Download as PDF, PPTX
Statistical Reasoning Lecture Note_Correlation.pdf
- 1. PMC 500 STATISTICAL REASONINGIN EDUCATION CORRELATION Dr. Nor Asniza Ishak Dr. Ahmad Zamri Khairani
- 2. Correlation •Thorne and Giesen(2003): •Correlation: - the degree of linear relationship between two variables •Example: Mathematics score and Physics score
- 3. Student Mathematics ScorePhysics 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 patternsdo 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 correlationbetween 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. Ø Creating ascatterplot 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 studyingHistory test (hour) The points in the scatterplot form a close approximation to a straight line slanting upward to the right
- 12. Time spent studyingHistory 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 doyou see in this set of points? n The 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 whichstudents turn in their test papers What pattern do you see in these points? n The points on the scatterplot form a random pattern n The random pattern of points indicates a zero correlation
- 19. Order in whichstudents turn in their test papers
- 20. The Pearson CorrelationCoefficient • A scatterplot shows the direction of the linear relationship between two variables n positive, negative or zero correlation • But, it does not provide an exact measure of the magnitude of the linear relationship between two variables n High, 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 Pearsonr 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 usingSPSS 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 n r 2 = 0.6922 = 0.479 n 47.9% of the variance in Y can be associated with the variance in X
- 27. Testing r forSignificance: 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 forSignificance: 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. Analyze the data using SPSS 4. Make a decision about the null hypothesis 5. Make a conclusion
- 29. Example 1 • Aresearcher 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: Statethe 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: Setthe 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: Makea decision about the null hypothesis Reject H because p < 0.01
- 36. Step 5: Makea conclusion
- 37. Example 2 • Aresearcher 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: Statethe 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: Setthe 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: Makea decision about the null hypothesis Fail to reject Ho because p > 0.05
- 44. Step 5: Makea conclusion
- 45.