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:
$$\begin{array}{c|ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}X&3&11&4&29&21&39&14&30&21&11&2&4&16&30&44&20&15&36&30&60&30&4&13&23&1&36&24&11&21&30&2&17&21&30&21&23&20&23&23&59&23&28&30&30&30&30&38&7&16&30&43&22&30&59&30&37&4&29&30&30&30&28&5&30&24&30&24&30&20&30&6&37&54&24&24&24&24&26&29&29&30&30&11&30&11&24&28&28&30&30&50&24&28&30&30&30&30&22&38&54&11&36&24&15&30&30&24&59&52&36&21&30&57&52&6&48&30&22&23&49&11&11&37&62&37&45&46&62&14&17&30&30&49&57&39&49&51&84&79&80&67&88&79&74&66&88&64&63&66&67&85&81&88&91&63&80&65&81&85&89&81&87&79&91&63&63&65&91&91&87&88&63&74\\Y&5255&5000&4900&4544&4522&4195&4109&4100&3900&3700&3309&3309&3309&3305&3300&3300&3220&3209&3209&3200&3200&3150&3150&3109&3101&3100&3100&3099&3000&2999&2900&2900&2900&2900&2809&2809&2800&2800&2722&2700&2700&2700&2700&2700&2700&2700&2671&2650&2650&2650&2622&2609&2609&2601&2601&2600&2600&2600&2600&2600&2600&2599&2550&2550&2531&2528&2520&2520&2510&2509&2503&2500&2500&2500&2500&2500&2500&2500&2500&2500&2500&2500&2499&2488&2450&2450&2450&2450&2450&2450&2422&2401&2400&2400&2400&2400&2388&2381&2320&2319&2309&2301&2301&2300&2300&2300&2299&2290&2260&2238&2222&2214&2210&2201&2201&2200&2188&2180&2160&2150&2131&2129&2124&2119&2100&2100&2100&2100&2100&2100&2100&2100&2098&2085&2050&2010&2010&2900&2800&2700&2600&2500&2350&2319&2047&4300&4050&3433&2910&2701&2560&2501&2400&2400&2400&2400&2400&2301&2300&2250&2212&2200&2200&2200&2119&2110&2107&2100&2100&2100&2089&2019&2000\end{array}$$The equation of the regression line is:
$$y~=~2929 ~-~ 7.392 \cdot x$$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 | 5255 | 15765 | 9 |
11 | 5000 | 55000 | 121 |
4 | 4900 | 19600 | 16 |
29 | 4544 | 131776 | 841 |
21 | 4522 | 94962 | 441 |
39 | 4195 | 163605 | 1521 |
14 | 4109 | 57526 | 196 |
30 | 4100 | 123000 | 900 |
21 | 3900 | 81900 | 441 |
11 | 3700 | 40700 | 121 |
2 | 3309 | 6618 | 4 |
4 | 3309 | 13236 | 16 |
16 | 3309 | 52944 | 256 |
30 | 3305 | 99150 | 900 |
44 | 3300 | 145200 | 1936 |
20 | 3300 | 66000 | 400 |
15 | 3220 | 48300 | 225 |
36 | 3209 | 115524 | 1296 |
30 | 3209 | 96270 | 900 |
60 | 3200 | 192000 | 3600 |
30 | 3200 | 96000 | 900 |
4 | 3150 | 12600 | 16 |
13 | 3150 | 40950 | 169 |
23 | 3109 | 71507 | 529 |
1 | 3101 | 3101 | 1 |
36 | 3100 | 111600 | 1296 |
24 | 3100 | 74400 | 576 |
11 | 3099 | 34089 | 121 |
21 | 3000 | 63000 | 441 |
30 | 2999 | 89970 | 900 |
2 | 2900 | 5800 | 4 |
17 | 2900 | 49300 | 289 |
21 | 2900 | 60900 | 441 |
30 | 2900 | 87000 | 900 |
21 | 2809 | 58989 | 441 |
23 | 2809 | 64607 | 529 |
20 | 2800 | 56000 | 400 |
23 | 2800 | 64400 | 529 |
23 | 2722 | 62606 | 529 |
59 | 2700 | 159300 | 3481 |
23 | 2700 | 62100 | 529 |
28 | 2700 | 75600 | 784 |
30 | 2700 | 81000 | 900 |
30 | 2700 | 81000 | 900 |
30 | 2700 | 81000 | 900 |
30 | 2700 | 81000 | 900 |
38 | 2671 | 101498 | 1444 |
7 | 2650 | 18550 | 49 |
16 | 2650 | 42400 | 256 |
30 | 2650 | 79500 | 900 |
Step 2: Find the sum of every column:
$$ \sum{X} = 6662 ~,~ \sum{Y} = 457402 ~,~ \sum{X \cdot Y} = 16870886 ~,~ \sum{X^2} = 357062 $$Step 3: Use the following equations to find $a$ and $b$:
$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 457402 \cdot 357062 - 6662 \cdot 16870886}{ 173 \cdot 357062 - 6662^2} \approx 2929 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 173 \cdot 16870886 - 6662 \cdot 457402 }{ 173 \cdot 357062 - \left( 6662 \right)^2} \approx -7.392\end{aligned}$$Step 4: Substitute $a$ and $b$ in regression equation formula
$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~2929 ~-~ 7.392 \cdot x\end{aligned}$$Consider the following set of points: ${(-3 , -4), \, (2 , 3), \, (7 , 11)}$
a) Find the regression line for the given data points.
b) Plot the given points and the regression line.
The values of $X$ and their corresponding values of $Y$ are shown in the table below:
$$ \begin{array}{c|ccccc} X & ~1~ & ~2~ & ~3~ & ~4~ & ~5 \\ Y & ~4~ & ~8~ & ~9~ & ~11~& ~16 \end{array} $$Find a Pearson correlation coefficient.
Please tell me how can I make this better.