Question

find the correlation coefficient for the data set below.

x	1	2	3	4	5	6	7
y	3	2	4	0	1	5	7

Explanation:

Step1: Recall correlation - coefficient formula

The correlation coefficient $r$ is given by $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$, where $n$ is the number of data - points. Here, $n = 7$.
Let $x=\{1,2,3,4,5,6,7\}$ and $y = \{3,2,4,0,1,5,7\}$.
First, calculate $\sum x$:
$\sum x=1 + 2+3+4+5+6+7=\frac{7\times(7 + 1)}{2}=28$.

Step2: Calculate $\sum y$

$\sum y=3 + 2+4+0+1+5+7=22$.

Step3: Calculate $\sum xy$

$xy$ values are: $1\times3 = 3$, $2\times2 = 4$, $3\times4 = 12$, $4\times0 = 0$, $5\times1 = 5$, $6\times5 = 30$, $7\times7 = 49$.
$\sum xy=3+4+12+0+5+30+49 = 103$.

Step4: Calculate $\sum x^{2}$

$x^{2}$ values are: $1^{2}=1$, $2^{2}=4$, $3^{2}=9$, $4^{2}=16$, $5^{2}=25$, $6^{2}=36$, $7^{2}=49$.
$\sum x^{2}=1 + 4+9+16+25+36+49 = 140$.

Step5: Calculate $\sum y^{2}$

$y^{2}$ values are: $3^{2}=9$, $2^{2}=4$, $4^{2}=16$, $0^{2}=0$, $1^{2}=1$, $5^{2}=25$, $7^{2}=49$.
$\sum y^{2}=9+4+16+0+1+25+49 = 104$.

Step6: Substitute values into the formula

$n = 7$.
$n\sum xy=7\times103 = 721$, $(\sum x)(\sum y)=28\times22 = 616$.
$n\sum x^{2}=7\times140 = 980$, $(\sum x)^{2}=28^{2}=784$.
$n\sum y^{2}=7\times104 = 728$, $(\sum y)^{2}=22^{2}=484$.
$r=\frac{721 - 616}{\sqrt{(980 - 784)(728 - 484)}}=\frac{105}{\sqrt{196\times244}}=\frac{105}{\sqrt{47824}}=\frac{105}{218.7}$
$r\approx0.48$.

Answer:

Approximately $0.48$