Business mathametics and statistics b.com ii semester (2)

2
…
is ca
11
1
lled a
12
22
x of o
1
2
rd
… … … … …
1
… …
… …
… …
2 … …
ution
11
21
; In ge
12
22
neral
2 x
2 3
3 4
12=1+
UNIT 1 MATRICES
INTRODUCTION:
Matrices are one of the most powerful tool in mathematic. This mathematical tool provide a
means of storing large quantities of information’s in such a way that each piece can be identified
and manipulated matrix notation and operations are used in electronic spreadsheet programs for
personal computers, which in turn is used in different areas of business and science like cost
estimation, analyzing the result of an experiment. Matrices are also in genetics industrial
management, economics etc.
A rectangular array of mn numbers ,real or complex in the form of m horizontal lines [called
rows] n vertical lines [called columns] matri er m by n like;
…
…
…
…
An m.n matrix is usually denoted by A={aij}m.n.
Example; construct a 2x2 matrix whose elements are given by [i] aij=i+j [ii], aij=ixj
Sol a 2x2 matrix is written as
A= 2
[1] aij=i+j
a11=1+1=2 a 2=3 a21=2+1=3 a22=2+2=4
Therefore A=
[ii] aij=ixj
a11=1x1=1
a12=1x2=2

1 2
2 4
ng nly one c
atrix
2 0
0 3
s a sq
a21=2x1=2
a22=2x2=4
Therefore A=
Types of Matrices
Row matrix
A matrix having only one row is called a row matrix.
Example. A= [1 2 3] is a row matrix of order 1x3.
Column Matrix
A matrix havi o olumn is called a column matrix
4
Example. A= 5
6
Is a column matrix of order 3x1.
Square Matrix
A matrix in which the number of rows are equal to the number of columns is called a square
matrix. A matrix of order (nxn) is called a square matrix of order n or an n rowed matrix. A
matrix of order mxn where m# n is called a rectangular matrix.
Diagonal matrix.
A diagonal m i uare matrix that has zero everywhere except on the main diagonal.
Example. A=

ents
3 0
0 3
are eq
a uni
1 0
0 1
t matr
atrix
0
0 0
h of w
is a
mber o
2 1
4
ows an
and
olum
5
8
Thus
are c
O =
Is a diagonal matrix of order 2x2.
B=[5]1x1 is a diagonal matrix of order 1x1. The lines along which the diagonal elements lie is
called principal leading diagonal elements.
Scalar Matrix
A diagonal matrix is called scalar matrix in which every non diagonal element is zero and all the
diagonal elem ual.
Example. A=
Is a scalar matrix of order 2x2 .
Unit Matrix
A square matrix in which every non diagonal elements is zero and the main diagonal elements is
unity is called ix.
Example. I2 =
Is aunit matrix of order 2x2.
Zero Matrix
A m eac hose elements is zero is called zero matrix. Example
0 0
0
zero matrix of order 2x3.
Comparable Matrices
Two matrices are said to be comparable matrices if they are of same order i.e. they have same
nu f r d c ns.
A=
3
5 6
B=
6 7
9 1
omparable matrices.

me are equal.
and B=
Exa mpl
a
e
re
Equal Matrices
Two matrices are said to be equal matrices if they are of same order and corresponding
ele nts
1 2
A= 3 4
5 6
equal matrices.
Algebra of Matrices
Matrices involves the following operations namely addition of matrices, subtraction,
multiplication of a matrix by a scalar and multiplication of matrices.
Addition of Matrices
Let A and B be two comparable matrices then their sum A+B is the matrix obtained by adding
the corresponding elements of A and B is same as that of A+ B. IN genral for A=[aik]mxn and
B=[bij]mxnthen A+B=[aij+bij]mxn
Properties of Matrix addition and subtraction
(i) Commutatively: for any two matrices A and B of same order A+B=B+A
(ii) Associativity: if A, B and C are three matrices of same order than[A+B]+C=A+[B+C]
(iii) Additive identity: the null matrices is the identity element for the matrices additions
A+0=0+A=A
(iv) Additive Inverse; for any matrix A=[aij]m x n then there exists a unique matrix -A=[-
aij]m x n such that A+[-A]=0=[-A]+A
Let A and B be two comparable matrices. Then A-B is defined as A- B =A+[-B].
Thus the matrices A-B is obtained by subtracting from each element ofA, The
Corresponding element of B
If A=[aij]mxnand B=[bij]mxnthen A- B=[cij]mxn
wherecij=aij– bijfor all i and j,

f A an
+ ƛ
d B are
.
j]mxn
2 3
4
[bij]n×
and
en efined as
xamp
2x1
4 1
B=
2 +
2 +
2 1
1
1 +
1 + 1
3
3
3 +
3 +
1 +
1 +
1
1
+
+
x2 +
2 +
2 0
x1
1
2 +
4 +
0
9
=
2 0
6 + 1 4 + 3 + 3 6 + 3 + 1
4 + 0 8 + 2 + 0 12 + 2
Multiplication of a Matrix by a Scalar;
The product of a scalar I.e. a number ƛ and a matrix A=[aij]mxnis denoted by ƛ
is defined as ƛ A=[ ƛ aij]m.n
Properties of scalar multiplication
i) I two matrices of same order and ƛ is a scalar then ƛ [A+ B] = ƛ
ii) If A is a matrix and ƛ and µ are any two scalar,then
a. (ƛ+ µ)A= ƛA+ µA.
b. ƛ (µA)= (ƛµ)A.
c. (1)A = A.
d. (-1)A=-A.
Multiplication of Matrices
Multiplication of matrices is conformableif and only if the number of columns in the first is
equal to the number, of rows in the second matrix. Thus AB is defined if the number, of columns
in A is equal to number, of rows in B.
If the order of the matrix A bem×n and that of matrix B be n×p then the order of the product AB
is m×p.
Thus if A= [ai B= p th ABis d
AB=[cij]m×p.
E le; A=
1
1 2 3
1
3
=
3 1 2 3 1 2 3 1
2 0 4 2 4
10 10
= 8 10 14

a1a2
−
11
21
− −
an expres
12
22
lled t
1
2
he
− − − −
1 2
− −
− −
− −
− −
The process of multiplication in matrices is sometimesreferredas row by column multiplication
Propertiesof Matrices Multiplication
1. Multiplication of matrices is non-commutative in general.
2. MatricesMultiplication is associative if a b & c are three conformable matrices then (AB) C =
A (BC).
3. Mar mal is distributive with respect to addition of matrices.
4. If AB is null matrices i.e. AB=O it does not necessarily mean that A or B should be null
matrices.
5. If I is the identity matrices of order n , it has the property that it can commute with every
square matrices of order n i.e. if A is square matrices of order n then I A = AI = A
Transpose of Matrices
If A = {aij} be a mxn matrix then the matrix obtained by interchanging the rows into columns of
A is called the transpose of A. Transpose of A is denoted by A’
Determinants
Corresponding to each square matrix
1 a12 a13 − − − a13
A=
1 a22 a23 − − − a2n
− − − − − − − − −
an1 an2 an3 − − − ann
There is associated sion ca Determinant of A , denoted by det A or |A|
−
Written as det A =
−
−
−

f a de
11
21
term
12
22
inan
matrix o
11
21
f o
12
22
rder 2x
= a11 a
4 3
22
= 8 -6
find the
4
−3 5
V
=
alue
8
A matrix is an arrangement of numbers and so it has no fixed value , while each determinant has
fixed value.
Value of a determinant of order 1
Let A = [a] be a matrix of order 1 then , determinant of a is defined to be = a.
Value o t of order 2
Let A=
Be a 2 , then the determinant of A is defined as
|A| = 22 – a21 a12
Example = = 2.
Example of m
Where =
=> 20+3m = 8
=> 3m= 8-20
=> 3m= -12
=> m= -4
For finding the value of a determinant of order 3 we need the following definitions
Minor of aij in |A|
The minor of an element aij in |A| is defined as the value of the determinant obtained deleting ith
row and jth column of |A| and is denoted by Mij.

amp
1
2
3
le
1
1
Find
12
22
the
13
23
1 32 33
mino
22
32
23
33
a23
21
31
23
33
= a21 aly M12
21
31
=
22
32
= a21 a
the
1
4
1
d c
ors of t
5 6
3 1
he elem
= 5 – 1
Cofactors of aij in |A|
The cofactor cij of an element aij is defined as cij = (-1)i+j . Mij
Ex rs and cofactors of the elements of the determinant
∆=
Solution the minor M11 of a11 is given by M11 =
=>a22 a33 - a32
Similar 33-a31 a23
M13 = 32 - a31 a22
Similarly we may obtain the minors of each remaining elements. Now if we denote the cofactors
of aij by Cij then
C11 = (-1)1+1 M11 = a22 a33 - a32 a23
C12 = (-1)1+2 M12 = -M12 = a31 a23 – a21 a33
C13 = (-1)1+3 M13 = M13=a21 a33 - a31 a23
Similarly we may the cofactors of remaining elements
Find minors an ofactors of each of the elements of
2 3
|A| = 5 6
3 1
The min ents of |A| are given by
M11 = 8 = -13

4 6
1 1
= 4 – 6
4 5
1 3
= 12 –
2 3
3 1
= 2 – 9
1 3
1 1
= 1 –3
1 2
1 3
= 3 – 2
2 3
5 6
= 12 –
1 3
4 6
= 6 – 1
1 2
4 5
= 5 – 8
M12 = = -2
M13 = 5 = 7
M21 = = -7
M22 = = -2
M23 = = 1
M31 = 15 = -3
M32 = 2 = -6
M33 = = -3
The cofactors of the corresponding elements of |A| are
C11=(-1)1+1 M11 = M11=-13
C12 =(-1)1+2 M12 =-(-2) M11=2
C12 =(-1)1+3 M13 = M13=7
C21 =(-1)2+1 M21 =-(-7) =7
C22 =(-1)2+2 M22 = M22=-2
C23 =(-1)2+3 M23 = -1
C31 =(-1)3+1 M31 = -3
C32 =(-1)3+2 M32 –(-6)=6
C33 =(-1)3+3 M33 = -3

Exp
1
2
3
an
1
1
ding
12
22
the g
13
23
1 32
i
33
12 + a13 c13 =
ven
=a1
2
3
2
2
a
23
33
1 M
- a1
1 – a
2
3
12
1
1
M1
23
33
2 + a1
+
M13
2
3
1
1
22
32 =
orde
+
−
+
the
1
2
3
e of
pand al
3 4
-2
4 5
ong f
x
2
3
irst r
4
+
5
ow
3x
2
3
3
4
Value of a Determinant
The value of a determinant is the sum of the product of elements of a (row or a column) with
their corresponding cofactors.
Expansion of a Determinant
determinant by first row
1.( its cofactor) + a12 . .( its cofactor)+ a13 .( its cofactor) = a11 c11 + a12
c 1 1 3
= a11 2 a13 a11(a22 a33 – a32 a23 ) –
a12 (a21 a33 – a31 a23) + a13 (a21 a32 – a31 a22)
Above determinant may be expanded by any row or column.
Note By Expanding a determinant by any row or column using minors we should keep in view
the following symbols for a determinant of r 3
− +
+ −
− +
Find val
2 3
|A|= 3 4
4 5
Ex
1x
=1(15-16) -2 (10-12)+3(8-9)
=1(-1)-2(-2)+3(-1)=-1+4-3 =0

a1a2
−
an1
a1
a1
−
an1
A1
A1
−
a s conside
Properties of Determinants
1. The value of a determinant remains unaltered by interchanging its rows or columns.
2. If two rows or columns of a determinant are interchanged the sign of the determinant is
changed but its numerical value remains unaltered.
3. If two rows or columns of a determinant are identical then the value of determinant is
zero.
4. If every element of any row or column of a determinant is multiplied by the same
quantity then the value of whole determinant is also multiplied by the same quantity.
5. If each element of any row or column of a determinant consists of two terms then the
determinant can be expressed as the sum of two determinant of the same order.
Adjoint of a Matrix
The adjoint of a square matrix A is the matrix obtained by replacing the elements of the
transpose matrix A’ by the corresponding cofactors of the replaced elements.
Let A=
Then A’=
1 a12 − − − a1n
1 a22 − − − a2n
− − − − − − − − −
an2 − − − ann
1 a21 − − − an1
2 a22 − − − a2n
− − − − − − − − −
a2n − − − ann
And adj A=
1 A21 − − − An1
2 A22 − − − An2
− − − − − − − − −
A1n A2n − − − Ann
Where Aij is the cofactors of aij in |A’| i.e cofactors of the replaced elements
Ex mple let’ r the matrix
1 2 0
A= −2 3 1
3 1 4

− 2 31
4
1
1
e hav
|
e A(adj A)=
)=(1/| |
(adj A) A = |A|I
)A=1 ---
equati
|
ons we get
) and1/|
| )A.
Then A’ = 2 3 1
0 1
Therefore adj A =
1 − 8 2
1 4 − 1
−11 5 7
Which is obtained by replacing the elements of A’ by the corresponding cofactors.
An Important relation between a matrix A and its adjoint
If A be a square matrix of order n and I, the unit matrix of same order then A(adj A)=(adj A) A
= |A|I
Inverse of a matrix
If A be a square matrix of order n and there exits another square matrix B of same order such that
A. B = B.A = I
Where I is unit matrix of order n then B is called inverse of A and is denoted by A-1 thus A A-
1= A-1 A=I
A square matrix A is set to be singular or nonsingular according as
|A| = 0 or |A| ≠ 0 respectively.
Inverse of a matrix in terms of its Adjoint
Let AB an invertible square matrix then by definition
A A-1= A-1 A=I ------- 1
Also w or
A(1/| -----------2
From above two
A A-1 = A(
A-1 A= (1/|

| .
3
3
−
2
=
ofac
8
−4
f the
=
.
−2
2 =
−2
2 =
−1
− =
3
2 =
3
2 =
−1
8 =
Either of which leads to A-1 = 1/|
Example Find the inverse of the matrix A=
− 10 − 1
−2 8 2
2 − 4 − 2
Solution |A|=
− 10 − 1
2 8 2
− 4 − 2
3(-16 +8)+10(4 -4)-1(8-16)
=3(-8)-1(-8) = -24 +8 = -16 ≠ 0
Therefore A is non singular, A-1 exists
Now c tors o elements of |A| are
A11=
2
2 -16+8 =-8
A12= -
2
− 2 0
.
A13=
8
− 4 -8
.
A21= -
0 − 1
4 − 2
.
-16
A22=
− 1
− 2 -4
.
A23= -
− 10
− 4 -8
.
A31=
0 1
2 -12
.

− 13
− =
3
− =
−8
= 0
−8
16 −
−12 −
| (
4 − 8
4 4
)
= 0
1
cate
25
20
econo
and
ly
1
erable ov
arges
25
20
e to e
15
80
50
h party is
=
25
20x
⎢
A32= - 2 2 -4
.
A33=
− 10
2 8 4
.
Therefore adj A =
−8 0 − 8
−
− 16 − 12
− 4 − 4
− 8 4
Therefore A-1 = 1/|
A-1 1/-16
−8 − 16 − 12 ⎡ ⎤
0 − 4 − 4 ⎥
⎢ ⎥
−8 − 8 4
⎣ − ⎦Application of Matrices
A company employs 50labourers from different categories of different age groups as
Category a.(20 – 22 )years b.(23 – 25)years c.(26-28)years
Party A 25 20 15
Party B 20 30 10
The rate of applicable to categories a, b &c areRs 1500 Rs 800& Rs500 respectively. Using
Matrices find which gory is mical pref er the other.
Solution Let C =
20 15
D =
30 10
500
800
500
Total ch payabl ac given by
00
CD =
20 15
0
30 10
0
x1500 + 20x800 + 15x500
1500 + 30x800 + 10x500

375
300
=
6
5
10
90
e in e
100
60
2
3
1
et A=
100
60
2
3
1
heref
200
120
0 800
=
0
440
80
=
00 + 16000 + 7500 00
00 + 24000 + 5000 00
Therefore party b is more economical as compared to party a.
Example A manufacturer produces three items x,y,z which he sells into markets annual sales are
indicated below.
1 10000 2000 18000
2 6000 20000 8000
If unit sale price of x,y,z are Rs2 Rs3 & Rs1.
Find total revenu ach market.
Solution L
00 2000 18000
00 20000 8000
And B =
T ore AB=
00 2000 18000
00 2000
=
00 + 6000 + 18000
00 + 60000 + 8000
00
000
Hence the total revenue in the market 1 is Rs 44000 & total revenue in market 2 is Rs 80000.
Example 3 A man has built three houses for business purpose. For completing these houses he
purchased martial from three stories. For the first time he purchased 40 trucks , 10 trucks & 20
trucks of sand and 5 trucks , 6 trucks and 7 trucks of cement respectively. The cost of cement
truck is Rs 2000 per truck and that of sand is Rs 600. Find the total amount paid by that man to
each store.

ut
4
t AB the
B be
200
60
he ma
t
4
mount paid
2000
600
by
=
t
8
ent stories
830
= 236
44
iven by
Sol ion Le matrix representing the trucks of cement and sand as
0 5
A= 10 6
20 7
Let t trix representing the price as
B=
0
0
Then otal a he man to differ is g
AB=
0 5
10 6
20 7
000 + 3000
2000 + 3600
40000 + 4200
00
00
200
Hence amount paid to
1st
store = Rs83000
2nd
store = Rs 23600 and
3rd
store = Rs 44200.

UNIT 2 DIFFERENTIAL CALCULAS
Introduction :
A symbol that represents exactly one number is called a constant and a symbol that represents
any one of the members in a set of numbers is called a variable. Thus 1,2,π etc are constants and
if x represents any number between 1 to 6 then x is a variable.
Function: If X and Y be to non-empty sets then a function f from the set X into set Y is
correspondence such that for each element of X there exits one element of Y.
This correspondence is generally denoted by f :X Y.
Domain and Range: Let Y = f(x) be a real function the domain of this function is defined to be
the set of all real numbers for which the function is defined and is denoted by D(f).i.e. D(f)= {x
£ R : f (x) is defined }.
Also the range of f is defined to be the set of all possible values of y satisfying y = f(x). Thus R (f)
= {f(x) £ R : x £ (D(f)}.
Testing for Function: It can be tested wether a given relation is a function are not by using
following tests:
1. In case of a function the first set i.e. the domain is fully used up.
2. In case of a function the first members of all the ordered pairs are different.
3. In case of a function each element of the first set has only one image in the second set.
Types of Functions
Constant Function: Let C be a fixed real number then a function f(x) = c for all x £ R is
known as constant function.
Identity Function: The function f defined by f (x) = x forall x £ R is known as identity Function.
Reciprocal Function: The function defined by f(x) = 1/x forall x ≠ 0 is called reciprocal
function.

Modulas Function: The defined by f(x) =x when x ≥ 0
= -x when x <0 is called modules function.
Polynomial Function: A function of the form P(x) = a0xn + a1 xn-1 + a2X n -2 +…...+ an-
2x+an where a0, a1,a2…….. an-1 are real numbers , a0 ≠ 0 and n is a non-negative integer is
called a polynomial function of degree n. polynomial of degree 1,2,3 are respectively called
linear quadratic, cubic polynomials.
The greatest integer Function: If x £ R then [x] is defined as the greatest integer not exceeding
x. For example we have [3.01]=3 [3.2] = 3 [-1.3]=-2 and [-2]=-2 etc.
Concept of limits and continuity of function:
Consider the function f(x) = x2– 1/x-1 the function is not defined at x=1 because f(1) = 0/0
which is meaningless.
Now f(x) = (x-1)(x+1)/x-1 = x+1 only when x ≠ 1
If we give to x a value not exactly 1 but slightly more than 1 then clearly the value of function is
slightly more than 2. Now if we go on decreasing the value and take it nearer to 1.then clearly
the value of f(x) will come nearer to 2 as shown below:
If x =1.1 then f(x) =2.1
If x =1.01 then f(x) =2.01
If x =1.001 then f(x) =2.001
- - - - - - - -
- - - - - - - -
If x= 1.00001 then f(x)=2.00001
Thus as the value of x approaches to 1 the value of f(x) approaches to 2 and is written as x  1
then f(x)2
Similarly if we give to x a value slightly less than 1 the value of f(x) is slightly less than 2.

Now if we go on increasing this value and take it nearer to 1 then the value of f(x) will come
nearer to 2 as shown below
If x=0.9 then f(x)=1.9
If x=0.99 then f(x)=1.99
If x=0.999 then f(x)=1.999
- - - - - - - -
- - - - - - - -
If x= 0.99999 then f(x) =1.99999
- - - - - - - -
- - - - - - - -
Thus in this case also as x1 then f(x)2
We express this fact as lt x1 x2 – 1/x-1 =2
Limit
We say that lt x a f(x) = l,if whenever xa then f(x)l
Example lt x2 x3
-8 /x2
-4
Solution Put x = 2+h so that h0 as x2
lt x2 x3
-8 /x2
-4 =lth0 (2+h)3
-8 / (2+h)2
-4
=lt h0 8+12h +6h2
+h3
-8/2+4h+h2
-4
=lt h0 h(12+6h+h2)/h(4+h) = lt h0 12+6h+h2
/4+h
= 12/4 =3

Continuity: Afunction f(x) is set to be continuous at x = a if
1. Lt xa f(x) exists.
2. f(a) is defined i.e f(x) has a definite value at x=a.
3. lt xa f(x) =f(a).
thus the function f(x) is said to be continuous at x = a if lt xa-
f(x) =lt xa+
f(x) =f(a).
If a function f(x) is not continuous at x= a then function is discontinuous at x = a. In that case x =
ais the point of discontinuity
Kinds of discontinuity.
1. Discontinuity of first Kind: A function f(x) is said to have discontinuityof first kind at x=a if
xa-
f(x) and lt xa+
f(x) both exits but not equal to one another.
2. Discontinuity of Second Kind: A function f(x) is said to have discontinuity of second kind at
x=a if neither lt xa-
f(x) nor lt xa+
f(x) exits.
3. Mixed Discontinuity: A function f(x) is said to have mixed discontinuity at x= a if only one
of the limits lt xa-
f(x) or lt xa+
f(x) exits.
4. RemovableDiscontinuity: A function f(x) is said to have removable discontinuity at x= a lt
xa-
f(x) = lt xa+
f(x) ≠ f(a).
In this case discontinuity can be removed by redefining the function at x= a.
Graphical Meaning of Discontinuity: Graphically a function is said to be continuous at a point
if the graph of the function has no break at that point.
Show that the function f(x) = -x2
when x ≤ 0
= x2
when x >0 is continuous at x = 0
Solution when x =0 f(x) = -x2
=>f(0)=0
Lt x0- f(x) = lt x0-(-x2
) =0
Lt x0+ f(x) = lt x0+(x2
)=0
Therefore lt x 0- f(x) = ltx0+f(x)

Both the limits exits and are equal therefore lt x0 f(x) =0 =f(0)
Implies f(x) is continuous at x = 0
Differentiation: It is one of the most and fundamental operation in calculas. The theory of
differentiation has been developed on the basis of the concept of limits and continuity of a
function. The operation consists of finding out the rate of change of the dependent variable with
respect to independent variable. The ratio thus measured is known as the derivative or
differential coefficient of a function.
Definition: Let Y = f(x) be a function. Let ∂x be small increment in x and ∂y be the
corresponding increment in value of y then
lt∂x0 ∂y/∂x is known as derivative of y with respective x and is denoted by dy/dx.
i.edy/dx = lt ∂x 0 ∂y/∂x = lt ∂x 0 f(x+∂x)-f(x)/∂x.
Rules of Differentiation: Let u,v,w….. be the functions of a single variable x whose derivatives
exits then.
1. d(k)/dx =0 i.edifferentiation of constant is zero.
2. d(ku)/dx = k(du/dx).
3. d(k1 u+k2v)/dx = k1 du/dx + k2 dv/dx.
4. d(u.v)/dx = udv/dx+vdu/dx. (Product rule).
5. d (u/v)/dx = vdu/dx –u dv/dx /v2
(Quotient Rule)
6. if y = f(t) and t = øx then dy/dx = dy/dt . dt/dx (Chain Rule for differentiation function of
a function)
7. dy/dx . dx/dy = 1
8. Let u =f(x) and v=g(x) be two function of a single variable x then derivative with
respective v is denoted by du/dv = du/dx /dv/dx this is called derivative of one function
with respective to other.
Derivative of some simple functions:
1. f(x) = xn

Here y= xn
therefore y + ∂y = (x + ∂x)n
on subtracting ∂y =( x+∂x)n
- xn
∂y/∂x = ( x+∂x)n
- xn
/∂x
dy/dx = lt∂x 0 ( x+∂x)n
- xn
/x+∂x - x
= nxn-1
lt za zn
– an / x-a = nan-1
Therefore d(xn
)/dx = nxn-1
2. f(x)= c where c is constant here y =c
y+∂y=c
∂y=y+∂y-y=c-c =0 ∂y/∂x = 0
Lt ∂x 0 ∂y/∂x = 0
dy/dx =0
d(c)/dx =0
3. f(x)= ex
here y = ex
therefore y+∂y = ex
+∂x
∂y = ex
+∂x - ex
= ex
(e∂x
-1)
Therefore ∂y /∂x = ex
(e∂x
-1) /∂x
Therefore dy/dx = lt∂x0 ∂y/∂x = ex
lt∂x0 e∂x
-1/∂x = ex
x 1 = ex
d(ex
)/dx = ex
4. f(x)=x1/3
y=x1/3
y+∂y = (x+∂x)1/3
∂y/∂x = (x+∂x)1/3
– x1/3
/ ∂x
dy/dx =lt∂x0 (x+∂x)1/3
– x1/3
/x+ ∂x –x
=1/3 x-2/3
since lt xa xn
– an
/x-a = nan-1
5. F(x)=ax

=
Y=ax
therefore y+∂y = ax+∂x
∂y = ax+∂x
-ax
= ax
(a∂x
-1)
∂y/∂x = ax
(a∂x
-1)/∂x
dy/dx lt∂x0 ∂y/∂x =lt∂x0a
x
(a
∂x
-1)/∂x
=ax
log .
d(ax
)/dx =ax
log .
Role of Differentiation in Economics:
Elasticity of demand: Law of demand is regarding qualitative relationship between price and
quantity demanded. According to this law there is inverse relationship between price and
quantity demand. But elasticity of demand is regarding quantitative relation between price and
quantitatesdemanded. It deals with the ratio of percentage change it quantitate demanded due to
percentage change in price. Since quantity demanded is dependent on price so this relation can
be written as q=f(p) where q is quantity demanded and is dependent on price p here price is
independent variable.
Elasticity of demand is defined as: Elasticity of demand = lt =∆p 0 ∆q/q / ∆p/p = ∆p0
∆q/∆q.p/q = -dq/dp.p/q where the negative sign implies inverse relation between quantity
demanded and price.
Example find elasticity of demand for p =xe x
Solution dp/dx= x ex
+ex
= ex
(x+1)
Elasticity of demand = -p/q dq/dp = -x ex
/x .1/ex
(x+1) = -1 /x+1
Where x is quantity.
Maxima and Minima: The technique of Maxima and Minima comes to our help to known the
maximum and minimum values that a given function can take.
Note: for maximum and minimum value of a function at a point

a) dy/dx =0
b) d2y/dx2
is negative
And for minimum value of a function at a point
a) dy/dx =0
b) d2y/dx2
is positive
Application of Maxima and Minima in Economics: Maxima and minima can be used to find
maximum profit or sale when profit is maximum when revenue and cost function are given. It
also helps us to maximize and minimize cost. Here we shall discuss the economic problems
relating to maximum and minimum or one variable only.
Example the cost function C=2x2
-300x+16000 find output where cost is minimum and the
minimum cost.
Solution the given function for minimization is C=2x2
-300x+16000 differentiation we get
dc/dx = 4x-300 for maximum or minimum dy/dx =0
=>4x-300=0 =>x=300/4=75
Second derivative d2y/dx2
=4 which is positive so the function is minimum at x=75
Minimum cost =2(752
)-300(75)+16000
=11250-22500+16000=4750
Problem relating to maximization of revenue
Example from the following price and quantity relationship find the level of output where the
total revenue is maximum the function is p = 100-4x
Solution we have p = 100-4x
Total revenue (TR)=px=(100-4x)x = 100x-4x2
Differentiating we get d(TR)dx = 100-8x
For maxima or minima dy/dx=0

So 100-8x=0 => 8x=100 =>x=12.5
Second derivative that is d2(TR)/dx2
= -8(negative)point of maxima
It is negative so total revenue function is maximum when q =12.5 and maximum revenue =
100-4(12.5) =100-50 = 50
UNIT 3 BASIC MATHEMATICS OF FINANCE
Introduction
If a person barrows some money from someone, then while making a payment to the lender the
barrower has to pay some extra money. This extra money is called the interest. The money
barrowed is called the principal along with the interest is called amount. The number of
years/months/days for which the money has been kept by the barrower is called the time. The
extra money is paid on Rs 100 per year is called the rate percent per annum.
Interest: The interest is of two types
1. Simple interest (SI) and 2. Compound Interest (CI)
When the interest is reckoned uniformly that is if it is Rs5 on Rs100 for 1 year then it is Rs 10
on Rs 100 for 2 years Rs 15 on Rs 100 for 3 years and so on. This is called simple Interest. On
the other hand, suppose one barrows Rs 100 at 5% per annum for 2 years then SI after one year
is Rs 5.Now after one year the principal becomes Rs 100+5 =105 and in second year the interest
on Rs on 105 is charged instead the interest on Rs 100 and so on this is called compound Interest.
Thus in case of SI the interest is charged on money barrowed uniformly while in case of CI the
interest after one Year ia added to the principal and then interest is charged on this amount. After
2 years total amount is taken as principal and the interest for the third year is calculated on this
amount and so on.
If P is the principal, R is the Rate percent per annum; T is the time in years then
i. SI = P x R x T / 100
ii. P=100xSI/RxT

iii. R=100xSI/PxT
Formula for finding the compound interest
C.I can be obtained by
C.I = A-P
=P(1+Rx.01)n
-P
Rate of Interest: The interest is paid according to agreement which is in the form of a rate per
unit of the principal barrowed .It can be half yearly as per the agreement. It is generally denoted
by R.
Part B: UNIT-I- UNIVARIATE ANALYSIS
Condensation of data is necessary in statistical analysis because of the fact that a large number of
big figure are not confusing to mind but also very difficult to analyze. This chapter is about using
figures known as summary measures to represent or summarize quantitative data. Because they
are used to describe sets of data they are also called descriptive measures. Thus we will focuses
on the techniques that can be used to study single variable. The type of data that these techniques
are intended to analyze is called univariate data because they consist of observed values of single
variables. The techniques themselves make up what is known as univariate analysis. Thus when
we have the data on single variable, we are said to have a univariate population.
Basically there are two types of measure used for statistical analysis of data:
a. Descriptive statistics and
b. Analytic statistic or statistical inference.
As far as analytical statistic is considered, it is beyond the scope of this study material. This unit
will primarily focus on Descriptive statistics.
All statistical measures devised to describe the data are called descriptive statistics. These may
again be of three types:

a. Those used for uni-variate analysis (like averages, dispersions, skewness etc.)
b. Those used for bi-variate analysis (such as simple correlation, simple regression etc.).
c. Those used for multi-variate analysis (like multiple-correlation, multiple regression,
factor analysis etc.).
Next unit deals with bivariate analysis while as multi-variate analysis is beyond the scope of this
study material.
Statistical Measure for Univariate Analysis
Where the data consists of measurement of only one variable, they are often presented either in
the form of a frequency table or a time series. In a frequency table one column gives observed
values of a random variable X and the other gives the frequency of each value. Frequency table
is commonly analyzed in terms of its four important characteristics viz. central tendency,
dispersion, skewness and kurtosis. On the other hand, in time series one column gives certain
units of time (e.g., hours, weeks, months, years, etc.) and the other gives the observed values of a
variable as it varies from one time period to another. Time series is analyzed in terms of its four
important components viz. trend, seasonal, variations, cyclical variation, and irregular variations.
A. Measures of Central Tendency
When we have huge amount of statistical data it will create confusion and we will be not in a
position to understand it. So it is necessary to condense them and make them understandable and
comparable for scientific treatment. For this purpose a central value, which represents the whole
mass of data is worked out. This value is called Central Tendency or an Average. These
measures are so called because they show a tendency of a distribution to concentrate at certain
values, usually somewhere in the center of the distribution. For this reason, an average is
frequently referred as measures of central tendency or Measures of location.
Averages
One of the powerful tools of analysis is to calculate a single average value that represents the
entire mass of data. The word average is very commonly used in day-day conversation. An
average is a single value which is considered as the most representative or typical value for a

given set of data. Such a value is neither the smallest no the largest value, but it is a number
whose value is somewhere in the middle of the group.
Objectives of Averaging
The two important objectives of averaging are as under:
a. To get one single value that describes the characteristics of the entire data.
b. To facilitate comparison.
Characteristics of a Good Average
The main characteristics of a good average are as under:
a. It should be easy to understand.
b. It should be simple to calculate.
c. It should be based on all the observation.
d. It should be rigidly defined.
e. It should have sampling stability.
Types of Statistical average
The two important types of Average are Mathematical Average (Arithmetic Mean, Harmonic
Mean, and Geometric Mean) and Positional Average (Median and Mode).
1. Arithmetic Mean
It is also known as the arithmetic Average, is the most common measure of central tendency. It is
obtained by adding the values of the items and dividing by the number of items.
Calculation of Arithmetic (Individual Observation) is computed by applying any of the following
two methods:
a. Direct Method
b. Shortcut Method
Direct Method
The formula for calculating Arithmetic mean using direct method is as under:

=
+ + + Σ
= , ℎ
= ℎ , = . ,
= . , , , …… . =
t ΣX.
=
arith
l be:
=
metic
+ , ℎ
an ca n also be
ℎ
calcula ted by taking any arbitrary origin in that case
i.e
the formula
= ( − ) =
Σ
Steps to Calculate Arithmetic Meanusing Direct Method
1. Add all the values of the variable X and ge
2. Find out the total number of items i.e. N.
3. Divide the total number of items by N i.e.
Shortcut Method
The me
shal
Steps to Calculate Arithmetic Meanusing Shortcut Method
1. Take an assumed Mean
2. Take the deviations from the assumed mean and denote by dx
3. Obtain the sum of these deviations i. e. Σdx
4. Find the total number of items (i.e. N)
5. Put the values in the formula and calculate Mean.
Calculation of Arithmetic Discrete Series is computed by applying any of the following two
methods:
a. Direct Method
b. Shortcut Method
Direct Method
The formula for calculating Mean is

= whe
ith the vari ble an
. . Σ
d obt
and
Follo
=
Shortcut
win g f
Σ
+
ethod
ormula is used
, ℎ
usin
=
g s
−
hortcut
,
method:
ℎ , .
eviations by dx
Multiply these d
)
The f
=
ormula for ca
re, f is frequency, X denotes any observation and N the sum of frequency.
Steps to Calculate Arithmetic Meanusing Direct Method (Discrete Series)
1. Multiply the frequency of each item w a ain the total denoted as ΣfX
2. Find out the sum of frequencies i.e. Σ
3. Divide the total i.e ΣfX by the sum of frequencies calculate Mean.
M
Steps to Calculate Arithmetic Meanusing shortcut Method (Discrete Series)
1. Take an assumed mean (A)
2. Take the deviation of the variable X from the assumed mean (dx =X-A) and denote
d
3. eviations (dx) by their respective frequencies (f) and obtain the total i.e.
Σ
4. Divide the total obtain in step (iii) by total frequencies ( i.e.
5. Put the values in the formula and obtain mean.
Calculation of Arithmetic-Continuous Series is computed by applying any of the following two
methods:
a. Direct Method
b. Shortcut Method
Direct Method
lculating Mean is

mid val
,
ue of
ℎ
each clas
1 =
s
2 = .
)N by findin
ach mid-v
.
Follo
=
wing f
+ , Where A i
tions (dx).
ply the respecti
The sum of the squared
less than the sum of the s
= ,then ΣX
iation
ed de
s o
via
Steps to Calculate Arithmetic Meanusing Direct Method (Continuous Series)
1. Find out the
(Mid value =
2. Find out g the total of the frequencies (N =Σ
3. Multiply e alue by the corresponding frequency to find out fX.
4. Find out Σ
5. Put the values in the formula and obtain Mean.
Shortcut Method
ormula is used when using shortcut method:
s assumed mean, dx= X - A (deviations of mid-points from assumed
mean, f is the frequency, N is total sum of frequencies.
Steps to Calculate Arithmetic Meanusing Shortcut Method (Continuous Series)
1. Find out mid-value of each class
2. Take an assumed mean (A)
3. From the mid-value of each class deduct the assumed mean i.e. (X-A) and find out
devia
4. Multi ve frequencies of each class by these deviations and obtain the total
i.e., Σ
5. Put the values in the formula and calculate Mean
Mathematical properties of Arithmetic Mean
The various important Mathematical properties of Arithmetic Mean are as under:
1. The sum of the deviations of the items from the actual mean is always Zero.
2. dev f the items from arithmetic mean is minimum i.e.,
quar tions of the items from any other value.
3. = N

4. If we have arithmetic mean and the number of items of two or more than two related
groups we can calculate the combined leverage.
5. The mean of all sums (Differences) of corresponding items in two series, (Number of
items being equal in the two series) is equal to the sum (or differences) of means of the
two series.
6. If a constant amount is added or subtracted or multiplied or divided, the Mean will also
be affected accordingly.
7. The Mean is not affected by any change in origin. This means that for any value of the
assumed mean, the value of the arithmetic mean remains the same.
8. The standard error of the arithmetic is always less than that of any other measure of
central tendency.
Merits and Demerits of Arithmetic Mean
The arithmetic mean is the most popular average in practice. It is due to the fact that it possesses
most of the characteristics of a good average. However, arithmetic mean unduly affected by the
presence of extreme values. Also, in open-end frequency distribution it is difficult to compute
mean without making assumptions regarding the size of the class interval of the open end classes.
The arithmetic mean is usually neither the most commonly occurring value nor the middle value
in the distribution and in extremely asymmetrical distribution it is not a good measure of central
tendency.
2. Median
Median is the measure of a central tendency which appears in the middle of an ordered sequence
of value. That is half of the observations in a set of data are smaller than it and half of the
observations are greater than it. The median is also called a positional average. The term position
refers to the place of a value in a series.
Calculation of Median in Individual Observations
To calculate the Median of an individual series, we have to first arrange data either in ascending
or descending order and then following formula is used to calculate Median:

) .
) ℎ ite
)th
) ℎ .
Median (M) = Size of the (
Note: In case the number of items in a series is odd, then the median is the middle value after the
items have arranged in either ascending or descending order. If the number of items is odd,
median is obtained as the arithmetic mean of the middle observations.
Calculation of Median in Discrete Series
The discrete series involve frequencies, in order to find out Median, we need to divide the total
frequency into two equal parts.
Steps
1. Arrange the data in ascending or in descending order.
2. Calculate cumulative frequencies
3. Find out the Median by applying the formula i.e. M = Size of the ( m.
4. Find out the total in the cumulative frequency column which is either equal to ( or
next higher than that.
5. Locate the value of the variable corresponding to the cumulative frequency. This value of
the variable is the value of the median.
Calculation of Median-Continuous Series
In continuous series, median cannot be located in a straight-forward method. In this case, the
median lie in class-interval i.e., between lower and upper limit of a class interval. For exact
value, we have to interpolate median with the help of a formula. In this case, like mean, we have
to assume that value in each class is uniformly distributed in the class-interval.
Steps
1. Arrange the data in ascending order.
2. Calculate Cumulative frequency
3. Apply the formula, Median = size of the (

Now lo
or ne
this
On the class inte
+ ∗
rval is
wher
apply the fo
= Lower
cumu
all cl
=
lative frequency of t
asses lower than Med
he class preceding the med
ian class.
ℎ
ian class o
4. ok at the cumulative frequency column and find the total which is either equal to
xt higher than that and ascertain the value of the class interval corresponding to
.
5. ce determined, then rmula
= e, M= Median, limit of the median class, cf =
r sum of the frequencies of
C= Class interval of the median class
Merits and Demerits of Median
The Median is superior to arithmetic mean in certain respects. It is useful in case of open-end
distribution and also it is not influenced by the presence of extreme values. In fact when extreme
values are present in series, the median is a more satisfactory measure of central tendency than
the mean.
However, Median is a positional average; its value is not determined by each and every
observation. Also Median cannot be used for determining the combined median of two or more
groups. Also the median is less reliable average than the mean for estimation purposes since it is
more affected by sampling variations. Furthermore, the Median tends to be rather unsuitable
value if the number of observations is small.
3. Mode
Mode is the most typical or commonly observed value in a set of data. It refers to that value
which occurs most frequently in a distribution. Mode is easiest to compute since it is the value
corresponding to the highest frequency.
Calculation of Mode – (Individual Series)

g forn
∗
Mode
ulae i
Whe
=
ass
=
The value occurring maximum number of times is the modal value and this can be known by
inspection.
Calculation of Mode – (Discrete Series)
In discrete series, mode can be known either by inspection method or by grouping method.
Sometimes inspection method can give misleading result when the difference between the
frequencies preceding or succeeding the modal size is very small and the items are heavily
concentrated on either side. In that case it is desirable to apply grouping method by preparing a
Grouping table and an Analysis table to determine mode.
A grouping table has six columns.
Column I. The original frequencies are taken and the maximum frequency is encircled.
Column II Frequencies are added in Two’s
Column III Leave the first item and add the frequencies in two’s
Column IV The frequencies are added in three’s
Column V Leave the first frequency and add the remaining in three’s
Column VI Leave the first two frequencies and add the frequencies in three’s
In each case take the maximum total and put it in a circle. Once the grouping table is prepared,
an analysis table is drawn out of it. In all the six cases, maximum frequency is taken and entered
in the relevant box.
Calculation of in a Continuous Class
Followi m s used in calculating Mode:
Z = L+
re, L = Lower limit of the modal class,
Δ Difference between the frequency of the modal class and the frequency of the pre-modal
cl i.e. preceding class i.e., |f1 –f0| (ignoring signs)
Δ Difference between the frequency of the modal class and the frequency of the post-modal
class i.e. succeeding class i.e. |f1 –f2| (ignoring signs)
f0 = Frequency of the class preceding the modal class
f1 = Frequency of the modal class
f2 = Frequency of the class succeeding the modal class
i = The size of the modal class.

ther v
×
a lues)
×
mbolic
,
ally
ℎ
it is writt
,
e
…
as:
re
items
)
Merits and Demerits of Mode
Mode is not influenced by the extreme values and its value can be obtained in open-end
distribution without ascertaining the class limits of the open ends. Mode can be easily used to
describe qualitative phenomenon.
However, Mode is not rigidly defined measure as there are several formulae for calculating the
mode, all of which usually give somewhat different answers. Also the value of Mode cannot be
determined in the case of bimodal distribution.
Relationship between Mean, Median and Mode
A distribution in which the values of mean, median and mode coincide is known as symmetrical
distribution. Conversely stated, when the values of mean, median and mode are not equal the
distribution is known as asymmetrical or skewed distribution. Karl Pearson has expressed the
relationship as :
Mode = 3Median -2Mean
4. Geometric Mean
Geometric mean is the nth
root of the product of n items in a series. If there are two items, we
take the square root: if three, the cubes root: so on. If there are zeros or negative values in the
series, the geometric mean cannot be used (because it would be zero or negative regardless of the
size of o . Sy n
G.M = … . . . . fers various observation in the series.
Calculation of Geometric Mean(Individual Series)
In case of individual observation series, the procedure to calculate geometric mean is same as
that of arithmetic mean, the only difference in G.M is that we have to take sum total of log
values of all the and divide it by the number of items. The formula is as:
G.M = Antilog (

then ad
)
Calculation of Geometric Mean(Discrete and Continuous Series)
In case of discrete and continuous series, the frequencies are multiplied to the logarithmic values
of the items and ded. The formula is as:
G.M = Antilog (
Note: In the case of continuous series we have to first find out mid-points then apply formula
Merits and Demerits of Geometric Mean
Geometric Mean is highly useful in averaging ratios and the percentages and in determining rates
of increase and decrease. It is also capable of algebraic manipulation. For example, if the
geometric mean of two or more series and their number of observation are known, a combined
geometric mean can be easily calculated. However, compared to arithmetic mean, this average is
more difficult to compute and interpret. Also, geometric mean cannot be compared when there
are both negative and positive values in a series or one or more observations are having zero
values.
Properties of Geometric Mean
1. When each item of the data is replaced by the value of G.M., the product remains
unaffected.
2. The product of the ratios of the G.M to the item below or equal to it is the product of the
ratios of the item above the G.M.
3. It is relative value and is dependent on all items.
4. It is never larger than arithmetic mean.
5. The G.M of the product of corresponding items in two series is equal to product of their
G.M.
6. Combined G.M can be found provided we have G.M of different series given.
5. Harmonic Mean
Harmonic Mean is the reciprocal of the average of reciprocals of the values of items of a series.
The formulae for calculating Harmonic Mean is as:

⋯ ) ( )
ula is a
( )
H.M = =
(
Calculation of Harmonic Mean(Discrete Series and Continuous Series)
The form s:
H.M =
Note: In the case of continuous series we have to first find out mid-points then apply formula.
Uses of Harmonic Mean
The calculation of harmonic mean is done in special cases. It is useful for computing the average
rate of increase in losses of a public limited concern or the average price at which the goods have
been sold or the average speed at which journey has been performed. The rate usually expresses
a relationship between different units which can be expressed reciprocally.
Merit and Demerits of Harmonic Mean
The harmonic mean, like the arithmetic and geometric mean, is computed from all observations.
It is useful in special cases for average rates and is capable of for further algebraic treatment
However, harmonic mean cannot be computed when there are both positive and negative
observations or one or more observations have zero value.
B. Positional Measures orPartition Values
Median is a value which splits in two equal parts. While as Quartile divide the series into four,
Deciles into ten and Percentile into hundred.
i. Quartile: it divided the series into four equal parts. For any series there will be three
quartiles.
First or lower Quartile (Q1): It divides the distribution in such a way that one-fourth (25%) of
total items fall below it and three fourth (75%) fall above it.
Formula for Individual and Discrete Series

th
)
nuous
th
) it
polation of t
+ ×
he value
ating t
th
)
hich w
th
) ite
Q1 = Size of the ( item (in case of discrete series, we have to calculate cumulative
frequency)
Formula for Conti Series
Q1 = Size of the ( em. It will determine the size of class interval where Q1 falls. For
inter of Q1, the formula is as:
Q1=
Q1= Lower Quartile or First Quartile
L1 = Lower limit of class interval where Q1 lies
N = number of observations Cf
=cumulativefrequencies
f=simple frequency of Q1 group
C= class interval
Second Quartile (Q2)
Second Quartile (Q2) is Median (already shown)
Third Quartile (Q3)
Formula for calcul he (Q3) individual and discrete series is as:
Q3 = Size of the 3( item
Formula for calculating the (Q3) continuous series is as:
In continuous series, like median and first quartile, the actual value has to be interpolated from
the class interval w e get from following formula:
Q3 = Size of the 3( m. This will determine the class in which Q3 falls

rder to deter
( )
+ −
mine th
e the de
th
)
In o e actual value we have to apply the formula of interpolation
Q1=
ii. Deciles (D)
Deciles divide the series into 10 equal parts. For any series there are 9 deciles. It ranges from D1
to D9.
Decile in Individual and Discrete Series
Formula for determin cile in individual and discrete series is
as:
D1 = = Size of the ( item (in case of discrete series, we have to calculate cumulative
frequency)
iii. Percentile (P)
Percentile divides the series into 100 parts. For any series, there are 99 percentiles from P1 to P99.
Percentile in Individual and Discerte Series
It is calculated in the same way as that of Decile, only we have to change the denominator by
100.
Percentile in continuous Series
It is calculated in the same way as that of Decile, only we have to change the denominator by
100.
C. Measure of Dispersion
The average alone cannot adequately describe a set of observations, unless all the observations
are alike. It is necessary to describe the variability or dispersion of the observations. Also in two
or more distribution average value may be same but still there can be wide dispersion in the
formation of the distribution. Measures of dispersion help us in studying the important

(This for∓
characteristics of a distribution i.e., the extent to which the observations vary from one another
and from some average value.
Significance of Dispersion
Following are the importance of dispersion:
1. To determine the reliability of an average.
2. To serve as a basis for the control of the variability
3. To compare two or more series with regard to their variability
4. To facilitate the use of other statistical measures
Methods of studying Dispersion
The following are the important methods of studying dispersion;
i. Range
ii. Quartile Deviation
iii. Average Deviation
iv. Standard Deviation
v. Lorenz Curve
i. Range
Range as a measure of dispersion represents a difference between the values of extreme items i.e.,
the largest and the smallest items of the data under review.
Absolute range = Highest value – Lowest value (This formula is used for both discrete and
continuous series)
The relative measure corresponding to range is called the coefficient of range which is obtained
by applying the formula.
Coefficient of range = mula is used for both discrete and
continuous series)

ly 50% of the
Merits and Demerits of Range
The use of range is appropriate for certain types of data and certain purposes. Among these are
the ranges in temperature during the day or year, and the range in stock prices during some
period of time. It is also used in quality control check.
However range suffers from some limitation i.e. the inclusion of any single abnormal item
changes the range materially. The range does not take into account the distribution of item values
within its limits. The range varies too much from sample to sample taken at random from the
same population.
ii. Quartile Deviation
The quartile deviation also called the semi-interquartile range is the difference between the upper
and the lower quartile divided by 2 (or in other words, it is the middle item lying between the
quartile). If quartile deviation is very small, then it denotes small variation or large uniformity of
the middle items. A characteristics of the quartile deviation is the fact that within +- QD of the
median, approximate items are found.
Quartile Deviation =
Coefficient of Quartile Deviation =
Merits of Quartile Deviation
1. In certain respects it is superior to range as a measure of dispersion.
2. It has special utility in measuring dispersion in case of open-end distributions or one in
which the data may be ranked but measured quantitatively.
3. It is also useful in erratic or highly skewed distribution, where other measure of
dispersion would be warped by extreme values.
Limitations of Quartile Deviation
1. Quartile Deviation ignores 50% items, i.e., the first 25%. As the value of quartile
deviation does not depend upon every observation.
2. It is not capable of mathematical manipulation.
3. Its value is very much affected by sampling fluctuations.

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iii. Average Deviation
Average deviation or mean deviation is obtained by calculating the absolute deviations of each
observation from median or mean and then averaging these deviations by taking their arithmetic
mean.
deviation from median or mean by ignoring +- signs.
Coefficient of Meandeviation
The coefficient of mean deviation is calculated with the objective of comparison. It is calculated
by dividing mean deviation by the average used. If deviations are taken from mean we will
divide it by mean, if deviations are taken either from mode or median we will divide it by mode
or median.
Coefficient of deviation taken from Mean (MDX) =
Coefficient of deviation taken from Median (MDm)
Coefficient of deviation taken from Mode (MDz) =
Coefficient of Meandeviation (Discrete and continuous series)
The formula ting Coefficient of Mean deviation in case of discrete and continuous
series is MD=
Merits and Demerits of Average or Meandeviation
Mean deviation and its coefficient are used in statistical studies for judging the variability, and
thereby render the study of central tendency of a series more precise by throwing light on the
distinctiveness of an average. It is a better measure of variability than range as it takes into
consideration the values of all items of a series. Even then it is not a frequently used measure as
it is not amenable to algebraic process. The major drawback of this method is that the algebraic

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signs are ignored while taking deviations of the items. If the signs of the deviations are not
ignored, the net sum of the deviations will be zero. Hence this method may not give accurate
result.
iv. Standard Deviation
It is most widely used measure of dispersion of a series and is commonly denoted by the symbol
‘ ’ (pronounced as sigma). This was introduced by Karl Pearson in 1893. Standard deviation is
defined as the square-root of the average of squares of deviations, when such deviations for the
values of individual items in a series are obtained from the arithmetic average. It is worked out as
under:
n ( tual mean formula)
= med mean formula)
Calculation of Standard Deviation (Discrete and Continuous Series)
e f lculating standard deviation is as:
= ula)
= ed mean formula)
Coefficient of standard deviation or variation
When we divide the standard deviation by the arithmetic average of the series, the resulting
quantity is known as coefficient of standard deviation which happens to be a relative measure
and is often used for comparing with similar measure of other series. When this coefficient of
standard deviation is multiplied by 100, the resulting figure is known as coefficient of variation.
Sometimes, we work out the square of standard deviation, known as variance, which is
frequently used in the con f analysis of variation.
Coefficient of variation = 00

on a
ffereeff
Va
of di nt fact
ce =
or √
o
Merits of Standard Deviation
1. It is the best measure of variation because it is based on every item of the distribution.
2. It is possible to calculate the combined standard deviation of two or more groups.
3. It is prominently used in further statistical work. For example, in computing skewness,
correlation etc. standard deviation is made use of.
Demerits of Standard Deviation
1. Standard deviation is referred to only as an absolute measure of dispersion and thus it
cannot be used for comparing the two phenomena.
2. As compared to other measures it is difficult to compute.
3. It gives weights to extreme values and less to those which are near the mean. It is because
of the fact that the squares of the deviations which are big in size would be
proportionately greater than the squares of those deviations which are comparatively
small.
Properties of Standard Deviation
1. Combined standard deviation of two or more series can be calculated.
2. The standard deviation of first natural numbers can be obtained.
3. The sum of the squares of the deviation of all the observations from their arithmetic mean
is minimum.
4. Standard deviation is independent of change of origin but not scale.
v. Variance
Variance is the square of the standard deviation. The term was first coined by R.A Fisher in 1913.
The measure of variation is liable for further quantitative analysis. If we are dealing with a
phenomen ffected by a number of variables in that case variance helps us in separating the
ects rs
rian
= √

PART B: UNIT-II CORRELATION
This chapter focuses on the techniques that can be used to study the relationship between two
variables. The type of data that these techniques are intended to analyze is called bivariate data
because they consist of observed values of two variables. The techniques themselves make up
what is known as bivariate analysis. Thus when we have the data on two variables, we are said to
have a bivariate population and if the data happen to be on more than two variables, the
population is known as multivariate population. The scope of this chapter is mainly on bivariate
analysis.
Bivariate analysis is of great importance to business. The results of this sort of analysis have
indeed affected many aspects of business considerably. For example, the establishment of the
relationship between smoking and health problems transformed the tobacco industry. The
marketing strategies of many organizations are often based on the analysis of consumer
expenditure in relation to age or income. There are lots of examples on bivariate data analysis.
Thus the chapter will introduce us to some of the techniques that companies and other
organizations use to analyze the bivariate data.
Using bivariate data, we are generally interested in knowing:
• Whether there exists any relationship or association between the two variables.
• Whether one of the two variables is the cause and the other the effect or in other
words, to study the cause and effect relationship between the two variables.
The first question is answered by the use of correlation and association techniques and the
second question by the technique of regression.
A. Correlation:
Correlation analysis is the study of relationship between two variables. If two variables say ‘x’
and ‘y’ vary in such a way that a change in one is accompanied by a change in the other or in
other words an increase or decrease in the one is accompanied by an increase or decrease (vice-
versa) in the other than the variables are said to be correlated. For instance, there exists some
relationship between family income and expenditure on the luxury items. The relationship
between these two variables or more than two variables can be studied with the help of statistical
tool that is called correlation.

Types of correlation:
There are different types of correlation. Some of the important types are as:
• Positive and Negative
• Simple, Partial Multiple
• Linear and Non-linear
 Positive and Negative correlation
Whether correlation is positive or negative would depend upon the direction of change of the
variable. If both the variables are varying in the same direction, i.e., if one variable is increasing
the other on an average is also increasing, or, if one variable is decreasing the other on an
average is also decreasing, correlation is said to be positive. On the other hand, if the variables
are moving in opposite directions i.e. as one variable is increasing the other is decreasing or vice-
versa, correlation is said to be negative.
 Simple, Partial and Multiple correlation
When only two variables are considered it is a problem of single correlation. In the case of
partial correlation, two or more variables are taken into consideration assuming other variables to
be constant. In multiple correlations three or more variables are studied simultaneously.
 Linear and Non-linear correlation
When variation in the values of two variables have constant ratio, there will be linear correlation
between them. In non-linear correlation, the amount of change in one variable does not bear a
constant ratio to the amount of change in the other related variable.
Degree and Interpretation of Correlation Coefficient:
The range of correlation coefficient denoted by ‘r’ lies between two limits i.e. +1 and -1
• If ‘r’ > 0 it indicates positive correlation
• If ‘r’ < 0 it indicates negatives correlation
• If ‘r’ = 0 it indicates no correlation

Importance of the study of correlation
The study of correlation is of immense use in practical life because of the following reasons:
• Most of the variables show some kind of relationship and with the help of correlation
technique we can measure the degree of relationship existing between the variables.
• Correlation analysis contributes to the understanding of economic behavior, aids in
locating the critically important variables on which others depend. It may reveal to the
economist the connection by which disturbances spread and suggests the paths through
which stabilizing forces become effective.
• It helps in forecasting and planning because changes in variables and its impact can be
estimate beforehand.
• It helps us to reach at reliable conclusions about relationship of variables and uncertainty
is also reduced.
• In the field of industry and commerce, the correlation technique helps to make estimates
like sales, profit, costs, demand etc.
Correlation and Causation:
Correlation analysis helps us in determining the degree of relationship between two or more
variables, but it does not tell us anything about the cause and effect relationship. The
explanation of significant degree of correlation may be any one, or a combination of the
following factors:
• The correlation between the variables may be due to pure chance, this is called
spurious correlation.
• The correlated variables may be influenced by one or more other variables.
• Both the variables may be mutually influenced each other so that neither can be
designated as the cause and the other the effect.

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Limitation of correlation analysis
No doubt correlation analysis is one the most widely used, but at the same time it is also one of
the most widely abused statistical measures. It is abused in the sense that one sometime
overlooks the fact that correlation measures nothing but the strength of linear relationship and
that it does not necessarily imply a cause-effect relationship.
Methods of correlation
There are different methods of ascertaining the relationship between the two variables. These
methods are categorized as graphical methods and algebraic or mathematical methods. With the
help of graphical methods, we can visualize the relationship between the two variables. While as
using algebraic or mathematical method we can determine the extent or degree of relationship
between two variables.
Graphical Methods:
• Scatter Diagram
• Graphic method
 Scatter Diagram
This method is also known as dot diagram, datagram or scatter gram. Scatter diagram is one of
the simplest methods of diagrammatic representation of a bivariate distribution. It provides the
simplest tool of determining the correlation between two variables. The term scatter refers to the
dispersion or spread of the dots on the graph. Suppose we want to measure the heights and
weights of a certain number of people denoted the quantities by ‘x’ and ‘y’ and plot them on a
graph paper ndicular axes. If the origin of axes is taken at(x, y), where x, y
are the mea and y respectively, the points may be scattered all around the
origin. ∑( a measure of correlation between x and y. Following points
should be considered while interpreting the correlation between two variables through scatter
diagram:
i. If the points plotted are very close to each other, it shows high correlation,
otherwise poor correlation is expected.

ii. If the points on the diagram show upward or downward trend, it is a sign of
correlation. If in case no trend is depicted by the points, then the variables are
uncorrelated.
iii. If there is upward trend from left to right, the correlation is positive. On the other
hand if the points show a downward trend from left to right, the correlation is
negative.
iv. The correlation would be perfectly positive or equal to one if all the points on a straight
line starting from left bottom and moving upwards towards the right top. On the other hand,
the correlation would be perfect and negative if all the points lie on a straight line starting
from top left and falls to right bottom.
Merits of Scatter Diagram
i. It is the simplest method involving no mathematical calculations.
ii. Extreme items in the series have no impact on determining the correlation between
the two variables.
iii. The visual inspection of the diagram at the first instance helps everyone to make
estimate about the position.
iv. This method helps us to measure the best fit by free hand method and thus shows the
better approximation results as it is drawn on the graph.
Merits of Scatter Diagram
• This method is suitable only in case of small number of observations.
• Degree of correlation cannot be determined with this method.
 Graphical Method
This method is also known as correlogram or simple graph method. To find out correlation
between two variables ‘x’ and ‘y’, values of ‘x’ and ‘y’ are plotted on the graph and two curves
of both variables are obtained and we draw conclusions about the correlation by looking at graph.

If movement of these curves is in the same direction, correlation is said to be positive and if
movement is in opposite direction, correlation is said to be negative.
Merits of Graphical Method
• This method is easy to understand and does not involve tedious mathematical
calculations.
• It shows trend between the two variables for a period.
Demerits of Graphical Method
• Degree of correlation cannot be determined with this method.
 Algebraic or Mathematical Methods:
Following are the important algebraic methods of correlation:
i. Karl Pearson’s Coefficient of Correlation
ii. Spearman’s Rank Correlation Coefficient
iii. Concurrent Deviation method
i. Karl Pearson’s Coefficient of Correlation
Originated by Karl Pearson about 1900, the coefficient of correlation describes the strength of
the relationship between two variables. Denoted by ‘r’, it is often referred to as Pearson’s ‘r’ and
as the Pearson’s product moment correlation coefficient. It can assume value from -1 to +1
inclusive. A correlation coefficient of +1 indicates a perfect positive correlation, -1 indicates
perfect negative correlation. If there is absolutely no correlation between two variables, the
Pearson’s ‘r’ is zero.
Assumptions of Karl Pearson’s Coefficient of Correlation
• There must exist a linear relationship between two variables.
• The cause and effect relation should exist between two variables.

• Two variables are affected by many independent causes and from a normal
distribution.
Properties of the Coefficient of Correlation
• The formula is based upon the arithmetic mean and standard deviation.
• The value of ‘r’ lies within the range of +1 and -1.
• The value of ‘r’ is independent of change of scale and origin of the variable x and y.
• The value of ‘r’ is the geometric mean of two regression coefficients.
• Probable and standard error can be calculated.
Merits of the Karl Pearson’s Coefficient of Correlation
• It takes into account all the observations of the series.
• It provides numerical measurement of coefficient of correlation.
• This method measures both degree as well as direction of the correlation between the
variables at a time.
• Karl Pearson’s coefficient of correlation is a pure number independent of units.
Therefore, the comparison between the series can be done easily.
• Karl Pearson’s coefficient of correlation technique can easily be applied for higher
algebraic treatment.
Limits of the Karl Pearson’s Coefficient of Correlation
• The correlation coefficient always assumes linear relationship regardless of the fact
whether the assumption is true or not.
• The value of the coefficient is unduly affected by the extreme values.
• As compared to other methods of finding correlation, this method is more time
consuming.

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Calculation of Karl Pearson’s Coefficient of Correlation
The calculation of Karl Pearson’s Coefficient of Correlation can be divided into parts:
• In case of individual series or ungrouped data
• In case of grouped data
Karl Pearson’s Coefficient of Correlation in case of ungrouped data is calculated by the
following three methods:
• Direct Method (Actual Mean Method)
• Product Moment method
• Shortcut Method (Assumed average method)
 Direct Method
This method takes into account deviations from the actual mean of the series and the following
f mula :
=
.
=
.
e riance of x and y
ns deviations in X series from its actual mean
ns deviations in Y series from its actual mean
Standard deviation of X series
Standard deviation of Y series
N = No. of Observations
In above
∑ = Covariance of X and Y

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Therefore = is the direct method to find ‘r’.
The above formula is simple to calculate and easy to understand, as it does not require
calculations of standard deviation of both of the series
Steps to Solve Questions with Direct Method
1. f X and Y series (i.e. calculate
2. X and Y series from their respective series (i.e.
every individual items of X and Y series we have to calculate
the deviations.
3. these deviation in X and Y series and find e s
ns individually of both of the series (i.e. Σ Σ(
4. Mul eviation of X series with its corresponding individual devi
of for all the items of series and find out its sum i.e. {Σ
Σ [(
5. Finally put all the values in the above formula to obtain ‘r’.

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Limitation of Direct Method
This method is lengthy and time consuming process, as true means and deviations of both of the
series has to be calculated first. Moreover the values of standard deviations of two series are also
to be known. To overcome this problem, we can use another formula known as product moment
formula that does involve any calculation of standard deviations.
 Product Moment Method
Use following Formula to ca ‘r’ usi uct ent Method
−( ) − ( )
N is the number of pairs of values in X and Y series
Steps to Solve Questions with Product Moment Me
1. Calculate the sum of X and Y series separately
2. Calculate the sum of squares of X and Y series separately
3. Calculate the sum of product of the corresponding values of X and Y series (
4. Finally put all the values in the formula to obtain ‘r’.
Shortcut Method (Assumed Average Method)
When actual means are in fractions, say that actual means of X and Y series are 45.82 and
63.984, than the calculation of coefficient of correlation by the method discussed above would
involve lot of calculations and involve a plenty of time. To overcome this problem, short cut
method/Assumed mean method is quite useful. Following formula is used to calculate coefficient
of correlation using of sh tcut method:
.
=
− ( ) − ( )
rs to deviations of X series from an assumed mean i.e. (X - A),
rs to deviations of Y series from an assumed mean i.e. (Y - A),

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Σ rs to the sum of the product of the deviation of X and Y series from their assumed
Σ ers to the sum of the product of the deviation of X series from an assumed mean.
Σ ers to the sum of the product of the deviation of Y series from an assumed mean.
Steps to Solve Questions with Shortcut Method (Assumed Average Method)
1. ut the devi f ries from an med mean and show these deviations by
find out Σ ls ut the de rom an assumed mean and
show the find out
2. Next sq s and find out
3. Multiply mmate this to obtain Σ
Coefficient of Correlation in Grouped Data
When we have large number of items or observations, the data needs to be classified into two
way frequency distribution called bivariate frequency table or correlation table. The class
tervals for X series are listed in rows at the left of the table and those for Y series in the column
adings. F a is used to calculate coefficient of correlation for grouped data:
=
Steps to Calculate Coefficient of Correlation in Grouped Data
1. Find out t series and show these d ns by ilarly find out
the deviations ow these deviations by
2. Multiply th the respective frequency (f) of eac d write the
figure on the left hand upper corner of each cell.
3. Su te all the values as c ted in 2 step and denote it by Σf
4. M the frequencies of able X by the deviations of X and obtain the total
Σf similarly obtain Σf

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5. Take the squares of the de s of the series X and multi em by the respective
frequencies and obtain Σf imilarly square the deviatio e Y series andmultiply them by their respective frequencies and obtain Σf
ii. Charles Spearman’s Coefficient of Correlation
This method of finding out co-variability between two variables was developed by the British
psychologist Charles Edward Spearman in 1904. This measure is especially useful when we
come across such variables which are incapable of quantitative measurements, for example
honesty, intelligence, hard work etc. These variables are qualitative in nature and in such cases
we rank individuals in order of merit for their characteristics. Therefore, Spearman’s coefficient
of correlation is good measure in cases where abstract quantity of one group is correlated with
that of the other. The main objective here is to determine the extent to which the two sets of
rankings are similar or dissimilar. In Spearma coefficient of correlation, we take the
difference in ranks, squaring them and finding out aggregate of the squared differences. This
fficient nted by the Greek letter Rho ( and the formula used for its computation is:
= 1
W epresents Spearman’s coefficient
Σ notes the sum of the squared differences between pairs of ranks and N the number of
pairs of observations. The value of this coefficient always lies between +1 and -1.
In Rank correlation, we have three types of cases:
1. When ranks are given
2. When ranks are not given
3. When ranks are equal
When ranks are given
When ranks are given, the steps required to compute rank correlation are as:

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1. Take the difference of the two ranks, i.e. ( nd denote these difference by d.
2. Square these differences and obtain the total Σ
3. Put all the values in the above formula and obtain
When ranks are not given
When actual data is given and not the ranks, in such cases we have to first assign the ranks.
Ranks can be assigned by taking either the highest value as 1 or lowest value as 1. After that we
have to follow the same above mentioned steps.
When ranks are equal
In some cases it is necessary to assign equal ranks to two or more entries. In such a case, we have
to give each entry an av rank. Thus, if two individuals are ranked equal say at fifth place,
they are en the e, 5.5, while if three are ranked equal at fifth place they are given
the rank here equal ranks e entries, an adjustment in the above
formula c g the rank coeffi is made. The adjustment consists of
adding o the value of Σ nds for the number of items with
common ranks. If there is more than one such group of items with common rank, this value is
added as many times the num he i t case be written as:
= 1 −
{6 + + ⋯}
Merits of the Rank Method
1. This method is simpler to understand and easier to apply as compared to the Karl
Pearson’s method.
2. Where the data are of a qualitative nature like honesty, beauty, intelligence, etc. this
method can be used with great advantage.
3. This is the only method that can be used where we are given the ranks and not the actual
data.
4. Even where actual data are given, rank method can be applied for ascertaining the degree
of correlation roughly.

Limitations of the Rank Method
1. This method cannot be used for finding out correlation in a grouped frequency
distribution.
2. Where the number of observations exceeds 30, the calculations become quite tedious.
PART B: UNIT-II: REGRESSION ANALYSIS
As already discussed, correlation show how strong the linear relationship between two variables
might be but it doesn’t tell us exactly what that relationship is. So if we need to know about the
way in which two variables are related or the impact of one variable on other, we have to use the
other part of basic bivariate analysis, i.e. regression analysis. Regression actually means going
backwards. This technique was first developed by the genetics pioneer Francis Galton, who
wanted a way of representing how the height of children was genetically restrained or ‘regressed’
by the height of their parents.
B. RegressionAnalysis
Regression is the determination of a statistical relationship between two or more variables. The
simplest form of this technique is simple linear regression (which is often abbreviated to SLR).
In simple regression, we have only two variables, one variable (defined as independent) is the
cause of the behavior of another one (defined as dependent variable). Regression can only
interpret what exists physically i.e., there must be a physical way in which independent variable
(x) can affect dependent variable (y). It enables us to find the straight line most appropriate for
representing the connection between two sets of observed values. Because the line that we ‘fit’ to
our data can be used to represent the relationship it is rather like an average in two dimensions, it
summarizes the link between the variables. Simple linear regression is called simple because it
analyses two variables, it is called linear because it is about finding a straight line.
Types of RegressionModels
 Simple and Multiple Regression Models.
 Linear and Non-linear Regression Models.

p ula n pa me
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Simple and Multiple Regression Models
If a regression model characterizes the relationship between a dependent variable (y) and only
one independent variable (y), then such a regression model is known as simple regression model.
If more than one independent variable is associated with a dependent variable, then such
regression model is known as multiple regression models.
Linear and Non-linear Regression Models
If the value of a dependent variable (y) in a regression model tends to increase in direct
proportion to an increase in direct proportion to an increase in the value of independent variables,
then such a regression model is called a linear model. The simplest form of linear relationship as
a straight line. The straight line (linear) regression model can be expressed with respect to the
op tio ra ters a and b as:
= + +
a and b are constant values. Where a = y-intercept, that represents average value of the
dependent variables y when x =0
b = slope of the regression line that represents the expected change in the value of y (either
ive or negative) for a unit change in the value of x.
error term that represents the amount of variation of an individual value of y from its
expected value for a given value of x about the regression line.
RegressionEquations/Estimating Lines
Regression lines are based on regression equation. These are also known as estimating equations.
These are algebraic expression of regression lines. As there are two regression lines, so, there are
two regression equations i.e. the regression equation of X and Y which shows the variation in the
value of X for given changes in Y. The regression equation of Y on X describe the changes in the
values of Y for given changes in X. So in bivariate series, we consider two lines of regression. If
two regression lines are identical (on straight line) then correlation coefficient varies between -1
and +1.

Methods of Drawing RegressionLines
The regression lines can be drawn by two methods as given below:
 Free Hand Curve Method
 The Method of Least Squares
Free Hand Curve Method
This method is also known as the method of Scatter Diagram. This is a very simple method of
constructing regression lines. At the same time it is crude and very rough and rarely used method
of drawing regression lines. In this method, the value of paired observations of the variable are
plotted on the graph paper. It takes the shape of scatter diagram over the graphic range of X axis
and Y axis. The independent variable is taken on vertical axis. However this method is crude and
very rough and rarely used method of drawing regression lines.
The Method of Least Squares
In this method the line drawn through the plotted points in such a way that the sum of the squares
of the deviations of the actual Y values from the computed Y values is the minimum or the least.
A line fitted by this method is called line of best fit.
Methods of Calculating RegressionEquation or Derivation of RegressionLines
Following are the two methods to form the two regression equations, that is, equation for Y on X
and for X on Y.
1. Regression equations through normal equations
2. Regression equation through regression co-efficient
RegressionEquations through Normal Equations
The two main equations generally used in regression analysis are:
i. Y on X
ii. X on Y
For Y on X, the equation is

Fo X Y, t
+ b
…
ven e
=
qu
(
ation
−
Y =
)
a
= +
r on he equation is
= +
a and b are constant values. Where a = y-intercept, that represents average value of the
dependent variables y when x =0. In case of Y on X it is an estimated value of Y when X is zero
and vice-versa in case of X on Y
b = slope of the regression line that represents the expected change in the value of y (either
positive or negative) for a unit change in the value of x. It is also known as regression coefficient
and is denoted by byx for Y on X and bxy for X on Y.
RegressionEquation of Y on X
The regression equation of Y on X can be written as Y = a +bX………(.i)
To arrive at two normal equations summate (Σ) equation ( i) Σy =Na ΣX………..(ii)
Now multiply the whole equation (ii) by X, we get ΣXY = a ΣX + b Σ …..(iii)
Equation ii and iii are called normal equation.
Similarly we can calculate for RegressionEquation X on Y
RegressionEquation through RegressionCoefficients
Following are the main methods to calculate regression coefficient Y on X (byx) or X on Y (bxy)
i. Taking deviations from the actual mean
ii. Taking deviation from assumed mean
iii. Applying formula in case of grouped data (continuous series)
Taking deviations from the actual mean
Gi +bX is written as
Y-

= and =
……
…
ΣEq. (
=
ii) red
th
uces
us the =
− = ( − )
regress ion
Si ly can
=
Regr
−
essio
=
n e uatio
−
n of
) w
X
her
n Y c
Taking Y- Y-
In that case we get y =bx
As we know that two normal equations are:
ΣY =Na + b ΣX
ΣXY = a ΣX + b Σ
Writing them in terms of x and y, we get
Σy =a + b Σx… ………(i)
Σxy = a Σx + b Σ ……….(ii)
Now deviations are taken from actual means, in that case
Σx = 0 and Σy = 0
Therefore eq. (i) will be reduced to
Na = 0 or a = 0
Σxy =
Y on X can be written as
milar we calculate RegressionEquation X on Y i.e.
=
= × =
q o an be written as:
( e is the regression coefficient of X on Y and is denoted by byx

And
−
Regr
=
essi n eq
−
uatio
) w
n o
her
Y o
The f
−
ormu
=
lae o calc
−
ulat
) ℎ
e regression coefficient of X on Y is as:
:
×
)
f ped D
/
ata
/
ntin
/
ous
/×
/
/
ies
/ /
/ /
×
/
/
o f n X can be written as:
( e is the regression coefficient of Y on X and is denoted by bxy
Taking Deviation from Assumed MeanX and Y
t
(
−
=
(
=
−
Where dx =(X-A) and dy = (Y-A).
Similarly in case of Yon X, we can calculate the regression coefficient, only denominator is
changed, while as numerator will remain same.
In this method it is needed to find the value of b only. In this method, the regression coefficients
are to be found before solving the regression equation.
Applying Formula in Case o Grou (Co u Ser )
−
=
( )
×
−
Similarly
−
=
( )
×
−
Here ix = class interval of x-variable, iy = class interval of y-variable.
Application of RegressionAnalysis
1. It is used in estimation or prediction of unknown variables.
2. It has greater applicability in establishing relationship between two variables.

e correl
=
ation coeffic
×
ients is
sio n
t have
coefficien
+
e sam
ts bxy a
≥
e sign (ei
nd
3. The regression analysis provides regression co-efficient which are generally used in
calculation of co-efficient of correlation and the square of co-efficient of correlation is
called the coefficient of determination which measure the degree of association that exists
between two variables.
4. It is used in calculating the error involved in estimating the error involved in using the
regression line as a basis for estimation.
Limitation of RegressionAnalysis
Regression analysis suffers from following limitations i.e.
1. It is based on the assumption of linear relationship.
2. The linear relationship between the variables can only be ascertained within limits.
3. The calculation of regression equations is presumed on a static condition of a relationship
between the variables.
Properties of RegressionCoefficient
1. Th the geometric mean of two regression coefficients i.e.
r=
2. Regression coefficients are independent of origin but not of scale
3. If one regression coefficient is greater than one, then other regression must be less than
one, because the value of correlation coefficient cannot exceed one.
4. Both regression coefficients mus th ther positive or negative).
5. The arithmetic mean of regres byx is more than or equal to the
correlation coefficient, r that is

PART- B: UNIT-III: INDEX NUMBERS & TIME SERIES ANALYSIS
This chapter focuses on the price indices and basic time series analysis and is designed to
summarize sets of bivariate data in which one of the variables is time. In this chapter we will
learn to measure changes over time so as to adjust figures for the effects of inflation, analyze
time series and predict future values of time series.
Index Numbers:
When series are expressed in same units, we can use averages for the purpose of comparison, but
when the units in which two or more series are expressed happen to be different, statistical
averages cannot be used to compare them. In such situations we have to rely upon some relative
measurement which consists in reducing the figures to a common base. Once such method is to
convert the series into the series of index numbers. This is done when we express the given
figures as percentages of some specific figure on a certain data. We can, thus, define an index
number as a number which is used to measure the level of a given phenomenon as compared to
the level of the same phenomenon at some standard date.
. In economic sphere, index numbers are often termed as ‘economic barometers measuring the
economic phenomenon in all its aspects either directly by measuring the same phenomenon or
indirectly by measuring something else which reflects upon the main phenomenon.
.
Merits and Demerits of Index Numbers
The use of index number weights more as a special type of average, meant to study the changes
in the effect of such factors which are incapable of being measured directly. It is helpful in
framing policies and comparing the living standards. It is useful tool for prediction and deflation.
However, Index numbers are only approximate indicators and as such give only a fair idea of
changes but cannot give an accurate idea. Chances of error also remain at one point or the other
while constructing an index number.
But one must always remember that index numbers measure only the relative changes. Changes
in various economic and social phenomena can be measured and compared through index
numbers. Different indices serve different purposes. Specific commodity indices are to serve as a

divid
Symb o
=
lically
×
al of
ℎ
base year
Σ =
prices and the quotient is multiplied by100.
, &
= .
measure of changes in the phenomenon of that commodity only. Index numbers may measure
cost of living of different classes of people.
Classification of Index Numbers
Index Numbers can be classified in terms what they measure. In economics the classification is
based on:
i. Price
ii. Quantity
iii. Cost of living
iv. Value
v. Special purpose
Methods of constructing Index Numbers
The index numbers can be constructed through the following two methods:
i. Unweighted Index Numbers and ii. Weighted Index Numbers
There are two types Unweighted index numbers i.e.
a. Simple Aggregative Method
b. Simple Average of Relative Method
Simple Aggregative Method
This is the simplest method of constructing index numbers. When this method is used to
construct a price index, the total of current year prices for the various commodities in question is
ed by the tot
100
Σ
However this method suffers from two main limitations i.e. first the units in which prices of
commodities are given affect the price index. Second no consideration is given to the relative
importance of commodities.
Simple Average of Relative Method
When this method is used to construct a price index, first of all price relatives are obtained for
the various items included in the index and then an average of these relatives is obtained using
any one of the measures of central tendency i.e. arithmetic mean, median, mode, geometric mean

formu
=
monic
s use
m
d fo
× 10
can
ompu
ℎ
ting inde x number.
. ℎ .
then th
=
e
Σ
Σ
r’s co
ce in
×
dex is g
100
index i
=
s:
Σ
Σ
metho d t
× 100
and har ean. When arithmetic mean is used for averaging the relatives, following
la i r c
Σ 0
Similarly we calculate the index using Median, Mode, Geometric Mean and Harmonic Mean.
The main importance of this method is that it does not influence the index because equal
importance is given to all the items. The index is not influenced by the units to which prices are
quoted or by the absolute level of individual prices. Relatives are pure numbers and are therefore
independent of the origin.
Weighted Index Number:
Weighted Index Numbers are constructed by following two types:
Weighted Aggregative Index Number and Weighted Average of Relative Index
Weighted AggregativeIndex Number
In this method appropriate weights are given to different commodities to show their relative
importance in the group. For price index numbe nstruction, quantity weights are used. If W
is the weight attached to commodity, pri iven by:
WW
The important weighted aggregative of this method are:
1. Laspeyres or Base Year Method
2. Paasche or Current Year Method
3. Dorbish and Bowley’s Method
4. Fisher’s Ideal Method
5. Marshall-Edgeworth Method
6. Kelly’s Method
Laspeyres Method
In this he base year quantities are taken as weights. The formula for constructing the
Paasche or Current Year Method

index i
=
s:
Σ
Σ
ethod t
× 100
metic
of b o
=
th pe
w
Paas
=
e inde
+
. Symb
×
The
=
sher’
Σ
Σ
deal i
s id
nd
ea
+
x is th
ndex
e
is
×
ormu la
=
thod
or co
+
also
struc
)
the
tin
×
+ )
T.L Ke
=
Metho
ey has
)
d
s
×
)
sted f
ℎ
ollowing
=
m
+
thod
In this m he currentyear quantities are taken as weights. The formula for constructing the
Dorbish and Bowley’s Method
Dorbish and Bowley has suggested simple arith mean of two indices i.e. Laspeyres and
Paasche so as to take into account the influence riods i.e. current as well as base
periods. The formula for constructing the index is: here L = Laspeyres index and P =
ch x olically it is written as:
2
Fisher’s Ideal Method
100
Fisher’s I e geometric mean of the Laspeyres and Paasche indices.
Fi l i given by the formula:
Σ
100
Σ
Marshall-Edgeworth Method
In this me current year as well as base year prices and quantities are considered. The
f f n g the index is:
Σ(
100
Σ(
Kelly’s
ll ugge e for constructing the index
Σ(
100
Σ( 2
Weighted Average of relativeIndexNumber

rithm et
=
multip
c or geo
, ℎ
metric mea
=
e qu
n.
×
rmul
=
a of this method is as:
ℎ ℎ
Time
interc
×
h
Reve
ange d,
=
test: Thi
hen the res
ℎ
1
= =
.
ves the
×
val
=
gi
.
ue ratio:
1
ℎ = ℎ
= ℎ
,
rent
t per
×
io
eriod
d 0 o
×
n
at 1 o
base p
=
Unlike in weighted aggregative method price relatives were not computed. However, like
unweighted relatives method it is also possible to compute weighted average relative. Therefore,
in this method the price relatives are found by dividing the current year’s price by the base year
price and lying th otient by 100. For purposes of averaging we may use either the
a i The fo
Σ
100 &
Σ
Test of Adequacy of Index Number Formula
Formulae for constructing index number should fulfill criteria specified in the test of adequacy so
that error in measuring index number is reduced to the minimum.
1. Unit test: This test requires that the formula for constructing an index should be
independent of the units in which variables are specified. Hence, average, which is used
should be relative and not absolute.
2. rsal s test requires that if the time subscript of index formula is
t ulting index should be the reciprocal of the original index i.e.
3. Factor reversal test: This test is satisfied if index of price multiplied by an index of
quantity with the same base and current years same coverage and weights of commodities
Vol. of index = Laspeyre, Paasche and Marshall-Edgeworth index method do not
satisfy this test. Fisher index method satisfies this test.
4. Circular test: This test is an extension of time reversal test. This test is satisfied of an
index is constructed with cur p n base period 0, and for current period 2
on base period 1, and for curren eriod 2 and we get the result of one
if all
these indices are multiplied. 1

Consumer Price Index Numbers or Standard of Living Index
Index numbers are generally designed to represent the average change in prices paid by the
ultimate consumer for specialized goods and services over the period of time. As these index
numbers are related with change in prices, which the ultimate consumers would have to pay for
their consumption pattern, so these index numbers are also known as cost of living index
numbers, price of living index numbers or retail price index numbers. These index numbers
measure the effects of living conditions of different classes of people (consumers) for any change
in the level of prices over the period of time. For constructing the consumer price index numbers
the same procedure is adopted as in the construction of wholesale price index numbers.
Method to Construct Consumer Price Index Numbers
Following two methods are used to calculate the Consumer Price Index:
1. Aggregate Expenditure Method (This method is same as that of Laspeyre’s Method)
2. Family Budget Method (This method is same as that of weighted average relative index)
Time series analysis:
In the context of economic and business researches, we may obtain quite often data relating to
some time period concerning a given phenomenon. Such data is labeled as ‘Time Series’. More
clearly it can be stated that series of successive observations of the given phenomenon over a
period of time are referred to as time series. In short time series refers to such a series in which
one variable is time and the analysis of such data is known as time series analysis.
Importance of Time Series Analysis
1. To study the past behavior of the variable under study and to the causes and directions of
the fluctuations.
2. It helps in making a comparison between the behaviors of different time series when the
data are recorded systematically.
3. It is helpful in predicting the future behavior of a particular variable.
4. It is also useful in forecasting the business cycle or trade cycles.

Limitations of Time Series Analysis
1. The time series data is available in huge quantity but the right kind of data that is required
for analysis is difficult to have.
2. The individual observations in the time series are a composite of many factors which may
be pulling together or in a reverse direction at any point of time.
3. The different forces which affect the economic data are not regular in their operations.
The influence of factors like climate, customs and traditions is not regular.
Components of Time Series
It is generally observed that the values of a time series show various types of fluctuations over a
period of time which are caused by multiple forces. These fluctuations are also called the
variations, the pattern, the movements, the element of time series or the components of time
series. Followings are the basic components of a time series.
(i) Secular trend or long term trend that shows the direction of the series in a long period of time.
The effect of trend (whether it happens to be a growth factor or a decline factor) is gradual, but
extends more or less consistently throughout the entire period of time under consideration.
Sometimes, secular trend is simply stated as trend (or T).
(ii) Short time oscillations i.e., changes taking place in the short period of time only and such
changes can be the effect of the following factors:
(a) Cyclical fluctuations (or C) are the fluctuations as a result of business cycles and are
generally referred to as long term movements that represent consistently recurring rises and
declines in an activity.
(b) Seasonal fluctuations (or S) are of short duration occurring in a regular sequence at specific
intervals of time. Such fluctuations are the result of changing seasons. Usually these fluctuations
involve patterns of change within a year that tends to be repeated from year to year. Cyclical
fluctuations and seasonal fluctuations taken together constitute short-period regular fluctuations.
(c) Irregular fluctuations (or I), also known as Random fluctuations, are variations which take
place in a completely unpredictable fashion. All these factors stated above are termed as

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