CORRELATION
One of the statistical concepts that is most related to this type of analysis is the correlation
coefficient. The correlation coefficient is the unit of measurement used to calculate the intensity
in the linear relationship between the variables involved in a correlation analysis, this is easily
identifiable since it is represented with the symbol r and is usually a value without units which
is located between 1 and -1. In other words, correlation examines the relationship between pairs
of scores. The greater the association between the two variables, the more accurate we can
predict the relationship between the variables. Correlation between two variables can be either
a positive correlation, a negative correlation, or no correlation. Following are a few methods
to calculate correlation
1. Pearson’s product moment (r)
2. Rank-order correlations (Spearman’s ‘p’ or ‘rho’)
3. Biserial correlation
4. Tetrachoric Correlation
5. Multiple correlation
•
Positive correlation: A positive correlation between two variables means both the variables
move in the same direction. An increase in one variable leads to an increase in the other variable
and vice versa. For example, happiness leads to increase psychological well-being. Positive
correlation is denoted by + (plus)
•
Negative correlation: A negative correlation between two variables means that the variables
move in opposite directions. An increase in one variable leads to a decrease in the other variable
and vice versa. For example, stress decreases performance. Negative correlation is denoted by
– (minus)
•
Weak/Zero correlation: No correlation exists when one variable does not affect the other. For
example, there is no correlation between the number of years of school a person has attended
and the letters in his/her name.
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Following are the interpretation
± 0.00
± 0.20
±0 .40
± 0.70
𝑡𝑜
𝑡𝑜
𝑡𝑜
𝑡𝑜
±
±
±
±
Indifferent or negligible relationship
Low correlation or very slight relationship
Substantial or marked relationship
High correlation
0.20
0.40
0.70
1.00
PEARSON’ PRODUCT MOMENT METHOD: In 1896, Karl Pearson, scientist associated
with Galton’s laboratory, developed a rigorous mathematical treatment of these matters. Galton
was the first person to use the symbol ‘r’ for a simple correlation coefficient. The term product
moment is borrowed from physics. It refers to the measure of the strength of a linear association
between two variables — denoted by r.
Formula:
X
Y
X2
Y2
Deviation of each score from the mean on test X
Deviation of each score from the mean on test Y
Squared deviation scores on test X
Squared deviation scores on test Y
For example: Following are the marks of 5 students obtained in test X and test Y. Determine
whether correlation exists between marks on test X and test Y
A
50
40
Test x
Test y
B
60
60
C
30
40
X
(X-M)
10
20
-10
0
-20
Y
(Y-M)
0
20
0
-10
-10
D
40
30
E
20
30
X2
Y2
XY
100
400
0
100
400
1000
0
400
0
0
100
600
0
400
0
0
200
600
Subject
Test X
Test Y
A
B
C
D
E
Total
Mean
50
60
30
40
20
200
40
40
60
40
30
30
200
40
STEP 1:
subtract test score of test X minus Mean (e.g., 50 - 40 = 10)
2
STEP 2:
STEP 3:
STEP 4:
STEP 5:
STEP 6:
subtract test score of test Y minus Mean (e.g., 40 - 40 = 0)
Square the values obtained in X to get X2 values (e.g., 10 * 10 = 100)
Square the values obtained in Y to get Y2 values (e.g., 0 * 10 = 0)
Multiple the values obtained in X with Y to get XY values (e.g., 10 * 0 = 0)
Apply Person Correlation formula and Substitute the values.
STEP 7:
STEP 8:
r = .77
(high correlation)
STEP 9:
Interpretation: The obtained score of .77 indicates that there is a high
correlation found Test X and Test Y.
Sums for practice
A)
B)
score in
score in
Students English
Maths
A
40
78
B
21
70
C
25
60
D
31
55
E
38
80
F
47
66
C)
sl.no Stock A
1
2
3
4
5
Stock B
45
50
53
58
60
9
8
8
7
5
D)
Sl no:
Sl
no:
1
2
3
4
5
College A College B
10
5
13
10
15
15
17
20
19
25
1
2
3
4
Group A
Group B
40
99
25
79
22
69
54
89
3
Answers
A
B
C
D
r = 0.35
r = - 0.90
r = 0.064
r = -2.19
4