Topics
Sets, Relations, and Functions
- Sets and Their Representations
- Classification of Sets
- Empty Set (Null or Void Set)
- Finite and Infinite Sets
- Equal Sets
- Subsets
- Power Set
- Universal Set
- Venn Diagrams
- Union of Sets
- Intersection of Sets
- Disjoint Sets
- Difference of Sets
- Complement of a Set
- Practical Problems on Union and Intersection of Two Sets
- Algebra of Real Functions
- Algebraic Operations on Functions
Complex Numbers and Quadratic Equations
- Introduction of Complex Number
- Concept of Complex Numbers
- Complex Numbers as Ordered Pairs of Reals
- Representation of Complex Numbers
- Argand Plane and Polar Representation
- Algebraic Operations of Complex Numbers
- Properties of Conjugate, Modulus and Argument (or Amplitude) of Complex Numbers
- Square Root of a Complex Number
- Triangle Inequality
- Integral Powers of Iota
- Rotational Theorem of Complex Number
- Cube Root of Unity
- Geometry of Complex Numbers
- Demoiver's Theorem
- Powers of Complex Numbers
- Sum and Product of Root
- Nature of Roots of a Quadratic Equation
- Formation of a Quadratic Equation with Given Roots
- Quadratic Formula (Shreedharacharya's Rule)
- Condition for Common Roots
- Maximum and Minimum Value of Quadratic Equation
- Quadratic Expression in Two Variables
- Solution of Quadratic Inequalities
Matrices and Determinants
- Concept of Matrices
- Operations on Matrices>Scalar Multiplication
- Operations on Matrices> Addition and Subtraction of Matrices
- Operations on Matrices>Scalar Multiplication
- Operations on Matrices> Matrix Multiplication
- Types of Matrices
- Concept of Determinant
- Properties of Determinants
- Evaluation of Determinants
- Application of Determinants to Coordinate Geometry
- Adjoint and Inverse of a Matrix
- Inverse of a Square Matrix by the Adjoint Method
- Test of Consistency
- Application of Determinants to Coordinate Geometry
- Transpose of a Matrix
- Symmetric and Skew Symmetric Matrices
- Multiplication of Two Determinants
- Minors and Co-factors
- Some Special Cases of Matrix
- Rank of a Matrix
Permutations and Combinations
- Permutations
- Fundamental Principles of Counting
- Combination
- Meaning of P (n,r) and C (n,r)
- Simple Applications of Permutations and Combinations
- Factorials
- Division and Distribution of Objects
- Dearrangement Theorem
- Sum of Numbers
- Important Result About Point
Mathematical Inductions
- Mathematical Induction
- Principle of Mathematical Induction
- Motivation
- Simple Applications of Mathematical Induction
Binomial Theorem and Its Simple Applications
- Introduction of Binomial Theorem
- Binomial Theorem for Positive Integral Indices
- General and Middle Terms
- Properties of Binomial Coefficient with Simple Application
- Expansion of Binomial
- Coefficient of Any Power of 'X'
- Greatest Term
- Independent Term
- Particular Term from End in Binomial Expansion
- Greatest Binomial Coefficients
- Number of Terms in the Expansion of (x + y + z)n
- Multinomial Theorem
- Infinite Series
- Binomial Theorem for Any Index (Without Proof)
Sequence and Series
- Sequence, Series, and Progression
- Relation Between Arithmetic Mean (A.M.), Geometric Mean (G.M.), Harmonic Mean (H.M.)
- Arithmetico-geometric Sequence
- Arithmetic Progression (A.P.)
- Geometric Progression (G. P.)
- Harmonic Progression (H. P.)
- Insertion of Arithmetic
- Geometric Mean
- Sum to' n' Terms of a Geometric Progression
- Some Special Sequences
Limit, Continuity, and Differentiability
- Concept of Limits
- Limits by Factorisation, Substitution and Rationalisation
- Algebra of Limits
- Limits of Polynomials and Rational Functions
- Limits of Logarithmic Functions
- Limits of Exponential Functions
- Limits of Trigonometric Functions
- Inverse Functions
- Graphs of Simple Functions
- Continuous and Discontinuous Functions
- Concept of Differentiability
- Differentiation of the Sum, Difference, Product, and Quotient of Two Functions
- Derivatives of Composite Functions
- Derivative of Inverse Trigonometric Function
- Derivative of Inverse Trigonometric Function
- Exponential and Logarithmic Functions
- Exponential and Logarithmic Functions
- Derivatives of Composite Functions
- Differentiation of Implicit Functions
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Mean Value Theorem
- Simple Problems on Applications of Derivatives
- Rate of Change of Quantities
- Increasing and Decreasing Functions
- Maxima and Minima
- Tangents and Normals
- Limits Using L-hospital's Rule
- Evaluation of Limits
- Infinite Series
- Successive Differentiation
- nth Derivative of Standard Functions
- Algebra of Derivative (Leibnitz or Product Rule)
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Approximations
Integral Calculas
- Integration
- Integration as an Inverse Process of Differentiation
- Fundamental Integrals Involving Algebraic Functions
- Fundamental Integrals Involving Trigonometric Functions
- Fundamental Integrals Involving Exponential Functions
- Fundamental Integrals Involving Logarithms Functions
- Methods of Integration> Integration by Substitution
- Methods of Integration> Integration by Parts
- Methods of Integration> Integration Using Partial Fraction
- Methods of Integration>Integration Using Trigonometric Identities
- Integrals of Some Particular Functions
- Definite Integral as the Limit of a Sum
- Fundamental Theorem of Integral Calculus
- Properties of Definite Integrals
- Evaluation of Definite Integrals
- Area Bounded by Two Curves
- Area Under Simple Curves
- Integration of Some Special Irrational Algebraic Functions
- Summation of Series by Integration
Diffrential Equations
- Introduction to Ordinary Differential Equations
- Formation of Ordinary Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Solution of a Differential Equation
- Solutions of Linear Differential Equation
- Methods of Solving First Order, First Degree Differential Equations
- Forms of Solving Differential Equations> Variable Separable
- Forms of Solving Differential Equations> Homogeneous Differential Equations
- Forms of Solving Differential Equations>Linear Differential Equations
- Solution by Inspection Method
Co-ordinate Geometry
- Brief Review of Cartesian System of Rectanglar Co-ordinates
- Distance Formula
- Section Formula in Coordinate Geometry
- Locus
- Translation of Axes
- Concept of Slope (or, gradient)
- Parallel and Perpendicular Lines
- Intercepts on Axes
- Various Forms of the Equation of a Line
- Intersection of Two Lines
- Angle Between Two Lines
- Conditions for Concurrence of Three Lines
- Distance in Lines (Point & Parallel Lines)
- Equations of Line in Different Forms
- Equations of Internal and External by Sectors of Angles Between Two Lines Co-ordinate of the Centroid, Orthocentre, and Circumcentre of a Triangle
- Equations of Internal and External by Sectors of Angles Between Two Lines Co-ordinate of the Centroid, Orthocentre, and Circumcentre of a Triangle
- Equation of Family of Lines Passing Through the Point of Intersection of Two Lines
- Equations of a Circle in Standard Form
- Equations of a Circle in General Form
- Equation of a Circle When the Endpoints of a Diameter Are Given
- Point of Intersection of a Line and a Circle
- Equation of Tangent and Condition of Tangency
- Fundamentals of Conic Sections
- Parabola and its types
- Hyperbola and its Types
- Ellipse and its Types
- Condition for Y = mx + c to Be a Tangent and Point(s) of Tangency
- Results of Triangle
- Various Forms of Equation of a Line
- Shortest Distance Between Two Lines
- Bisector of the Angle Between the Two Lines
- Distance in Lines (Point & Parallel Lines)
- Foot of the Perpendicular
- Position of a Point with Respect to a Line
- Pedal Points
- Pair of Straight Lines
- Advanced Concept of Circle
- Standard Equation of a Circle
Three Dimensional Geometry
- Foundation of Coordinate Geometry
- Coordinates of a Point in Space
- Distance Between Two Points
- Section Formula in Coordinate Geometry
- Direction Ratios, Direction Cosine & Direction Angles in Vector
- Direction Cosines and Direction Ratios of a Line
- The Angle Between Two Intersecting Lines
- Skew Lines
- Shortest Distance Between Two Lines
- Equations of Line in Different Forms
- Intersection of the Line and Plane
- Coplanarity of Two Lines
- Angle Between Two Lines
- Projection of a Point on a Line
- Projection of a Line Segment Joining Two Points
- Equation of a Straight Line in Cartesian and Vector Form
- Concept of Slope (or, gradient)
- Distance in Lines (Point & Parallel Lines)
- Shortest Distance Between Two Lines
- Different Forms of Equation of a Plane
- Angle Between Two Planes
- Angle Between Line and a Plane
- Distance Between Two Parallel Planes
- Position of Point and Line wrt a Plane
- Projection of a Line on a Plane
Vector Algebra
- Basic Concepts of Vector Algebra
- Vector Analysis
- Vector Operations>Addition and Subtraction of Vectors
- Components of Vector in Algebra
- Multiplication of Vectors>Scalar Product(Dot Product)
- Scalar Triple Product
- Vector Triple Product
- Basic Concepts of Vector Algebra
- Section Formula in Coordinate Geometry
- Linear Dependent and Independent Vectors
- Position Vector of a Point in a Space
- Modulus of a Vector
- Collinearity of Three Points
- Coplanarity of Three Vectors and Four Points
- Vector Inequality
- Product of Two Vectors > Scalar (Dot) Product
- Projection of a Vector Along Any Other Vector
- Area of a Parallelogram
- Geometric Interpretation of the Area of a Triangle
Statistics and Probability
- Measures of Discretion
- Arithmetic Mean
- Mean of Grouped Data
- Basic Concept of Median
- Basic Concept of Mode
- Standard Deviation
- Variance
- Mean Deviation
- Geometric Mean
- Harmonic Mean (H.M.)
- Coefficient of Variation
- Addition Theorem of Probability
- Multiplication Theorem of Probability
- Bayes’ Theorem
- Probability using Binomial Distribution
- Concept of Probability
- Elementary Types of Events and Properties of Probability
- Odds in Favour and Against
- Boole's Inequality
- Demorgan's Law
- Independent Events
- Conditional Probability
- Probability Distribution of Discrete Random Variables
- Poisson Distribution
Trigonometry
- Trigonometric Identities (Square Relations)
- Trigonometric Equations
- Trigonometric Functions
- Properties of Inverse Trigonometric Functions > Self-adjusting Property
- Angles of Elevation and Depression
- Circular System
- Trigonometric Ratios
- Domain and Range of Trigonometric Functions
- Trigonometric Functions of Allied Angels
- Conditional Trigonometric Identities
- Greatest and Least Value of Trigonometric Expressions
- Solution of Trigonometric Equations (Solution in the Specified Range)
- Intervals for Inverse Trigonometric Functions
- Infinite Series of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions > Negative Argument Property
- Properties of Inverse Trigonometric Functions > Complementary Property
- Properties of Inverse Trigonometric Functions > Conversion Property
- Properties of Inverse Trigonometric Functions > Addition & Subtraction Formula for Inverse Tangent
- Properties of Inverse Trigonometric Functions > Double-angle Property
- Properties of Inverse Trigonometric Functions > Triple-angle Property
- Properties of Inverse Trigonometric Functions > Addition–Subtraction Formula for Inverse Sine & Cosine
Mathematical Reasoning
- Mathematical Reasoning
- Introduction of Validating Statements
- Mathematically Acceptable Statements
- Statements and Truth Values in Mathematical Logic
- Tautology, Contradiction, and Contingency
- Logical Connective
- Truth Tables
- Logical Equivalance
- Quantifier, Quantified and Duality Statements in Logic
- Converse, Inverse and Contrapositive of the Conditional Staternent
- Negative of a Compound Statement
- Algebra of Statements
Linear Inequality
- Linear Inequations
- Method of Solving a Linear Inequality
- System of Inequations
- Inequalities of Various Functions
Properties of Triangles
- Properties of Triangle
- Solutions of Triangle>Polar Co-Ordinates
- Geometrical Concepts Related to a Circle
- Advanced Regular Polygon
- Angles of Elevation and Depression
Estimated time: 20 minutes
- Definition: Median
- Formula: Odd Number of Observations
- Formula: Even Number of Observations
- Example 1
- Example 2
- Example 3
- Key Points Summary
CISCE: Class 6
Definition: Median
Median is the value of the middle-most observation(s). The median is a measure of central tendency which gives the value of the middle-most observation in the data.
CISCE: Class 6
Formula: Odd Number of Observations
If the number of data points (n) is odd, the median is,
Median = `((n+1)/2)^(th)` term
CISCE: Class 6
Formula: Even Number of Observations
If n is even, the median is the average of the values at positions
Median = Average of `(n/2)^(th)` and `(n/2+1)^(th)` values
CISCE: Class 6
Example 1
Question: Find the median of:
(i) 9, 7, 6, 14, 10, 4 and 11; (ii) 2, 5, 9, 4, 12, 3, 7, 4, 10 and 7.
Solution:
(i) 9, 7, 6, 14, 10, 4 and 11;
Steps
-
Arranged in ascending order.
4 6 7 9 10 11 14 -
The middle term(s) are directly identified by counting.
4 6 7 9 10 11 14
Clearly, middle term is 9; therefore, the median = 9
(ii) 2, 5, 9, 4, 12, 3, 7, 4, 10 and 7
Steps
-
Arranged in ascending order.
2 3 4 4 5 7 7 9 10 12 -
The middle term(s) are directly identified by counting.
2 3 4 4 5 7 7 9 10 12 - Number of data = 10, which is even
The average of two middle values (for even)|
⇒ The two middle data = 5 and 7
∴ Median = Average of 5 and 7 = `"5 + 7"/"2"` = 6
CISCE: Class 6
Example 2
Question: The following are scores obtained by 11 players in a cricket match 7, 21, 45, 12, 56, 35, 25, 0, 58, 66, 29. Find the median score.
Solution:
Let us arrange the values in ascending order.
0,7,12,21,25,29,35,45,56,58,66
The number of values = 11 which is odd
Median = `((n+1)/2)^(th)` term
Median = `((11+1)/2)^(th)` value
= `(12/2)^(th)` value = 6th value = 29
CISCE: Class 6
Example 3
Example 8.9 For the following ungrouped data: 10, 17, 16, 21, 13, 18, 12, 10, 19, 22. Find the median.
Solution:
Arrange the values in ascending order.
10, 10, 12, 13, 16, 17, 18, 19, 21, 22.
The number of values = 10
Median = Average of `(n/2)^(th)` and `(n/2+1)^(th)` values
Median = Average of `(10/2)^(th)` and `(10/2+1)^(th)` values
= Average of 5th and 6th values
= `(16 + 17)/2 = 33/2 = 16.5`
CISCE: Class 6
Key Points Summary
Definition: The middle value in an ordered set of data.
Essential Rule: Always sort the data (ascending or descending) first.
Outlier Resistant: The median is not affected by extreme values (outliers), unlike the mean.
Odd vs. Even:
- Odd (n): The median is the single middle data point.
- Even (n): The median is the average of the two middle data points.
Example Question 1
Find the median of the data: 24, 36, 46, 17, 18, 25, 35
We arrange the data in ascending order, we get 17, 18, 24, 25, 35, 36, 46 Median is the middle observation.
Therefore 25 is the median.
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Related QuestionsVIEW ALL [8]
The median of the following incomplete table is 92. Find the missing frequencies:
| C.I. | 30 – 50 | 50 – 70 | 70 – 90 | 90 – 110 | 110 – 130 | 130 – 150 | Total |
| f | 6 | ? | 18 | 20 | ? | 10 | 80 |
Ten students appeared for a test in Mathematics and Statistics and they obtained the marks as follows:
| Sr. No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Marks in Mathematics | 42 | 38 | 36 | 32 | 23 | 25 | 35 | 37 | 25 | 23 |
| Marks in Statistics | 22 | 26 | 29 | 34 | 50 | 45 | 23 | 28 | 32 | 36 |
If the median will be the criteria, in which subject, the level of knowledge of the students is higher?
From the following distribution, determine median graphically.
| Daily wages (in ₹) | No. of employees |
| Above 300 | 520 |
| Above 400 | 470 |
| Above 500 | 399 |
| Above 600 | 210 |
| Above 700 | 105 |
| Above 800 | 45 |
| Above 900 | 7 |
The following is the data of pocket expenditure per week of 50 students in a class. It is known that the median of the distribution is ₹120. Find the missing frequencies.
| Expenditure per week (in ₹) |
0 – 50 | 50 – 100 | 100 – 150 | 150 –200 | 200 –250 |
| No. of students | 7 | ? | 15 | ? | 3 |
Find the missing frequency given that the median of the distribution is 1504.
| Life in hours | No. of bulbs |
| 950 – 1150 | 20 |
| 1150 – 1350 | 43 |
| 1350 – 1550 | 100 |
| 1550 – 1750 | - |
| 1750 – 1950 | 23 |
| 1950 – 2150 | 13 |
