Answer to Question #157562 in Statistics and Probability for Akash Buragohain

Question #157562

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

X - 105, 120, 95, 150, 130

Y - 100, 115, 110, 135, 115


1
Expert's answer
2021-01-25T15:02:51-0500

The coefficient of correlation is calculated with the formula:


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

xˉ=xin=105+120+95+150+1305=120\bar x = \frac{\sum x_i}{n}=\frac{105+120+95+150+130}{5}= 120

yˉ=yin=100+115+110+135+1155=115\bar y = \frac{\sum y_i}{n}=\frac{100+115+110+135+115}{5}= 115

r=(105120)(100115)+(120120)(115115)+(95120)(110115)+(150120)(135115)+(130120)(115115)((105120)2+(120120)2+(95120)2+(150120)2+(130120)2)((100115)2+(115115)2+(110115)2+(135115)2+(115115)2)=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)}}= =(15)(15)+0+(25)(5)+3020+0((15)2+02+(25)2+302+102)((15)2+02+(5)2+202+02)=225+125+600(225+625+900+100)(225+25+400)=9501850650=0.866=\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


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