Statistics Definitions > Spearman Rank Correlation / Spearman’s Rho
What is Spearman Rank Correlation / Spearman’s Rho?
The Spearman rank correlation coefficient, rs, is the nonparametric version of the Pearson correlation coefficient. Your data must be ordinal, interval or ratio. Spearman’s returns a value from -1 to 1, where:
+1 = a perfect positive correlation between ranks
-1 = a perfect negative correlation between ranks
0 = no correlation between ranks.
The formula for the Spearman rank correlation coefficient when there are no tied ranks is:
The scores for nine students in physics and math are as follows:
Physics: 35, 23, 47, 17, 10, 43, 9, 6, 28
Mathematics: 30, 33, 45, 23, 8, 49, 12, 4, 31
Compute the student’s ranks in the two subjects and compute the Spearman rank correlation.
Step 1: Find the ranks for each individual subject. I used the Excel rank function to find the ranks. If you want to rank by hand, order the scores from greatest to smallest; assign the rank 1 to the highest score, 2 to the next highest and so on:
Step 2: Add a third column, d, to your data. The d is the difference between ranks. For example, the first student’s physics rank is 3 and math rank is 5, so the difference is 3 points. In a fourth column, square your d values.
Step 4: Sum (add up) all of your d-squared values.
4 + 4 + 1 + 0 + 1 + 1 + 1 + 0 + 0 = 12. You’ll need this for the formula (the Σ d22> is just “the sum of d-squared values”).
Step 5: Insert the values into the formula. These ranks are not tied, so use the first formula:
= 1 – (6*12)/(9(81-1))
= 1 – 72/720
The Spearman Rank Correlation for this set of data is 0.9.
Tied ranks are where two items in a column have the same rank. A couple of different formulas exist for dealing with tied ranks. Perhaps the easiest way is to use the mean of the tied ranks.
Check out our YouTube channel for hundreds of statistics help videos!