Finite Math Examples

\begin{array}{c} x & y \\ - 1 & 1 \\ 0 & 2 \\ 1 & - 3 \\ 2 & - 14 \\ 3 & - 31 \end{array}

$\begin{array}{c} x & y \\ - 1 & 1 \\ 0 & 2 \\ 1 & - 3 \\ 2 & - 14 \\ 3 & - 31 \end{array}$

Step 1

The linear correlation coefficient measures the relationship between the paired values in a sample.

r = \frac{n (\sum_{}^{} x y) - \sum_{}^{} x \sum_{}^{} y}{\sqrt{n (\sum_{}^{} x^{2}) - {(\sum_{}^{} x)}^{2}} \cdot \sqrt{n (\sum_{}^{} y^{2}) - {(\sum_{}^{} y)}^{2}}}

$r = \frac{n (\sum_{}^{} x y) - \sum_{}^{} x \sum_{}^{} y}{\sqrt{n (\sum_{}^{} x^{2}) - {(\sum_{}^{} x)}^{2}} \cdot \sqrt{n (\sum_{}^{} y^{2}) - {(\sum_{}^{} y)}^{2}}}$

Step 2

Sum up the

x

$x$ values.

\sum_{}^{} x = - 1 + 0 + 1 + 2 + 3

$\sum_{}^{} x = - 1 + 0 + 1 + 2 + 3$

Step 3

Simplify the expression.

\sum_{}^{} x = 5

$\sum_{}^{} x = 5$

Step 4

Sum up the

y

$y$ values.

\sum_{}^{} y = 1 + 2 - 3 - 14 - 31

$\sum_{}^{} y = 1 + 2 - 3 - 14 - 31$

Step 5

Simplify the expression.

\sum_{}^{} y = - 45

$\sum_{}^{} y = - 45$

Step 6

Sum up the values of

x \cdot y

$x \cdot y$ .

\sum_{}^{} x y = - 1 \cdot 1 + 0 \cdot 2 + 1 \cdot - 3 + 2 \cdot - 14 + 3 \cdot - 31

$\sum_{}^{} x y = - 1 \cdot 1 + 0 \cdot 2 + 1 \cdot - 3 + 2 \cdot - 14 + 3 \cdot - 31$

Step 7

Simplify the expression.

\sum_{}^{} x y = - 125

$\sum_{}^{} x y = - 125$

Step 8

Sum up the values of

x^{2}

$x^{2}$ .

\sum_{}^{} x^{2} = {(- 1)}^{2} + {(0)}^{2} + {(1)}^{2} + {(2)}^{2} + {(3)}^{2}

$\sum_{}^{} x^{2} = {(- 1)}^{2} + {(0)}^{2} + {(1)}^{2} + {(2)}^{2} + {(3)}^{2}$

Step 9

Simplify the expression.

\sum_{}^{} x^{2} = 15

$\sum_{}^{} x^{2} = 15$

Step 10

Sum up the values of

y^{2}

$y^{2}$ .

\sum_{}^{} y^{2} = {(1)}^{2} + {(2)}^{2} + {(- 3)}^{2} + {(- 14)}^{2} + {(- 31)}^{2}

$\sum_{}^{} y^{2} = {(1)}^{2} + {(2)}^{2} + {(- 3)}^{2} + {(- 14)}^{2} + {(- 31)}^{2}$

Step 11

Simplify the expression.

\sum_{}^{} y^{2} = 1171

$\sum_{}^{} y^{2} = 1171$

Step 12

Fill in the computed values.

r = \frac{5 (- 125) - 5 \cdot - 45}{\sqrt{5 (15) - {(5)}^{2}} \cdot \sqrt{5 (1171) - {(- 45)}^{2}}}

$r = \frac{5 (- 125) - 5 \cdot - 45}{\sqrt{5 (15) - {(5)}^{2}} \cdot \sqrt{5 (1171) - {(- 45)}^{2}}}$

Step 13

Simplify the expression.

r = - 0.91406188

$r = - 0.91406188$