• Home
  • Courses
  • Online Test
  • Contact
    Have any question?
    +91-8287971571
    contact@dronstudy.com
    Login
    DronStudy
    • Home
    • Courses
    • Online Test
    • Contact

      Class 11 MATHS – JEE

      • Home
      • All courses
      • Class 11
      • Class 11 MATHS – JEE
      CoursesClass 11MathsClass 11 MATHS – JEE
      • 1.Sets, Relation and Functions
        12
        • Lecture1.1
          Introduction to sets, Description of sets 32 min
        • Lecture1.2
          Types of Sets, Subsets 39 min
        • Lecture1.3
          Intervals, Venn Diagrams, Operations on Sets 37 min
        • Lecture1.4
          Laws of Algebra of Sets 26 min
        • Lecture1.5
          Introduction to sets and its types, operations of sets, Venn Diagrams 28 min
        • Lecture1.6
          Functions and its Types 38 min
        • Lecture1.7
          Functions Types 17 min
        • Lecture1.8
          Cartesian Product of Sets, Relation, Domain and Range 40 min
        • Lecture1.9
          Sum Related to Relations 04 min
        • Lecture1.10
          Sums Related to Relations, Domain and Range 22 min
        • Lecture1.11
          Chapter Notes – Sets, Relation and Functions
        • Lecture1.12
          NCERT Solutions – Sets, Relation and Functions
      • 2.Trigonometric Functions
        28
        • Lecture2.1
          Introduction, Some Identities and Some Sums 16 min
        • Lecture2.2
          Some Sums Related to Trigonometry Identities, trigonometry Functions Table and Its Quadrants 35 min
        • Lecture2.3
          NCERT Sums Ex.3.3 (Q.1-5)Based on Trigometry table and Their Quadrants, Trigonometry Identities of Sum and Diff. of two Angles 21 min
        • Lecture2.4
          NCERT Sums Ex-3.2 Based on Trigonometry Function of Lower & Higher Angles 22 min
        • Lecture2.5
          NCERT Sums Ex-3.3 (Q.6 – 10) Based on Radian Angles 11 min
        • Lecture2.6
          NCERT Sums Ex-3.3 (Q.11-13)Based on Trigonometry Identities 16 min
        • Lecture2.7
          NCERT Sums Ex-3.3 (Q. 14)Based on Trigonometry Identities 14 min
        • Lecture2.8
          NCERT Sums Ex-3.3 (Q.16) Based on Trigonometry Identities 05 min
        • Lecture2.9
          NCERT Sums Ex-3.3 (Q.17 -21) Based on Trigonometry Identities 12 min
        • Lecture2.10
          NCERT Sums Ex-3.4 (Q. 1 – 9), Trigonometry Equation 25 min
        • Lecture2.11
          Sums Based on Trigonometry Equations 24 min
        • Lecture2.12
          Sums Based on Trigonometry Equations 11 min
        • Lecture2.13
          Sums Based on Trigonometry Equations 11 min
        • Lecture2.14
          Sums Based on Trigonometry Equations 17 min
        • Lecture2.15
          Equations Having two Variable Angle which satisfy both equations 10 min
        • Lecture2.16
          Trigonometrical Identities-Some important relations and Its related Sums 16 min
        • Lecture2.17
          Sums Related to Trigonometrical Identities 18 min
        • Lecture2.18
          Properties of Triangles and Solution of Triangles-Sine formula, Napier Analogy and Sums 17 min
        • Lecture2.19
          Relation Between Degree and Radian, Quadrant and NCERT Sum Ex.3.1, 3.2 41 min
        • Lecture2.20
          Trigonometric Functions Table 09 min
        • Lecture2.21
          Some Trigonometric Identities and its related Sums 42 min
        • Lecture2.22
          Sums Related to Trigonometrical Identities 19 min
        • Lecture2.23
          Sums Related to Trigonometrical Identities 41 min
        • Lecture2.24
          Sums Related to Trigonometrical Identities 23 min
        • Lecture2.25
          Trigonometry Equations 44 min
        • Lecture2.26
          Sum Based on Trigonometry Equations 07 min
        • Lecture2.27
          Sums Based on Trigonometry function of Lower Angle 03 min
        • Lecture2.28
          Chapter Notes – Trigonometric Functions
      • 3.Mathematical Induction
        5
        • Lecture3.1
          Introduction to PMI 25 min
        • Lecture3.2
          NCERT Solution of EX- 4.1 14 min
        • Lecture3.3
          NCERT Solution of EX- 4.1 22 min
        • Lecture3.4
          NCERT Solution of EX- 4.1 13 min
        • Lecture3.5
          Chapter Notes – Mathematical Induction
      • 4.Complex Numbers and Quadratic Equation
        15
        • Lecture4.1
          Introduction, Nature of Roots, Numbers, Introduction of i 27 min
        • Lecture4.2
          Sum Related to Relations, Real and Imaginary part of C-N, Conjugate of a C-N 33 min
        • Lecture4.3
          Absolute value or Modulus of a C-N and Related Sums 29 min
        • Lecture4.4
          Sums Related To Multiplicative Inverse 29 min
        • Lecture4.5
          Polar Form of a C-N 32 min
        • Lecture4.6
          Sums Related To Polar Form 32 min
        • Lecture4.7
          Square Roots of C-N and its Related Sums 28 min
        • Lecture4.8
          De Moivris Theorem and its related Sums 31 min
        • Lecture4.9
          Introduction, Nature of Roots, Numbers, Introduction of i and its Sums, Real and Imaginary Part of C-N 35 min
        • Lecture4.10
          Sums Related to Real and Imaginary Part of C-N and Operations on C-N 13 min
        • Lecture4.11
          Sums Related To Multiplicative Inverse 07 min
        • Lecture4.12
          Sums Related To Multiplicative Inverse and Modulus and Argument of a C-N 33 min
        • Lecture4.13
          Polar form of a C-N, Nature of Roots 38 min
        • Lecture4.14
          Sums Based on Roots of Quadratic Equations, Sums of Polar form 10 min
        • Lecture4.15
          Chapter Notes – Complex Numbers and Quadratic Equation
      • 5.Linear Inequalities
        4
        • Lecture5.1
          Introduction, Solve some Linear Inequalities and its Graph 42 min
        • Lecture5.2
          Solve some Linear Inequalities and its Graph 12 min
        • Lecture5.3
          Solve some Linear Inequalities and its Graph and Introduction-Permutations and Combinations 35 min
        • Lecture5.4
          Chapter Notes – Linear Inequalities
      • 6.Permutations and Combinations
        5
        • Lecture6.1
          NCERT Sums Ex-7.3, Equation 41 min
        • Lecture6.2
          NCERT Sums Ex-7.3, Equation 02 min
        • Lecture6.3
          Combination and NCERT Sums Ex-7.4 40 min
        • Lecture6.4
          NCERT Sums Ex-7.1 & 7.2 22 min
        • Lecture6.5
          Chapter Notes – Permutations and Combinations
      • 7.Binomial Theorem
        19
        • Lecture7.1
          Introduction to Binomial Theorem 21 min
        • Lecture7.2
          Binomial General Expansion and Their Derivations and its Related Sums 22 min
        • Lecture7.3
          Pascal’s Triangle Theorem, Addition of Two Expansion, NCERT Sums Ex-8.1 26 min
        • Lecture7.4
          Sums of Miscellaneous Exercise and Ex-8.1, Finding the Any Term from nth Term 42 min
        • Lecture7.5
          NCERT Sums Ex-8.1 14 min
        • Lecture7.6
          NCERT Sums Ex-8.1 04 min
        • Lecture7.7
          NCERT Sums Ex-8.2, Middle Term 21 min
        • Lecture7.8
          NCERT Sums Ex-8.2, Middle Term Related Sums 08 min
        • Lecture7.9
          To Find the Coefficient of X^r in the Expansion of (X+A)^n, NCERT Sums Ex-8.2 and Miscellaneous Ex. 40 min
        • Lecture7.10
          NCERT Sums Ex-8.2 10 min
        • Lecture7.11
          To Find the Sum of the Coefficients in the Expansion of (1+x)^n and its Related Sums 27 min
        • Lecture7.12
          Sums Related to Binomials Coefficients 24 min
        • Lecture7.13
          Binomial Theorem for any Index and its Related Sums 27 min
        • Lecture7.14
          Introduction to Binomial Theorem, General Term in the Expansion of (x+a)^n. 39 min
        • Lecture7.15
          NCERT Sums Ex-8.1 & 8.2, Pascals’ Triangle, pth Term from End 24 min
        • Lecture7.16
          Sums related to Finding the Coefficient, NCERT Sums Ex-8.2, Middle Term 40 min
        • Lecture7.17
          Sums Related to Middle Term 17 min
        • Lecture7.18
          Sums Related to Coefficient of the Any Term 31 min
        • Lecture7.19
          Chapter Notes – Binomial Theorem
      • 8.Sequences and Series
        14
        • Lecture8.1
          Introduction, A.P., nth Term and Sum of nth Term, P Arithmetic Mean B/w a and b, Sum Based on Fibonacci Sequence 27 min
        • Lecture8.2
          NCERT Sums Ex-9.2 37 min
        • Lecture8.3
          NCERT Sums Ex-9.2 18 min
        • Lecture8.4
          NCERT Sums Ex-9.2, Geometric Progression -Introduction, nth term, NCERT Sums Ex-9.3 39 min
        • Lecture8.5
          NCERT Sums Ex-9.3 16 min
        • Lecture8.6
          Sum of n term of G.P., NCERT Sums Ex-9.3 40 min
        • Lecture8.7
          NCERT Sums Ex-9.3 08 min
        • Lecture8.8
          NCERT Sums Ex-9.3, Insert P Geometrical Mean B/w a and b 36 min
        • Lecture8.9
          NCERT Sum Ex-9.3 17 min
        • Lecture8.10
          NCERT Sum Ex-9.3 09 min
        • Lecture8.11
          Some Special Series, NCERT Sum Ex-9.4 36 min
        • Lecture8.12
          NCERT Sum Ex-9.4 02 min
        • Lecture8.13
          NCERT Sum Ex-9.4 18 min
        • Lecture8.14
          Chapter Notes – Sequences and Series
      • 9.Properties of Triangles
        2
        • Lecture9.1
          Sine 7 Cosine Rule, Projection Formulae, Napier’s Analogy, Incircle, Some Sums 41 min
        • Lecture9.2
          Angle of Elevations and Depression. and Its Related Sums 13 min
      • 10.Straight Lines
        30
        • Lecture10.1
          Introduction, Equation of Line, Slope or Gradient of a line 24 min
        • Lecture10.2
          Sums Related to Finding the Slope, Angle Between two Lines 22 min
        • Lecture10.3
          Cases for Angle B/w two Lines, Different forms of Line Equation 23 min
        • Lecture10.4
          Sums Related Finding the Equation of Line 27 min
        • Lecture10.5
          Sums based on Previous Concepts of Straight line 32 min
        • Lecture10.6
          Parametric Form of a Straight Line 16 min
        • Lecture10.7
          Sums Related to Parametric Form of a Straight Line 16 min
        • Lecture10.8
          Sums Based on Concurrent of lines, Angle b/w Two Lines 45 min
        • Lecture10.9
          Different condition for Angle b/w two lines 04 min
        • Lecture10.10
          Sums Based on Angle b/w Two Lines 36 min
        • Lecture10.11
          Equation of Straight line Passes Through a Point and Make an Angle with Another Line 09 min
        • Lecture10.12
          Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 15 min
        • Lecture10.13
          Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 17 min
        • Lecture10.14
          Finding the Distance of a point from the line 34 min
        • Lecture10.15
          Sum Based on Finding the Distance of a point from the line and B/w Two Parallel Lines 33 min
        • Lecture10.16
          Sums Based on Find the Equation of Bisector of Angle Between two intersecting Lines 44 min
        • Lecture10.17
          Sums Based on Find the Equation of Bisector of Angle Between two intersecting Lines 02 min
        • Lecture10.18
          Introduction, Distance B/w Two Points, Slope, Equation of Line 32 min
        • Lecture10.19
          NCERT Sums Ex-10.1 43 min
        • Lecture10.20
          NCERT Sums Ex-10.1 29 min
        • Lecture10.21
          NCERT Sums Ex-10.1 & 10.2 43 min
        • Lecture10.22
          NCERT Sums Ex-10.2 30 min
        • Lecture10.23
          NCERT Sums Ex-10.2 41 min
        • Lecture10.24
          NCERT Sums Ex-10.2 & 10.3 21 min
        • Lecture10.25
          NCERT Sums Ex- 10.3 (Reduce the Equation into intercept Form, Normal form) 42 min
        • Lecture10.26
          NCERT Sums Ex-10.3 21 min
        • Lecture10.27
          NCERT Sums Ex-10.3 (Equation of Parallel line, Perpendicular Line of given line, Sums Based of Angle B/w Two Lines) 42 min
        • Lecture10.28
          NCERT Sums Ex-10.3 09 min
        • Lecture10.29
          NCERT Sums Ex-10.3 26 min
        • Lecture10.30
          Chapter Notes – Straight Lines
      • 11.Conic Sections
        21
        • Lecture11.1
          Introduction, General Equation of second Degree, Parabola, Sums based on Finding Equation of Parabola 41 min
        • Lecture11.2
          Sums Based on Equation of Parabola, Four Forms of Parabola-Form (i) 30 min
        • Lecture11.3
          Sums Based on Four Forms of Parabola-Form (i) 32 min
        • Lecture11.4
          Four Forms of Parabola-Form (ii), (iii) (iv) 13 min
        • Lecture11.5
          Sums Based on Four forms of Parabola 18 min
        • Lecture11.6
          Position of a Point with Respect to Parabola and its Sums 43 min
        • Lecture11.7
          Circles-Introduction, Different Cases for Circle Equations, NCERT Sums Ex-11.1 16 min
        • Lecture11.8
          NCERT Sums Ex-11.1 40 min
        • Lecture11.9
          Circle Important Point Revise, Intersection of Axes, NCERT Sums Ex-11.1 11 min
        • Lecture11.10
          NCERT Sums Ex-11.1 44 min
        • Lecture11.11
          Parabola- Introduction, General Equation , Sums, Some Important Concepts for Parabola 12 min
        • Lecture11.12
          Different Form of Parabola, NCERT Sum Ex-11.2 13 min
        • Lecture11.13
          NCERT Sum Ex-11.2 34 min
        • Lecture11.14
          Ellipse-Introduction, General Equation, NCERT Sums Ex-11.3 36 min
        • Lecture11.15
          NCERT Sums Ex-11.3 02 min
        • Lecture11.16
          NCERT Sums Ex-11.3 23 min
        • Lecture11.17
          Hyperbola-Introduction, NCERT Sums Ex-11.4 12 min
        • Lecture11.18
          NCERT Sums Ex-11.4 25 min
        • Lecture11.19
          Chapter Notes – Conic Sections Circles
        • Lecture11.20
          Chapter Notes – Conic Sections Ellipse
        • Lecture11.21
          Chapter Notes – Conic Sections Parabola
      • 12.Coordinate Geometry
        8
        • Lecture12.1
          Introduction to Rectangular Cartesian Coordinate Geometry (2D), Distance b/w two points 23 min
        • Lecture12.2
          Cartesian Coordinate of points 32 min
        • Lecture12.3
          Questions rel to cartesian coordinate of points 25 min
        • Lecture12.4
          Section Formula – Case 1, Case 2 24 min
        • Lecture12.5
          Problem Solving 26 min
        • Lecture12.6
          Centeroid, Incenter, Circumcenter of a triangle 30 min
        • Lecture12.7
          Locus Problems 17 min
        • Lecture12.8
          Problem Solving 21 min
      • 13.Three Dimensional Geometry
        3
        • Lecture13.1
          Introduction to 3D 18 min
        • Lecture13.2
          Numerical problems 14 min
        • Lecture13.3
          Chapter Notes – Three Dimensional Geometry
      • 14.Limits And Derivatives
        12
        • Lecture14.1
          Introduction to limits 42 min
        • Lecture14.2
          EX-13.1 16 min
        • Lecture14.3
          Questions based on algebra of limits 41 min
        • Lecture14.4
          Limits of a polynomial 12 min
        • Lecture14.5
          rational function 37 min
        • Lecture14.6
          trigo function 21 min
        • Lecture14.7
          Introduction to Derivatives 37 min
        • Lecture14.8
          Ex-13.2 22 min
        • Lecture14.9
          Algebra of derivatives 38 min
        • Lecture14.10
          Derivative of polynomial 13 min
        • Lecture14.11
          trigo function 11 min
        • Lecture14.12
          Chapter Notes – Limits And Derivatives
      • 15.Mathematical Reasoning
        3
        • Lecture15.1
          What is statement ? Special word and phrases, negation of statement , Compound statement , and & or in compound statement , truth table Solving the problems of Ex- 14.1 , 14.2 25 min
        • Lecture15.2
          Solving Ex-14.3, Ex-14.4, Implications, Validating statements, Ex-14.5, Direct method 24 min
        • Lecture15.3
          Chapter Notes – Mathematical Reasoning
      • 16.Statistics
        5
        • Lecture16.1
          Mean, Median, Mode, Range, Mean Deviation Solution of Ex-15.1 27 min
        • Lecture16.2
          Mean Deviation about Mean & Median, Ex-15.2, Mean and Variance, Standard deviation 35 min
        • Lecture16.3
          Ex-15.2 , Variance and Standard deviation 09 min
        • Lecture16.4
          Ex-15.3, Analysis of frequency distribution, comparison of two frequency distribution with same mean 23 min
        • Lecture16.5
          Chapter Notes – Statistics
      • 17.Probability
        3
        • Lecture17.1
          Outcomes & sample space, Ex. 16.3 19 min
        • Lecture17.2
          Ex.16.3, Probability of an event, Algebra of event 38 min
        • Lecture17.3
          Chapter Notes – Probability
      • 18.Binary Number
        2
        • Lecture18.1
          Binary numbers, Conversion of Binary to Decimal and Decimal to binary 45 min
        • Lecture18.2
          Addition, Subtraction, Multiplication, Division 02 min

        Chapter Notes – Statistics

        Statistics is the science of collection, organization, presentation, analysis, and interpretation of the numerical data.

        Useful Terms

        1.Limit of the Class

        The starting and end values of each class are called the Lower and Upper limits.

        2.Class Interval

        The difference between the upper and lower boundaries of a class is called class interval or size of the class.

        3.Primary and Secondary Data

        The data collected by the investigator himself is known as the primary data, while the data collected by a person, other than the investigator is known as the secondary data.

        4.Variable or Variate

        A characteristic that varies in magnitude from observation to observation. e.g., weight, height, income, age, etc, are variables.

        5.Frequency

        The number of times an observation occurs in the given data is called the frequency of the observation.

        6.Discrete Frequency Distribution

        A frequency distribution is called a discrete frequency distribution if data are presented in such a way that exact measurements of the units are clearly shown.

         

        7.Continuous Frequency Distribution

        A frequency distribution in which data are arranged in class groups which are not exactly measurable.

        8.Cumulative Frequency Distribution

        Suppose the frequencies are grouped frequencies or class frequencies. If however, the frequency of the first class is added to that of the second and this sum is added to that of the third and so on, then the frequencies, so obtained are known as cumulative frequencies (cf).

        Graphical Representation of Frequency Distributions

        1. Histogram To draw the histogram of a given continuous frequency distribution, we first mark off all the class intervals along X-axis on a suitable scale. On each of these class intervals on the horizontal axis, we erect (vertical) a rectangle whose height is proportional to the frequency of that particular class, so that the area of the rectangle is proportional to the frequency of the class.

        If however, the classes are of unequal width, then the height of the rectangles will be proportional to the ratio of the frequencies to the width of the classes.

        1. Bar Diagrams In bar diagrams, only the length of the bars are taken into consideration. To draw a bar diagram, we first mark equal lengths for the different classes on the axis, i.e., X- axis.

        On each of these lengths on the horizontal axis, we erect (vertical) a rectangle whose heights is proportional to the frequency of the class.

        1. Pie Diagrams Pie diagrams are used to represent a relative frequency distribution. A pie diagram consists of a circle divided into as many sectors as there are classes in a frequency distribution.

        The area of each sector is proportional to the relative frequency of the class. Now, we make angles at the centre proportional to the relative frequencies.

        And in order to get the angles of the desired sectors, we divide 360° in the proportion of the various relative frequencies. That is,

        Central angle = [Frequency x 360° / Total frequency]

        1. Frequency Polygon To draw the frequency polygon of an ungrouped frequency distribution, we plot the points with abscissae as the variate values and the ordinate as the corresponding frequencies. These plotted points are joined by straight lines to obtain the frequency polygon.

        1. Cumulative Frequency Curve (Ogive) Ogive is the graphical representation of the cumulative frequency distribution. There are two methods of constructing an Ogive, viz (i) the ‘less than’ method (ii) the ‘more than’ method.

         

        Measures of Central Tendency

        Generally, the average value of distribution in the middle part of the distribution, such type of values are known as measures of central tendency.

        The following are the five measures of central tendency

        1. Arithmetic Mean
        2. Geometric Mean
        3. Harmonic Mean
        4. Median
        5. Mode

        Arithmetic Mean

        The arithmetic mean is the amount secured by dividing the sum of values of the items in a series by the number.

        Arithmetic Mean for Unclassified Data

        If n numbers, x1, x2, x3,….., xn, then their arithmetic mean

        Arithmetic Mean for Frequency Distribution

        Let f1, f2 , fn be corresponding frequencies of x1, x2,…, xn. Then,

        Arithmetic Mean for Classified Data

        Class mark of the class interval a-b, then x = a + b / 2

        For a classified data, we take the class marks x1, x2,…, xn of the classes as variables, then arithmetic mean

        Step Deviation Method

        where, A1 = assumed mean ui = xi – A1 / h

        fi = frequency

        h = width of interval

        Combined Mean

        If x1, x2,…, xr be r groups of observations, then arithmetic mean of the combined group x is called the combined mean of the observation

        A = n1 A1 + n2A2 +….+ nrAr / n1 + n2 +…+ nr Ar = AM of collection xr

        nr = total frequency of the collection xr

        Weighted Arithmetic Mean

        If w be the weight of the variable x, then the weighted AM Aw = Σ wx / Σ w

        Shortcut Method

        Aw = Aw‘ + Σ wd / Σ w, Aw‘ = assumed mean

        Σ wd = sum of products of the deviations and weight

        Properties of Arithmetic Mean

        1. Mean is dependent of change of origin and change of scale.
        2. Algebraic sum of the deviations of a set of values from their arithmetic mean is zero.
        3. The sum of the squares of the deviations of a set of values is minimum when taken about mean.

        Geometric Mean

        If x1, x2,…, xn be n values of the variable, then G = n√x1, x2,…, xn

        or G = antilog [log x1 + log x2 + … + log xn / n]

        For Frequency Distribution

         

        Harmonic Mean (HM)

        The harmonic mean of n items x1, x2,…, xn is defined as

        If their corresponding frequencies f1, f2,…, fn respectively, then

        Median

        The median of a distribution is the value of the middle variable when the variables are arranged in ascending or descending order.

        Median (Md) is an average of position of the numbers.

        Median for Simple Distribution

        Firstly, arrange the terms in ascending or descending order and then find the number of terms n.

        1. If n is odd, then (n + 1 / 2)th term is the median.
        2. If n is even, then there are two middle terms namely (n / 2)th and (n / 2 + 1)th terms. Hence, Median = Mean of (n / 2)th and (n / 2 + 1)th terms.

        Median for Unclassified Frequency Distribution

        1. First find N / 2, where N = Σ fi.
        2. Find the cumulative frequency of each value of the variable and take value of the variable which is equal to or just greater than N / 2
        3. This value of the variable is the median.
        4. Median for Classified Data (Median Class)

        If in a continuous distribution, the total frequency be N, then the class whose cumulative frequency is either equal to N / 2 or is just greater than N / 2 is called median class.

        For a continuous distribution, median Md = l + ((N / 2 – C) / f) * h

        where, l = lower limit of the median class f = frequency of the median class

        N = total frequency = Σ f

        C = cumulative frequency of the class just before the median class h = length of the median class

        Quartiles

        The median divides the distribution in two equal parts. The distribution can similarly be divided in more equal parts (four, five, six etc.). Quartiles for a continuous distribution is given by

        Q1 = l + ((N / 4 – C) / f) * h Where, N = total frequency

        l = lower limit of the first quartile class f = frequency of the first quartile class

        C = the cumulative frequency corresponding to the class just before the first quartile class h = the length of the first quartile class

        Similarly, Q3 = l + ((3N / 4 – C) / f) * h

        where symbols have the same meaning as above only taking third quartile in place of first quartile.

        Mode

        The mode (Mo) of a distribution is the value at the point about which the items tend to be most heavily concentrated. It is generally the value of the variable which appears to occur most frequently in the distribution.

        1. Mode for a Raw Data

        Mode from the following numbers of a variable 70, 80, 90, 96, 70, 96, 96, 90 is 96 as 96 occurs maximum number of times.

        2 For Classified Distribution

        The class having the maximum frequency is called the modal class and the middle point of the modal class is called the crude mode.

        The class just before the modal class is called pre-modal class and the class after the modal class is called the post-modal class.

        Mode for Classified Data (Continuous Distribution) Mo = l + (f0 – f1 / 2 f0 – f1 – f2) x h

        Where, 1 = lower limit of the modal class f0 = frequency of the modal class

        f1 = frequency of the pre-modal class f2 = frequency of the post-modal class h = length of the class interval

        Relation between Mean, Median and Mode

        1. Mean — Mode = 3 (Mean — Median)
        2. Mode = 3 Median — 2 Mean

        Symmetrical and Skew distribution

        A distribution is symmetric, if the same number of frequencies is found to be distributed at the same linear teance on either side of the mode. The frequency curve is bell shaped and A = Md = Mo

        In anti-symmetric or skew distribution, the variation does not have symmetry.

        1. If the frequencies increases sharply at beginning and decreases slowly after modal value, then it is called positive skewness and A > Md > Mo.

        1. If the frequencies increases slowly and decreases sharply after modal value, the skewness is said to be negative and A < Md < Mo.

        Measure of Dispersion

        The degree to which numerical data tend to spread about an average value is called the dispersion of the data. The four measure of dispersion are

        1. Range
        2. Mean deviation
        3. Standard deviation
        4. Square Deviation

         

        Range

        The difference between the highest and the lowest element of a data called its range. i.e., Range = Xmax – Xmin

        ∴ The coefficient of range = Xmax – Xmin / Xmax + Xmin

        It is widely used in statistical series relating to quality control in production.

        1. Inter quartile range = Q3 — Q1
        2. Semi-inter quartile range (Quartile deviation)

        ∴ Q D = Q3 — Q1 / 2

        and coefficient of quartile deviation = Q3 — Q1 / Q3 + Q1

        1. QD = 2 / 3 SD

        Mean Deviation (MD)

        The arithmetic mean of the absolute deviations of the values of the variable from a measure of their Average (mean, median, mode) is called Mean Deviation (MD). It is denoted by δ.

        1. For simple (discrete) distribution δ = Σ |x – z| / n

        where, n = number of terms, z = A or Md or Mo

        1. For unclassified frequency distribution δ = Σ f |x – z| / Σ f
        2. For classified distribution δ = Σ f |x – z| / Σ f

        Here, x is for class mark of the interval.

        1. MD = 4 / 5 SD
        2. Average (Mean or Median or Mode) = Mean deviation from the average / Average Note The mean deviation is the least when measured from the median.

        Coefficient of Mean Deviation

        It is the ratio of MD and the mean from which the deviation is measured. Thus, the coefficient of MD

        = δ A / A or δ M d / M d or δ M o / M o

        Standard Deviation (σ)

        Standard deviation is the square root of the arithmetic mean of the squares of deviations of the terms from their AM and it is denoted by σ.

        The square of standard deviation is called the variance and it is denoted by the symbol σ2.

        1. For simple (discrete) distribution

        1. For frequency distribution

        1. For classified data

        Here, x is class mark of the interval.

        Shortcut Method for SD σ =  where, d = x — A’ and A’ = assumed mean

        Standard Deviation of the Combined Series

        If n1, n2 are the sizes, X1, X2 are the means and σ1, σ2 are the standard deviation of the series, then the standard deviation of the combined series is

        Effects of Average and Dispersion on Change of origin and Scale

        Change of origin Change of scale

        Mean Dependent Dependent

        Median Not dependent Dependent

        Mode Not dependent Dependent Standard Deviation Not dependent Dependent Variance Not dependent Dependent

        Important Points to be Remembered

        1. The ratio of SD (σ) and the AM (x) is called the coefficient of standard deviation (σ / x).
        2. The percentage form of coefficient of SD i.e., (σ / x) * 100 is called coefficient of variation.
        3. The distribution for which the coefficient of variation is less is called more consistent.
        4. Standard deviation of first n natural numbers is √n2 – 1 / 12
        5. Standard deviation is independent of change of origin, but it is depend on change of scale.

         

        Root Mean Square Deviation (RMS)

        The square root of the AM of squares of the deviations from an assumed mean is called the root mean square deviation. Thus,

        1. For simple (discrete) distribution

        S = √Σ (x – A’)2 / n, A’ = assumed mean

        1. For frequency distribution S = √Σ f (x – A’)2 / Σ f

        if A’ — A (mean), then S = σ

        Important Points to be Remembered

        1. The RMS deviation is the least when measured from AM.
        2. The sum of the squares of the deviation of the values of the variables is the least when measured from AM.
        3. σ2 + A2 = Σ fx2 / Σ f
        4. For discrete distribution f =1, thus σ2 + A2 = Σ x2 / n.
        5. The mean deviation about the mean is less than or equal to the SD. i.e., MD ≤ σ

        Correlation

        The tendency of simultaneous variation between two variables is called correlation or covariance. It denotes the degree of inter-dependence between variables.

        Perfect Correlation

        If the two variables vary in such a manner that their ratio is always constant, then the correlation is said to be perfect.

        Positive or Direct Correlation

        If an increase or decrease in one variable corresponds to an increase or decrease in the other, then the correlation is said to the negative.

        Negative or Indirect Correlation

        If an increase of decrease in one variable corresponds to a decrease or increase in the other, then correlation is said to be negative.

        Covariance

        Let (xi, yi), i = 1, 2, 3, , n be a bivariate distribution where x1, x2,…, xn are the values of variable x and y1, y2,…, yn those as y, then the cov (x, y) is given by

        where, x and y are mean of variables x and y.

        Karl Pearson’s Coefficient of Correlation

        The correlation coefficient r(x, y) between the variable x and y is given r(x, y) = cov(x, y) / √var (x) var (y) or cov (x, y) / σx σy

        If (xi, yi), i = 1, 2, … , n is the bivariate distribution, then

        Properties of Correlation

        (i) – 1 ≤ r ≤ 1

        1. If r = 1, the coefficient of correlation is perfectly positive.
        2. If r = – 1, the correlation is perfectly negative.
        3. The coefficient of correlation is independent of the change in origin and scale.
        4. If -1 < r < 1, it indicates the degree of linear relationship between x and y, whereas its sign tells about the direction of relationship.
        5. If x and y are two independent variables, r = 0
        6. If r = 0, x and y are said to be uncorrelated. It does not imply that the two variates are independent.
        7. If x and y are random variables and a, b, c and d are any numbers such that a ≠ 0, c ≠ 0, then

        r(ax + b, cy + d) = |ac| / ac r(x, y)

        1. Rank Correlation (Spearman’s) Let d be the difference between paired ranks and n be the number of items ranked. The coefficient of rank correlation is given by

        ρ = 1 – Σd2 / n(n2 – 1)

        1. The rank correlation coefficient lies between – 1 and 1.
        2. If two variables are correlated, then points in the scatter diagram generally cluster around a curve which we call the curve of regression.
        3. Probable Error and Standard Error If r is the correlation coefficient in a sample of n pairs of observations, then it standard error is given by

        1 – r2 / √n

        And the probable error of correlation coefficient is given by (0.6745) (1 – r2 / √n).

        Regression

        The term regression means stepping back towards the average.

         

        Lines of Regression

        The line of regression is the line which gives the best estimate to the value of one variable for any specific value of the other variable. Therefore, the line of regression is the line of best fit and is obtained by the principle of least squares.

        Regression Analysis

        1. Line of regression of y on x,

        y — y = r σy / σx (x – x)

        1. Line of regression of x and y, x – x = r σx / σy (y — y)
        2. Regression coefficient of y on x and x on y is denoted by

        byx = r σy / σx, byx = cov (x, y) / σ2x and byx = r σx / σy, bxy = cov (x, y) / σ2y

        1. Angle between two regression lines is given by

        1. If r = 0, θ = π / 2 , i.e., two regression lines are perpendicular to each other.
        2. If r = 1 or — 1, θ = 0, so the regression lines coincide.

        Properties of the Regression Coefficients

        1. Both regression coefficients and r have the same sign.
        2. Coefficient of correlation is the geometric mean between the regression coefficients.
        3. 0 < |bxy byx| le; 1, if r ≠ 0 i.e., if |bxy|> 1, then | byx| < 1
        4. Regression coefficients are independent of the change of origin but not of scale.
        5. If two regression coefficient have different sign, then r = 0.
        6. Arithmetic mean of the regression coefficients is greater than the correlation coefficient.
        Prev Ex-15.3, Analysis of frequency distribution, comparison of two frequency distribution with same mean
        Next Outcomes & sample space, Ex. 16.3

        Leave A Reply Cancel reply

        Your email address will not be published. Required fields are marked *

        All Courses

        • Backend
        • Chemistry
        • Chemistry
        • Chemistry
        • Class 08
          • Maths
          • Science
        • Class 09
          • Maths
          • Science
          • Social Studies
        • Class 10
          • Maths
          • Science
          • Social Studies
        • Class 11
          • Chemistry
          • English
          • Maths
          • Physics
        • Class 12
          • Chemistry
          • English
          • Maths
          • Physics
        • CSS
        • English
        • English
        • Frontend
        • General
        • IT & Software
        • JEE Foundation (Class 9 & 10)
          • Chemistry
          • Physics
        • Maths
        • Maths
        • Maths
        • Maths
        • Maths
        • Photography
        • Physics
        • Physics
        • Physics
        • Programming Language
        • Science
        • Science
        • Science
        • Social Studies
        • Social Studies
        • Technology

        Latest Courses

        Class 8 Science

        Class 8 Science

        ₹8,000.00
        Class 8 Maths

        Class 8 Maths

        ₹8,000.00
        Class 9 Science

        Class 9 Science

        ₹10,000.00

        Contact Us

        +91-8287971571

        contact@dronstudy.com

        Company

        • About Us
        • Contact
        • Privacy Policy

        Links

        • Courses
        • Test Series

        Copyright © 2021 DronStudy Pvt. Ltd.

        Login with your site account

        Lost your password?

        Modal title

        Message modal