Variance and Standard Deviation
Last Updated :
14 Apr, 2025
Variance and Standard Deviation are the important measures used in Mathematics and statistics to find meaning from a large set of data. The different formulas for Variance and Standard Deviation are highly used in mathematics to determine the trends of various values in mathematics. Variance is the measure of how the data points vary according to the mean, while standard deviation is the measure of the central tendency of the distribution of the data.
Variance and Standard DeviationThe major difference between variance and standard deviation is in their units of measurement. Standard deviation is measured in a unit similar to the units of the mean of data, whereas the variance is measured in squared units.
Variance
Variance is defined as "The measure of how far the set of data is dispersed from their mean value". Variance is represented with the symbol σ2. In other words, we can also say that the variance is the average of the squared difference from the mean.
Properties of Variance
Various properties of the Variance of the group of data are,
- As each term in the variance formula is firstly squared and then their mean is found, it is always a non-negative value, i.e. mean can be either positive or can be zero, but it can never be negative.
- Variance is always measured in squared units. For example, if we have to find the variance of the height of the students in a class, and if the height of the student is given in cm, then the variance is calculated in cm2.
There are two formulas for Variance, that are:
- Population Variance
- Sample Variance
The mathematical formula to find the variance of the given data is,
\bold{\sigma^2 = \sum_{i=1}^n{\frac{(x_i - \bar{x})^2}{n}}}
Where,
- σ2 is the Variance of the Population,
- N is the Number of Observation in the Population,
- Xi is the ith observation in the Population, and
- x̄ is the mean of the Population.
This formula is also called the Population variance formula as it is used for finding the variation in the population data.
Also, The other formula for finding the variance is the sample variance formula, which is discussed in the image.
Standard Deviation
How far our given set of data varies from. the mean of the data is measured in standard deviation. Thus, we define standard deviation as the "spread of the statistical data from the mean or average position". We denote the standard deviation of the data using the symbol σ.
We can also define the standard deviation as the square root of the variance.
Properties of Standard Deviation
Various properties of the Variance of the group of data are,
- Standard Deviation is the square root of the variance of the given data set. It is also called root mean square deviation.
- Standard Deviation is a non-negative quantity i.e., it always has positive values or zero values.
- If all the values in a data set are similar then the Standard Deviation has a value close to zero. Whereas if the values in a data set are very different from each other,, then standard deviation has a high positive value.
There are two formulas for the standard deviation listed as follows:
- Population Standard Deviation
- Sample Standard Deviation
The mathematical formula to find the standard deviation of the given data is,
s = \sqrt{\frac{\sum_{i=1}^n (x_i - x̄)^2}{n}}
Where,
- σis the Standard Deviation of the Population,
- N is the Number of Observation in the Population,
- Xi is the ith observation in the Population, and
- X̄ is the mean of the Population
This formula is also called the Population standard deviation formula as it is used for finding the standard deviation in the population data.
,i.e. sample variance formula is discussed in the image below,
Learn more about, Standard Deviation Formula
Relation between Standard Deviation and Variance
Variance and Standard deviation are the most common measure of the given set of data. They are used to find the deviation of the values from their mean value or the spread of all the values of the data set.
- Variance is defined as the average degree through which all the values of a given data set deviate from the mean value.
- Standard Deviation is the degree to which the values in a data set are spread out concerning, the mean value.
The relationship between Variance and Standard Deviation is discussed below.
Variance = (Standard Deviation)2
OR
√(Variance) = Standard Deviation
Variance vs Standard deviation
The differences between Standard Deviation and Variance are discussed in the table below.
Standard Deviation | Variance |
|---|
Standard Deviation is defined as the square root of
the variance. | Variance is defined as the average of the squared
differences from the mean. |
Standard deviation provides a measure of the
typical distance between data points and the mean. | Variance provides a measure of the average
squared distance between data points and the mean. |
| It is represented by the Greek symbol σ. | It is represented by a square of the Greek symbol sigma.
i.e. σ2. |
| It has the same unit as the data set. | Its unit is the square of the unit of the data set. |
| It represents the volatility in the market or given data set. | It represents the degree to which the average return
varies according to the long-term change in the market. |
Solved Questions on Variance and Standard Deviation
Question 1: Find the variance and standard deviation of all the possibilities of rolling a die.
Solution:
All possible outcomes of rolling a die are {1; 2; 3; 4; 5; 6}.
This data set has six values (n) = 6
Before finding the variance, we need to find the mean of the data set.
Mean, x̅ = (1+2+3+4+5+6)/6 = 3.5
We can put the value of data and mean in the formula to get;
σ2 = Σ (xi – x̅)2/n
⇒ σ2 = [(1-3.5)2 + (2-3.5)2 + (3-3.5)2 + (4-3.5)2 + (5-3.5)2 + (6-3.5)2]/6
⇒ σ2 = (6.25+2.25+0.25+0.25+2.25+6.25)/6
Variance (σ2) = 2.917
Now,
Standard Deviation (σ) = √ (σ2)
⇒ Standard Deviation (σ) = √(2.917)
⇒ Standard Deviation (σ) = 1.708
Question 2: Find the variance and standard deviation of all the even numbers less than 10. , Find the variance and standard deviation of all the even numbers less than 10.
Even Numbers less than 10 are {2, 4, 6, 8}
This data set has five values (n) = 4
Before finding the variance, we need to find the mean of the data set.
Mean, x̅ = (2+4+6+8)/4= 5
We can put the value of data and mean in the formula to get;
σ2 = Σ (xi – x̅)2/n
⇒ σ2 = [(2-5)2 + (4-5)2 + (6-5)2 + (8-5)2]/4
⇒ σ2 = (9 + 1 + 1 + 9 )/5 = 20/4
Variance (σ2) = 5
Now, Standard Deviation (σ) = √ (σ2)
⇒ Standard Deviation (σ) = √(5)
⇒ Standard Deviation (σ) = 2.238
Practice Problems - Variance and Standard Deviation
Try solving the given question to grab the idea of what we just learned :
Question 1: Find the variance and standard deviation of the following data
Question 2: Find the standard deviation of the following data:
Question 3: Find the variance of the first 69 natural numbers.
Answer 1 : Mean x̄ = 6
- Population Variance = 8.5
- Standard Deviation = √(8.5) = 2.91
Answer 2 : Mean x̄ = 12
- Population Variance = 48.5
- Thus, Standard Deviation = √48.5 = 6.96
Answer 3 : Variance = 396.67
People Also Read:
Similar Reads
CBSE Class 11 Maths Notes CBSE Class 11 Maths Revision Notes have been designed in the most basic and detailed format possible, covering nearly all domains such as differential calculus, arithmetic, trigonometry, and coordinate geometry. We know how hard it gets when you shift to an altogether new grade where subjects are no
15+ min read
Chapter 1: Sets
Representation of a SetSets are defined as collections of well-defined data. In Math, a Set is a tool that helps to classify and collect data belonging to the same category. Even though the elements used in sets are all different from each other, they are all similar as they belong to one group. For instance, a set of dif
8 min read
Types Of SetsIn mathematics, a set is defined as a well-defined collection of distinct elements that share a common property. These elements— like numbers, letters, or even other sets are listed in curly brackets "{ }" and represented by capital letters. For example, a set can include days of the week. The diffe
13 min read
Universal SetsUniversal Set is a set that has all the elements associated with a given set, without any repetition. Suppose we have two sets P = {1, 3, 5} and Q = {2, 4, 6} then the universal set of P and Q is U = {1, 2, 3, 4, 5, 6}. We generally use U to denote universal sets. Universal Set is a type of set that
6 min read
Venn DiagramVenn diagrams are visual tools used to show relationships between different sets. They use overlapping circles to represent how sets intersect, share elements, or stay separate. These diagrams help categorize items, making it easier to understand similarities and differences. In mathematics, Venn di
14 min read
Operations on SetsSets are fundamental in mathematics and are collections of distinct objects, considered as a whole. In this article, we will explore the basic operations you can perform on sets, such as union, intersection, difference, and complement. These operations help us understand how sets interact with each
15+ min read
Union of SetsUnion of two sets means finding a set containing all the values in both sets. It is denoted using the symbol '∪' and is read as the union. Example 1:If A = {1, 3. 5. 7} and B = {1, 2, 3} then A∪B is read as A union B and its value is,A∪B = {1, 2, 3, 5, 7}Example 2:If A = {1, 3. 5.7} and B = {2, 4} t
12 min read
Chapter 2: Relations & Functions
Cartesian Product of SetsThe term 'product' mathematically refers to the result obtained when two or more values are multiplied together. For example, 45 is the product of 9 and 5.To understand the Cartesian product of sets, one must first be familiar with basic set operations such as union and intersection, which are appli
7 min read
Relations and FunctionsIn mathematics, we often deal with sets of numbers or objects and the ways they are connected. Two important concepts that help us describe these connections are relations and functions.A relation is simply a connection between two sets of objects. Think of it as a rule that pairs elements from one
3 min read
Domain and Range of RelationsThe domain is the set of all possible input values (the "x" values), and the range is the set of all possible output values (the "y" values) in a relation.For any two non-empty sets A and B, we define the relation R as the subset of the Cartesian product of A × B where each member of set A is relate
8 min read
Piecewise FunctionPiecewise Function is a function that behaves differently for different types of input. As we know a function is a mathematical object which associates each input with exactly one output. For example: If a function takes on any input and gives the output as 3. It can be represented mathematically as
11 min read
Range of a FunctionFunctions in math can be thought of as vending machines. Given the money in the form of input, they give some cans or cookies in return. Similarly, functions take some input numbers and give us some output. It can be said that, in real life, Everything can be formulated and solved with the help of f
7 min read
Chapter 3: Trigonometric Functions
Chapter 4: Principle of Mathematical Induction
Chapter 5: Complex Numbers and Quadratic Equations
Complex NumbersComplex numbers are an essential concept in mathematics, extending the idea of numbers to include solutions for equations that don't have real solutions. Complex numbers have applications in many scientific research areas, signal processing, electromagnetism, fluid dynamics, quantum mechanics, and v
12 min read
Algebra of Real FunctionsThe algebra of real functions refers to the study and application of algebraic operations on functions that map real numbers to real numbers. A function can be thought of as a rule or set of rules which map an input to an output knows as its image. It is represented as x ⇢ Function ⇢ y. A real funct
5 min read
Algebraic Operations on Complex NumbersA complex number is a number that includes both a real and an imaginary part. It is written in the form:z = a + biWhere:a is the real part,b is the imaginary part,i is the imaginary unit, satisfying i2 = −1.Algebraic operations on complex numbers follow specific rules based on their real and imagina
7 min read
Polar Representation of Complex NumbersComplex numbers, which take the form z = x + yi, can also be represented in a way that highlights their geometric properties. This alternative representation is known as the polar form. The polar representation of a complex number expresses it in terms of its magnitude (modulus) and direction (argum
9 min read
Absolute Value of a Complex NumberThe absolute value (also called the modulus) of a complex number z = a + bi is its distance from the origin in the complex plane. The absolute value tells you how far a number is from zero, regardless of its direction (positive or negative).It is denoted as ∣z∣ and is given by the formula:|z| = \sqr
7 min read
Conjugate of Complex NumbersIn the world of mathematics, complex numbers are one of the most important discoveries by mathematicians as they help us solve many real-life problems in various fields such as the study of electromagnetic waves, engineering, and physics.The Conjugate of a Complex Number is also a complex number obt
6 min read
Imaginary NumbersImaginary numbers are numbers as the name suggests are the number that is not real numbers. All the numbers real and imaginary come under the categories of complex numbers. Imaginary numbers are very useful in solving quadratic equations and other equations whose solutions can not easily be found us
9 min read
Chapter 6: Linear Inequalities
Compound InequalitiesCompound Inequalities are the combination of two or more inequalities. These inequalities are combined using two conditions that are AND, and OR. These conditions have specific meanings and they are solved differently. The inequities in compound inequalities are individually solved using normal rule
10 min read
Algebraic Solutions of Linear Inequalities in One VariableA linear inequality is a mathematical expression involving an inequality symbol (<, >, ≤, or ≥) and a linear expression. Unlike linear equations, which give a specific solution, linear inequalities define a range of possible solutions.Example: 2x+3>5 In this case, the inequality indicates t
8 min read
Graphical Solution of Linear Inequalities in Two VariablesWe know how to formulate equations of different degree, and it is used a lot in real life, but the question arises, is it always possible to convert a situation into an equation? Sometimes we get statements like, the number of Covid cases per day in Delhi has reached more than 10,000. This phrase “L
8 min read
Solving Linear Inequalities Word ProblemsWe are well versed with equations in multiple variables. Linear Equations represent a point in a single dimension, a line in a two-dimensional, and a plane in a three-dimensional world. Solutions to linear inequalities represent a region of the Cartesian plane. It becomes essential for us to know ho
10 min read
Chapter 7: Permutations and Combinations
Fundamental Principle of CountingThe fundamental principle of counting is a basic concept used to determine the total number of possible outcomes in a situation where there are multiple independent events. It allows us to count a large number of possibilities without needing to list each one individually.For example, consider guess
11 min read
PermutationIn Mathematics, Permutation is defined as a mathematical concept that determines the number of possible arrangements for a specific set of elements. therefore, it plays a big role in computer science, cryptography, and operations research. For example, take a set {1, 2, 3}:All Permutations taking al
15+ min read
CombinationsCombination is a way of choosing items from a set, (unlike permutations) when the order of selection doesn't matter. In smaller cases, it's possible to count the number of combinations. Combination refers to the mixture of n things taken k at a time without repetition.Example: For set S = {a, b, c},
8 min read
Chapter 8: Binomial Theorem
Chapter 9: Sequences and Series
Sequences and SeriesA sequence is an ordered list of numbers following a specific rule. Each number in a sequence is called a "term." The order in which terms are arranged is crucial, as each term has a specific position, often denoted as an​, where n indicates the position in the sequence.For example:2, 5, 8, 11, 14,
10 min read
General and Middle Terms - Binomial Theorem - Class 11 MathsBinomial theorem or expansion describes the algebraic expansion of powers of a binomial. According to this theorem, it is possible to expand the polynomial "(a + b)n" into a sum involving terms of the form "axzyc", the exponents z and c are non-negative integers where z + c = n, and the coefficient
7 min read
Arithmetic SeriesAn arithmetic series is the sum of the terms of an arithmetic sequence, where an arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Or we can say that an arithmetic progression can be defined as a sequence of numbers in which for every pair of
5 min read
Arithmetic SequenceAn arithmetic sequence or progression is defined as a sequence of numbers in which the difference between one term and the next term remains constant.For example: the given below sequence has a common difference of 1.1 2 3 4 5 . . . n ⇑ ⇑ ⇑ ⇑ ⇑ . . . 1st 2nd 3rd 4th 5th . . . nth TermsThe Arithmetic
8 min read
Geometric Progression or GPGeometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio.For Example, the sequence given below forms a GP with a common ratio of 2 1 2 4 8 16 . . . n⇑ ⇑ ⇑ ⇑ ⇑ . . . 1st 2nd 3rd 4th 5th . . . nt
12 min read
Geometric SeriesIn a Geometric Series, every next term is the multiplication of its Previous term by a certain constant, and depending upon the value of the constant, the Series may increase or decrease.Geometric Sequence is given as: a, ar, ar2, ar3, ar4,..... {Infinite Sequence}a, ar, ar2, ar3, ar4, ....... arn {
3 min read
Arithmetic Progression and Geometric ProgressionArithmetic Progression and Geometric Progression: The word "sequence" in English means a collection of some numbers or objects in such a way that it has a first member, a second member, and so on. Sequences can be of anything, for example. - January, February, .... is the sequence of months in a yea
10 min read
Special Series in Maths - Sequences and Series | Class 11 MathsSpecial Series: A series can be defined as the sum of all the numbers of the given sequence. The sequences are finite as well as infinite. In the same way, the series can also be finite or infinite. For example, consider a sequence as 1, 3, 5, 7, … Then the series of these terms will be 1 + 3 + 5 +
10 min read