# Standard Deviation

For sets X and Y, is it true that

$$s_{XY}=s_{X}\overline{Y}+s_{Y}\overline{X},$$​

where $$s$$ represents the standard deviation and $$XY$$ is the set containing $$x_{i}y_{i}$$?

First off, I assume you mean sample deviation? Standard deviation is a constant.

I doubt the result is true in general. However, there may be a specific case where the formula holds.

Do you have any idea what X and Y might be distributed as? Where does the problem arise?

matt grime
Homework Helper
What are X and Y, and what does XY really mean? How can x_iy_i make sense, since this will completely depend on how one labels elements of the sets (apparently they're sets) X and Y? It also presupposes that |X|=|Y| too.

Simfish
Gold Member
Hi there. Remember me? :)

I think it means that the standard deviation of the union of X and Y is equal to the standard deviation of x * the mean of Y + the standard deviation of y * the mean of X.

Uggh I don't know how to prove those - have to go to some statistics textbooks...

Anyways, if Y mean = 100 and Y SD = 0, and X mean = 0 and X SD = 0, then the formula would compute a combined SD of 0. But then your combined sample has both elements of 0 and 100, and it must have a standard deviation. So the formula is not universally true.

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