Chapter 1, Problem 22SE

Prove that the sum of the deviations of a set of measurements about their mean is equal to zero; that is,

To determine

Prove that the sum of deviations about the mean of a set of observations is zero, that is, ${\displaystyle \sum _{i=1}^n\left(y_i-\overline{y}\right)}=0$.

Explanation of Solution

The formula for the mean of a set of n observations is, $\overline{y}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{y}_i}$.

Now, consider the following calculation, starting from the left-hand-side of the given expression:

$\begin{array}{c}{\displaystyle \sum _{i=1}^n\left(y_i-\overline{y}\right)}={\displaystyle \sum _{i=1}^n{y}_i}-{\displaystyle \sum _{i=1}^n\overline{y}}\\ =n\left(\frac{1}{n}{\displaystyle \sum _{i=1}^n{y}_i}\right)-n\overline{y}\\ =n\overline{y}-n\overline{y}\\ =0.\end{array}$

Thus, the left-hand-side is 0, which is the right-hand-side of the given expression.

Hence, it is proved that ${\displaystyle \sum _{i=1}^n\left(y_i-\overline{y}\right)}=0$.