Question

Calculate the Karl Pearson’s coefficient of correlation from the following pairs of values and interpret the result: Values of X 12 9 8 10 11 13 7 Values of Y 14 8 6 9 11 12 3  

Answer 1

$$\bar{X}=\dfrac{\sum_iX_i}{n}=\dfrac{70}{7}=10$$

$$\bar{Y}=\dfrac{\sum_iY_i}{n}=\dfrac{63}{7}=9$$

$$SS_{XX}=\sum_i(X_i-\bar{X})^2=\sum_iX_i^2-n\cdot\bar{X}^2$$

$$=728-7(10)^2=28$$

$$SS_{YY}=\sum_i(Y_i-\bar{Y})^2=\sum_iY_i^2-n\cdot\bar{Y}^2$$

$$=651-7(9)^2=84$$

$$SS_{XY}=\sum_i(X_i-\bar{X})(Y_i-\bar{Y})=\sum_iX_iY_i-n\cdot\bar{X}\bar{Y}$$

$$=676-7(10)(9)=46$$

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}=\dfrac{46}{\sqrt{28}\sqrt{84}}\approx$$

$$\approx0.948504$$

$0.7<r\leq1$ means a strong positive correlation.