Notation In this Chapter

$$\begin{array}{ll} \begin{array}{rl} S_{xx}&=\sum_{i=1}^{n}{(x_{i}-\bar{x})^{2}}=\sum_{i=1}^{n}{x_{i}^{2}}-n\cdot{\bar{x}^{2}}=\sum_{i=1}^{n}{x_{i}^{2}}-\frac{\big(\sum_{i=1}^{n}{x_{i}}\big)^{2}}{n}\\ S_{yy}&=\sum_{i=1}^{n}{(y_{i}-\bar{y})^{2}}=\sum_{i=1}^{n}{y_{i}^{2}}-n\cdot{\bar{y}^{2}}=\sum_{i=1}^{n}{y_{i}^{2}}-\frac{\big(\sum_{i=1}^{n}{y_{i}}\big)^{2}}{n}\\ S_{xy} & =\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=\sum_{i=1}^{n}x_{i}y_{i}-n\cdot{\bar{x}\bar{y}} =\sum_{i=1}^{n}x_{i}y_{i}-\frac{\big(\sum_{i=1}^{n}{x_{i}}\big)\big(\sum_{i=1}^{n}{y_{i}}\big)}{n}\\ & =\sum_{i=1}^{n}y_{i}(x_{i}-\bar{x})=\sum_{i=1}^{n}x_{i}(y_{i}-\bar{y})\\ \sigma^{2}&=Var(X)=\frac{S_{xx}}{df}\\ \end{array}\\ Var(Y)=\frac{S_{yy}}{df} \end{array}$$