A correlation analysis.ppt 2018

Ravindra Kumar Saini
Deptt of Management of Studies,
COER School of Management
Roorkee, Uttrakhand

1. The Concept of Correlation
2. Types of Correlation
3.

This concept
was first
propounded
by French
space
scientist A.
Bravis .A. Bravis (1811-63)

Correlation technique
in the form of graphical
presentation was first
investigated by
Sir Francis Galton
(Father of
Regression
Analysis)
Sir Francis Galton
16 Feb. 1822 – 17
Jan. 1911

Karl Pearson
(27 March 1857 – 27 April
1936)
Karl Pearson was
English Mathematician
who has been credited
with establishing the
discipline of
Mathematical Statistics.
In 1911 he founded the
world's first university
statistics department at
University College
London.

 Correlation is a statistical technique which measures the
degree and direction of relationship between two or more
than two variables.
 Correlation is a statistical method used to determine the
degree to which two variables are related.
 Correlation, a statistic that measures the strength and
direction of a linear relationship between two quantitative
variables.
 Correlation study the relatedness between two or more than
two variables.

A. Degree of Relationship: Degree of relationship may be
Perfect, very Strong, strong, moderate, weak, or none.
B. Direction of Relationship:
1. Positive (Direct) Relationship: variables move
in same direction.
2. Negative (Inverse) Relationship: variables
move in opposite directions.
‘r’ (Correlation) ranges in value from –1.0 to +1.0
-1.0 0.0
+1.0
Perfect Negative No Relation Perfect
Positive

S. No. Degree of
correlation
Positive Negative
1 Perfect +1 -1
2
Very Strong
Above +.9 and up to
+.99
Below -.90 and up to -.99
3 Strong Above + .75 and up to
+.9
Below -.75 and up to -.90
4 Moderate Above +.25 and up to
+.75
Below -.25 and up to-.75
5 Low (Weak) Above 0 and up to +.25 Below -0 and up to -.25
6 Absence 0 0

Degree and Direction of Relationship
LRF, QSL, QSD, QF

A correlation analysis.ppt 2018

Positive or Direct Correlation: If two variables
move together in same direction, the correlation
between them is said to be positive or direct
correlation.
X 10 12 15 18 20
Y 15 20 22 25 37
X 80 70 60 40 30
Y 50 44 30 20 10

X 100 90 60 40 30
Y 10 20 30 40 50
X 20 30 40 60 80
Y 40 30 22 15 10

 Linear Correlation: correlation said to be linear if
the proportionate change in the value of the two
variables constant.
 Exp. If x and y are two variables, then in linear
correlation is always constant. In linear correlation the
graph drawn with the pair of values of x and y is always
a straight line.
x 10 20 30 40 50
y 20 40 60 80 100
y
x
∆
∆

 Simple Correlation: The correlation between two variables
known as simple correlation.
Multiple or partial Correlation: When more than two
variables are studied this is the problem either multiple
correlation or partial correlation.
 Partial Correlation: more than two variables are involved
but the relationship is studied between two variables only,
keeping other variables constant.
 Multiple Correlation: When three or more variables are
studied it is called multiple correlation. Three or more
variables are studied simultaneously.

 A. Graphical Methods:
 Scatter Diagram Method
The Graphic Method
 B. Mathematical Method:
 Karl Pearson’s coefficient of Correlation
Spearman’s Ranking Method
Concurrent Deviation method

Scatter Diagram Then Value of r
There is no linear
correlation
between two
variables
r = o
Y
X

There is linear
prefect degree
positive
correlation
between two
variables
r = +1
Y
X

There is no linear
correlation
between two
variables
r = -1
Y
X

There is positive
correlation between
two variables but with
limited degree
0<r<1
(Always between 0
and 1)
Y
X

There is negative
correlation between
two variables but with
limited degree
r = -1<r<0
(Always between -1
and 0)
Y
X

1. Both the curves drawn on the graph are moving
in the same direction (either upwards or
downwards) correlation is said to be positive.
2. If curves are moving in the opposite directions,
correlation is said to be negative.
3. If the unpredictable fluctuations in the curves
showing no similarity, there may be no
correlation or it may of low degree correlation.
1. Both the curves drawn on the graph are moving
in the same direction (either upwards or
downwards) correlation is said to be positive.
2. If curves are moving in the opposite directions,
correlation is said to be negative.
3. If the unpredictable fluctuations in the curves
showing no similarity, there may be no
correlation or it may of low degree correlation.

Karl Pearson, was a great biologist and statistician.
He suggested a mathematical method for measuring
the value of linear relationship between two variables.
It is most widely used method in practice. It is
denoted by ‘r’ .
The formula for calculating ‘r’ is
( , ) ( )( )
( . . )( . . ) x y
Cov X Y X X Y Y
r
S D ofX S D ofY nσ σ
∑ − −
= =

var
x yd d
Co iance
N
∑
=
2
. . x
X
d
S D or
N
σ
∑
=
2
. . y
y
d
S D or
N
σ
∑
=
Finally, following formula is used to find the value
of correlation:
2 2
x y
x y
d d
r
d d
∑
=
∑ ×∑

2 2 2 2
( )( )
( ) ( )
x y x y
x x y y
N d d d d
r
N d d N d d
or
∑ − ∑
=
∑ − ∑ ∑ − ∑
2 2 2 2
( )( )
( ) ( )
N XY X Y
r
N X X N Y Y
∑ − ∑ ∑
=
∑ − ∑ ∑ − ∑

Name of
Teacher
Performance (y) Experience in
Years (x)

S. No Name of Teacher Grade Points
(FDP)
Experience
(In Years)
1 Dr. Sidhharth Jain 9 10
2 Mr. Arvind Kumar 6.5 5
3 Mr. Ashutosh Shukla 8.5 7
4 Ms. Supriya 8 7
5 Mr. Raghav Janwani 6.5 2
6 Mr. Akhil Dangwal 7.5 8
7 Mr. Ajay Sharma 6 5
8 Dr. Himadri Phukan 7.5 10
9 Ms. Divya Mishra 8 8
10 Dr. Maridula 8.5 8

Experience
in years (x)
Points (y)
X2
Y2 XY
10 9 100 81 90
5 6.5 25 42.25 32.5
7 8.5 49 72.25 59.5
7 8 49 64 56
2 6.5 4 42.25 13
8 7.5 64 56.25 60
5 6 25 36 30
10 7.5 100 56.25 75
8 8 64 64 64
8 8.5 64 72.25 68
10 9 100 81 90
7 8.5 49 72.25 59.5
∑X= 87 ∑Y= 93.5 ∑x2 = 693 ∑y2=739.75 ∑XY = 697.5

2 2 2 2
( )( )
( ) ( )
N XY x Y
r
N X X N Y Y
∑ − ∑ ∑
=
∑ − ∑ ∑ − ∑
235.5
317.2668
r =
8370 8134.5
(8316 7569) 8877 8742.25
r
−
=
− −

Result: There is high degree of positive correlation
between experience and Grade points.
Application of Ms Excel: above value can be calculated
with the help of Ms Excel.
235.5
(747 134.75
r =
×
235.5
100658.30
r =
235.5
317.2668
r =
0.742277r =

1. There is a linear relationship.
2. There may be existence of more than two
variables.
3. There is a cause and effect relationship between
two variables
4. The value of Karl Pearson’s coefficient of
correlation always between +1 and -1 can not be
greater than 1 in any case.

1.The value of r always lies between +1 and -1
2. It is not affected by change of origin or change of
scale.
3.It is a relative measure. It does not have any unit.

It is used to determine the limits coefficient of
correlation. It is a very old measure of testing the
reliability of ‘r’. The formula for calculating PE of
Karl Pearson’s coefficient of correlation is as
follows:
2
.6745(1 )
Pr ( . .)
r
obableError P E
N
−
=

S. No.
Value of Correlation
(r)
then
1. r< P.E.
There is no evidence of
correlation between two variables
2. r>6P.E. Correlation will be considered
significant
3. P.E.<r<6P.E. cannot say anything about the
significance of r
4.
Note:
A. The number of pairs of observation must be large other
wise result may be wrong.
B. Study should be based on random sampling.

P.E. can determine limits within which r of the total
population or other samples selected from the
same population at random will lie between
minimum limit and maximum limit with a
probability of 50%.
Maximum (Upper) limit = r+P.E.
Minimum limit (lower) = r-P.E.
P.E. can determine limits within which r of the total
population or other samples selected from the
same population at random will lie between
minimum limit and maximum limit with a
probability of 50%.
Maximum (Upper) limit = r+P.E.
Minimum limit (lower) = r-P.E.

Correlation
Constant
Variable
Order of
Coefficient of r
r12 zero
Zero order
coefficient
r12.3 or r23.1 or
r31.2
One
First order
coefficient

Meaning: Coefficient of determination explain the
percentage of variation in the in the dependent
variable Y, that can be expressed in terms of the
independent variable.
Coefficient of Determination (r2
) =
Explained
Variance
Total Variance

Explained Variance: variations in series Y due
to variance in series X known is explained
variance.
Unexplained Variance: such variations in Y
series are called unexplained which are not due to
variations in X series.
Total variance =
Explained Variance + Unexplained
variance

Exp.
Suppose coefficient of correlation between X and y
is +.742277 then coefficient of determination will
be .81it indicates that:
1. 55.10% variations in the Y series are due to X
Variable.
2. 44.90% variations in Y series are due to other
variables.
Coefficient of non-determination =
2
1 r−

Theoretical work on t- distribution was
done by W. S. Gosset (1876-1937). The t-
distribution is used when sample size is
30 or less than 30 and population is
normally distributed.
2
r
t
1 r
n 2
=
−
−

Sample statistics estimate the
Population parameters like:
1. M tries to estimate μ
2. r tries to estimate ρ (“rho” –
Greek symbol not “p”)
r correlation for a sample based
on a the limited observations we
have
ρ actual correlation in population
the
true correlation
Population :
Correlation is ρ
‘rho’
Population :
Correlation is ρ
‘rho’
Sample:
Correlation
is ‘r’
Sample:
Correlation
is ‘r’

Is there evidence of a linear relationship between experience of a
teacher and grade at the .05 level of significance?
2
r
t
1 r
n 2
=
−
−
2
.742277
1 .(.742277)
12 2
=
−
−
0.4490
.7422
242
77
0
98
1
=
.742277
0.0449
=
0.2119019
.742277
3 5
35
. 03t = =

0
tα/2
α/2=.025
2 2
r .742277
t 3.503
1 r 1 .(.742277)
n 2 8 2
= = =
− −
− −
Conclusion:
There is evidence of a
linear relationship
between Experience
of a Teacher and
Grade points at the
5% level of
significance
Decision: Reject
H0
Reject H0Reject H0
α/2=.025
-tα/2
Do not reject H0
-2.228
2.228 3.503
d.f. = 12-2
= 10
Acceptance
Area

A correlation analysis.ppt 2018

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A correlation analysis.ppt 2018