Find the correlatin coefficient r using bivariate data.

Given:
The given bivariate data is(1,6),(3,2), and (2,4).
Calculation:
First determine ${\displaystyle \sum x_i{y}_i,\sum x_i\text{and}\sum y_i}$.
${\displaystyle x_i\to }$ first coordinates of the ordered pairs.
${\displaystyle y_i\to }$ second coordinates of the ordered pairs.
${\displaystyle \sum x_i{y}_i=1\cdot 6+3\cdot 2+2\cdot 4=20}$
${\displaystyle \sum x_i=1+3+2=6}$
${\displaystyle \sum x_i^2=1^2+3^2+2^2=14}$
${\displaystyle \sum y_i=6+2+4=12}$
${\displaystyle \sum y_i^2=6^2+2^2+4^2=56}$
Find the sample variance ${\displaystyle s^2}$ using the formula ${\displaystyle s^2=\frac{\sum x_i^2-\frac{\sum x_i^2}{n}}{n-1}}$
${\displaystyle s_x^2=\frac{14-\frac{6^2}{3}}{3-1}=\frac{14-12}{2}=1}$
${\displaystyle s_y^2=\frac{56-\frac{{12}^2}{3}}{3-1}=\frac{56-48}{2}=4}$
Find sample standard deviation.
${\displaystyle s_x=\sqrt{1}=1}$
${\displaystyle s_y=\sqrt{4}=2}$
For covariance ${\displaystyle s_{xy}usingf\text{or}\mu s_{xy}=\frac{\sum x_i{y}_i-\frac{\sum x_i-\sum y_i}{n}}{n-1}}$.
Where n = number of ordered pairs.
n=3
${\displaystyle s_{xy}=\frac{20-\frac{6\cdot 12}{3}}{3-1}}$
${\displaystyle =\frac{20-\frac{72}{3}}{3-1}}$
${\displaystyle =\frac{20-24}{2}}$
${\displaystyle =-\frac{4}{2}=-2}$
Find the correlation coefficient r using the formula ${\displaystyle r=\frac{s_{xy}}{s_x{s}_y}}$.
${\displaystyle r=-\frac{2}{1\cdot 2}=-1}$
Hence the value of correlation coefficient r = -1.