1. Overview and interpretation
Correlation
Overview and interpretation

2. Making a Scatterplot Line up the data in columns
(eliminate missing data)
Plot the student’s score on each variable
Bill
Adapted from Wiersma, W., & Jurs, S. G. (1990). Educational measurement and testing (2nd ed.). Needham Heights, MA: Allyn and Bacon.

3. Inspect the scatterplot
The correlation coefficient (Pearson r) can only be interpreted for linear relationships
These are all examples of linear relationships
The strength of the correlations vary
Shavelson, R. J. (1996). Statistical reasoning for the behavioral sciences (Third ed.). Needham Heights, MA: Allyn & Bacon.

4. Inspect the scatterplot (2)
If you see these types of distributions, you are dealing with a curvilinear relationship

5. Inspect the scatterplot (3)
Students who seem to be ‘out on their own’ in the scatter plot are called outliers
Including outliers in the calculation can change the relationship
outlier

6. Pearson r correlation coefficient
Range from -1.0 (perfect inverse correlation) to +1.0 (perfect correlation)
The sign (+, -) shows the direction of the relationship
The number shows the strength of the relationship (regardless of sign)
No relationship is 0.0

7. The formula
Note that there are other equivalent formulas also possible.

8. Assumptions of Pearson correlation
Each pair of scores is independent
Each set of scores is normally distributed
The relationship between scores is linear

9. Interpreting correlation
Correlation merely shows a relationship between two variables, not the meaning of the relationship
Correlation is not causation
Statistical significance does not imply importance
Statistical significance merely indicates that the correlation strength is greater than one would expect by chance

10. Statistical significance of r
(Cody & Smith, 1997)
Imagine a population with a zero correlation
Now, sample 10 points from this population
The resulting sample would probably have a non-zero correlation

11. Statistical significance
If a correlation is much larger than what one would expect by chance, it is considered to be significant
Significant does not mean important or strong
Significant merely means that the size of the correlation coefficient is larger than would be expected by a chance sampling from a zero correlation population

12. Determining significance
Most statistical software packages will automatically flag significant correlations
If checking by hand, compare the r value with the appropriate table
2-tailed decision at alpha = .05 is common
If the value of r is equal to or larger than the value in the table, the correlation is significant

13. Decide the level of certainty that you want
This is the table from the back of a statistics book
Find the number corresponding to your N – 2
Check to see if your correlation coefficient is as large or larger than the one in the table

14. Coefficient of determination
The coefficient of determination (r2) is a measure of the shared variance between the two variables
(Shavelson, 1996)

15. Potential problems in correlation analysis
restriction of range
correlation of TOEFL, GRE, etc. with grade point average
skewedness
test too easy or too difficult
attribution of causality
variable must be correlation to claim that they are causally related, but correlation alone is not sufficient to prove causality

16. Point-biserial correlation
Used to correlate a dichotomous variable with a continuous variable
In testing, used to correlate a person’s performance on an item (correct, incorrect) with their total test score
Used as an index of item discrimination

17. Point-biserial formula
IF for item
1 – IF for item
Mean on the test for people who got item correct
Mean on the test for people who got item incorrect
Standard deviation for test

18. TAP output Number Item Disc. # Correct # Correct Point Adj.
Item Key Correct Diff. Index in High Grp in Low Grp Biser. Pt Bis
Item 01 (2 ) (0.93) 3 (0.21)
Item 02 (4 ) (0.87) 4 (0.29)
Item 03 (4 ) (1.00) 4 (0.29)
Item 04 (3 ) (0.93) 3 (0.21)
Item 05 (2 ) (1.00) 7 (0.50)
Item 06 (1 ) (0.87) 2 (0.14)
Item 07 (3 ) (0.93) 7 (0.50)
Item 08 (4 ) (1.00) 3 (0.21)
Item 09 (4 ) (0.93) 2 (0.14)
Item 10 (4 )# (0.87) 9 (0.64)