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Variance and Standard Deviation

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NCERT Mathematics Textbook for Class 11

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Variance and Standard Deviation
3-dimensional coordinate geometry
3D Geometry
A Line Segment
A Mind-Reading Game
Absolute maxima and Absolute minima
Addition and Subtraction of Algebraic Expressions
Addition of Integers
Addition of Numbers with Decimals
Addition of Vectors
Adjoint and Inverse of a Matrix
Adjoint of a Matrix
Age
Algebra
Algebra
Algebra
Algebra of Complex Numbers
Algebraic Expressions
Algebraic Identities
Algebraic Methods of Solving a Pair of Linear Equations
Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation
Altitudes Of A Triangle
Analytic Functions
Analytic functions
Analytical Geometry
Angle Between Curves
Angle Subtended by a Chord at a Point
Angle Subtended by an Arc of a Circle
Angle Sum Property Of A Triangle
Angle between Two Planes
Angle between a Line and a Plane
Angle between two lines
Angles
Angles – ‘Acute’, ‘Obtuse’ and ‘Reflex’
Angles – ‘Right’ and ‘Straight’
Application of Integrals
Application of calculus
Application of derivatives
Applications of Compound Interest Formula
Applications of Conics
Applications of Derivatives
Applications of Determinants and Matrices
Approximations
Area
Area Of A Parallelogram
Area between Two Curves
Area of Rhombus
Area of a Polygon
Area of a Triangle
Area of a Triangle - by Heron’s Formula
Area of the region bounded
Area under Simple Curves
Areas Related to Circle
Areas Related to Circles
Areas of Sector and Segment of a Circle
Arithmetic Mean
Arithmetic Operations
Arithmetic Progression
Arithmetic Progression and Quadratic Equations
Arithmetic Progression and Variance
Arithmetic Progressions
Arithmetic and Geometric Progressions
Average
Average Speed
Averages
Axiomatic Approach to Probability
Banking (Recurring Deposit Accounts)
Basic Algebra
Basic Geometrical Ideas
Basics of Accounting
Basics of Probability
Bayes' Theorem
Bernoulli Trials and Binomial Distribution
Binary Operations
Binary operations
Binomial Distribution
Binomial Expansion
Binomial Theorem for Positive Integral Indices
Binomial theorem
Boat and Stream
Boolean Algebra
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Calculus
Calculus
Cartesian Product of Sets
Cartesian System
Cauchy's Integral Theorem
Cauchy’s integral formula
Chain Rule
Chance and Probability
Charge Given On Borrowed Money Or Simple Interest
Checking For Parallel Lines
Circle
Circle Graph or Pie Chart
Circle and Parabola Geometry
Circles
Circles, Chords and Tangents
Circular Motion
Circular Permutation
Classification of Triangles
Co-ordinate Geometry
Coefficient of Variation
Collinearity of points
Combinations
Combinatorics
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Common Factors and Common Multiples
Comparing Decimals
Comparing Fractions
Comparing Numbers
Complement of a Set
Complex Analysis
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Complex Numbers and Quadratic Equations
Complex numbers
Composition of Functions and Invertible Function
Compound Interest
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Congruence
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Continuity
Continuity
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Continuity of a function
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Coordinate Geometry
Coordinate Geometry
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Core Mathematics
Counting functions
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Cyclic Quadrilaterals
Data Interpretation
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Decimals
Definite Integral
Definite Integral
Definite Integrals
Definite and indefinite integrals
Derivatives
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Derivatives of Functions in Parametric Forms
Determinant
Determinants
Determinants
Differentiability
Differential Calculus
Differential Equations
Differential Equations
Differential equations
Differentiation
Differentiation by taking log
Differentiation of Parametric Functions
Direct Proportion
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Distance Formula
Distance and angle measurement
Distance between Two Lines
Distance between parallel lines
Distance between point and line
Distance between points
Distance of a Point From a Line
Distance of a Point from a Plane
Divergence theorem
Divisibility Rules
Divisibility and Factors
Divisibility and Remainder
Division Of Decimal Numbers
Division Of Fractions
Division Of Integers
Division of Algebraic Expressions Continued (Polynomial ÷ Polynomial)
Domain of a Function
Domain of a Relation
Double and triple integrals
Eccentricity of a conic
Eigenvalues
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Eigenvectors
Elementary Mathematics
Elementary Operation (Transformation) of a Matrix
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Equal Chords and their Distances from the Centre
Equal Sets
Equation of Circle
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Equation of a Plane
Equations
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Euclid’s Definitions, Axioms and Postulates
Euler’s Theorem
Evaluation of Definite Integrals by Substitution
Evaluation of Limits
Evaluation of derivatives
Event
Exponential and Logarithmic Functions
Exponents
Exponents and Powers
Exterior Angle Of A Triangle And Its Property
Faces, Edges And Vertices
Factorisation
Factorisation of Polynomials
Factors and Multiples
Family of Lines
Financial Mathematics
Finding the Square of a Number
Finite and Infinite Sets
Fourier series
Fraction
Function
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Functions
Fundamental Theorem of Calculus
GST
General Equation of a Line
General and Particular Solutions of a Differential Equation
Geometric Progression
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Geometrical Meaning of the Zeroes of a Polynomial
Geometry
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Graphical Method of Analysis
Graphical Method of Solution of a Pair of Linear Equations
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Harmonic Progression
Harmonic Progressions
Height and Distance
Heights and Distances
Highest Common Factor
Horizontal and vertical lines
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Hyperbolic Functions
Ideal gas equation
Increasing and Decreasing Functions
Indefinite Integrals
Independent Events
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Inequalities
Installments
Integers
Integral Calculus
Integral Equations
Integrals of Some Particular Functions
Integrating Factor
Integration
Integration
Integration and Area Calculation
Integration and Trigonometry
Integration by Partial Fractions
Integration by Parts
Interest
Intersecting Cylinders
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Intersecting Lines and Non-intersecting Lines
Intersection of Planes
Introduction to Graphs
Introduction to Trigonometry
Introduction: Plane Figures And Solid Shapes
Inverse Proportion
Inverse Trigonometric Functions
Inverse Trigonometric Functions
Inverse Trigonometric Functions
Inverse of a Function
Inverse of a Matrix
Invertible Matrices
Irrational Numbers
Kinds of Quadrilaterals
Kinematics
Knowing our Numbers
LCM and HCF
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Large Numbers in Practice
Laurent's Series
Laws of Exponents
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Like Fractions
Limit and Continuity
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Limits and derivations
Line Symmetry And Rotational Symmetry
Line integrals and Green's theorem
Linear Algebra
Linear Algebra
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Matrix
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Maxima & Minima
Maxima and Minima
Mean
Mean Deviation
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Mean and Variance of Random variables
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Mean, median, mode and standard deviation
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Methods of Integration
Methods of Solving First Order, First Degree Differential Equations
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Mixtures & Alligations
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Mode of Grouped Data
Multiplication Of Decimal Numbers
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Multiplication Theorem on Probability
Multiplication of a Vector by a Scalar
Multiplying a Monomial by a Monomial
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Multiplying a Polynomial by a Polynomial
Natural Number
Nature of Roots
Need For Rational Numbers
Nets For Building 3-D Shapes
Normal distribution
Number System
Number Systems
Number of Tangents from a Point on a Circle
Number of integer solutions
Numerical Methods
Odd one Out
Operations On Rational Numbers
Operations on Real Numbers
Operations on Relations
Operations on Sets
Optimization
Order and Degree of Differential Equation
Order and Degree of a Differential Equation
Order of Matrix
Organisation of Data
Orthogonal Complement
Orthogonal Polynomials
Pair of Linear Equations in Two Variables
Pairs Of Lines
Parabola
Parallel Lines
Partial Differential Equations
Partial Fractions
Partnership
Partnership Accounts
Percentage
Percentage – Another Way Of Comparing Quantities
Percentages
Perimeter
Permutation and Combination
Permutations
Permutations and Combinations
Perpendicular Lines
Perpendicular from the Centre to a Chord
Pictograph
Pipe and Cistern
Plane
Plane Figures
Playing with Numbers
Points
Poisson distribution
Polar Coordinates
Pole and Polar
Polygons
Polynomials
Polynomials in One Variable
Population Growth Calculation
Power Series
Power series solutions for ordinary points
Prices Related To An Item Or Buying And Selling
Prime and Composite Numbers
Principle of Mathematical Induction
Probability
Probability
Probability
Probability Distribution
Probability and Statistics
Probability distributions
Probability of Random Experiments
Problem on Trains
Product of Matrices
Product of Two Vectors
Profit and Loss
Properties Of Addition And Subtraction Of Integers
Properties Of Division Of Integers
Properties Of Multiplication Of Integers
Properties of Determinants
Properties of Inverse Trigonometric Functions
Properties of Lines
Properties of Rational Numbers
Properties of Triangles
Properties of Vectors
Properties of a Parallelogram
Proportion
Propositional Logic
Pythagorean Theorem
Quadratic Equation
Quadratic Equations
Quadratic Polynomials
Quadrilaterals
Race
Random Experiments
Random Variables
Random Variables and its Probability Distributions
Range of a Function
Rank of a Matrix
Rate of Change of Quantities
Rate of Change of Velocity
Ratio
Ratio and Proportion
Rational Equations
Rational Number
Rational Numbers
Rational or Irrational Numbers
Real Numbers
Real Numbers and their Decimal Expansions
Recalling Ratios and Percentages
Rectangle
Reducing Equations to Simpler Form
Regression analysis
Related Angles
Related Rates and Surface Area
Relations
Relations
Relations and Functions
Relations and Geometry
Relations and functions
Relationship between Zeroes and Coefficients of a Polynomial
Revisiting Irrational Numbers
Right-Angled Triangles And Pythagoras Property
Rotation of Axes
Rotational Symmetry
SI & CI
Second Order Derivative
Second Order Derivatives
Section Formula
Sequence and Series
Sequence and Series
Sequence and series
Sequences and Series
Sequences and Series
Sequences and Series of real numbers
Series
Set Theory
Set Theory & Algebra
Set Theory and Geometry
Set Theory and Logic
Set Theory and types of Sets
Sets
Sets
Sets and Relations
Sets and their Representations
Shares and Dividends
Shortest Distance Between Skew Lines
Significant figures
Similarity of Triangles
Simple Interest
Simple Ratios
Simplification
Simplifying the Dividend
Slope of a line
Solution of Differential Equations
Solution of a Linear Equation
Solution of a Quadratic Equation by Factorisation
Solutions of Differential Equations
Solutions of Linear Algebraic Equations
Solving Equations having the Variable on both Sides
Some Applications
Some Applications of Trigonometry
Some More Criteria for Congruence of Triangles
Some More Interesting Patterns
Some Problems on HCF and LCM
Some Properties of Definite Integrals
Some Properties of a Triangle
Some Special Parallelograms
Spanning Trees
Special Functions
Special Terms of Binomial Expansion
Speed, Time and Distance
Square
Square Roots
Square Roots of Decimals
Square and Square Roots
Standard deviation
Statements
Statistics
Statistics
Statistics and Probability
Straight lines
Strictly increasing or strictly decreasing function
Subsets
Subtraction of Decimals
Subtraction of Integers with the help of a Number Line
Sum Of The Lengths Of Two Sides Of A Triangle
Sum of First n Terms of an AP
Sum of the Measures of the Exterior Angles of a Polygon
Surds and Indices
Surface Area of Cube, Cuboid and Cylinder
Surface Area of a Combination of Solids
Surface Area of a Right Circular Cone
Surface Area of a Sphere
Surface Areas and Volumes
Surface integral
Symmetric and Skew Symmetric Matrices
System of Linear Equations
Systems of Linear Equations
Tangent to a Circle
Tangents and Normals
Tax
Tests for Divisibility of Numbers
The Empty Set
The Fundamental Theorem of Arithmetic
The Idea of a Variable
The Mid-point Theorem
The Modulus and the Conjugate of a Complex Number
Three Dimensional Geometry
Three Dimensional Geometry
Time Series Analysis
Time Speed Distance
Time and Work
Time, Speed and Distance
Topology
Transpose of a Matrix
Transpose of a Matrix
Triangles
Trigonometric Equations
Trigonometric Functions
Trigonometric Functions of Sum and Difference of Two Angles
Trigonometric Identities
Trigonometric Identities
Trigonometric Ratios
Trigonometric Ratios of Some Specific Angles
Trigonometry
Trigonometry
Types of Functions
Types of Matrices
Types of Relations
Understanding Quadrilaterals
Union and Intersection of Sets
Unit Digit
Unit Vectors
Unitary Method
Universal Set
Use Of Bar Graphs With A Different Purpose
Use Of Percentages
Use of Exponents to Express Small Numbers in Standard Form
Using Decimals
Variance
Various Forms of the Equation of a Line
Vector Algebra
Vector Algebra
Vector Calculus
Vector basics
Vector space
Vectors
Vectors
Vectors
Vectors in plane and space
Venn Diagrams
Venn Diagrams
Viewing Different Sections Of A Solid
Volume and Capacity
Volume of Cube, Cuboid and Cylinder
Volume of a Combination of Solids
Volume of a Right Circular Cone
Volume of a Sphere
What is Factorisation?
Whole Numbers
angle between two lines
applications of integrals
argand plane
axiomatic approach to probability
binomial distribution
binomial expansion formula
cartesian products of sets
chords
circle
complex numbers
composite of functions
conjugate of a complex number
coordinates of a point in space
distance
distance and displacement
distance between two points
frequency distributions
fundamental principle of counting
general and middle terms
general equation of a line
geometric progression
graphical solution of linear inequalities in two variables
homogeneous differential equation
inequalities
integral
introduction to three dimensional geometry
limits and derivatives
limits of trigonometric functions
linear inequalities
linear inequalities in one variable
lpp graphical method
mathematical reasoning
matrix transformation
measurement of angles
nth Term of an AP
permutations and combinations
probability meaning
range
relationship between a.m. and g.m.
sections of a cone
sequences
sets
solution of system of linear inequalities in two variables
speed and velocity
types of differential equations
types of functions
types of matrices
types of relations
types of sets
types of vectors
validating statements
x-intercepts and y-intercepts

List of top Mathematics Questions on Variance and Standard Deviation

An analysis of monthly wages paid to workers in two firms A and B, belonging to the same industry, gives the following results:

-	Firm A	Firm B
No. of wage earners	586	648
Mean of monthly wages	Rs 5253	Rs 5253
Variance of the distribution of wages	100	121

(i) Which firm A or B pays larger amount as monthly wages?

(ii) Which firm, A or B, shows greater variability in individual wages?

From the data given below state which group is more variable, A or B?

X	35	54	52	53	56	58	52	50	51	49
Y	108	107	105	105	106	107	104	103	104	101

From the data given below state which group is more variable, A or B?

Marks	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Group A	9	17	32	33	40	10	9
Group B 1	10	20	30	25	43	15	7

Find the mean, variance and standard deviation using short-cut method.

Height in cms	70-75	75-80	80-85	85-90	90-95	95-100	100-105	105-110	110-115
No. of children	3	4	7	7	15	9	6	6	3

Find the mean and variance for the following frequency distribution.

Classes	0-30	30-60	60-90	90-120	120-150	150-180	180-210
Frequencies	2	3	5	10	3	5	2

Find the mean and standard deviation using short-cut method.

$x_i$	60	61	62	63	64	65	66	67	68
$f_i$	2	1	12	29	25	12	10	4	5

Find the mean and variance for the data

$x_i$	62	93	97	98	102	104	106
$f_i$	3	2	3	2	6	3	3

Find the mean and variance for the data

$x_i$	6	10	14	18	24	28	30
$f_i$	2	4	7	12	8	4	3

Find the mean and variance for the first 10 multiples of 3.

Find the mean and variance for the first n natural numbers.

The variance of first 25 natural numbers is

The variance of first n natural numbers is :

The variance of the numbers $2, 3, 11 $ and $x$ is $\frac{49}{4}$. Find the value of $x$.

The observation which occur most frequently is known as :

The mean of two samples of size 200 and 300 were found to be 25, 10 respectively. Their standard deviations were 3 and 4 respectively. The variance of combined sample of size 500 is

Suppose a population A has 100 observations 101, 102, ............., 200 and another population B has 100 obsevrations 151, 152, ................ 250. If $V_A$ and $V_B$ represent the variances of the two populations, respectively then $\frac{V_A}{V_B}$ is

Standard deviation for first $10$ natural numbers is

\(x_i\)	60	61	62	63	64	65	66	67	68
\(f_i\)	2	1	12	29	25	12	10	4	5

\(x_i\)	6	10	14	18	24	28	30
\(f_i\)	2	4	7	12	8	4	3