91Ó°ÊÓ

To find the linear correlation coefficient by the definition.

The linear correlation coefficient using the definition is

$r=\frac{\frac{1}{n-1}∑\left(x_i-\stackrel{¯}{x}\right)\left(y_i-\stackrel{¯}{y}\right)}{{\mathrm{σ}}_x{\mathrm{σ}}_y}$

The standard deviations are

$$\begin{gathered} {\mathrm{σ}}_x=\sqrt{\frac{1}{n-1}∑(x-\stackrel{¯}{x}{)}^2}, \\ {\mathrm{σ}}_y=\sqrt{\frac{1}{n-1}∑(y-\stackrel{¯}{y}{)}^2} \end{gathered}$$

Calculation:

Make a table of values

Find the standard deviations

$$\begin{gathered} {\mathrm{σ}}_x=\sqrt{\frac{1}{n-1}∑(x-\stackrel{¯}{x}{)}^2} \\ =\sqrt{\frac{1}{4-1}\left(6\right)} \\ =1.414 \end{gathered}$$

$$\begin{gathered} {\mathrm{σ}}_y=\sqrt{\frac{1}{n-1}∑(y-\stackrel{¯}{y}{)}^2} \\ =\sqrt{\frac{1}{4-1}\left(104\right)} \\ =5.88 \end{gathered}$$

Find the correlation coefficient

$$\begin{gathered} r=\frac{\frac{1}{n-1}∑\left(x_i-\stackrel{¯}{x}\right)\left(y_i-\stackrel{¯}{y}\right)}{{\mathrm{σ}}_{\stackrel{¯}{x}}{\mathrm{σ}}_y} \\ =\frac{\frac{1}{4-1}(-18)}{(1.414)(5.88)} \end{gathered}$$

Hence,

The linear coefficient by the definition$=-0.721$

To find the linear correlation coefficient by the formula.

The linear correlation coefficient using the formula

$r=\frac{∑xy-\frac{{∑}_x{∑}_y}{n}}{\sqrt{\left(∑x^2-\frac{∑x^2}{n}\right)\left(∑y^2-\frac{∑y^2}{n}\right)}}$

Calculation:

Make a table of values

Find the correlation coefficient

$$\begin{gathered} r=\frac{∑xy-\frac{∑x{∑}^n}{n}}{\sqrt{\left(∑x^2-\frac{{\left(∑x\right)}^2}{n}\right)\left(∑y^2-\frac{{\left(∑y\right)}^2}{n}\right)}} \\ =\frac{42-\frac{240}{4}}{\sqrt{(42-(144/4\left)\right)(204-(20\sqrt{4}\left)\right)}} \end{gathered}$$

Hence,

The linear coefficient by the formula$=-0.721$

Hence, both the results are the same.