Chapter 4.5, Problem 67ES

To determine

To prove: The estimating result from the least squares principle is $\overline{x}=\frac{1}{n}\left(x_1+x_2+\mathrm{...}+x_n\right)$ , that is, $\overline{x}$ is the arithmetic average of the observed values. The least square principle states that the estimate $\overline{x}$ should be chosen to minimize $S={\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{...}+{\left(x_n-\overline{x}\right)}^2$ , which is the sum of the squares of the deviations between the estimate $\overline{x}$ and the measured values.