Minimizing the sum of absolute difference of each point from the average does it minimize the variance?

Question

I have a problem in which there is a set of $N$ points. Each point has a set of possible weight (lets call them $W_i$ where $i \in [1,N])$. My goal is to select the correct weight $W_i$ for each point $i$ so that the variance of the weights is minimized. If I find the selection of weights $W_i$ so that the function $f=\sum_{i=1}^{N} \mid W_i - \mu \mid$, where $\mu$ is the mean value of the $W_i$, is minimized, does that $ \Rightarrow $ that the variation of the $W_i$ is also minimized? Is there a way to formally prove it? Sorry for my bad terminology and use of English. Any help will be greatly appreciated :)