Is it possible to rewrite this equation using Einstein Summation?

Question

Can we rewrite the below linked equation as: $$\overline{x} = \frac{x_i}{n} $$

Or is it necessary to have two variables and indices?

Here's the equation

Answer 1

You could define $a:=(1,\dots,1)$ and then write $\bar x=\frac1n a_i x_i$. In general, Einstein is best used for expressions involving linear operations, that is, linear operators applied to vectors, or more generally, products of tensors. If you can rewrite an expression using linear operations, like the scalar product used above, you can use Einstein notation.

The reason why you need such operations is, as you said, that the notation only makes you take sums when an index comes up twice in a product. As for why the notation was chosen this way: exactly because physicists use it to describe linear operations. That's what it's made for.