Spearman's rank correlation, when it beats Pearson, and what each of them is actually measuring.
Six points on a perfect straight line, so both coefficients start at 1. Drag the last point away and watch what happens to each.
Spearman only looks at the order. Moving a point further out does not change which one is biggest.
| Pearson r | Spearman rs | |
|---|---|---|
| Uses | the actual values | only the ranks |
| Detects | linear association only | any monotonic trend |
| Outliers | very sensitive | barely affected |
| r = 1 means | exactly on a straight line | the order matches perfectly |
Monotonic means "always going the same way". It does not have to be straight. A curve that rises the whole time has Spearman exactly 1 and a Pearson well below it, which is the second demonstration above.
Rank each variable separately, then work out Pearson's coefficient on the ranks. That is all it is.
| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| rank of x | 1 | 2 | 3 | 4 | 5 | 6 |
| y | 1 | 4 | 9 | 16 | 25 | 36 |
| rank of y | 1 | 2 | 3 | 4 | 5 | 6 |
The ranks match exactly, so rs = 1, even though the points are nowhere near a straight line and Pearson gives 0.979.
Equal values share the average rank. Two values tied for 1st and 2nd both get rank 1.5, and the next one is rank 3. Skipping to rank 2 and 3 is the common slip, and it changes the answer.
1. Six points lie exactly on y = x² for x = 1 to 6. What is rs?
2. Three data values are 7, 7 and 9. What rank does each 7 get?
3. A scatter rises steadily except for one point far off to the right. Which coefficient should you quote?
Both coefficients come from technology. The marks are in choosing the right one and justifying it, which is a sentence about the shape of the scatter or about an outlier.
Ranking correctly, including averaging ties.
Never delete an awkward point to make Pearson look better. Use Spearman and say why, which is the whole reason this sub-topic exists.
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