Given three vectors a, b, and c, is it possible that correlations between a and b, a and c, and b and c are all negative? I.e. is this possible?

corr(a,b)<0corr(a,c)<0corr(b,c)<0

**Answer**

It is possible if the size of the vector is 3 or larger. For example

a=(−1,1,1)b=(1,−9,−3)c=(2,3,−1)

The correlations are

cor(a,b)=−0.80...cor(a,c)=−0.27...cor(b,c)=−0.34...

We can prove that for vectors of size 2 this is not possible:

cor(a,b)<02(∑iaibi)−(∑iai)(∑ibi)<02(a1b1+a2b2)−(a1+a2)(b1b2)<02(a1b1+a2b2)−(a1+a2)(b1b2)<02(a1b1+a2b2)−a1b1+a1b2+a2b1+a2b2<0a1b1+a2b2−a1b2+a2b1<0a1(b1−b2)+a2(b2−b1)<0(a1−a2)(b1−b2)<0

The formula makes sense: if a1 is larger than a2, b2 has to be larger than b1 to make the correlation negative.

Similarly for correlations between (a,c) and (b,c) we get

(a1−a2)(c1−c2)<0(b1−b2)(c1−c2)<0

Clearly, all of these three formulas can not hold at the same time.

**Attribution***Source : Link , Question Author : Antti A , Answer Author : Community*