Correlation and regression calculator

solution

You entered the following data:

$$\begin{array}{c|cccccccc}X&3&26&17&4&10&23&5&1\\Y&22300&23001&23301&27000&28200&32001&38300&53000\end{array}$$

The equation of the regression line is:

$$y~=~36281 ~-~ 484.8 \cdot x$$

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.

$X$	$Y$	$X\cdot Y$	$X \cdot X$
3	22300	66900	9
26	23001	598026	676
17	23301	396117	289
4	27000	108000	16
10	28200	282000	100
23	32001	736023	529
5	38300	191500	25
1	53000	53000	1

Step 2: Find the sum of every column:

$$ \sum{X} = 89 ~,~ \sum{Y} = 247103 ~,~ \sum{X \cdot Y} = 2431566 ~,~ \sum{X^2} = 1645 $$

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

$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 247103 \cdot 1645 - 89 \cdot 2431566}{ 8 \cdot 1645 - 89^2} \approx 36281 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 8 \cdot 2431566 - 89 \cdot 247103 }{ 8 \cdot 1645 - \left( 89 \right)^2} \approx -484.8\end{aligned}$$

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

$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~36281 ~-~ 484.8 \cdot x\end{aligned}$$