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CORRELATION ANALYSIS
Content
•
•
Type of correlation
Scatter diagram
•
The covariance
•
The correlation co-efficient
•
Probable Error
•
Spearman’s rank correlation
Formulae
1
...
Correlation co-efficient
a)
b)
c)
d) When deviations are taken from actual mean
...
Probable error
4
...
2
...
There is a positive correlation between the variables if it as negative
then negative correlation
...
3
...
4
...
TYPES OF CORRELATION
Let
be the two variables under study
...
e
...
e
...
These three situations can be stated as follows statistically (i)
positively correlated (ii) negatively correlated (iii) no-correlation
...
Let and be the two
variables under study with observation each
...
Y-Values
25
20
15
Y-Values
10
5
0
0
2
4
6
8
10
12
14
Figure 1
Fig 1 represents that there is a positive correlation between the variables
2
and
...
Y-Values
17
...
5
16
Y-Values
15
...
5
0
1
2
3
4
5
6
7
8
9
Figure 3
In Fig 3 we are not able to recognize a pattern, i
...
, there is no correlation between
and
...
3
THE COVARIANCE
Covariance states only the nature of the relationship between the variables, but we also want to know
how far these variables are associated with each other, we use correlation co-efficient to find the same
...
Correlation co-efficient was defined by Karl Pearson in 1890
...
e
...
You are given the following data relating to aptitude scores and productivity index
Aptitude score (X)
9
18
18
20
20
23 Productivity
index (Y) 33
23
33
42
29
32 Find the co-efficient of
correlation between aptitude scores and productivity index
...
PROBABLE ERROR AND ITS SIGNIFICANCE
Probable error of correlation co-efficient is a measure of reliability of , which is given by the formula
...
, correlation coefficient is definitely not significant
...
5
, correlation co-efficient is
EXERCISES
1
...
Age of husband : 20
22
23
25
25
28
Age of wife :
18
20
22
24
21
26
29
26
30
25
30
27
34
29
2
...
No
...
passed :
480 300
450
560
450
300
3
...
Price (in shs)
22 23 24 25 26 27 28 29 30
Demand (in 000 units)
: 18 19 19 16 17 16 16 15 13 11
4
...
Calculate co-efficient of correlation using Karl Pearson’s method
...
Given below are the monthly incomes and their net saving of a sample of 10 supervisory staff
belonging to a firm
...
Employee No
...
From the following data calculate the correlation co-efficient
...
From the following details find the value of
and
...
From the following data, compute the co-efficient of correlation between
series
series
No
...
10
...
11
...
(i)
(ii)
Comment on the value of
...
How do you interpret when it is
?
(a) When
, it means is proportion to , it means as increases also increases
...
This ensures perfect negative association
...
13
...
Age (years):
15
16
17
18
19
20
No
...
A company wanted to assess the impact of and expenditure
table presents data for past 8 years
...
Comment on the value
...
From the following data, compute the co-efficient of correlation between
series
series
No
...
Find the correlation co-efficient
...
following
17
...
(ii)
(iii)
Comment on the value of
...
In correlation co-efficient we used the actual value to measure the degree of relationship; here we use
the rank of the given data
...
Ranks are obtained by arranging the data in order of their
merits
...
Example 1
Calculate rank correlation from the following
...
This common rank is the average of the ranks which these items would have attained if
they were slightly different from each other and the next item gets the rank next to the ranks (actual rank)
...
This adjustment factor is added to
values
...
Marks in statistics :
15 20 28 12
Marks in Accountancy : 40 30 50 30
is the number of items repeated ) for each reapeated
40
20
60
10
20 80
30 60
Solution
15
40
7
3
4
16
20
30
5
...
5
0
...
4
12
36
8
5
3
9
40
20
3
7
-4
16
60
10
2
8
-6
36
20
30
5
...
5
0
...
5
In series: 20 is repeated 2 times, the actual ranks will be 5 and 6
the average is 5
...
Here
In Y series : 30 is repeated 3 times, it shares 4th, 5th and 6th position 30 gets the average
...
Ten competitors in a beauty contest are ranked by three judges in the following order
...
Find rank correlation co-efficient for the following data and give your comments
...
Find the rank correlation co-efficient and comment
Marks in mathematics : 60 70 50 40 80 90
Marks in statistics :
64 58 72 44 86 80
85 96
95 81
4
...
Marks in statistics : 84 56 89 58 59 67 74 78
Marks in law :
38 69 56 58 63 78 87 77
5
...
Father : 65
60
67
63
74
Son :
49
45
57
40
60
Obtain the rank correlation co-efficient and comment on the results
...
Calculate the rank correlation
...
Height of fathers and sons are given in inches
...
Compute the rank correlation between I
...
Personal:
A
B
C
D
E
F
I
...
Ten competitors in a beauty contest are ranked by three judges in the following order
...
10
...
Q’s of 7 father and their eldest sons
...
Q :
90
98
103
104
105
110
114
Sons I
...
11
...
and
2
...
Find
3
...
and
...
e
correlation) and the extend to which they are correlated
...
i
...
Regression methods are meant to
determine these functional relationships
...
In regression analysis there are two types of variable; namely the dependent and independent variable
...
If curve is straight line, it is called the line of regression
...
Let us consider the relationship between two variable and
...
(i)
Line of regression on : Here is the dependent variable of is independent variable and
this line gives the best estimate for the value of for any specified value of
...
The regression equation for on is given by
In the above equations
for any specified value
and are constants and are obtained by least squares principle
...
”
Regression Equation of on is
...
Actual
Estimated
is minimized by partially differentiating it Value Value
zero
...
and respective and equating them to
These equations are known as normal equations
...
Regression equation of on :
The regression equation of on
is
Here we have to minimize
Similarly here we get
As the normal equations and the values of
Example
Find the regression equation
:
2
4
6
:
5
15
20
Solution
2
4
6
8
Total
20
The normal equation for
on for the following data
...
25
10
225
60
400
120
625
200
1275
390
on
...
The regression co-efficient is a measure of change in
dependent variable corresponding to a unit change in independent variable
...
3 we get
Multiplying and dividing by
for
we get
on
Similarly
We get
for
for
on
on
13
By
for
on
Now the two regression equations are
for on
for
on
Computing of regression co-efficient
Regression co-efficient for
on
is given by
Let
Be the deviation from the original units then
If the deviations are taken from actual means then
Similarly for on
we have
Let
Be the deviation from the original units then
If the deviations are taken from actual means then
Example 1
You are given the following
Mean
10
15
14
S
...
Regression equation of
on
...
The range of regression co-efficient is
to
2
...
Property hold the same for regression co-efficient on
...
If one of the regression co-efficient is greater than unity the other one is less than unity
...
e, if
then
5
...
16
DISTINCTION BETWEEN CORRELATION AND REGRESSION ANALYSIS
As we have discussed about these two analysis separately, now let us distinguish between them
...
It studies the relationship between two
variables
2
...
Correlation analysis is confined only to
the study of linear relationship
4
...
It studies the nature of covariance
...
The relationship may be purely a
chance and it may not have practical
relevance
There is perfect relationship and it has
practical relevance
6
...
e
Regression co-efficient is not symmetric i
...
Correlation co-efficient is a relative
measure and its range is -1 to 1
Regression co-efficient is an absolute measure
and its range is
to
Exercise
1
...
Age
56
42
72
36
63
47
55
49
38
42
68
60
on
and
on from the following table and estimate the blood pressure
Blood pressure
147
125
160
118
149
128
150
145
115
140
152
155
17
2
...
9
7
expenditure
on annual profit
...
Estimate the profit for
and
expenditure of Shs 8 thousand
...
Calculate the two lines of regression for the following data
...
You are given the following in formation about advertisement and sales
...
Exp
...
D
5
25
Correlation co-efficient = 0
...
Calculate the two regression equations
...
Find the likely sales when advertisement expenditure is Shs 25 million
5
...
of pairs = 12
A
...
50
125
...
D
13
...
85
Summation of products of corresponding deviations from respective means = 2176
...
6
...
Mean of A = 39
...
5
Std
...
of A = 10
...
Dev
...
8
Co-efficient of correlation between A and B =0
...
18
7
...
Mean
20
10
S
...
Find the two regression equations
...
If
Mean
25
22
S
...
10
...
25
43
28
46
35
49
32
41
31
36
36
32
29
31
38
30
34
33
32
39
Compute co-efficient of correlation between and
19
11
...
of students
: 800 600 900
700
500
400
No
...
To study the effect of rain on yield of wheat the following results were obtained
...
D
Yield :
800
12
Rainfall :
50
2
Estimate the yield when the rainfall is 80 inches
...
Find the two regression equations, regression co-efficient of correlation from the following figures