Correlation and regression calculator

Enter two data sets and this calculator will find the equation of the regression line and correlation coefficient. The calculator will generate a step by step explanation along with the graphic representation of the data sets and regression line.

solution

You entered the following data:

$$\begin{array}{c|cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}X&18&4&23&17&12&4&17&4&4&11&4&22&17&4&4&19&1&12&23&17&23&29&4&17&13&12&4&1&4&12&17&12&17&17&17&17&23&4&17&4&25&22&25&29&26&27&27&4&1&19&17&15&12&4&12&17&19&11&19&23&28&28&29&4&4&24&26&28&23&3&11&4&17&17&12&17&17&9&9&12&17&11&24&14&19&1&5&10&12&12&14&17&17&17&17&22&1&17&9&1&1&1&1&1&1&1&1&1&1&1&1&4&4&27\\Y&3401&4100&3402&3210&3100&3150&3408&3350&3300&3218&3311&4115&3421&3800&3200&3450&3300&3150&3530&3512&3502&2500&3200&3430&3320&3200&3609&3210&3198&3110&3150&3120&3210&3205&3200&3820&3350&3400&3310&3100&3200&3450&3610&3208&3664&3320&4100&3910&3700&3352&3350&3305&3325&3200&3016&3343&3311&3420&3510&3500&3208&3200&3450&3200&3200&4000&3250&3350&3602&3300&3410&4010&3400&3325&3210&3221&3308&3410&3150&3187&3218&3100&3451&3431&3451&3301&3930&3200&3220&3205&3420&3420&3350&3620&3350&3421&3200&3260&4305&3405&3300&3219&3212&3210&3219&3400&3300&3250&3230&3300&3520&3300&3350&4388\end{array}$$

The equation of the regression line is:

$$y~=~3334 ~+~ 3.436 \cdot x$$

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.

$X$	$Y$	$X\cdot Y$	$X \cdot X$
18	3401	61218	324
4	4100	16400	16
23	3402	78246	529
17	3210	54570	289
12	3100	37200	144
4	3150	12600	16
17	3408	57936	289
4	3350	13400	16
4	3300	13200	16
11	3218	35398	121
4	3311	13244	16
22	4115	90530	484
17	3421	58157	289
4	3800	15200	16
4	3200	12800	16
19	3450	65550	361
1	3300	3300	1
12	3150	37800	144
23	3530	81190	529
17	3512	59704	289
23	3502	80546	529
29	2500	72500	841
4	3200	12800	16
17	3430	58310	289
13	3320	43160	169
12	3200	38400	144
4	3609	14436	16
1	3210	3210	1
4	3198	12792	16
12	3110	37320	144
17	3150	53550	289
12	3120	37440	144
17	3210	54570	289
17	3205	54485	289
17	3200	54400	289
17	3820	64940	289
23	3350	77050	529
4	3400	13600	16
17	3310	56270	289
4	3100	12400	16
25	3200	80000	625
22	3450	75900	484
25	3610	90250	625
29	3208	93032	841
26	3664	95264	676
27	3320	89640	729
27	4100	110700	729
4	3910	15640	16
1	3700	3700	1
19	3352	63688	361

Step 2: Find the sum of every column:

$$ \sum{X} = 1480 ~,~ \sum{Y} = 385213 ~,~ \sum{X \cdot Y} = 5030407 ~,~ \sum{X^2} = 27770 $$

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

$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 385213 \cdot 27770 - 1480 \cdot 5030407}{ 114 \cdot 27770 - 1480^2} \approx 3334 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 114 \cdot 5030407 - 1480 \cdot 385213 }{ 114 \cdot 27770 - \left( 1480 \right)^2} \approx 3.436\end{aligned}$$

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

$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~3334 ~+~ 3.436 \cdot x\end{aligned}$$

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Script name : correlation-and-regression-calculator

Form values: 18 4 23 17 12 4 17 4 4 11 4 22 17 4 4 19 1 12 23 17 23 29 4 17 13 12 4 1 4 12 17 12 17 17 17 17 23 4 17 4 25 22 25 29 26 27 27 4 1 19 17 15 12 4 12 17 19 11 19 23 28 28 29 4 4 24 26 28 23 3 11 4 17 17 12 17 17 9 9 12 17 11 24 14 19 1 5 10 12 12 14 17 17 17 17 22 1 17 9 1 1 1 1 1 1 1 1 1 1 1 1 4 4 27 , 3401 4100 3402 3210 3100 3150 3408 3350 3300 3218 3311 4115 3421 3800 3200 3450 3300 3150 3530 3512 3502 2500 3200 3430 3320 3200 3609 3210 3198 3110 3150 3120 3210 3205 3200 3820 3350 3400 3310 3100 3200 3450 3610 3208 3664 3320 4100 3910 3700 3352 3350 3305 3325 3200 3016 3343 3311 3420 3510 3500 3208 3200 3450 3200 3200 4000 3250 3350 3602 3300 3410 4010 3400 3325 3210 3221 3308 3410 3150 3187 3218 3100 3451 3431 3451 3301 3930 3200 3220 3205 3420 3420 3350 3620 3350 3421 3200 3260 4305 3405 3300 3219 3212 3210 3219 3400 3300 3250 3230 3300 3520 3300 3350 4388 , reg , g , , , , Regression line X = [ 18 4 23 17 12 4 17 4 4 11 4 22 17 4 4 19 1 12 23 17 23 29 4 17 13 12 4 1 4 12 17 12 17 17 17 17 23 4 17 4 25 22 25 29 26 27 27 4 1 19 17 15 12 4 12 17 19 11 19 23 28 28 29 4 4 24 26 28 23 3 11 4 17 17 12 17 17 9 9 12 17 11 24 14 19 1 5 10 12 12 14 17 17 17 17 22 1 17 9 1 1 1 1 1 1 1 1 1 1 1 1 4 4 27 ] , Y = [ 3401 4100 3402 3210 3100 3150 3408 3350 3300 3218 3311 4115 3421 3800 3200 3450 3300 3150 3530 3512 3502 2500 3200 3430 3320 3200 3609 3210 3198 3110 3150 3120 3210 3205 3200 3820 3350 3400 3310 3100 3200 3450 3610 3208 3664 3320 4100 3910 3700 3352 3350 3305 3325 3200 3016 3343 3311 3420 3510 3500 3208 3200 3450 3200 3200 4000 3250 3350 3602 3300 3410 4010 3400 3325 3210 3221 3308 3410 3150 3187 3218 3100 3451 3431 3451 3301 3930 3200 3220 3205 3420 3420 3350 3620 3350 3421 3200 3260 4305 3405 3300 3219 3212 3210 3219 3400 3300 3250 3230 3300 3520 3300 3350 4388 ]

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