Find the Linear Correlation Coefficient table[[x,y],[5,19],[7,17],[10,16],[15,12],[19,7]]

$\begin{array}{c} x & y \\ 5 & 19 \\ 7 & 17 \\ 10 & 16 \\ 15 & 12 \\ 19 & 7 \end{array}$

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.

$\sum_{}^{} x = 5 + 7 + 10 + 15 + 19$

Step 3

Simplify the expression.

$\sum_{}^{} x = 56$

Step 4

Sum up the $y$ values.

$\sum_{}^{} y = 19 + 17 + 16 + 12 + 7$

Step 5

Simplify the expression.

$\sum_{}^{} y = 71$

Step 6

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

$\sum_{}^{} x y = 5 \cdot 19 + 7 \cdot 17 + 10 \cdot 16 + 15 \cdot 12 + 19 \cdot 7$

Step 7

Simplify the expression.

$\sum_{}^{} x y = 687$

Step 8

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

$\sum_{}^{} x^{2} = {(5)}^{2} + {(7)}^{2} + {(10)}^{2} + {(15)}^{2} + {(19)}^{2}$

Step 9

Simplify the expression.

$\sum_{}^{} x^{2} = 760$

Step 10

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

$\sum_{}^{} y^{2} = {(19)}^{2} + {(17)}^{2} + {(16)}^{2} + {(12)}^{2} + {(7)}^{2}$

Step 11

Simplify the expression.

$\sum_{}^{} y^{2} = 1099$

Step 12

Fill in the computed values.

$r = \frac{5 (687) - 56 \cdot 71}{\sqrt{5 (760) - {(56)}^{2}} \cdot \sqrt{5 (1099) - {(71)}^{2}}}$

Step 13

Simplify the expression.

$r = - 0.98533815$

Find the Linear Correlation Coefficient table[[x,y],[5,19],[7,17],[10,16],[15,12],[19,7]]

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