Topics
Sets and Functions
Trigonometric Functions
 Concept of Angle
 Introduction of Trigonometric Functions
 Signs of Trigonometric Functions
 Domain and Range of Trigonometric Functions
 Trigonometric Functions of Sum and Difference of Two Angles
 Trigonometric Equations
 Truth of the Identity
 Negative Function Or Trigonometric Functions of Negative Angles
 90 Degree Plusminus X Function
 Conversion from One Measure to Another
 180 Degree Plusminus X Function
 2X Function
 3X Function
 Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications
 Graphs of Trigonometric Functions
 Transformation Formulae
 Values of Trigonometric Functions at Multiples and Submultiples of an Angle
 Sine and Cosine Formulae and Their Applications
Relations and Functions
 Cartesian Product of Sets
 Relation
 Concept of Functions
 Some Functions and Their Graphs
 Algebra of Real Functions
 Ordered Pairs
 Equality of Ordered Pairs
 Pictorial Diagrams
 Graph of Function
 Pictorial Representation of a Function
 Exponential Function
 Logarithmic Functions
 Brief Review of Cartesian System of Rectanglar Coordinates
Sets
 Sets and Their Representations
 The Empty Set
 Finite and Infinite Sets
 Equal Sets
 Subsets
 Power Set
 Universal Set
 Venn Diagrams
 Intrdouction of Operations on Sets
 Union Set
 Intersection of Sets
 Difference of Sets
 Complement of a Set
 Practical Problems on Union and Intersection of Two Sets
 Proper and Improper Subset
 Open and Close Intervals
 Operation on Set  Disjoint Sets
 Element Count Set
Algebra
Binomial Theorem
Sequence and Series
Linear Inequalities
Complex Numbers and Quadratic Equations
Permutations and Combinations
 Fundamental Principle of Counting
 Concept of Permutations
 Concept of Combinations
 Introduction of Permutations and Combinations
 Permutation Formula to Rescue and Type of Permutation
 Smaller Set from Bigger Set
 Derivation of Formulae and Their Connections
 Simple Applications of Permutations and Combinations
 Factorial N (N!) Permutations and Combinations
Principle of Mathematical Induction
Coordinate Geometry
Straight Lines
Introduction to Threedimensional Geometry
Conic Sections
 Sections of a Cone
 Concept of Circle
 Introduction of Parabola
 Standard Equations of Parabola
 Latus Rectum
 Introduction of Ellipse
 Relationship Between Semimajor Axis, Semiminor Axis and the Distance of the Focus from the Centre of the Ellipse
 Special Cases of an Ellipse
 Eccentricity
 Standard Equations of an Ellipse
 Latus Rectum
 Introduction of Hyperbola
 Eccentricity
 Standard Equation of Hyperbola
 Latus Rectum
 Standard Equation of a Circle
Calculus
Limits and Derivatives
 Intuitive Idea of Derivatives
 Introduction of Limits
 Introduction to Calculus
 Algebra of Limits
 Limits of Polynomials and Rational Functions
 Limits of Trigonometric Functions
 Introduction of Derivatives
 Algebra of Derivative of Functions
 Derivative of Polynomials and Trigonometric Functions
 Derivative Introduced as Rate of Change Both as that of Distance Function and Geometrically
 Limits of Logarithmic Functions
 Limits of Exponential Functions
 Derivative of Slope of Tangent of the Curve
 Theorem for Any Positive Integer n
 Graphical Interpretation of Derivative
 Derive Derivation of x^n
Mathematical Reasoning
Mathematical Reasoning
Statistics and Probability
Statistics
 Measures of Dispersion
 Concept of Range
 Mean Deviation
 Introduction of Variance and Standard Deviation
 Standard Deviation
 Standard Deviation of a Discrete Frequency Distribution
 Standard Deviation of a Continuous Frequency Distribution
 Shortcut Method to Find Variance and Standard Deviation
 Introduction of Analysis of Frequency Distributions
 Comparison of Two Frequency Distributions with Same Mean
 Statistics Concept
 Central Tendency  Mean
 Central Tendency  Median
 Concept of Mode
 Measures of Dispersion  Quartile Deviation
 Standard Deviation  by Short Cut Method
Probability
notes
By using stepdeviation method, it is possible to simplify the procedure.
Let the assumed mean be ‘A’ and the scale be reduced to `1/h` times (h being the width of classintervals). Let the stepdeviations or the new
values be `y_i`.
i.e. `y_i = (x_i  A) /h or x_i = A +hy_i` ...(1)
we know that \[\bar{x} =\frac{\displaystyle\sum_{i=1}^{n} f_ix_i}{N} \]
Replacing xi from (1) in (2), we get
\[\bar{x} =\frac{\displaystyle\sum_{i=1}^{n} f_i(A + hy_i)}{N} \]
=\[\frac {1}{N} (\displaystyle\sum_{i=1}^{n}
f_i A + = \displaystyle\sum_{i=1}^{n}
hf_iy_i )\] = =\[\frac {1}{N} (A \displaystyle\sum_{i=1}^{n}
f_i + h = \displaystyle\sum_{i=1}^{n}
f_i y_i) \]
=A .\[\frac{N}{N} + h \frac {\displaystyle\sum_{i=1}^{n} f_iy_i}{N} \]
(because \[\displaystyle\sum_{i=1}^{n} f_i\] = N )
Thus `bar x = A + h bar y` ...(3)
Now Variance of the variable x, \[\sigma_x^2 =\frac {1}{N} \displaystyle\sum_{i=1}^{n}
f_i (x_i  \bar x)^2 \]
=\[\frac {1}{N} \displaystyle\sum_{i=1}^{n}
f_i (A +hy_i  A  h \bar y)^2 \] (Using (1) and (2))
=\[\frac {1}{N} \displaystyle\sum_{i=1}^{n}
f_i h^2 (y_i  \bar y)^2 \]
= \[\frac {h^2}{N} \displaystyle\sum_{i=1}^{n}
f_i (y_i \bar y)^2 = h^2 × \] variance of the variable y_{i}
i .e.`sigma_x^2 = h^2 sigma_x^2`
or `σ _x = hσ _y` ...(4)
From (3) and (4), we have
σ_{x} = \[{\frac{h}{N}}\sqrt{N\displaystyle\sum_{i=1}^{n} f_i y_i ^2  (\displaystyle\sum_{i=1}^{n} f_i y_i) ^2 } \] ...(5)
Shaalaa.com  Shortcut method for Standard Deviation
Related QuestionsVIEW ALL [4]
From the prices of shares X and Y below, find out which is more stable in value:
X 
35 
54 
52 
53 
56 
58 
52 
50 
51 
49 
Y 
108 
107 
105 
105 
106 
107 
104 
103 
104 
101 
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?
Find the mean and standard deviation using shortcut 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, variance and standard deviation using shortcut method
Height in cms 
No. of children 
7075 
3 
7580 
4 
8085 
7 
8590 
7 
9095 
15 
95100 
9 
100105 
6 
105110 
6 
110115 
3 