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:
$$\begin{array}{c|cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}X&48.91&49.10&45.92&38.34&33.75&33.62&37.04&41.65&46.59&45.09&49.20&47.12&59.47&60.30&59.63&47.60&49.76&48.24&53.27&66.15&80.54&91.16&95.96&98.17&105.37&102.71&99.74&101.58&102.59&97.49&98.42&92.72&96.38&102.33&107.65&105.03&96.56&91.97&93.46&97.23&92.05&97.49&91.82&88.91&86.24&92.19&96.49&88.06&84.96&86.53&104.87&103.02&107.07&98.48&98.83&100.36&93.19&79.20&88.81&95.70\\Y&95.42&95.88&93.05&94.58&98.22&99.65&98.75&100.21&97.02&96.48&95.85&97.44&95.66&96.99&94.71&98.66&95.32&95.00&90.65&88.41&87.02&86.05&82.78&81.52&79.81&80.40&79.53&80.25&79.72&81.40&80.19&80.66&80.26&80.32&82.14&81.54&83.38&83.40&81.81&83.17&82.00&79.23&79.87&80.16&79.99&80.03&81.22&82.71&81.75&83.13&78.85&79.14&78.79&79.42&80.52&78.49&76.31&79.08&74.17&74.04\end{array}$$The equation of the regression line is:
$$y~=~109.8 ~-~ 0.3018 \cdot x$$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.
$X$ | $Y$ | $X\cdot Y$ | $X \cdot X$ |
48.91 | 95.42 | 4666.9922 | 2392.1881 |
49.10 | 95.88 | 4707.708 | 2410.81 |
45.92 | 93.05 | 4272.856 | 2108.6464 |
38.34 | 94.58 | 3626.1972 | 1469.9556 |
33.75 | 98.22 | 3314.925 | 1139.0625 |
33.62 | 99.65 | 3350.233 | 1130.3044 |
37.04 | 98.75 | 3657.7 | 1371.9616 |
41.65 | 100.21 | 4173.7465 | 1734.7225 |
46.59 | 97.02 | 4520.1618 | 2170.6281 |
45.09 | 96.48 | 4350.2832 | 2033.1081 |
49.20 | 95.85 | 4715.82 | 2420.64 |
47.12 | 97.44 | 4591.3728 | 2220.2944 |
59.47 | 95.66 | 5688.9002 | 3536.6809 |
60.30 | 96.99 | 5848.497 | 3636.09 |
59.63 | 94.71 | 5647.5573 | 3555.7369 |
47.60 | 98.66 | 4696.216 | 2265.76 |
49.76 | 95.32 | 4743.1232 | 2476.0576 |
48.24 | 95.00 | 4582.8 | 2327.0976 |
53.27 | 90.65 | 4828.9255 | 2837.6929 |
66.15 | 88.41 | 5848.3215 | 4375.8225 |
80.54 | 87.02 | 7008.5908 | 6486.6916 |
91.16 | 86.05 | 7844.318 | 8310.1456 |
95.96 | 82.78 | 7943.5688 | 9208.3216 |
98.17 | 81.52 | 8002.8184 | 9637.3489 |
105.37 | 79.81 | 8409.5797 | 11102.8369 |
102.71 | 80.40 | 8257.884 | 10549.3441 |
99.74 | 79.53 | 7932.3222 | 9948.0676 |
101.58 | 80.25 | 8151.795 | 10318.4964 |
102.59 | 79.72 | 8178.4748 | 10524.7081 |
97.49 | 81.40 | 7935.686 | 9504.3001 |
98.42 | 80.19 | 7892.2998 | 9686.4964 |
92.72 | 80.66 | 7478.7952 | 8596.9984 |
96.38 | 80.26 | 7735.4588 | 9289.1044 |
102.33 | 80.32 | 8219.1456 | 10471.4289 |
107.65 | 82.14 | 8842.371 | 11588.5225 |
105.03 | 81.54 | 8564.1462 | 11031.3009 |
96.56 | 83.38 | 8051.1728 | 9323.8336 |
91.97 | 83.40 | 7670.298 | 8458.4809 |
93.46 | 81.81 | 7645.9626 | 8734.7716 |
97.23 | 83.17 | 8086.6191 | 9453.6729 |
92.05 | 82.00 | 7548.1 | 8473.2025 |
97.49 | 79.23 | 7724.1327 | 9504.3001 |
91.82 | 79.87 | 7333.6634 | 8430.9124 |
88.91 | 80.16 | 7127.0256 | 7904.9881 |
86.24 | 79.99 | 6898.3376 | 7437.3376 |
92.19 | 80.03 | 7377.9657 | 8498.9961 |
96.49 | 81.22 | 7836.9178 | 9310.3201 |
88.06 | 82.71 | 7283.4426 | 7754.5636 |
84.96 | 81.75 | 6945.48 | 7218.2016 |
86.53 | 83.13 | 7193.2389 | 7487.4409 |
Step 2: Find the sum of every column:
$$ \sum{X} = 4792.08 ~,~ \sum{Y} = 5142.2 ~,~ \sum{X \cdot Y} = 400513.4553 ~,~ \sum{X^2} = 416473.7478 $$Step 3: Use the following equations to find $a$ and $b$:
$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 5142.2 \cdot 416473.7478 - 4792.08 \cdot 400513.4553}{ 60 \cdot 416473.7478 - 4792.08^2} \approx 109.8 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 60 \cdot 400513.4553 - 4792.08 \cdot 5142.2 }{ 60 \cdot 416473.7478 - \left( 4792.08 \right)^2} \approx -0.3018\end{aligned}$$Step 4: Substitute $a$ and $b$ in regression equation formula
$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~109.8 ~-~ 0.3018 \cdot x\end{aligned}$$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.