Orders of data in Divided Differences and Lagrangian Interpolation

Question

As we know that the order of data points i.e. x values do not matter in Divided Differences and The Lagrangian Interpolation. Why is that? What happens if we arrange them in order? better interpolating polynomial value can be obtained?

Answer 1

Note that

$$f(x)=\sum_{j=1}^Ny_k\prod_{\begin{smallmatrix}k=1\\ k\neq j\end{smallmatrix}}^N\frac{x-x_k}{x_j-x_k}$$

is unchanged by a permutation of the $(x_i,y_i)$, since addition and multiplication are commutative.