Statistics Examples

\begin{array}{c} x & p \\ 0 & 0.2 \\ 1 & 0.3 \\ 2 & 0.1 \\ 3 & 0.4 \end{array}

$\begin{array}{c} x & p \\ 0 & 0.2 \\ 1 & 0.3 \\ 2 & 0.1 \\ 3 & 0.4 \end{array}$

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 = 0 + 1 + 2 + 3

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

Step 3

Simplify the expression.

\sum_{}^{} x = 6

$\sum_{}^{} x = 6$

Step 4

Sum up the

y

$y$ values.

\sum_{}^{} y = 0.2 + 0.3 + 0.1 + 0.4

$\sum_{}^{} y = 0.2 + 0.3 + 0.1 + 0.4$

Step 5

Simplify the expression.

\sum_{}^{} y = 1

$\sum_{}^{} y = 1$

Step 6

Sum up the values of

x \cdot y

$x \cdot y$ .

\sum_{}^{} x y = 0 \cdot 0.2 + 1 \cdot 0.3 + 2 \cdot 0.1 + 3 \cdot 0.4

$\sum_{}^{} x y = 0 \cdot 0.2 + 1 \cdot 0.3 + 2 \cdot 0.1 + 3 \cdot 0.4$

Step 7

Simplify the expression.

\sum_{}^{} x y = 1.7

$\sum_{}^{} x y = 1.7$

Step 8

Sum up the values of

x^{2}

$x^{2}$ .

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

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

Step 9

Simplify the expression.

\sum_{}^{} x^{2} = 14

$\sum_{}^{} x^{2} = 14$

Step 10

Sum up the values of

y^{2}

$y^{2}$ .

\sum_{}^{} y^{2} = {(0.2)}^{2} + {(0.3)}^{2} + {(0.1)}^{2} + {(0.4)}^{2}

$\sum_{}^{} y^{2} = {(0.2)}^{2} + {(0.3)}^{2} + {(0.1)}^{2} + {(0.4)}^{2}$

Step 11

Simplify the expression.

\sum_{}^{} y^{2} = 0.30000001

$\sum_{}^{} y^{2} = 0.30000001$

Step 12

Fill in the computed values.

r = \frac{4 (1.7) - 6 \cdot 1}{\sqrt{4 (14) - {(6)}^{2}} \cdot \sqrt{4 (0.3) - {(1)}^{2}}}

$r = \frac{4 (1.7) - 6 \cdot 1}{\sqrt{4 (14) - {(6)}^{2}} \cdot \sqrt{4 (0.3) - {(1)}^{2}}}$

Step 13

Simplify the expression.

r = 0.40000004

$r = 0.40000004$

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