Number of terms (n) = 6
Using the formula,
GM = (P)^{1/n}
GM = (46080)^{1/6}
GM = 5.98
Example 2: Calculate the geometric mean of the sequence, 4, 8, 12, 16, 20.
Solution:
Given,
- Sequence, 4, 8, 12, 16, 20
Product of terms (P) = 4 Ã— 8 Ã— 12 Ã— 16 Ã— 20 = 122880
Number of terms (n) = 5
Using the formula,
GM = (P)^{1/n}
GM = (122880)1/5
GM = 10.42
Example 3: Calculate the geometric mean of the sequence, 5, 10, 15, 20.
Solution:
Given,
- Sequence, 5, 10, 15, 20
Product of terms (P) = 5 Ã— 10 Ã— 15 Ã— 20 = 15000
Number of terms (n) = 4
Using the formula,
GM = (P)^{1/n}
GM = (15000)^{1/4}
GM = 11.06
Example 4: Find the number of terms in a sequence if the geometric mean is 32 and the product of terms is 1024.
Solution:
Given,
- Product of terms (P) = 1024
- GM of terms = 32
Using the formula,
GM = (P)^{1/n}
â‡’ 1/n = log GM/log P
â‡’ n = log P/log GM
â‡’ n = log 1024/log 32
â‡’ n = 10/5
â‡’ n = 2
Example 5: Find the number of terms in a sequence if the geometric mean is 8 and the product of terms is 4096.
Solution:
Given,
- Product of terms (P) = 4096
- GM of terms = 8
Using the formula,
GM = (P)^{1/n}
â‡’ 1/n = log GM/log P
â‡’ n = log P/log GM
â‡’ n = log 4096/log 8
â‡’ n = 12/3
â‡’ n = 4
Example 6: Find the number of terms in a sequence if the geometric mean is 4 and the product of terms is 65536.
Solution:
Given,
- Product of terms (P) = 65536
- GM of terms = 4
Using the formula,
GM = (P)^{1/n}
â‡’ 1/n = log GM/log P
â‡’ n = log P/log GM
â‡’ n = log 65536/log 4
â‡’ n = 16/2
â‡’ n = 8
Example 7: Find the number of terms in a sequence if the geometric mean is 16 and the product of terms is 16777216.
Solution:
Given,
- Product of terms (P) = 16777216
- GM of terms = 16
Using the formula we have,
GM = (P)^{1/n}
â‡’ 1/n = log GM/log P
â‡’ n = log P/log GM
â‡’ n = log 16777216/log 16
â‡’ n = 24/4
â‡’ n = 6
Practice Questions on Geometric Mean
Q1. Calculate the geometric mean of the sequence, 15, 25, 35, 45.
Q2. What is the geometric mean of 7 and 28?
Q3. Find the number of terms in a sequence if the geometric mean is 22 and the product of terms is 655360.
Q4. What is the geometric mean of 4 and 25?
FAQs on Geometric Mean
1. What is Geometric Mean?
Geometric mean of numbers is time measure of the central tendency that is used to find the central values of the given data set. It is found by taking the product of all the given value and then taking the nth roots of the number.
2. What is Formula for Geometric Mean?
The formula to calculate the geometric mean is added below, suppose we gave n numbers x_{1}, x_{2}, …. x_{n} then the geometric mean formula is,
GM = (x_{1} Ã— x_{2} Ã— … Ã— x_{n})1/n
3. What is AM, GM Inequality?
The AM and GM inequality is the inequality that states that, AM is always greater than equal to the GM. This is represented as,
A.M â‰¥ G.M
4. What is Relation Between AM, GM and HM?
For given n numbers the relation between AM, GM and HM is,
GM^{2} = AM Ã— HM
5. What is Geometric Mean in Statistics?
The geometric mean in statistics is the average multiple of all the value of the given numbers. Geometric mean is found by taking the multiple of all the number and then taking the n th root of the number.
6. What is the Geometric Mean of 4 and 6?
The geometric mean of 4 and 6 is 4.89 (approx).
7. What is the Geometric Mean of 4 and 66?
The geometric mean of 4 and 16 is 8.
8. What is the Geometric Mean of 9 and 4?
The geometric mean of 9 and 4 is 6.
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