slmna1-11 ecob 07 correlation goutam
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CLASS XI
ECONOMICS
Basic Concepts
CORRELATION
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Introduction
Many times, we come across problems which involve
two or more variables.
Example: Rainfall and production of rice
Road accidents and number of cars
Sales and profit etc.
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We find that there is always some relationship between
the two variables.
When one variable changes, the other also changes in
the same or in the opposite direction, we say that the
two variables are correlated.
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A single number that describes the degree of relationship
between two variables.
Correlation
Relation between income and
consumption. With rise in
income consumption increasesand vice-versa
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Determines the degree of relationship between
variables. By knowing one variable other
variables can be chalked out.
Helps in measuring the relationship between
the two variables
Significance
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Helps in formation of laws and concepts in economictheory
Economists establish relationship between the
variables like demand and supply, price level etc.
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Helps in framing policies
Helps in business activities to take fruitful decisions
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REMEMBER
Correlationdoe
sn'tmeasurec
auseandeffect
relation.
Itmeasuresonly
degreeandinte
nsityof
relationship.Exampl
e:Lowrainfallisrelated
tolow
agriculturalprod
uctivity.Butlow
production
maybeduetoo
therreasonssu
chaspoor
qualityofseeds,
traditionalmetho
dsof
agricultureetc.
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Negative correlationPositive correlation
Types
OX
Y Y
OX
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Positive correlation: Variables move together insame direction
Example: Advertising and sales
Negative correlation: Variables move in opposite
direction
Example: Higher the price of petrol less will be its
demand
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The value of correlation (r) always lies between1 to + 1 ( 1 < r < + 1).
Value of r lies between 0 and 1 Positive correlation
Value of r lies between 0 and -1 Negative
correlation
r = 0 No correlation
Properties
-1 0 +1
Perfect
Negative
Correlation
No Correlation Perfect
Positive
Correlation
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Methods of Studying Correlation
Scatter Diagram Spearmans Rank
Correlation
Karl Pearsons
Coefficient of
Correlation
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Simplest way of determining the relationship between two
variable in a special type ofdotted chart
Scatter Diagram
Y
XO
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1
2
3
4
5
Positive Correlation
Negative Correlation
Perfect Negative Correlation
Perfect Positive Correlation
No Correlation
Types of Scatter Diagram
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As the value of one variable increases, the
value of other variable also increases.
Example: Temperature and sale of cold drinks.
Positive Correlation
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Example: Watching TV and marks scored.
Students spending more time watching TV tend to
score less marks in class and vice-versa.
As the value of one variable increases, the value ofother variable also decreases.
or
Two variables move in different direction
Negative Correlation
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Change in two variables in equal proportion in
the same direction
Y
XO
Perfect Positive Correlation
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Change in two variables in equal proportion
in an inverse direction
Perfect Negative Correlation
Y
XO
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The points are randomly scattered on graph
Example: Rainfall in India and production of cars in
Germany
Ra
infa
ll
Production of cars
No Correlation
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Your Turn
Following are the details of heights and weights of 5students of a class, draw a scatter diagram and
determine the form of association.
Weight(Kg)
50 65 60 50
Height(inches)
62 72 70 58
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A mathematical method for
measuring the linear relationship
between the variable X and Y
It indicates the quantitative
relationship between two
variables.Karl Pearson, a pioneer of statistics,
developed ideas of correlation and
regression that have been widely
applied across different branches of
science.
Karl Pearsons Coefficient of
Correlation
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1. Direct method
2. Indirect method
2 2
=
xyr
x y
( ) ( )
( ) ( )2 2
2 2
=
dx dy dxdy
Nr
dx dy dx dy
N N
Methods of Calculation
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Example: Calculate the correlation between the
weights and heights of 9 students by directmethod.
Weight(Kg)
48 49 50 51 52 53 54 55 56
Height(cm)
100 105 105 104 110 115 125 130 132
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Weight (X) Height (Y)
48
49
50
51
52
53
54
55
56
100
105
105
104
110
115
125
130
132
Calculate mean for X and Y
468
9
52
XX
N
=
=
=
1026
9
114
YY
N
=
=
=
468X= 1026Y=
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X Y
48
49
50
51
52
53
54
55
56
100
105
105
104
110
115
125
130
132
- 4
- 3
- 2
- 1
0
1
2
3
4
- 14
- 9
- 9
- 10
- 4
1
11
16
18
56
27
18
10
0
1
22
48
72
16
9
4
1
0
1
4
9
16
196
81
81
100
16
1
121
256
324
x y
254xy=
114
,
X = 52
=
Here
Y
= x X X = y Y Y2y2x
21176y =2 60x =
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2 2
254
60 1176
95
r
.
xy
x y
=
=
=
Positive correlation between weight and height
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Example: Calculate Karl Pearsons correlation
between price and demand by indirect method.
Price (Rs) 14 16 17 18 19 20 21 22 23
Demand(Quantity)
84 78 70 75 66 67 62 58 56
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Price(X)
Demand(Y)
14
16
17
18
19
20
21
22
23
84
78
70
75
66
67
62
58
56
- 6
- 4
- 3
- 2
-1
0
1
2
3
14
8
0
5
- 4
-3
-8
-12
-14
20( )
dx X A
A
=
= 70( )
dy Y A
A
=
=
12d x=
14d y=
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- 6
- 4
- 3
- 2
-1
0
1
2
3
14
8
0
-5
- 4
-3
-8
-12
-14
36
16
9
4
1
0
1
4
9
196
64
0
25
16
9
64
144
196
-84
-32
0
10
4
0
-8
-24
-42
2
8 0d x=2
714d y = 176.d yd y=
dx X A= dy Y A=
14d y= 12d x=
2d x 2d y .dx dy
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( ) ( )
( ) ( )
2 2
2 2
2 2
12 14176
9
12 1480 7149 9
. -
( ) ( )( )
dx dy
dx dy Nr
dx dy dx dy
N N
r
=
=
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176 (18.66)
144 19680 714
9 9
176 (18.66)
64 692.33
194.66
8 26.31
194.66210.48
.92
.
r
r
r
r
r
Itisacaseofstrongnegativecorrelation
=
=
=
=
=
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Calculate correlation of coefficient between
variable X and Y.
Your Turn
Variable (X) 6 2 4 9 1 3 5 8
Variable (Y) 13 8 12 15 9 10 11 16
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Charles Edward Spearman, an
English psychologist Known for
work in statistics and for
Spearman's rank correlation
coefficient.
Helps in calculating the correlationofqualitative variables
Based on ranks of items rather than
their actual values
Can be used even when actual
values are unknown
Spearmans Rank Correlation
Example: To know the correlation
between honesty and wisdom, one
can use this method by assigning
ranks to items.
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When ranks are given
When ranks are not given
2
3
61k Dr
N N=
Formula for Different Cases
2
N Number of pairs of observations
D Total of squares of the
differences of corresponding ranks
=
=
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2 3 3
1 1 2 2
3
1 1
612 12
1
( ) ( ) ....
k
D m m m mr
N N
m Number items of equal ranks
+ + + =
=
When ranks are equal or repeated
Formula for Different Cases
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In a singing competition, two judges accorded following
ranks to 10 contestant.
Judge A 10 8 5 3 6 1 2 9 7 4
Judge B 10 6 5 4 7 9 8 2 1 3
When ranks are given
JUDGE AJUDGE BD =R1 R2 D2
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1 2
10
8
5
3
6
1
2
9
7
4
10
6
5
4
7
9
8
2
1
3
0
2
0
-1
-1
-8
-6
7
6
1
0
4
0
1
1
64
36
49
36
1
2
192D =
Here,
R1=Row 1
R2= Row 2
2
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2
3
3
61
6 1921
10 10
11521
990
1 1 16
0 16
( )
.
.
k
Dr
N N=
=
=
=
=
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Calculate Spearmans coefficient of correlation between
marks assigned to 7 students by tow judges in a poem
competition.
Judge A 25 12 40 20 8 15 10
Judge B 12 10 18 16 6 25 15
When ranks are not given
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Judge A Rank
(R1)
JudgeB
Rank
(R2) D = R1- R2 D2
25
12
40
20
8
15
10
2
5
1
3
7
4
6
12
10
18
16
6
25
15
5
6
2
3
7
1
4
-3
-1
-1
0
0
3
2
9
1
1
0
0
9
4
224D =
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3
3
3
61
6(24)1
7 7
1441
7 7
1441
336
k
Dr
N N=
=
=
=
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k
1 0.43
r 0.57
Positive correlation
=
=
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Your Turn
Marks of 10 students in Hindi and English are given below.
Find the correlation between the two subjects.
Hindi 80 38 95 30 74 84 91 60 66 40
English 85 50 92 58 70 65 88 56 52 46
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Marks of 7 students in accounts and statistics out
of 50 marks are given.
Subject Marks
Accounts 40 42 35 40 47 42 30
Statistics 38 45 42 35 30 40 35
When ranks are equal or repeated
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Accounts Statistics R1 R2 D = R1- R2 D2
37
42
35
40
47
42
30
38
45
42
35
30
40
35
3
5.5
2
4
7
5.5
1
4
7
6
2.5
1
5
2.5
-1
-1.5
-4
1.5
6
0.5
-1.5
1
2.25
16
2.25
36
0.25
2.25
260D =
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2 3 3
1 1 2 2
3
3 3
3
3
1 16
12 121
1 16 60 2 2 2 2
12 121
7 7
1 16 60 6 6
12 121
7 7
( ) ( ) ....
,
( ) ( ) ....
( ) ( ) ....
k
D m m m m
r
N N
Here m number of items of equal ranks
+ + + =
=
+ + + =
+ + + =
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[ ]
1 16 60
2 21
336
6 611
336
3661
336
1 1 08
0 08
.
.r
+ + =
=
=
=
=
Negative correlation
N FINGER TIPS
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Now, on your FINGER TIPS
Qualitative Variables:Those variables which cannot
be measured such as bravery, wisdom, beauty etc.
Correlation: A single number that describes the
degree of relationship between two variables. When
both the variables move in same direction they are
said to the positively correlated and when move in
opposite direction, it is called negative correlation.
Scatter Diagram: It is a graphic method of studying
correlation.
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Ranking: Allotment of rank on the basis of ascending
or descending order
Negative correlation:When the two variables movein opposite direction, it is called negative correlation.With an increase in the value of one variable there is
a decrease in value of other.
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