Question

Calculate karl pearson's coefficient of correlation from the following data :

X - 105, 120, 95, 150, 130

Y - 100, 115, 110, 135, 115

Answer 1

The coefficient of correlation is calculated with the formula:

$$r=\frac{\sum (x_i-\bar x)(y_i - \bar y)}{\sqrt{\sum (x_i-\bar x)^2\sum (y_i - \bar y)^2}}$$

$\bar x = \frac{\sum x_i}{n}=\frac{105+120+95+150+130}{5}= 120$

$\bar y = \frac{\sum y_i}{n}=\frac{100+115+110+135+115}{5}= 115$

$r=\frac{(105-120)(100-115)+(120-120)(115 - 115)+(95-120)(110 - 115)+(150-120)(135 - 115)+(130-120)(115 - 115)}{\sqrt{((105-120)^2+(120-120)^2+(95-120)^2+(150-120)^2+(130-120)^2)((100 - 115)^2+(115 - 115)^2+(110 - 115)^2+(135 - 115)^2+(115 - 115)^2)}}=$ $=\frac{(-15)(-15)+0+(-25)(-5)+30\cdot20+0}{\sqrt{((-15)^2+0^2+(-25)^2+30^2+10^2)((-15)^2+0^2+(-5)^2+20^2+0^2)}} =\frac{225+125+600}{\sqrt{(225+625+900+100)(225+25+400)}} =\frac{950}{\sqrt{1850\cdot650}}=0.866$

Answer: 0.866