Correlation and regression calculator

solution

You entered the following data:

$$\begin{array}{c|ccccccccccccccccccccccccc}X&4&5&23&19&26&29&1&18&3&19&23&27&19&25&17&29&1&1&3&29&29&25&26&1&12\\Y&2910&2701&2560&2501&2400&2400&2400&2400&2400&2301&2300&2250&2212&2200&2200&2200&2119&2110&2107&2100&2100&2100&2089&2019&2000\end{array}$$

The equation of the regression line is:

$$y~=~2331 ~-~ 2.894 \cdot x$$

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	2910	11640	16
5	2701	13505	25
23	2560	58880	529
19	2501	47519	361
26	2400	62400	676
29	2400	69600	841
1	2400	2400	1
18	2400	43200	324
3	2400	7200	9
19	2301	43719	361
23	2300	52900	529
27	2250	60750	729
19	2212	42028	361
25	2200	55000	625
17	2200	37400	289
29	2200	63800	841
1	2119	2119	1
1	2110	2110	1
3	2107	6321	9
29	2100	60900	841
29	2100	60900	841
25	2100	52500	625
26	2089	54314	676
1	2019	2019	1
12	2000	24000	144

Step 2: Find the sum of every column:

$$ \sum{X} = 414 ~,~ \sum{Y} = 57079 ~,~ \sum{X \cdot Y} = 937124 ~,~ \sum{X^2} = 9656 $$

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

$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 57079 \cdot 9656 - 414 \cdot 937124}{ 25 \cdot 9656 - 414^2} \approx 2331 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 25 \cdot 937124 - 414 \cdot 57079 }{ 25 \cdot 9656 - \left( 414 \right)^2} \approx -2.894\end{aligned}$$

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

$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~2331 ~-~ 2.894 \cdot x\end{aligned}$$