1. Joint Distributions AND CORRELATION Coefficients (Part 3)

2. Textbook Credits Textbook
Shavelson, R.J. (1996). Statistical reasoning for the behavioral sciences (3rd Ed.). Boston: Allyn & Bacon.
Supplemental Material
Ruiz-Primo, M.A., Mitchell, M., & Shavelson, R.J. (1996). Student guide for Shavelson statistical reasoning for the behavioral sciences (3rd Ed.). Boston: Allyn & Bacon.

3. Overview
Joint Distributions
Correlation Coefficients

4. Joint Distributions and Correlation Coefficients
Correlational studies answer the question
“What is the relationship of variable X and variable Y?” or
“How are scores on one measure (X) associated with scores on another measure(Y)?”
First, we want to summarize the scores, and
Second, examine the relationship between the scores on the two measures
First step: Arrange the scores to represent them in the form of a joint distribution (the representation of a pair of scores for each subject)
Second step: Summarize the relationship represented by the JD with a single number we call correlation coefficient(a descriptive statistic that represents the magnitude of the relation, 0 to |1|, and the direction of the relation, + or -).

5. The Psychological Belief Scale and Student Achievement
Research Example
The Psychological Belief Scale and Student Achievement
Intuition and prior experience suggest that it is easier to learn from teachers who have the same beliefs as the students
Prediction(intuition):
Students with similar beliefs as their instructors will earn the highest scores on exams
Exam scores should decrease as the difference in the students’ and instructors’ beliefs increases
Study: 3 introductory Psych. classes at 3 different colleges with 7 students each, with variable X representing a Belief Score and Y representing an Exam Score
What method to use?
General: Combine the data and for all 3 classes examine one overall average X with Y?
More Specific: Examine X and Y in each class separately?
Are the data consistent with predictions?

6. The Psychological Belief Scale and Student Achievement
Research Example
The Psychological Belief Scale and Student Achievement
Test 2 types of belief approaches: Humanistic(H) & Behavioristic(B)
Example: The central focus of the study of human behavior should be
The specific principles that apply to unique individuals(H)
The general principles that apply to all individuals(B)
Instructors and students received the belief scale beginning of course
Behavioristic orientation on the belief scale indicative by high scores
Humanistic orientation on the belief scale indicative by low scores

7. Joint Distribution: Tabular Representation
Behavioristic orientation on the belief scale indicative by high scores
Humanistic orientation on the belief scale indicative by low scores
Achievement(exam) score: students’ total scores earned in all class exams

8. Joint Distribution: Tabular Representation
Divided into 3 classes with 3 columns each. Take Class 1 as example:
Low belief scores are associated with moderately high exam scores(subjects 1 & 2)
Moderate belief scores are associated with high exam scores (subjects 3, 4, & 7)
High belief scores are associated with low exam scores (subjects 5 & 6)

9. Relationship of Student’s Belief & Exam Scores
Lines represent relationship between belief scale scores & exam scores
The magnitude of students’ scores differ from one class to the next as each instructor
gave a different exam
So all exam scores were converted to standard scores showing how far above (+) or
below (-) the class average a particular exam falls

10. Scatterplot of Student’s Belief & Exam Scores
A graphical representation of a JD showing pairs of each subject’s scores

11. Scatterplots for 3 classes & Instructor’s Belief Score
Comparison of Scatterplots for each of the 3 classes in the study
Curvilinear Relationship
Linear Relationship
Suspect Outlier

12. Correlation Coefficients: Linear Relationships

13. Properties of Linear Correlation Coefficients
The coefficient can take values from to
- A correlations of indicates a very strong negative relationship between X & Y
- A correlation of indicates a very strong positive relationship between X & Y
- A correlation of 0 indicates that there is no linear relationship between X & Y
The sign indicates the direction of the relationship between 2 variables
A positive relationship means:
- Low scores on X go with low scores on Y
- High scores on X go with high scores on Y(As X scores , Y scores )
A negative relationship means:
- Low scores on X go with high scores on Y
- High scores on X go with low scores on Y(As X scores , Y scores )

14. Determining the Correlation Coefficient Magnitude
Scatterplot characteristics are indicative of slope and data clustering:
- Correlation is 0 if slope is horizontal & vertical slope is undefined
- The clustering of data points determines the magnitude of correlation
- Tight clustering means the magnitude of the correlation coefficient is high
- Lose clustering means the magnitude of the correlation coefficient is low

15. SAT & GPA Relationships
Scatterplot characteristics are indicative of slope and data clustering:
- Correlation is 0 if slope is vertical or horizontal

16. SAT & GPA Relationships
Developing Statistics
- Student’s #1 deviation score on the SAT is:
- Student’s #1 deviation score on the GPA is:
- Student 1 earned scores below the mean for both SAT and GPA
𝒙=𝑿− 𝑿 =𝟒𝟓𝟎−𝟓𝟕𝟕=−𝟏𝟐𝟕
𝒚=𝒀− 𝒀 =𝟐.𝟒𝟎−𝟑.𝟏𝟎=−𝟎.𝟕𝟎

17. SAT & GPA Minitab Results
Descriptive Statistics: SAT(X), GPA(Y)
Total
Variable Count Mean StDev Variance Sum
SAT(X)
GPA(Y)

18. Covariance of SAT & GPA Scores
Measuring how two sets of deviation go together or covary
- Student’s #1 covariance (cross product)is:
- Note: When |x| and |y| are large  xy is large (students 1 & 5)
- Note: When |x| and |y| are small  xy is small (students 2, 3, &4)
- Covariance:
- Pearson product-moment correlation coefficient measures the strength with X and Y
- Correlation coefficient:
𝑪𝒐𝒗𝒙𝒚= 𝒙𝒚 𝑵−𝟏 = 𝟐𝟏𝟑.𝟔𝟓 𝟒 =𝟓𝟑.𝟒𝟏
𝒄𝒐𝒓𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏 𝑿, 𝒀 =𝒓𝒙𝒚= 𝑪𝒐𝒗𝒙𝒚 𝒔𝒙𝒔𝒚 = 𝟓𝟑.𝟒𝟏 𝟏𝟐𝟔.𝟎𝟖 𝟎.𝟒𝟖 =𝟎.𝟖𝟗
𝒙𝒚= −𝟏𝟐𝟕 −𝟎.𝟕𝟎 =𝟖𝟖.𝟗𝟎

19. Covariance of SAT & GPA Scores
Measuring how two sets of deviation go together or covary
- Student’s #1 covariance (cross product)is:
- Note: When |x| and |y| are large  xy is large (students 1 & 5)
- Note: When |x| and |y| are small  xy is small (students 2, 3, &4)
- Covariance:
- Pearson product-moment correlation coefficient measures the strength with X and Y
- Correlation coefficient:
𝒙𝒚= −𝟏𝟐𝟕 −𝟎.𝟕𝟎 =𝟖𝟖.𝟗𝟎
𝑪𝒐𝒗𝒙𝒚= 𝒙𝒚 𝑵−𝟏 = 𝟐𝟏𝟑.𝟔𝟓 𝟒 =𝟓𝟑.𝟒𝟏
𝒄𝒐𝒓𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏 𝑿, 𝒀 =𝒓𝒙𝒚= 𝑪𝒐𝒗𝒙𝒚 𝒔𝒙𝒔𝒚 = 𝟓𝟑.𝟒𝟏 𝟏𝟐𝟔.𝟎𝟖 𝟎.𝟒𝟖 =𝟎.𝟖𝟗
Minitab Results
Covariances: SAT(X), GPA(Y)
SAT(X) GPA(Y)
SAT(X)
GPA(Y)
Correlations: SAT(X), GPA(Y)
Pearson correlation of SAT(X) and GPA(Y) = 0.888

20. Correlation Between SAT & GPA Scores
Looking at the scatterplot to validate the correlation findings
- A linear relationship with a positive slope indicates a positive correlation
- The absolute magnitude 0.89 provides an index of the relationship strength(-1to +1)
- Points cluster closely about an imaginary line validating the relationship magnitude

21. Minitab Output: SAT & GPA Scores
Looking at the scatterplot to validate the correlation findings
- A linear relationship with a positive slope indicates a positive correlation
- The absolute magnitude 0.89 provides an index of the relationship strength(-1to +1)
- Points cluster closely about an imaginary line validating the relationship magnitude

22. Excel Output: SAT & GPA Scores

23. Excel Output SAT & GPA Scores

24. The Squared Correlation Coefficient
The squared correlation coefficient is the coefficient of determination
- It is the amount of variability that can be explained between X & Y
Recall: The larger |rxy| is, the stronger the relationship between X & Y
We previously found that: So
Now we want to convert to percentage of variance
- Tells us the percentage that X shares with Y in terms of variability to one another
- The % of variance in Y and X that can be explained is:
𝒓𝒙𝒚= 𝑪𝒐𝒗𝒙𝒚 𝒔𝒙𝒔𝒚 = 𝟓𝟑.𝟒𝟏 𝟏𝟐𝟔.𝟎𝟖 𝟎.𝟒𝟖 =𝟎.𝟖𝟗
𝒓 𝟐 𝒙𝒚= 𝟎.𝟖𝟗 𝟐 =𝟎.𝟕𝟗𝟐𝟏
𝒓 𝟐 𝒙𝒚 ×𝟏𝟎𝟎= 𝟎.𝟕𝟗𝟐𝟏 ×𝟏𝟎𝟎=𝟕𝟗.𝟐𝟏%

25. Percentage of Variance
Pictorial representation of the % of variance in exam scores accounted for by the variability in belief scores (computed from class 3 data)
Variability in X
Variability in Y

26. Spearman Rank Correlation Coefficient
Non-linear (curvilinear) monotonic increasing or decreasing functions
Monotonically decreasing f
Monotonically increasing f

27. Spearman Rank Correlation Coefficient
Example: Y is a monotonically increasing function of X

28. Spearman Rank Correlation Coefficient
Rank ordering the data for both X & Y and graph
- The converted ordered graph is now linear
- We can now compute the Pearson correlation coefficient for ranks between X & Y

29. Spearman Rank Correlation Coefficient

30. Sources of Misleading Correlation Coefficients
Too much confidence can lead to misleading interpretations
- Restriction of the range of values on one of the variables may reduce the magnitude
of the correlation coefficient

31. Sources of Misleading Correlation Coefficients
Too much confidence can lead to misleading interpretations
- Use of extreme groups may inflate the correlation coefficient

32. Sources of Misleading Correlation Coefficients
Too much confidence can lead to misleading interpretations
- Combining groups with different means on one or both variables may have an
unpredictable effect on the correlation coefficient

33. Sources of Misleading Correlation Coefficients
Too much confidence can lead to misleading interpretations
- Extreme scores (Outliers) may have a marked effect on the correlation coefficient,
especially if the sample size is small

34. Sources of Misleading Correlation Coefficients
Too much confidence can lead to misleading interpretations
- A curvilinear relationship between X and Y may account for a near-zero correlation
coefficient
No systematic relationship
Curvilinearly related: Use the eta (h) ratio coefficient measurement instead of the Pearson correlation coefficient

35. Correlation and Causality
Correlation does not imply causality
Many possible interpretations of a correlation coefficient:
Most common problem inferring causality from correlation: Selectivity!
X: beliefs
Y: Achievement
Z: Knowledge gained from related courses

36. Practice Exercises Part 3 Practice Exercises
Select a hypothetical product or a process and create some test data of your choice (plausible, no more than 10) as shown in textbook/class
Show your type of experimental approach
Create a detailed table of frequency distributions
Display your data with different types of graphs
Calculate the measures of central tendency and variability
Calculate the Z-score(s) and indicate the relative position in the normal distribution.
Provide any other pertinent information as a result
Part 3 Practice Exercises
Represent your joint distribution data in a tabular form
Create a scatterplot of your data
Create a covariance table (as table 6-4) and calculate the covariance
Calculate the correlation of the two variables
Calculate the R squared value and explain your findings as a result

37. Comments/Questions ?