Independence of X̄ for IID random variables My question is that: Is $\overline X$ independent of $X_1 - \overline X$, given that $X_1,\ldots,X_n$ are IID random variables? 
I was thinking that it is independent, but i don't know the logic behind it. 

An example showing that they are in fact dependent.
if $\bar{X}$ indicates the sample mean then they are not independent. For example take $N=2$ and the $X_i$ to be independent $\operatorname{Bernoulli} (p)$. Then $\bar{X}=(X_1 + X_2)/2$. Suppose we condition on $\bar{X}=0$. Then $X_1 - \bar{X} = 0$. This shows that they are dependent.
