Finite Math Examples

\begin{array}{c} x & y \\ 1 & 0 \\ 3 & 2 \\ 5 & 4 \\ 7 & 6 \end{array}

$\begin{array}{c} x & y \\ 1 & 0 \\ 3 & 2 \\ 5 & 4 \\ 7 & 6 \end{array}$

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 + 3 + 5 + 7

$\sum_{}^{} x = 1 + 3 + 5 + 7$

Step 3

Simplify the expression.

\sum_{}^{} x = 16

$\sum_{}^{} x = 16$

Step 4

Sum up the

y

$y$ values.

\sum_{}^{} y = 0 + 2 + 4 + 6

$\sum_{}^{} y = 0 + 2 + 4 + 6$

Step 5

Simplify the expression.

\sum_{}^{} y = 12

$\sum_{}^{} y = 12$

Step 6

Sum up the values of

x \cdot y

$x \cdot y$ .

\sum_{}^{} x y = 1 \cdot 0 + 3 \cdot 2 + 5 \cdot 4 + 7 \cdot 6

$\sum_{}^{} x y = 1 \cdot 0 + 3 \cdot 2 + 5 \cdot 4 + 7 \cdot 6$

Step 7

Simplify the expression.

\sum_{}^{} x y = 68

$\sum_{}^{} x y = 68$

Step 8

Sum up the values of

x^{2}

$x^{2}$ .

\sum_{}^{} x^{2} = {(1)}^{2} + {(3)}^{2} + {(5)}^{2} + {(7)}^{2}

$\sum_{}^{} x^{2} = {(1)}^{2} + {(3)}^{2} + {(5)}^{2} + {(7)}^{2}$

Step 9

Simplify the expression.

\sum_{}^{} x^{2} = 84

$\sum_{}^{} x^{2} = 84$

Step 10

Sum up the values of

y^{2}

$y^{2}$ .

\sum_{}^{} y^{2} = {(0)}^{2} + {(2)}^{2} + {(4)}^{2} + {(6)}^{2}

$\sum_{}^{} y^{2} = {(0)}^{2} + {(2)}^{2} + {(4)}^{2} + {(6)}^{2}$

Step 11

Simplify the expression.

\sum_{}^{} y^{2} = 56

$\sum_{}^{} y^{2} = 56$

Step 12

Fill in the computed values.

r = \frac{4 (68) - 16 \cdot 12}{\sqrt{4 (84) - {(16)}^{2}} \cdot \sqrt{4 (56) - {(12)}^{2}}}

$r = \frac{4 (68) - 16 \cdot 12}{\sqrt{4 (84) - {(16)}^{2}} \cdot \sqrt{4 (56) - {(12)}^{2}}}$

Step 13

Simplify the expression.

r = 1

$r = 1$