Geometric Mean is a statistical measure used to calculate the central tendency used to find the central value of the data set in statistics especially when dealing with growth rates or percentages. This method is great for comparing values that change over time, like investment returns or population growth.
Various types of mean 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.
What is Geometric Mean?
Geometric Mean is defined as the nth root of the product of "n" number of 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 the Geometric Mean is shown below:
∛(3×9×27) = ∛(729) = 9
Thus, the geometric mean is the measure of the central tendency that is used to find the central value of the data set.
The formula used to calculate the geometric mean of the given values is added below. Suppose we are given 'n' numbers x1, x2, x3, ..., xn then its geometric mean formula will be,
Another geometric mean formula is,
GM = Antilog (∑ log xk)/n
where,
- ∑log xk is Logarithm Value of sum of all Values in a Sequence
- n is the Number of values in the Sequence
Suppose x1, x2, x3, x4, ......, xn 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 = √(x1 × x2 × x3 × ... × xn)
GM = (x1 × x2 × x3 × ... × xn)1/n
Taking log on both sides of the equation we get,
log GM = log (x1 × x2 × x3 × ... × xn)1/n
Using, log formula log ab = b log a,
log GM = (1/n) log (x1 × x2 × x3 × ... × xn)
Using property, log (ab) = log a + log b,
log GM = (1/n) (log x1 + log x2 + log x3 + ... + log xn)
log GM = (∑ log xk)/n
Taking antilog on both sides we get,
GM = Antilog (∑ log xk)/n
Geometric Mean of Two Numbers
Suppose we are given two numbers 'a' and 'b' then the geometric mean formula for two numbers is :
GM of (a, b) = √(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 = √(4×16) = √(64) = 8
Thus, the GM of 4 and 16 is 8
Difference Between Arithmetic Mean and 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 = (x1 + x2 + ... + xn)/n | Geometric Mean Formula, GM = (x1 × x2 × ... × xn)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 = 4×6×10×8 = 1920 GM = (1920)1/4 = 6.2 |
How to Find the Geometric Mean?
Here are the simple steps to find the geometric mean of a set of numbers:
- Step 1: Multiply all the numbers together.
- Step 2: Count the total number of values (n).
- Step 3: Take the n-th root of the product.
The geometric mean of two numbers is found using the geometric mean formula, GM = √(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 = √(36.4) = √(144)
g = 12
Thus, the geometric mean of 36 and 4 is 12.
Example : Find the geometric mean of the numbers 2, 4, 8, and 16.
Solution
Given numbers: 2, 4, 8, 16
n = 4
Multiply the numbers together:
2 × 4 × 8 × 16 = 1024.
Take the 4th root (since there are 4 numbers):
∜1024 = 5.5 (approximately).
The geometric mean of 2, 4, 8, and 16 is approximately 5.5.
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 = √(ab)...(ii)
HM = 2ab/(a+b)...(iii)
from (i), (ii) and (iii)
HM = GM2/AM
GM2 = AM × HM
This is the required AM, GM and HM inequalities.
Application of Geometric Mean
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,
Examples problems on Geometric Mean
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 × 4 × 6 × 8 × 10 × 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 × 8 × 12 × 16 × 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,
Product of terms (P) = 5 × 10 × 15 × 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
⇒ 1/n = log GM/log P
⇒ n = log P/log GM
⇒ n = log 1024/log 32
⇒ n = 10/5
⇒ 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
⇒ 1/n = log GM/log P
⇒ n = log P/log GM
⇒ n = log 4096/log 8
⇒ n = 12/3
⇒ 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
⇒ 1/n = log GM/log P
⇒ n = log P/log GM
⇒ n = log 65536/log 4
⇒ n = 16/2
⇒ 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
⇒ 1/n = log GM/log P
⇒ n = log P/log GM
⇒ n = log 16777216/log 16
⇒ n = 24/4
⇒ 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?
Q5. A company has seen its yearly revenue grow by 10%, 15%, and 20% over the past three years. What is the geometric mean of the growth rate over these three years?
Q6. A gardener is planting three types of flowers with heights measured in centimeters as follows: 12 cm, 18 cm and 30 cm. Calculate the geometric mean of the heights of the flowers.
Q7. A researcher measured the weights (in kg) of five different fruits: 0.5, 0.75, 1.0, 1.5 and 2.0. What is the geometric mean of these weights?
Q8. If the geometric mean of three numbers is 10 and one of the numbers is 5, what is the product of the other two numbers?
Q9. A certain stock has returns of 8%, 12% and 20% over three consecutive years. What is the geometric mean of the stock's returns?
Q10. The prices of three products are $20, $50 and $80. Calculate the geometric mean of the prices to determine the average price level.
Answer Key:
- 25.97
- 14
- 4
- 10
- 12.16%
- 18.83 cm
- 0.89 kg
- 50
- 13.86%
- $40.00
Check: Harmonic Mean Formula
Conclusion
The geometric mean is a valuable tool for finding the average of numbers, especially when dealing with growth rates, ratios, or values that vary greatly. Unlike the arithmetic mean, it provides a more accurate reflection of data that involves multiplication or compounding. Whether calculating average returns on investments, comparing ratios, or understanding population growth, the geometric mean offers a practical solution for real-world problems.
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