Can anyone prove the following inequality:

$\displaystyle

\sum_{i=1}^N y_{i}^2/n_{i} - \frac{\left(\sum_{i=1}^N y_{i}\right)^2}{n} \ge 0,

$

where $\displaystyle n=\sum_{i=1}^N n_{i},\ \ n_{i} \ge 0$, and $\displaystyle y_{i}$ is a real quantity (can be both negative or positive)?

It appears that the inequality is valid (even used random numbers), but can't see how to prove it.