Correlation and regression calculator

solution

You entered the following data:

$$\begin{array}{c|ccccccccc}X&4&6&6&13&14&15&18&20&28\\Y&23433&23100&27600&18800&17600&25000&21800&23800&19200\end{array}$$

The equation of the regression line is:

$$y~=~24868 ~-~ 189.3 \cdot x$$

The graph of the regression line is:

explanation

We will find an equation of the regression line in 4 steps.

Step 1: Find $X \cdot Y$ and $X^2$ as it was done in the table below.

$X$	$Y$	$X\cdot Y$	$X \cdot X$
4	23433	93732	16
6	23100	138600	36
6	27600	165600	36
13	18800	244400	169
14	17600	246400	196
15	25000	375000	225
18	21800	392400	324
20	23800	476000	400
28	19200	537600	784

Step 2: Find the sum of every column:

$$ \sum{X} = 124 ~,~ \sum{Y} = 200333 ~,~ \sum{X \cdot Y} = 2669732 ~,~ \sum{X^2} = 2186 $$

Step 3: Use the following equations to find $a$ and $b$:

$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 200333 \cdot 2186 - 124 \cdot 2669732}{ 9 \cdot 2186 - 124^2} \approx 24868 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 9 \cdot 2669732 - 124 \cdot 200333 }{ 9 \cdot 2186 - \left( 124 \right)^2} \approx -189.3\end{aligned}$$

Step 4: Substitute $a$ and $b$ in regression equation formula

$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~24868 ~-~ 189.3 \cdot x\end{aligned}$$