Question

what is the correlation coefficient for the data shown in the table? 0 1 -1 5 10

Explanation:

Step1: Recall correlation coefficient formula

The formula for the correlation coefficient $r$ is $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$. Let $x = [15,10,5,0]$ and $y=[0,5,10,15]$, and $n = 4$.

Step2: Calculate sums

$\sum x=15 + 10+5 + 0=30$, $\sum y=0 + 5+10 + 15=30$, $\sum xy=(15\times0)+(10\times5)+(5\times10)+(0\times15)=0 + 50+50+0 = 100$, $\sum x^{2}=15^{2}+10^{2}+5^{2}+0^{2}=225 + 100+25+0=350$, $\sum y^{2}=0^{2}+5^{2}+10^{2}+15^{2}=0 + 25+100+225=350$.

Step3: Substitute into formula

$n(\sum xy)=4\times100 = 400$, $(\sum x)(\sum y)=30\times30 = 900$, $n\sum x^{2}=4\times350=1400$, $(\sum x)^{2}=30^{2}=900$, $n\sum y^{2}=4\times350 = 1400$, $(\sum y)^{2}=30^{2}=900$. Then $r=\frac{400-900}{\sqrt{(1400 - 900)(1400 - 900)}}=\frac{- 500}{\sqrt{500\times500}}=\frac{-500}{500}=-1$.

Answer:

-1