Summation: $\sum_{i=1}^{n} x_i^2$

Question

How can I write the expression below in alternative way?

$\sum_{i=1}^{n} x_i^2$

Where:

$\sum_{i=1}^{n} x_i = n\bar x $

Answer 1

\begin{align} \sum_{i=1}^N x_i^2 & = \sum_{i=1}^N (x_i - \bar x + \bar x)^2 \\ & = \sum_{i=1}^N (x_i - \bar x)^2 - 2 \bar x (x_i - \bar x) + \bar x^2 \\ & = \sum_{i=1}^N (x_i - \bar x)^2 + N \bar x^2 \\ & = N(\sigma^2 + \bar x^2) \end{align} is one way to rewrite the sum of squares.