1. Design and Data Analysis in Psychology II
3.1 RELATIONSHIP BETWEEN TWO QUANTITATIVE VARIABLES: PEARSON CORRELATION COEFFICIENT
Design and Data Analysis in Psychology II
Susana Sanduvete Chaves
Salvador Chacón Moscoso

2. DEFINITION
rXY
Coefficient useful to measure covariation between variables: in which way changes in a variable are associated to the changes in other variable.
Quantitative variables (interval or ratio scale).
Linear relationship EXCLUSIVELY.
Values: -1 ≤ rXY ≤ +1.
Interpretation:
+1: perfect positive correlation (direct association).
-1: perfect negative correlation (inverse association).
0: no correlation.

3. Perfect positive correlation: rxy = +1 (difficult to find in psychology)

4. Positive correlation: 0 < rxy < +1

5. Perfect negative correlation: rxy = -1 (difficult to find in psychology)

6. Negative correlation: -1 < rxy < 0

7. No correlation

8. Formulas
Raw scores
Deviation scores
Standard scores

9. Example X: Y:
Calculate rxy in raw scores.
Calculate rxy in deviation scores.
Calculate rxy in standard scores.

10. Example: scatter plot

11. Example : calculation of rxy in raw scores2 1 4 6 24 16 36 8 48 64 10 80
100 12
120
144 14
168
196 13
208
256
169 18
180
324 20 22
440
400
484
110
104
1390
1540
1342

12. Example : calculation of rxy in raw scores

13. Example : calculation of rxy in deviation scores2 1 -9
-9.4
84.6 81
88.36 4 6 -7
-4.4
30.8 49
19.36 8 -5
-2.4 12 25
5.76 10 -3
-0.4
1.2 9
0.16 -1
1.6
-1.6
2.56 14 3
4.8 16 13 5
2.6
6.76 18 7
-2.8 20 22
11.6
104.4
134.56
110
104
246
330
260.4

14. Example : calculation of rxy in deviation scores

15. Example : calculation of rxy in standard scoresZx Zy
ZxZy 2 1
-1.567
-1.842
2.886 4 6
-1.218
-0.862
1.051 8
-0.870
-0.470
0.409 10
-0.522
-0.078
0.041 12
-0.174
0.314
-0.055
0.174
-0.014 14
0.522
0.164 16 13
0.870
0.510
0.443 18
1.218
-0.096 20 22
1.567
2.273
3.561
110
104
8.391

16. Example : calculation of rxy in standard scores

17. Significance
Does the correlation coefficient show a real relationship between X and Y, or is that relationship due to hazard?
Null hypothesis  H0: rxy = 0. The correlation coefficient is drawn from a population whose correlation is zero (ρXY = 0).
Alternative hypothesis  H1: The correlation coefficient is not drawn from a population whose correlation is different to zero (ρXY ).

18. Significance Formula: Interpretation:
 Null hypothesis is rejected. The correlation is not drawn from a population whose score ρxy = 0. Significant relationship between variables exists.
 Null hypothesis is accepted. The correlation is drawn from a population whose score ρxy = 0.
Significant relationship between variables does not exist.
Exercise: conclude about the significance of the example.

19. Significance: example
Conclusions: we reject the null hypothesis with a maximum risk to fail of The correlation is not drawn from a population whose score ρxy = 0. Relationship between variables exists.

20. Other questions to be considered
Correlation does not imply causality.
Statistical significance depends on sample size (higher N, likelier to obtain significance).
Other possible interpretation is given by the coefficient of determination , or proportion of variability in Y that is ‘explained’ by X.
The proportion of Y variability that left unexplained by X is called coefficient of non-determination:
Exercise: calculate the coefficient of determination and the coefficient of non-determination and interpret the results.

21. Coefficient of determination: example
. 70.4% of variability in Y is explained by X.
. 29.6% of variability in Y is not explained.

22. Which is the final conclusion?
Significant effect
Non-significant effect
High effect size (≥ 0.67)
The effect probably exists
The non-significance can be due to low statistical power
Low effect size (≤ 0.18)
The statistical significance can be due to an excessive high statistical power
The effect probably does not exist

23. Which is the final conclusion?
Significant effect
Non-significant effect
High effect size (≥ 0.67)
The effect probably exists
The non-significance can be due to low statistical power
Low effect size (≤ 0.18)
The statistical significance can be due to an excessive high statistical power
The effect probably does not exist