Correlational Analysis is the statistical tool that is used
to describe the degree to which one variable is linearly related to
another. The degree of relationship can
be measured and represented quantitatively by the coefficient of correlation.
The Correlation Scale
1.
Low correlation – the changes in one variable
cannot be expected to signify change in the other, almost negligible
relationship (-0.10 – 0.20)
2.
Weak correlation – the change in one variable
may not be expected from a change in the other, definite but small relationship
(-0.49— -0.20, 0.21-0.49)
3.
Moderate Correlation – the change in one
variable is expected from a change in the other, substantial relationship
(-0.69 - -0.50, 0.50-0.79)
4.
High correlation – the change in one variable is
expected with reliability from a change in the other, marked relationship
(-0.70 – -0.89, 0.80 – 0.95)
5.
Very high correlation – the change in one
variable is expected to happen with greater reliability from a change in the
other, (-0.90- -1.00, 0.96 – 1.00)
The Pearson Product Moment Correlation Coefficient
r = pearson product –moment of
correlation coefficient
Sum XY = the total sum of the products of
x variable and y variable
X bar = mean of the independent variable
Y bar = mean of the dependent variable
Sx = Standard deviation of the
independent variable
Sy = standard deviation of the
independent variable
n = number of pairs
Test of Significance for r
Where
r = correlation coefficient
n = number of pairs
Degree of freedom = n -2
1.The adviser of a Grade 7 class wanted
to know if the students’ achievement test scores in Mathematics and Physics are
related. He took the test results of the top 10 students of her class and
presented the scores in a table. The
data indicated below: (alpha 0.05)
|
Mathematics (X)
|
46
|
45
|
43
|
33
|
31
|
31
|
29
|
26
|
25
|
24
|
|
Physics (Y)
|
48
|
46
|
42
|
40
|
36
|
35
|
37
|
38
|
44
|
42
|
2. A Class adviser was interested in
determining the relationship between the Mathematics grades and Chemistry
grades of the top 10 junior students.
She gathered the following data: (alpha = 0.01)
|
Mathematics Grade(X)
|
84
|
89
|
84
|
86
|
84
|
84
|
87
|
94
|
88
|
85
|
|
Chemistry Grade(Y)
|
85
|
88
|
83
|
86
|
85
|
88
|
89
|
93
|
84
|
84
|
3.
How is the academic performance of senior students related to their age?
Use 1 percent level of significance
|
Age
|
16
|
17
|
15
|
17
|
16
|
19
|
17
|
16
|
16
|
19
|
17
|
18
|
20
|
18
|
17
|
16
|
14
|
16
|
18
|
19
|
16
|
18
|
16
|
18
|
20
|
17
|
18
|
18
|
19
|
20
|
17
|
18
|
16
|
16
|
18
|
|
Final Grade
|
86
|
88
|
85
|
80
|
80
|
79
|
78
|
83
|
75
|
77
|
80
|
79
|
79
|
78
|
76
|
81
|
77
|
81
|
82
|
78
|
76
|
78
|
79
|
75
|
75
|
75
|
75
|
78
|
76
|
76
|
82
|
80
|
78
|
80
|
78
|
The Point Biserial Coefficient
Some situations involve variables where one is continuous
while the other is dichotomous and is assumed to be discrete. An example of this is the situation when the
two variables under consideration are the mathematics achievement test scores
of the students and gender. The achievement test score is continuous while
gender, which is either male or female, is a nominal dichotomous variable. The nominal dichotomous variable classifies
attribute into two mutually exclusive classifications. Other examples of nominal dichotomous
variables are attitudes (positive-negative) and responses (yes-no, true-false,
agree-disagree).
rpb = point biserial coefficient
X1 = mean of the continous variable of
one group
X0 = mean of the continous variable of
the other group
N1 = number of cases in one group
N0 = number of cases in the other group
N = total number of cases, N1 + N0
Sx = standard deviation of all measures
in the continuous variable
1.
A mathematics teacher wanted to know if
the students’ mathematics achievement test scores are related to their
gender. He randomly selected test
results of 10 students where she came up with six females and four males.
(alpha = 0.05)
|
Gender(male = 1, female = 0)
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
|
Math Achievement test scores
|
48
|
45
|
40
|
35
|
41
|
40
|
33
|
30
|
30
|
25
|
2.
The table below presents the performance
ratings of teachers who have passed and have not passed the licensure
examination for teachers. Using 5
percent level of significance, test if the teachers’ ratings are related to the
results of the licensure examination for teachers . Let 1 = passed and 0 = failed.
|
LET Results
|
1
|
1
|
0
|
0
|
1
|
0
|
1
|
|
Performance Rating
|
87
|
87
|
85
|
85
|
90
|
85
|
90
|
3.
Below is another group of composite scores on a
test battery and with each score the number of individuals who did (1) or did
not complete(0) a program of training.
|
Composite score
|
9
|
8
|
7
|
6
|
5
|
4
|
3
|
2
|
1
|
10
|
|
Completion
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
1
|
Kendall’s Coefficient of
Concordance (W)
Kendall’s tau, is an alternative
measure of relationship to the Spearman’s rho.
This test statistic, which was developed by Kendall, is used when the
data of the three or more variables are given in ordinal measures. It determines the degree of agreement in the
ranks given by three or more judges.
Test of Significance
1.
What
significant agreement exists between the respondents of principals, master
teachers, and classroom teachers on the factors that may increase the teachers’
morale? Alpha = 0.05
|
Factors
|
Ranks by Principal
|
Master Teachers
|
Teachers
|
|
Involvement in decision-making
|
1
|
1
|
1
|
|
Adequate instructional material
|
3
|
3
|
2
|
|
Less non-teaching assignments
|
4
|
4
|
4
|
|
Transparency in promotion
|
2
|
2
|
3
|
|
Smaller class size
|
5
|
5
|
5
|
2.
Calculate the correlation of concordance of the
ranks of the judges on the singing performances of 10 contestants.
|
Contestants
|
Judge1
|
Judge 2
|
Judge 3
|
Judge 4
|
|
1
|
9
|
10
|
10
|
9
|
|
2
|
8
|
6
|
7
|
5
|
|
3
|
3
|
1
|
2
|
4
|
|
4
|
10
|
9
|
8
|
10
|
|
5
|
4
|
5
|
1
|
3
|
|
6
|
2
|
3
|
4
|
1
|
|
7
|
7
|
8
|
9
|
6
|
|
8
|
5
|
4
|
6
|
7
|
|
9
|
1
|
2
|
3
|
2
|
|
10
|
6
|
7
|
5
|
8
|
Alpha = 0.05
3.
Four judges (parole board members) rank eight
convicts on “parole readiness”. By using
the coefficient of concordance, indicate the degree of consistency of the
judges.
|
Convict
|
Judge 1
|
Judge 2
|
Judge 3
|
Judge 4
|
|
1
|
1
|
1
|
1
|
1
|
|
2
|
2
|
4
|
3
|
2
|
|
3
|
3
|
3
|
2
|
4
|
|
4
|
4
|
2
|
4
|
3
|
|
5
|
5
|
6
|
5
|
5
|
|
6
|
6
|
5
|
6
|
7
|
|
7
|
7
|
7
|
8
|
6
|
|
8
|
8
|
8
|
7
|
8
|
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