اگر $\sum\limits_{{}}^{{}}{y_{i}^{2}}=1375\,\,,\,\,\sum\limits_{{}}^{{}}{x_{i}^{2}}=360\,\,,\,\,\sum\limits_{{}}^{{}}{{{x}_{i}}}{{y}_{i}}=500\,\,,\,\,\,\sum\limits_{{}}^{{}}{{{y}_{i}}}=75\,\,\,,\,\,\,\sum\limits_{{}}^{{}}{{{x}_{i}}}=40\,\,,\,\,n=5$ باشد، ضریب همبستگی کدام است؟
نمایش پاسخ
$\begin{align}
& r=\frac{\sum\limits_{{}}^{{}}{{{x}_{i}}{{y}_{i}}-\frac{(\sum\limits_{{}}^{{}}{{{x}_{i}})(\sum\limits_{{}}^{{}}{{{y}_{i}})}}}{n}}}{\sqrt{\left[ \sum\limits_{{}}^{{}}{x_{i}^{2}-\frac{(\sum\limits_{{}}^{{}}{{{x}_{i}}{{)}^{2}}}}{n}} \right]\left[ \sum\limits_{{}}^{{}}{y_{i}^{2}-\frac{(\sum\limits_{{}}^{{}}{{{y}_{i}}{{)}^{2}}}}{n}} \right]}} \\
& r=\frac{500-\frac{\left( 40 \right)\left( 75 \right)}{5}}{\sqrt{\left[ 360-\frac{{{\left( 40 \right)}^{2}}}{5} \right]\left[ 1375-\frac{{{\left( 75 \right)}^{2}}}{5} \right]}}=\frac{-100}{100}=-1 \\
\end{align}$