1. Correlations: Correlation Coefficient:
A succinct measure of the strength of the relationship between two variables (e.g. height and weight, age and reaction time, IQ and exam score).

2. There are various types of correlation coefficient, for different purposes:
1. Pearson's "r":
Used when both X and Y variables are
(a) continuous;
(b) (ideally) measurements on interval or ratio scales;
(c) normally distributed - e.g. height, weight, I.Q.
2. Spearman's rho:
In same circumstances as (1), except that data need only be on an ordinal scale - e.g. attitudes, personality scores.

3. r is a parametric test: the data have to have certain characteristics (parameters) before it can be used.
rho is a non-parametric test - less fussy about the nature of the data on which it is performed.

4. Correlations vary between:
+1 (perfect positive correlation: as X increases, so does Y):

5. ... and -1 (perfect negative correlation: as X increases, Y decreases, or vice versa).
r = 0 means no correlation between X and Y: changes in X are not associated with systematic changes in Y, or vice versa.

6. Calculating Pearson's r: a worked example: Do students who perform well on one statistics test also perform well on another?

7. Student:
Test 1 (X):
Test 2 (Y): X2 Y2 XY A 37 75
1369
5625
2775 B 41 78
1681
6084
3198 C 48 88
2304
7744
4224 D 32 80
1024
6400
2560 E 36
1296
2808 F 30 71
900
5041
2130 G 40
1600
3000 H 45 83
2025
6889
3735 I 39 74
1521
5476
2886 J 34
1156
2516
N=10
ΣX = 382
ΣY =776
ΣX2 = 14876
ΣY2 = 60444
ΣXY = 29832

8. Using our values (from the bottom row of the table:)
ΣX = 382
ΣY =776
ΣX2 = 14876
ΣY2 = 60444
ΣXY = 29832 ( )
776
382 *
29832 - 10 r = ( ) ( ) æ
382 2 ö æ
776 2 ö ç
14876 - ÷ * ç
60444 - ÷ ç ÷ ç ÷ è ø è ø 10 10

9. ( )
7455 .
391
253 80
188 40
226 60
283 r
60217
60444
14592
14876 20
29643
29832 = * -
r is This is a positive correlation: students who score highly on the first test tend to score highly on the second (and vice versa).

10. How to interpret the size of a correlation:
r2 is the "coefficient of determination". It tells us what proportion of the variation in the Y scores is associated with changes in X.
e.g., if r is 0.2, r2 is 4% (0.2 * 0.2 = = 4%).
Only 4% of the variation in Y scores is attributable to Y's relationship with X. Thus, knowing a person's Y score tells you essentially nothing about what their X score might be.

11. Our correlation of 0.75 gives an r2 of 56%.
An r of 0.9, gives an r2 of (0.9 * 0.9 = .81) = 81%.
Note that correlations become much stronger the closer they are to 1 (or -1).
Correlations of .6 or -.6 (r2 = 36%) are much better than correlations of .3 or -.3 (r2 = 9%), not merely twice as strong!

12. Spearman's rho:
Measures the degree of monotonicity rather than linearity in the relationship between two variables - i.e., the extent to which there is some kind of change in X associated with changes in Y:
Hence, copes better than Pearson's r when the relationship is monotonic but non-linear - e.g.:

13. Spearman's rho - worked example:
Is there a correlation between the number of vitamin treatments a person has, and their score on a memory test?

14. Subj:
No.vitamin teatments (X):
Memory test score (Y):
Vitamin treatment ranks (X):
Memory ranks (Y): D
(= X-Y) D2 A 2 22 1 +1 B 34 -1 C 3 36
3.5
+0.5
0.25 4 49 5 E 42
-0.5 F 6 57 7 G 82
7.5
-1.5
2.25 H 8
N = 8
ΣD2 = 6.0

15. OR

16. Step 1: assign ranks to the raw data, for each variable separately.
Rules for ranking:
(a) Give the lowest score a rank of 1; next lowest a rank of 2; etc.
(b) If two or more scores are identical, this is a "tie": give them the average of the ranks they would have obtained had they been different.
The next score that is different, gets the rank it would have had if the tied scores had not occurred.

17. e.g.:
raw score
"original"rank
actual rank:
Rank for the tied scores is (2+3)/2 = 2.5
raw score
actual rank:
Rank for the tied scores is (2+3+4)/3 = 3

18. Step 2:
Subtract one set of ranks from the other, to get a set of differences, D.
Step 3:
Square each of these differences, to get D2.
Step 4:
Add up the values of D2 , to get ΣD2. Here, ΣD2 = 8.5. N = 8.

19. Step 5:

20. rho = 0.93.
There is a strong positive correlation between the number of vitamin treatments a person has, and their memory test score.
Pearson's r on the same data = 0.86.

21. Using SPSS to obtain scatterplots: (a) simple scatterplot:
Graphs > Scatter/Dot...

22. Analyze > Regression > Curve Estimation...
Using SPSS to obtain scatterplots: (b) scatterplot with regression line:
Analyze > Regression > Curve Estimation...
"Constant" is the intercept, "b1" is the slope

23. Using SPSS to obtain correlations:
Analyze > Correlate > Bivariate...