Chapter 1.3, Problem 11E

The following results on summations will help us in calculating the sample variance s². For any constant c,

1. a ${\displaystyle \underset{i=1}{\overset{n}{\sum }}c}=nc.$
2. b ${\displaystyle \underset{i=1}{\overset{n}{\sum }}c{y}_i}=c{\displaystyle \underset{i=1}{\overset{n}{\sum }}{y}_i}.$
3. c ${\displaystyle \underset{i=1}{\overset{n}{\sum }}(x_i+y_i)}={\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{y}_i}.$

Use (a), (b), and (c) to show that

$s^2=\frac{1}{n-1}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{(y_i-\bar{y})}^2}=\frac{1}{n-1}\left[{\displaystyle \underset{i=1}{\overset{n}{\sum }}{y}_i^2-\frac{1}{n}{\left({\displaystyle \underset{i=1}{\overset{n}{\sum }}{y}_i}\right)}^2}\right].$