Question

Find the covariance s_(xy) using bivariate data.

Bivariate numerical data
Find the covariance $$\displaystyle{s}_{{{x}{y}}}$$ using bivariate data.

2021-01-05
The given bivariate data is(1,6),(3,2),and(2,4).
Calculation:
First determine $$\displaystyle\sum{x}_{{i}}{y}_{{i}},\sum{x}_{{i}}{\quad\text{and}\quad}\sum{y}_{{i}}$$.
$$\displaystyle{x}_{{i}}\rightarrow$$ first coordinates of the ordered pairs.
$$\displaystyle{y}_{{i}}\rightarrow$$ second coordinates of the ordered pairs.
$$\displaystyle\sum{x}_{{i}}{y}_{{i}}={1}\cdot{6}+{3}\cdot{2}+{2}\cdot{4}={20}$$
$$\displaystyle\sum{x}_{{i}}={1}+{3}+{2}={6}$$
$$\displaystyle\sum{y}_{{i}}={6}+{2}+{4}={12}$$
For covariance $$\displaystyle{s}_{{{x}{y}}}{u}{\sin{{g}}}{f}{\quad\text{or}\quad}\mu{l}{a}{s}_{{{x}{y}}}=\frac{{\sum{x}_{{i}}{y}_{{i}}-\frac{{\sum{x}_{{i}}-\sum{y}_{{i}}}}{{{n}}}}}{{{n}-{1}}}$$.
Where n = number of ordered pairs.
n=3
$$\displaystyle{s}_{{{x}{y}}}=\frac{{{20}-\frac{{{6}\cdot{12}}}{{{3}}}}}{{{3}-{1}}}$$
$$\displaystyle=\frac{{{20}-\frac{{{72}}}{{{3}}}}}{{{3}-{1}}}$$
$$\displaystyle=\frac{{{2}-{24}}}{{2}}$$
$$\displaystyle=-\frac{{4}}{{2}}=-{2}$$
Hence $$\displaystyle{s}_{{{x}{y}}}=-{2}$$.