Finite Math Examples

$\begin{array}{c} x & y \\ 1 & 20 \\ 2 & 30 \\ 3 & 28 \\ 4 & 44 \\ 5 & 40 \\ 6 & 58 \\ 7 & 45 \\ 8 & 58 \\ 9 & 65 \\ 10 & 63 \\ 12 & 75 \\ 13 & 75 \\ 14 & 78 \\ 15 & 75 \\ 16 & 82 \end{array}$

Step 1

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

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

Step 2

Sum up the $x$ values.

$\sum_{}^{} x = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 12 + 13 + 14 + 15 + 16$

Step 3

Simplify the expression.

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

Step 4

Sum up the $y$ values.

$\sum_{}^{} y = 20 + 30 + 28 + 44 + 40 + 58 + 45 + 58 + 65 + 63 + 75 + 75 + 78 + 75 + 82$

Step 5

Simplify the expression.

$\sum_{}^{} y = 836$

Step 6

Sum up the values of $x \cdot y$ .

$\sum_{}^{} x y = 1 \cdot 20 + 2 \cdot 30 + 3 \cdot 28 + 4 \cdot 44 + 5 \cdot 40 + 6 \cdot 58 + 7 \cdot 45 + 8 \cdot 58 + 9 \cdot 65 + 10 \cdot 63 + 12 \cdot 75 + 13 \cdot 75 + 14 \cdot 78 + 15 \cdot 75 + 16 \cdot 82$

Step 7

Simplify the expression.

$\sum_{}^{} x y = 8286$

Step 8

Sum up the values of $x^{2}$ .

$\sum_{}^{} x^{2} = {(1)}^{2} + {(2)}^{2} + {(3)}^{2} + {(4)}^{2} + {(5)}^{2} + {(6)}^{2} + {(7)}^{2} + {(8)}^{2} + {(9)}^{2} + {(10)}^{2} + {(12)}^{2} + {(13)}^{2} + {(14)}^{2} + {(15)}^{2} + {(16)}^{2}$

Step 9

Simplify the expression.

$\sum_{}^{} x^{2} = 1375$

Step 10

Sum up the values of $y^{2}$ .

$\sum_{}^{} y^{2} = {(20)}^{2} + {(30)}^{2} + {(28)}^{2} + {(44)}^{2} + {(40)}^{2} + {(58)}^{2} + {(45)}^{2} + {(58)}^{2} + {(65)}^{2} + {(63)}^{2} + {(75)}^{2} + {(75)}^{2} + {(78)}^{2} + {(75)}^{2} + {(82)}^{2}$

Step 11

Simplify the expression.

$\sum_{}^{} y^{2} = 52250$

Step 12

Fill in the computed values.

$r = \frac{15 (8286) - 125 \cdot 836}{\sqrt{15 (1375) - {(125)}^{2}} \cdot \sqrt{15 (52250) - {(836)}^{2}}}$

Step 13

Simplify the expression.

$r = 0.9607815$