# Does the size of the sum of squares say anything about the size of original sum?

Suppose I am told I have N numbers and that the sum of the square of those numbers is S. Can anything be said about the sum of those original N numbers?

Cauchy-Schwarz gives $$\sum_{n=1}^N |x_n| \le \sqrt{N} \sqrt{\sum_{n=1}^N x_n^2}.$$
For a lower bound, we have $$\left(\sum_{n=1}^N |x_n|\right)^2 \ge \sum_{n=1}^N x_n^2.$$
So, $\sqrt{S} \le \sum_{n=1}^N |x_n| \le \sqrt{NS}$.
Without further information, these bounds cannot be improved. If all of the numbers are the same, then $\sum_{n=1}^N |x_n| = \sqrt{NS}$. If all of the numbers are zero except one of them, then $\sum_{n=1}^N |x_n| = \sqrt{S}$.