# Does zero Spearman's rho imply zero Covariance?

To the question of the title I would "intuitively" answer yes, by the following informal argument:
Covariance "measures the strength of linear association" (when scaled by the product of standard deviations) between two variables, while Spearman's rho "measures the strength of monotone association."

Linear association is a subset of monotone association (isn't it?), hence, when the measure of monotone association is zero, the measure of linear association should also be zero.

But I have learned my lesson (and so I am not a menace to society) about easy "intuitive" arguments in Statistics. And my attempts to examine this conjecture formally were not fruitful so far.

So: Does a zero Spearman's rho imply zero Covariance?

Can we formally prove it, or disprove it even by a counter-example?

UPDATE
This post provides also examples that there is no such relation

• Are you referring to the sample statistics? No (as the answers below show), but that is just as a normal sample median and sample mean can differ, though they target the same underlying parameter. Likewise these rank correlation statistics are consistent (asymptotically unbiased), and will tend to zero if the population cov is 0. Dec 7, 2020 at 7:35
• Covariance does not measure the strength of linear association stats.stackexchange.com/a/229703/3277 Dec 7, 2020 at 8:00
• @ttnphns Apologies, certainly it is measured by Pearson's correlation coefficient, but sometimes I consider this "widely known". I made an edit to avoid confusing readers. Dec 7, 2020 at 12:36

Counterexample:

X Y
1 500
2 1
3 2
4 3
5 4


For these values,

• Pearson's $$r \approx -0.70$$
• Spearman's $$\rho = 0$$

That single large Y value affects the covariance much more than it affects Spearman's rank correlation coefficient.

No. Easy to see why. The use case of rank correlation is when we’re not satisfied with Pearson correlation, e.g. with its proneness to fall for outliers. Therefore, clearly Spearman correlation should not correspond to Pearson correlation outcomes.

Sometimes zero spearman correlation coincides with zero Pearson correlation and consequently zero covariance, but it’s not a general case.

• But Spearman correlation could equal Pearson correlation. If the data have Pearson correlation of $\pm 1$ then so also does Spearman. Some cases with Pearson correlation of zero will have Spearman correlation zero too. I wouldn't rule out other coincidences. Dec 6, 2020 at 19:04
• Nick, the original question ask about implication: X -> Y. In this case, your answer is correct statement that it's possible the two values can be equal. However, it's an answer to another question. The Spearman correlation is the Pearson correlation applied to the ranks, and therefore makes statements about the ranks, and not the actual values involved. Dec 7, 2020 at 0:20
• Despite the edit to improve the answer, the remaining statement "clearly Spearman correlation should not correspond to Pearson correlation outcomes" is too strong. Dec 7, 2020 at 1:16
• @BarryDeCicco I'm commenting on what Aksakal said, not answering the question. I am aware of the definitions. Dec 7, 2020 at 2:03