Statistical Reasoning Lecture Note_Correlation.pdf

pmc 500

1. PMC 500
STATISTICAL REASONING IN EDUCATION
CORRELATION
Dr. Nor Asniza Ishak
Dr. Ahmad ZamriKhairani

2. Correlation
•Thorne and Giesen(2003):
•Correlation:
-the degree of linear relationship between two
variables
•Example:
Mathematics score and Physics score

3. Student Mathematics Score Physics Score
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
96
94
90
89
88
72
68
65
60
57
50
45
40
34
30
90
87
84
81
80
70
65
60
58
53
48
40
38
30
25
What patterns do you find in these data?

4. •What patterns do you find in these data?
•In general,
-students with high Mathematics scores also have high
Physics scores
-students with low Mathematics scores also have low
Physics scores
•These patterns show that there is a correlation between
Mathematics score and Physics score

5. Scatterplot
•The correlation between Mathematics score and Physics score can be
represented by a scatterplot
•A scatterplot is a visual representation of the relationship between
two variables
•Each point in a scatterplot represents the value for the X and Y
variables for a subject
•Example: A scatterplot of the relationship between Mathematics
score (X) and Physics score (Y)

7. ØCreating a scatterplot using SPSS:
-Create a data file
-Click Graphs > Chart Builder... > Scatter/Dot ...
-Drag Simple Scatter to Chart preview box
-Drag the Y variable to the Y-Axis? box
-Drag the X variable to the X-Axis? box
-Click OK

8. Types of Correlation (Giesen, 2003)
1.Positive correlation
2.Negative correlation
3.Zero correlation

9. 1.Positive Correlation
•In other words, a positive correlation between two variables, X and Y,
implies a direct linear relationship between the variables
-X increases in value, Y increases in value
-X decreases in value, Y decreases in value
•Example:
-Time spent studying for a History test and the score earned on
the test

11. Time spent studying History test (hour)
The points in the scatterplot form
a close approximation to a
straight line slanting upward to
the right

12. Time spent studying History test (hour)

13. 2. Negative Correlation
•A negative correlation between two variables is one in which:
-high scores on one variable are associated with low scores on the other variable
-low scores on one variable are associated with high scores on the other variable
•In other words, a negative correlation between two variables, X and Y, implies an inverse linear relationship between the variables
-X increases in value, Y decreases in value
-X decreases in value, Y increases in value
•Example:
-Living off campus and involvement in campus activities

15. What pattern do you see in this set of
points?
nThe points in the scatterplot form a
close approximation to a straight line
slanting upward to the left
Living off campus, X (%)

17. 3. Zero Correlation
•A zero correlation between two variables is one in which:
-a high score on one variable is just as likely to be associated
with a low score as it is with a high score on the other variable
-a low score on one variable is just as likely to be associated with
a high score as it is with a low score on the other variable
•In other words, a zero correlation between two variables, X and Y,
implies no linear relationship between the variables
•Example:
-Order in which students turn in their test papers and their test
scores

18. Order in which students turn in their test papers
What pattern do you see in these points?
nThe points on the scatterplot form a
random pattern
nThe random pattern of points indicates a
zero correlation

19. Order in which students turn in their test papers

20. The Pearson Correlation Coefficient
•A scatterplot shows the direction of the linear relationship
between two variables
npositive, negative or zero correlation
•But, it does not provide an exact measure of the magnitude of
the linear relationship between two variables
nHigh, moderate or low correlation
•Thus, the English statistician Karl Pearson (1857-1936)
developed a statistic called the Pearson correlation coefficient or
the Pearson r

21. •The Pearson r is a statistic that indicates the direction and magnitude of the linear relationship between two interval or ratio scaled variables
•Examples of interval or ratio scaled variables:
-test score, height, weight, age, income, time spent studying for an exam, etc.
•The Pearson correlation coefficient is represented by the small letter r
•The values of r may range from –1.00 to +1.00
•The ‘ + ’ and ‘ –’ signs represent direct and inverse linear relationships respectively
•The absolute values of a correlation coefficient represent the magnitude of the linear relationship
•E.g. r = 0.95 and r = -0.95 have the same magnitude but different directions

23. Computing r using SPSS
Create a data file
•Analyze>Correlate > Bivariate ...
•Double-click on the X variable to move it to the Variables box
•Double-click on the Y variable to move it to the Variables box
•Click OK

25. Coefficient of Determination
•The square of the correlation coefficient is called the
coefficient of determination
•Coefficient of determination
•The proportion of the variance in a variable that can be
associated with the variance in the other variable
•Is represented by r 2

26. Coefficient of Determination
Example:
nr = 0.692
nr 2 = 0.6922 = 0.479
n47.9% of the variance in Y
can be associated with the
variance in X

27. Testing r for Significance: Assumptions
•The Pearson correlation coefficient, r is used to determine whether
there is a significant correlation between two interval or ratio scaled
variables
•Assumptions of the Pearson correlation coefficient, r :
•The sample is chosen randomly from the population and the scores
for each of the participants should be independent of all other
participants’ scores.
•The scores for each variable are normally distributed in the
population

28. Testing r for Significance: 5 main steps in
hypothesis testing using SPSS
1.State the null and alternative hypotheses
2.Set the significance level for rejecting the null hypothesis
3.Analyzethe data using SPSS
4.Make a decision about the null hypothesis
5.Make a conclusion

29. Example 1
•A researcher want to find if there is a correlation between
Mathematics scores and Physics scores of 30 Form Four
students.
•Test the hypothesis of whether the correlation between
Mathematics scores and Physics scores is significant at the
0.01 level of significance.

30. Step 1: State the null and alternative
hypotheses
•Ho: There is no significant correlation between Mathematics
scores and Physics scores in the population.
•H1 : There is a significant correlation between Mathematics
scores and Physics scores in the population.

31. Step 2: Set the significance level for rejecting
the null hypothesis

32. Step 3: Analyzethe data using SPSS

34. Step 3: Analyzethe data using SPSS
•Create a data file
•Click Analyze> Correlate > Bivariate ...
•Double-click on the variable named Mathematics scores to move it to
the Variables box
•Double-click on the variable named Physics scores to move it to the
Variables box
•Click OK

35. Step 4: Make a decision about the null
hypothesis
Reject H because p < 0.01

36. Step 5: Make a conclusion

37. Example 2
•A researcher want to find if there is a correlation between
Malay Language scores and English Language scores of 30
students
•Test the hypothesis of whether the correlation between
Mathematics scores and Physics scores is significant at the
0.05 level of significance.

38. Step 1: State the null and alternative
hypotheses
•Ho: There is no significant correlation between Malay Language
scores and English Language scores in the population.
•H1 : There is a significant correlation between Malay Language scores
and English Language scores in the population.

39. Step 2: Set the significance level for rejecting
the null hypothesis

40. Step 3: Analyzethe data using SPSS

42. Step 3: Analyzethe data using SPSS
•Create a data file
•Click Analyze> Correlate > Bivariate ...
•Double-click on the variable named Mathematics scores to move it to
the Variables box
•Double-click on the variable named Physics scores to move it to the
Variables box
•Click OK

43. Step 4: Make a decision about the null
hypothesis
Fail to reject Hobecause p > 0.05

44. Step 5: Make a conclusion

45. Thank You
Enjoy Statistics!!!