Correlation and regression calculator

solution

You entered the following data:

$$\begin{array}{c|cccccccccccccccccccccccccccc}X&17&23&24&11&30&24&1&24&11&3&7&11&2&11&1&16&30&17&7&35&37&37&44&45&46&49&51&59\\Y&55600&28600&34600&30088&29088&42000&25855&35000&41220&33220&35645&29288&31111&24022&36000&34353&33300&33200&29088&23433&23100&27600&18800&17600&25000&21800&23800&19200\end{array}$$

The equation of the regression line is:

$$y~=~36297 ~-~ 259.6 \cdot x$$

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.

$X$	$Y$	$X\cdot Y$	$X \cdot X$
17	55600	945200	289
23	28600	657800	529
24	34600	830400	576
11	30088	330968	121
30	29088	872640	900
24	42000	1008000	576
1	25855	25855	1
24	35000	840000	576
11	41220	453420	121
3	33220	99660	9
7	35645	249515	49
11	29288	322168	121
2	31111	62222	4
11	24022	264242	121
1	36000	36000	1
16	34353	549648	256
30	33300	999000	900
17	33200	564400	289
7	29088	203616	49
35	23433	820155	1225
37	23100	854700	1369
37	27600	1021200	1369
44	18800	827200	1936
45	17600	792000	2025
46	25000	1150000	2116
49	21800	1068200	2401
51	23800	1213800	2601
59	19200	1132800	3481

Step 2: Find the sum of every column:

$$ \sum{X} = 673 ~,~ \sum{Y} = 841611 ~,~ \sum{X \cdot Y} = 18194809 ~,~ \sum{X^2} = 24011 $$

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

$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 841611 \cdot 24011 - 673 \cdot 18194809}{ 28 \cdot 24011 - 673^2} \approx 36297 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 28 \cdot 18194809 - 673 \cdot 841611 }{ 28 \cdot 24011 - \left( 673 \right)^2} \approx -259.6\end{aligned}$$

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

$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~36297 ~-~ 259.6 \cdot x\end{aligned}$$