PREPARED BY MRS SHALINI ,ASSISTANT PROF.

1. LOGO
CORRELATION
ANALYSIS
PRESENTED BY
SHALINI

2. Introduction
Correlation a LINEAR association between two
random variables
Correlation analysis show us how to determine
both the nature and strength of relationship
between two variables
When variables are dependent on time correlation
is applied
Correlation lies between +1 to -1

3. A zero correlation indicates that there is no
relationship between the variables
A correlation of –1 indicates a perfect negative
correlation
A correlation of +1 indicates a perfect positive
correlation

4. Types of Correlation
There are three types of correlation
Types
Type 1 Type 2 Type 3

5. Type1
Positive Negative No Perfect
If two related variables are such that when
one increases (decreases), the other also
increases (decreases).
If two variables are such that when one
increases (decreases), the other decreases
(increases)
If both the variables are independent

6. When plotted on a graph it tends to be a perfect
line
When plotted on a graph it is not a straight line
Type 2
Linear Non – linear

8. Type 3
Simple Multiple Partial
Two independent and one dependent variable
One dependent and more than one independent
variables
One dependent variable and more than one
independent variable but only one independent
variable is considered and other independent
variables are considered constant

9. Methods of Studying Correlation
Scatter Diagram Method
Karl Pearson Coefficient Correlation of
Method
Spearman’s Rank Correlation Method

10. 1 8 0
1 6 0
1 4 0
1 2 0
1 0 0
8 0
6 0
4 0
2 0
0
0 5 0 2 0 0 2 5 0
1 0 0 1 5 0
D r u g A (dose i n m g )
Symptom
Index
160
140
120
100
80
60
40
20
0
0 50 250
100 150 200
Drug B (dose in mg)
Symptom
Index
Very good fit Moderate fit
Correlation: Linear
Relationships
Strong relationship = good linear fit
Points clustered closely around a line show a strong correlation.
The line is a good predictor (good fit) with the data. The more
spread out the points, the weaker the correlation, and the less
good the fit. The line is a REGRESSSION line (Y = bX + a)

11. Coefficient of Correlation
A measure of the strength of the linear relationship
between two variables that is defined in terms of the
(sample) covariance of the variables divided by their
(sample) standard deviations
Represented by “r”
r lies between +1 to -1
Magnitude and Direction

12. -1 < r < +1
 The + and – signs are used for positive linear
correlations and negative linear
correlations, respectively

13. n Y2
( Y)2
X2
n ( X)2
n X
Y X Y
rxy
Shared variability of X and Y variables on the
top
Individual variability of X and Y variables on the
bottom

14. Interpreting Correlation
Coefficient r
 strong correlation: r > .70 or r < –.70
 moderate correlation: r is between .30 &
.70
or r is between –.30 and –.70
 weak correlation: r is between 0 and .30
or r is between 0 and –.30 .

16. Nominal by nominal
correlational approaches
16
• Contingency (or cross-tab) tables
– Observed
– Expected
– Row and/or column %s
– Marginal totals
• Clustered bar chart
• Chi-square
• Phi/Cramer's V

17. Contingency tables
●
Bivariate frequency tables Cell frequencies
(red) Marginal totals (blue)
●
●

18. Contingency table:
Example
RED = Contingency cells BLUE =
Marginal totals

19. Contingency table:
Example
Chi-square is based on the differences between
the actual and expected cell counts.

20. Example
Row and/or column cell percentages may also
aid interpretation e.g., ~2/3rds of smokers snore, whereas only
~1/3rd
of non-smokers snore.

21. The underlying purpose of
correlation is to help address the
question:
What is the
• relationship or
• degree of association or
• amount of shared variance or
• or co-relation
between two variables?
2
1
Purpose of correlation

22. IMPORTANCE OF
CORRELATION
 Correlation helps to study the association between two
variables.
 Coefficient of correlation is vital for all kinds of research
work.
 It helps in establishing Validity or Reliability of an evaluation
tool.
 It helps to ascertain the traits and capacities of pupils while
giving guidance or counselling.
 Correlation analysis helps to estimate the future values.

23. ? What would be your interpretation if the correlation
coefficient equal to
1) r = 0
Ans : There is no correlation between the variables
2) r = -1
Ans: negative perfect correlation
3) r =0.2
Ans: low positive correlation
4) r = 0.9
Ans: high positive correlation
5) r = -0.3
Ans: low negative correlation
6) r = - 0.8
Ans: High negative correlation

24. Phi-Coefficient
Conceptual Explanation

25. A Phi coefficient is a non-parametric
test of relationships that operates on
two dichotomous (or dichotomized)
variables.
Dichotomous
means that the
data can take on
only two values.

26. A Phi coefficient is a non-parametric
test of relationships that operates on
two dichotomous (or dichotomized)
variables.
Like –
• Male/Female
• Yes/No
• Opinion/Fact
• Control/Treatment

27. It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.

28. It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
What does
this mean?

29. It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
Here is an
example
Data Set

30. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.

31. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
Two Dichotomous
Variables

32. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A
B
C
D
E
F
G
H
I
J
K
L
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.

33. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1
B 1
C 1
D 2
E 2
F 1
G 2
H 2
I 2
J 1
K 1
L 2
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.

34. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.

35. 2
2
1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2
E 2 Male
F 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1

36. 2
2
1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2
E 2 Male
F 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1

37. 2
1
2
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2
H 2 Female
I 2
J 1 1
K 1 1
L 2 1

38. 2
1
2
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2
H 2 Female
I 2
J 1 1
K 1 1
L 2 1

39. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects
variables across a
2x2 matrix to
estimate whether
there is a non-
random pattern
across the four
cells in the
2x
2
matrix.
Single

40. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects
variables across a
2x2 matrix to
estimate whether
there is a non-
random pattern
across the four
cells in the
2x
2
matrix.
Single

41. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects
variables across a
2x2 matrix to
estimate whether
there is a non-
random pattern
oss the four cells in the
acr
2x
2
matrix.
Married

42. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects
variables across a
2x2 matrix to
estimate whether
there is a non-
random pattern
oss the four cells in the
acr
2x
2
matrix.
Married

43. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.

44. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.

45. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.

46. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
MARITAL
STATUS
Male Female
Married
Single

47. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
MARITAL
STATUS
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married
Single

48. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
MARITAL
STATUS
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married
Single

49. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married
Single
MARITAL
STATUS

50. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married 1
Single
MARITAL
STATUS

51. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
MARITAL
STATUS
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married 1
Single

52. It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
MARITAL
STATUS
GENDER
Male Female
Married 1
Single
Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1

53. It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
MARITAL
STATUS
GENDER
Male Female
Married 1
Single
Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1 1
C 2 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1

54. It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married 1
Single
Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1 1
C 2 1 1
D 2 2
E 2 2
F 3 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
MARITAL
STATUS

55. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
MARITAL
STATUS
Male Female
Married 1
Single
A 1 2
B 1 1 1
C 2 1 1
D 2 2
E 2 2
F 3 1 1
G 2 2
H 2 1
I 2 2
J 4 1 1
K
L
1
2
1
1

56. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
MARITAL
STATUS
Male Female
Married 1
Single
A 1 2
B 1 1 1
C 2 1 1
D 2 2
E 2 2
F 3 1 1
G 2 2
H 2 1
I 2 2
J 4 1 1
K
L
1
2
1
1
5

57. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
MARITAL
STATUS
Male Female
Married 1
Single
A 1 2
B 1 1 1
C 2 1 1
D 2 2
E 2 2
F 3 1 1
G 2 2
H 2 1
I 2 2
J 4 1 1
K
L
1
2
1
1
5

58. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married 1
Single 5
MARITAL
STATUS

59. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
MARITAL
STATUS
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married 1
Single 5

60. It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
MARITAL
STATUS
GENDER
Male Female
Married 1
Single 5
Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 1 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1

61. It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
MARITAL
STATUS
GENDER
Male Female
Married 1
Single 5
Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 1 2 2
E 2 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1

62. It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married 1
Single 5
Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 1 2 2
E 2 2 2
F 1 1
G 3 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
MARITAL
STATUS

63. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married 1 4
Single 5
MARITAL
STATUS

64. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
MARITAL
STATUS
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married 1 4
Single 5

65. Subjects Gender
1= Male
2= Female
Marital Status
1 = Single
2 = Married
A 1 2
B 1 1
C 1 1
D 2 2
E 2 2
F 1 1
G 2 2
H 2 1
I 2 2
J 1 1
K 1 1
L 2 1
It intersects variables
across a 2x2 matrix to
estimate whether there is
a non-random pattern
across the four cells in the
2x2 matrix.
GENDER
Male Female
Married 1 4
Single 5 2
MARITAL
STATUS

66. four cells in the
2x2 matrix.
MARITAL
STATUS
GENDER
Male Female
Married 1 4
Single 5 2

67. Similar to a parametric correlation
coefficient, the possible values of a Phi
coefficient range from -1 to 0 to +1.

68. A Phi coefficient of 0 would indicate
that there is
no systematic pattern across the 2x2 matrix.
GENDER
MARITAL
STATUS
Male Female
Married 3 3
Single 3 3

69. A Phi coefficient of 0 would indicate
that there is
no systematic pattern across the 2x2 matrix.
or
GENDER
MARITAL
STATUS
Male Female
Married 5 5
Single 1 1

70. A Phi coefficient of 0 would indicate
that there is
no systematic pattern across the 2x2 matrix.
A
or
GENDER
MARITAL
STATUS
Male Female
Married 5 5
Single 1 1
Being male or female does not make you any
more likely to be married or single

71. A positive Phi coefficient would
indicate that most of the data are in
the diagonal cells.
GENDER
Male Female
Married
Single
MARITAL
STATUS

72. In terms of how to interpret this value, here is a helpful rule of
thumb:
Value of r
-1.0 to -0.5 or 1.0 to 0.5
-0.5 to -0.3 or 0.3 to 0.5
-0.3 to -0.1 or 0.1 to 0.3
-0.1 to 0.1
Strength of relationship
Strong
Moderate
Weak
None or very weak
GENDER
MARITAL
Male Female
Married 4 1
STATUS Single 2
5
+.507
So, the interpretation would be, that there is a strong relationship
between marital status and gender with being male making it
more likely you are married and being female making it more likely
you are single.

73. • A researcher wishes to determine if a significant
relationship exists between the gender of the
worker and if they experience pain while
performing an electronics assembly task.

74. Two survey questions are
asked of the workers:

75. Two survey questions are
asked of the workers:
• “Do you experience pain while
performing the assembly task? Yes No”

76. Two survey questions are asked of the
workers:
• “Do you experience pain while
performing the assembly task? Yes No”
• “What is your gender?
Female Male”

77. Step 1: Null and Alternative Hypotheses
• Ho: There is no relationship
between the gender of the worker
and if they feel pain while
performing the task.
• H1: There is a significant
relationship between the gender of
the worker and if they feel pain
while performing the task.

78. Step 2: Determine dependent and
independent variables and their formats.
• Gender is the independent variable
• Feeling pain is the dependent variable
• Gender is a dichotomous variable
• Feeling pain is a dichotomous variable
In this study it can only
take on two variables:
1 = Feel Pain
2 = Don’t Feel Pain

79. Step 3: Choose test statistic
• Because we are investigating the relationship between
two dichotomous variables, the appropriate test statistic
is the Phi Coefficient

80. • The Phi Coefficient should be set up as follows:
– Box A contains the number of Males that said Yes to
the pain item (4)
– Box B contains the number of Females that said Yes to
the pain item (6)
– Box C contains the number of Males that said No to
the pain item (11)
– Box D contains the number of Females that said No to
the pain item (8)
Males Females Total
Y
es
No
1 12 13
13 2 15
Step 4: Run the Test
Total
12 14

81. Phi Coefficient Test
Formula
 
(bc  ad)
(efgh)

82. Phi Coefficient Test
Formula
Males Females Total
Yes - Pain
a = 12 b = 1 e = 13
No - Pain
c = 2 d = 13 f = 15
Total
g = 14 h =14
(𝑒𝑓𝑔
ℎ)
Φ = (𝑏𝑐 −𝑎𝑑)
=
𝟏∗𝟐 −(𝟏𝟐∗𝟏𝟑) 154.0
= = +.788
15∗13∗14∗14 195.5
The Result is the Same: there is a strong relationship between gender and
feeling pain with females feeling more pain than males.

83. Step 5: Conclusions
There is a strong relationship between gender and pain
with more females reporting pain than males.
Males Females
Yes - Pain 1 12
No - Pain 13 2

84. Step 5: Conclusions
There is a strong relationship between gender and pain
with more females reporting pain than males.
Males Females
Yes - Pain 1 12
No - Pain 13 2