The Geometric Mean (GM) is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values. In mathematics and statistics, measures of central tendencies describe the summary of whole data set values. The most important measures of central tendencies are mean, median, mode, and range. Among these, the mean of the data set provides the overall idea of the data. The mean defines the average of numbers in the data set. The different types of mean are Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM).
In this lesson, let us discuss the definition, formula, properties, and applications of geometric mean and also the relation between AM, GM, and HM with solved examples in the end.
|1.||What is Geometric Mean?|
|2.||Geometric Mean Formula|
|3.||Difference Between Arithmetic Mean and Geometric Mean|
|4.||Relation Between AM, GM, and HM|
|5.||FAQs on Geometric Mean|
What is Geometric Mean?
The Geometric Mean (GM) is the average value or mean which signifies the central tendency of the set of numbers by taking the root of the product of their values. Basically, we multiply the 'n' values altogether and take out the nth root of the numbers, where n is the total number of values. For example: for a given set of two numbers such as 8 and 1, the geometric mean is equal to √(8×1) = √8 = 2√2.
Thus, the geometric mean is also defined as the nth root of the product of n numbers. Note that this is different from the arithmetic mean. In the arithmetic mean, data values are added and then divided by the total number of values. But in geometric mean, the given data values are multiplied, and then you take the root with the radical index for the final product of data values. For example, if you have two data values, take the square root, or if you have three data values, then take the cube root, or else if you have four data values, then take the 4th root, and so on.
Geometric Mean Formula
The Geometric Mean (G.M) of a data set containing n observations is the nth root of the product of the values. Consider, if x₁, x₂, ..., xₙ are the observations, for which we aim to calculate the geometric mean. The formula to calculate the geometric mean is given below:
GM = √x₁ · x₂ · ... · xₙ
GM = (x₁ · x₂ · ... · xₙ)1/n
It is also represented as:
G.M. = √∏ᵢ₌₁ⁿ xᵢ
Taking logarithm on both sides,
log GM = log (x₁ · x₂ · ... · xₙ)1/n
= (1/n) log (x₁ · x₂ · ... · xₙ)
= (1/n) [log x₁ +log x₂ + ... + log xₙ]
= (∑ log xᵢ) / n
Therefore, geometric mean, GM = Antilog (∑ log xᵢ) / n
This is an alternative formula of GM (that represents the same formula as in the image).
Note: For any grouped data, G.M can be calculated using:
GM = Antilog (∑ f log xᵢ) / n, where n = f₁ + f₂ + ... + fₙ.
Difference Between Arithmetic Mean and Geometric Mean
Here is a table representing the difference between arithmetic and geometric mean.
|Arithmetic Mean||Geometric Mean|
|In the arithmetic mean, data values are added and then divided by the total number of values.||The geometric mean can be found by multiplying all the numbers in the given data set and take the nth root for the obtained result.|
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 set, 4, 10, 16, 24
Here n = 4
Therefore, the G.M = (4 ×10 ×16 × 24)1/4
G.M = 11.13
Relation Between AM, GM, and HM
Before we learn the relation between the AM, GM and HM, we need to know the formulas of all these 3 types of mean. Assume that “a” and “b” are the two number and the number of values = 2, then
AM = (a+b)/2
⇒ 1/AM = 2/(a+b) ……. (I)
GM = √(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), we get
HM = GM2 /AM
⇒GM2 = AM × HM
GM = √[ AM × HM]
Hence, the relation between AM, GM, and HM is GM2 = AM × HM. Therefore the square of the geometric mean is equal to the product of the arithmetic mean and the harmonic mean.
Let us also see why the G.M for the given data set is always less than the arithmetic mean for the data set. Let A and G be A.M. and G.M.
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
Geometric mean has many advantages over arithmetic mean and it is used in many fields. Some of the applications are as follows:
- It is used in stock indexes because many of the value line indexes which are used by financial departments make use of G.M.
- To calculate the annual return on the investment portfolio.
- The geometric mean is used in finance to find the average growth rates which are also known as the compounded annual growth rate (CAGR).
- Geometric Mean is also used in biological studies like cell division and bacterial growth rate etc.
Tips & Tricks on Geometric Mean:
Some of the tips and tricks on G.M are as follows:
- The G.M for the given data set is always less than the arithmetic mean for the data set.
- If each value in the data set is substituted by the G.M, then the product of the values remains unchanged.
- The ratio of the corresponding observations of the G.M in two series is equal to the ratio of their geometric means.
- The products of the corresponding items of the G.M in the two series are equal to the product of their geometric mean.
Given below is the list of topics that are closely connected to the Geometric Mean. These topics will also give you a glimpse of how such concepts are covered in Cuemath.
Geometric Mean Examples
Example 1: Find the geometric mean of 1,3,5,7,9
The GM is given as (x1 × x2 × x3...× xn)1/n
= (1 × 3 × 5 × 7 × 9)1/5
Answer: Therefore, GM = 3.936
Example 2: Find the geometric mean of the following data.
Weight of Table (Kg) Log x 45 1.653 60 1.778 48 1.681 100 2.000 65 1.813 Total 8.925
Solution: Here n = 5
GM = Antilog (∑ log xᵢ) / n
=Antilog 8.925 / 5
Note: Try finding the GM using the first column of the table and applying the formula (x₁ · x₂ · ... · xₙ)1/ and see whether you get the same answer.
Answer: Therefore the GM of the given data is 60.95
Example 3: Find the geometric mean of the following grouped data for the frequency distribution of weights.
Weights of Cellphones (g) No of Cellphones (f) 60-80 22 80-100 38 100-120 45 120-140 35 140-160 20 Total 160
Let us use log table for calculating the log values of "mid x".
Weights of Cellphones (g) No of Cellphones (f) Mid x Log x f log x 60-80 22 70 1.845 40.59 80-100 38 90 1.954 74.25 100-120 45 110 2.041 91.85 120-140 35 130 2.114 73.99 140-160 20 150 2.716 43.52 Total 160 324.2
From the given data, n = 160
We know that the G.M for the grouped data is
GM = Antilog (∑ f log xᵢ) / n
GM = Antilog (324.2 / 160)
GM = Antilog (2.02625)
Therefore, the GM =106.23
Answer: Therefore the geometric mean of the given data is 106.23
FAQs on Geometric Mean
What is the Definition of Geometric Mean?
The geometric mean of n number of data values is the nth root of the product of all the data values. This is a kind of average used like other means (like arithmetic mean).
What Is the Difference Between the Arithmetic Mean and Geometric Mean?
The arithmetic mean is defined as the ratio of the sum of given values to the total number of values. Whereas in geometric mean, we multiply the “n” number of values and then take the nth root of the product.
What is Geometric Mean in Statistics?
What Is the Geometric Mean of 2 and 8?
The geometric mean 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 formula for geometric mean = √(ab)
Therefore, GM = √(2×8)
GM =√16 = 4
Therefore, the geometric mean of 2 and 8 is 4.
What Is Geometric Mean Formula?
Why Is Geometric Mean Better Than Arithmetic?
The geometric mean differs from the arithmetic average, or arithmetic mean, in how it is calculated because it takes into account the compounding that occurs from period to period. Because of this, investors usually consider the geometric mean a more accurate measure of returns than the arithmetic mean.