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.
Similar Reads
Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields
5 min read
Basic Arithmetic
What are Numbers?Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world.Numbers are like tools that help us:Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show or
15+ min read
Arithmetic OperationsArithmetic Operations are the basic mathematical operations—Addition, Subtraction, Multiplication, and Division—used for calculations. These operations form the foundation of mathematics and are essential in daily life, such as sharing items, calculating bills, solving time and work problems, and in
9 min read
Fractions - Definition, Types and ExamplesFractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of a numerator (the top number), indicating how many parts are considered, and a denominator (the bottom number), showing the total number of equal parts the whole is divided into. For E
7 min read
What are Decimals?Decimals are numbers that use a decimal point to separate the whole number part from the fractional part. This system helps represent values between whole numbers, making it easier to express and measure smaller quantities. Each digit after the decimal point represents a specific place value, like t
10 min read
ExponentsExponents are a way to show that a number (base) is multiplied by itself many times. It's written as a small number (called the exponent) to the top right of the base number.Think of exponents as a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing t
9 min read
PercentageIn mathematics, a percentage is a figure or ratio that signifies a fraction out of 100, i.e., A fraction whose denominator is 100 is called a Percent. In all the fractions where the denominator is 100, we can remove the denominator and put the % sign.For example, the fraction 23/100 can be written a
5 min read
Algebra
Variable in MathsA variable is like a placeholder or a box that can hold different values. In math, it's often represented by a letter, like x or y. The value of a variable can change depending on the situation. For example, if you have the equation y = 2x + 3, the value of y depends on the value of x. So, if you ch
5 min read
Polynomials| Degree | Types | Properties and ExamplesPolynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.A polynomial can have one
9 min read
CoefficientA coefficient is a number that multiplies a variable in a mathematical expression. It tells you how much of that variable you have. For example, in the term 5x, the coefficient is 5 — it means 5 times the variable x.Coefficients can be positive, negative, or zero. Algebraic EquationA coefficient is
8 min read
Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari
14 min read
Properties of Algebraic OperationsAlgebraic operations are mathematical processes that involve the manipulation of numbers, variables, and symbols to produce new results or expressions. The basic algebraic operations are:Addition ( + ): The process of combining two or more numbers to get a sum. For example, 3 + 5 = 8.Subtraction (−)
3 min read
Geometry
Lines and AnglesLines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang
9 min read
Geometric Shapes in MathsGeometric shapes are mathematical figures that represent the forms of objects in the real world. These shapes have defined boundaries, angles, and surfaces, and are fundamental to understanding geometry. Geometric shapes can be categorized into two main types based on their dimensions:2D Shapes (Two
2 min read
Area and Perimeter of Shapes | Formula and ExamplesArea and Perimeter are the two fundamental properties related to 2-dimensional shapes. Defining the size of the shape and the length of its boundary. By learning about the areas of 2D shapes, we can easily determine the surface areas of 3D bodies and the perimeter helps us to calculate the length of
10 min read
Surface Areas and VolumesSurface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces.Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate the
10 min read
Points, Lines and PlanesPoints, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using coordinates. We define a line as a geometrical figure that is extended in both direction
14 min read
Coordinate Axes and Coordinate Planes in 3D spaceIn a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su
6 min read
Trigonometry & Vector Algebra
Trigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give
4 min read
Trigonometric Equations | Definition, Examples & How to SolveTrigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value. The goal is to find the values of the variable (usually an angle) that satisfy the equation.For example, a simple trigonometric equation might
9 min read
Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t
10 min read
Trigonometric FunctionsTrigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in
6 min read
Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function
11 min read
Inverse Trigonometric IdentitiesInverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. These functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inv
9 min read
Calculus
Introduction to Differential CalculusDifferential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.Some of the prerequisites for Differential Calculus include:Independent and Dependent Varia
6 min read
Limits in CalculusIn mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp
12 min read
Continuity of FunctionsContinuity of functions is an important unit of Calculus as it forms the base and it helps us further to prove whether a function is differentiable or not. A continuous function is a function which when drawn on a paper does not have a break. The continuity can also be proved using the concept of li
13 min read
DifferentiationDifferentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables.In simple terms, it is a way of finding how things change. Imagine you're driving a car and looking at how your spee
2 min read
Differentiability of a Function | Class 12 MathsContinuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f
11 min read
IntegrationIntegration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.Imagine you have a car driving along a road, and its speed changes over time. At some moments, it's going faster; at other moments, it's slower. If y
3 min read
Probability and Statistics
Basic Concepts of ProbabilityProbability is defined as the likelihood of the occurrence of any event. It is expressed as a number between 0 and 1, where 0 is the probability of an impossible event and 1 is the probability of a sure event.Concepts of Probability are used in various real life scenarios : Stock Market : Investors
7 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Descriptive StatisticStatistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat
5 min read
What is Inferential Statistics?Inferential statistics is an important tool that allows us to make predictions and conclusions about a population based on sample data. Unlike descriptive statistics, which only summarize data, inferential statistics let us test hypotheses, make estimates, and measure the uncertainty about our predi
7 min read
Measures of Central Tendency in StatisticsCentral tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence o
11 min read
Set TheorySet theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple,
3 min read
Practice