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Geometric Mean Formula
The area of Mathematics known as geometry concerns the dimensions, sizes, forms, and angles of a wide range of everyday objects. Geometry comes from the Ancient Greek terms “geo” and “metron,” which both imply “measuring.” There are twodimensional and threedimensional shapes in Euclidean geometry.
Flat shapes in plane geometry include 2dimensional shapes, including triangles, squares, rectangles, and circles. In solid geometry, threedimensional shapes like a cube, cuboids, cones, etc., are also referred to as solids. The coordinate geometry of points, lines, and planes is the foundation of fundamental geometry.
The Geometric Mean Formula of a set of numbers is calculated using a formula called the geometric mean, as the name suggests. Remember that the geometric mean (or GM) is a sort of mean that uses the product of the values of the numbers to show the central tendency of a set of numbers. It is described as the product of n numbers’ nth root. It should be emphasised that the Geometric Mean Formula cannot be determined from the arithmetic mean. Only a set of positive real numbers can be used in statistics to define the geometric mean. Calculating the centre frequency f0 of bandwidth using the Geometric Mean Formula is demonstrated by the example of BW=f2f1.
The geometric mean performs the best when reporting average growth rates, percentage changes, and inflation. The geometric mean is more accurate for these data types than the arithmetic mean because they are expressed as fractions.
While the arithmetic means is suitable for numbers independent of one another (such as test scores), the geometric mean is more suitable for dependent values, percentages, fractions, or data with a broad range. To study the Geometric Mean Formula, one can visit the Extramarks website or mobile application.
What is Geometric Mean?
The Geometric Mean Formula is an average value or means representing the central tendency of a group of numbers by taking the product of the roots of those values. In essence, where n is the total number of values, one must multiply all ‘n’ values and subtract the nth root of the numbers. For instance, the geometric mean for a pair of numbers, such as 8 and 1, is equivalent to (8 + 1) = 8 + (2 + 2).
The product of n numbers is thus the nth root of the geometric mean, another definition. Keep in mind that this is different from the arithmetic mean. Data values are summed and then divided by the total number of values to determine the arithmetic mean. In contrast, the given data values are multiplied by a geometric mean. The final product of the data values is then calculated by taking the root of the radical index. If one has two data values, for instance, take the square root. If one has three data values, take the cube root. The fourth root should be taken if one has four data values.
Geometric Mean Formula
The geometric mean is a mathematical term that refers to an average calculated by multiplying all of the integers in a sequence together and then computing their nth root, where n is the total number of values in the sequence. It can be understood as the product of n values’ nth root. The geometric mean indicates the central tendency of a group of numbers by computing the root of the product of their values. For instance, if two integers, 4 and 7, are given, the geometric mean is (4 + 7) = 28 + 2 + 7.
Difference Between Arithmetic Mean and Geometric Mean
Arithmetic involves adding data values, dividing by the total number of values, and multiplying every number in the provided data set. The geometric mean can be calculated. Then, the nth root of the resulting number is used to calculate the Geometric Mean Formula.
For example, the given data sets are
10, 15, and 20
Here, the number of data points = 3
Arithmetic mean or mean = (10+15+20)/3
Mean = 45/3 =15
For example, for data sets 4, 10, 16, 24
Here, n = 4
Therefore, the G.M = (4 ×10 ×16 × 24)1/4
= 153601/4
G.M = 11.13
Relation Between AM, GM, and HM
Students need to know the formulas of all these 3 types of the mean. Assume that “a” and “b” are the two numbers and the number of values = 2, then
AM = (a+b)/2
⇒ 1/AM = 2/(a+b) ……. (I)
G.M. = √(ab)
⇒GM2 = ab ……. (II)
HM= 2/[(1/a) + (1/b)]
⇒HM = 2/[(a+b)/ab
⇒ HM = 2ab/(a+b) ….. (III)
Now, substitute (I) and (II) in (III), and the result is
HM = GM2 /AM
⇒GM2 = AM × HM
Or else,
GM = √[AM × HM]
Hence, the relation between AM, G.M., and H.M. is GM2 = AM × H.M. Therefore, the square of the geometric mean is equal to the product of the arithmetic and harmonic mean.
Let us also see why the G.M. for the given data set is always less than the arithmetic means for the data set. Let A and G be A.M. and G.M.
So,
A = (a+b)/2 and G=√ab
Now let’s subtract the two equations
A−G = (a+b)/2 − √ab = (a+b−2√ab)/2 = (√a−√b)2/2 ≥ 0
A−G ≥ 0
This indicates that A ≥ G
Application of Geometric Mean
The geometric mean is utilised in many fields and has numerous advantages over the arithmetic mean. The following are a few examples of applications: It is employed in stock indexes as a large number of value line indexes, which financial departments utilise to determine the annual return on the investment portfolio, incorporate G.M. In finance, average growth rates, also referred to as the compounded annual growth rate, are determined using the geometric mean. Additionally, studies on biological processes like bacterial growth and cell division use geometric means.
Geometric Mean Examples
Example 1: Find the geometric mean of 1,3,5,7,9
Solution:
The GM is given as (x1 × x2 × x3…× xn)1/n
= (1 × 3 × 5 × 7 × 9)1/5
= (945)1/5
= 3.936
Answer: Therefore, GM = 3.936
Practice Questions on Geometric Mean
Problem 1. Find the geometric mean of the sequence 2, 4, 6, 8, 10, and 12.
Solution:
We have sequences 2, 4, 6, 8, 10, and 12.
Product of terms (P) = 2 × 4 × 6 × 8 × 10 × 12 = 46080
Number of terms (n) = 6
Using the Geometric Mean Formula we have,
GM = (P)1/n
= (46080)1/6
= 5.98
Problem 2. Find the geometric mean of the sequence 4, 8, 12, 16, 20.
Solution:
We have sequences 4, 8, 12, 16, and 20.
Product of terms (P) = 4 × 8 × 12 × 16 × 20 = 122880
Number of terms (n) = 5
Using the Geometric Mean Formula we have,
GM = (P)1/n
= (122880)1/5
= 10.42
Problem 3. Find the number of terms in a sequence if the geometric mean is 32, and the product of terms is 1024.
Solution:
We have,
Product of terms (P) = 1024
GM of terms = 32
Using the Geometric Mean Formula we have,
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
Problem 4. Find the number of terms in a sequence if the geometric mean is 4, and the product of terms is 65536.
Solution:
We have,
Product of terms (P) = 65536
GM of terms = 4
Using the Geometric Mean Formula we have,
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
Problem 5. Calculate the Geometric Mean Formula 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,
x1,x2,x3….xn−−−−−−−−−−−−√nx1,x2,x3….xnn
G.M.ofX=X¯¯¯¯=10×5×15×8×12−−−−−−−−−−−−−−−−−√5G.M.ofX=X¯=10×5×15×8×125
72000−−−−−√5720005
=
(72000)15(72000)15
= 9.36.
FAQs (Frequently Asked Questions)
1. What does the term "Geometric Mean Formula" mean?
The nth root of the product of all the data values is the Geometric Mean Formula of n number of data values. This average type is applied similarly to other techniques (like arithmetic mean).
2. What sets the Arithmetic Mean Apart from the Geometric Mean Formula?
The ratio of the total number of values to the sum of the supplied values is known as the arithmetic mean. In contrast, while calculating the Geometric Mean Formula, students multiply the “n” values before taking the product’s nth root.
3. Why Is Arithmetic Better Than Geometric Mean Formula?
The Geometric Mean Formula calculation is different from the arithmetic mean because it accounts for compounding that takes place over time. As a result, investors frequently believe that the geometric mean is a more accurate way to measure returns than the arithmetic mean.
4. Is it true that a Geometric Mean Formula is always a positive number?
The Geometric Mean Formula only holds for positive integers; it does not hold for negative numbers. It is typically used to indicate a group of numbers whose values are meant to be multiplied or are exponential, such as a group of growth rates. For instance, the growth in the global population or the interest rate on financial investment.
Calculating the Geometric Mean Formula is difficult for negative numbers like 0. However, there are several solutions to this problem, all of which require the translation or transformation of the negative numbers into a meaningfully positive similar value.
5. What do geometric applications mean?
The Geometric Mean Formula is used in many different industries and has several benefits. Application examples include Numerous value line indexes used by finance departments with the geometric mean as their base.
Additionally, stock indexes employ it. It is applied when calculating the portfolio’s yearly return. The average growth rates also called compounded annual growth rates, are computed in finance. It is used, among other things, in studies on bacterial growth and cell division.
6. What Is the Geometric Mean Formula of 2 and 8?
The Geometric Mean Formula of 2 and 8 can be calculated as
Let a = 2 and b = 8
Here, the number of terms, n = 2
If n =2, then the Geometric Mean Formula = √(ab)
Therefore, GM = √(2×8)
GM =√16 = 4
Therefore, the Geometric Mean Formula of 2 and 8 is 4.