16. Non-parametric Tests

# 16.7 Spearman Rank Correlation Coefficient

This is a rank alternative to the Pearson correlation coefficient that may be used when the assumption of normality is not met for hypothesis testing. It is defined by

where

where = rank of point and = rank of point .

To test versus use itself as the test statistic and from the Rank Correlation Coefficient Critical Values Table as the critical statistic. (Note that the Rank Correlation Coefficient Critical Values Table requires .) Reject if .

Example 16.9 : Determine if the Spearman correlation between two textbook ratings, data given below, is significant.

 Book rating 1 () rating 2 () rank rank A 4 4 2 1 1 1 B 10 6 5 2 3 9 C 18 20 7 8 -1 1 D 20 14 8 6 2 4 E 12 16 6 7 -1 1 F 2 8 1 4 -3 9 G 5 11 3 5 -2 4 H 9 7 4 3 1 1

Note the preliminary data reduction (ranking and rank differences, ) done to the right side of the table.

1. Hypothesis.

(Note that population values are inferred in the hypotheses statement.)

2. Critical statistic.

From the Rank Correlation Coefficient Critical Values Table with and find

3. Test statistic.

4. Decide.

so do not reject .

5. Interpretation.

There is no significant correlation between the ratings.