Question

Calculate karl pearsons coefficent of correlation for the following data ?

X - 105,111,104,112,118,98,116,103,106,112

Y- 62,64,53,60,72,56,68,60,69,65

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c} x & y \\ \hline 105 & 62 \\ \hline 111 & 64 \\ \hline 104 & 53 \\ \hline 112 & 60 \\\hline 118 & 72 \\ \hline 98 & 56 \\ \hline 116 & 68 \\ \hline 103 & 60 \\ \hline 106 & 69 \\ \hline 112 & 65 \\ \end{array}$$

$$\bar{x}=\dfrac{\sum_ix_i}{n}$$

$$=\dfrac{105+111+104+112+118+98+116+103+106+112}{10}$$

$$=108.5$$

$$\bar{y}=\dfrac{\sum_iy_i}{n}$$

$$=\dfrac{62+64+53+60+72+56+68+60+69+65}{10}$$

$$=62.9$$

$$S_{xx}=\sum_i(x_i-\bar{x})^2=\sum_ix_i^2-n\cdot\bar{x}^2=356.5$$

$$S_{xy}=\sum_i(x_i-\bar{x})(y_i-\bar{y})=\sum_ix_iy_i-n\cdot\bar{x}\bar{y}=245.5$$

$$S_{yy}=\sum_i(y_i-\bar{y})^2=\sum_iy_i^2-n\cdot\bar{y}^2=314.9$$

$$r=\dfrac{S_{xy}}{\sqrt{S_{xx}}\sqrt{S_{yy}}}=\dfrac{245.5}{\sqrt{356.5}\sqrt{314.9}}=0.7327$$

Strong correlation