# Spearman correlation and Pearson correlation

If $$X$$ and $$Y$$ has the Spearman correlation $$r_s=1$$, is it possible that the Pearson correlation between $$X$$ and $$Y$$ is 0?

Yes if by $$1$$ and $$0$$ you mean $$\approx 1$$ and $$\approx 0$$. Here is one example in R with a stupid outlier

x=c(1:1000)
y=c(1:999,-100000)

> cor(x,y,method="spearman")
 0.994006
> cor(x,y,method="pearson")
 0.03571434

• I suspect you could find an example with $r_s\approx 1$ and $r_p=0$ but not $r_s= 1$ and $r_p\approx 0$ Dec 7, 2022 at 10:41
• Can we construct a monotone function $Y=f(X)$ such that $f$ is monontone, But $Cov(X,Y)=0$? Dec 7, 2022 at 10:41
• @mathbeginner What do you think? Dec 7, 2022 at 10:43
• I mean that if $x$ and $y$ has the strictly monotone relationship, for example, y=x^3, can we have cov(x,y)=0 Dec 7, 2022 at 10:48
• A Spearman correlation of 1 implies that all data lie on a rising monotonic curve (in fact on many such curves). I can't see that is compatible with Pearson correlation zero. Dec 7, 2022 at 11:41