Question

Calculate the correlation coefficient of the following data: x 45 46 46 47 48 49 50 y 44 48 45 48 52 51 49

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2\\ \hline & 45 & 44 & 1980 & 2025 & 1936\\ & 46 & 48 & 2208 & 2116 & 2304\\ & 46 & 45 & 2070 & 2116 & 2025\\ & 47 & 48 & 2256 & 2209 & 2304\\ & 48 & 52 & 2496 & 2304 & 2704\\ & 49 & 51 & 2499 & 2401 & 2601\\ & 50 & 49 & 2450 & 2500 & 2401\\ Sum= & 331 & 337 & 15959 & 15671 & 16275\\ \end{array}$$ $$\bar{X}=\dfrac{1}{n}\sum_iX_i=\dfrac{331}{7}=47.2857$$

$$\bar{Y}=\dfrac{1}{n}\sum_iY_i=\dfrac{337}{7}=48.1429$$

$$SS_{XX}=\sum_i(X_i-\bar{X})^2=19.428571$$

$$SS_{YY}=\sum_i(Y_i-\bar{Y})^2=50.857143$$

$$SS_{XY}=\sum_i(X_i-\bar{X})(Y_i-\bar{Y})=23.714286$$

Correlation coefficient:

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}$$

$$=\dfrac{23.714286}{\sqrt{29.219.428571}\sqrt{50.857143}}=0.714863$$

We have a strong positive correlation.