Correlation and regression calculator

solution

You entered the following data:

$$\begin{array}{c|cccccccccccc}X&48.91&49.10&45.92&38.34&33.75&33.62&37.04&41.65&46.59&45.09&49.20&47.12\\Y&95.42&95.88&93.05&94.58&98.22&99.65&98.75&100.21&97.02&96.48&95.85&97.44\end{array}$$

The equation of the regression line is:

$$y~=~105.1 ~-~ 0.1908 \cdot x$$

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

Step 2: Find the sum of every column:

$$ \sum{X} = 516.33 ~,~ \sum{Y} = 1162.55 ~,~ \sum{X \cdot Y} = 49947.9957 ~,~ \sum{X^2} = 22602.3217 $$

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

$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 1162.55 \cdot 22602.3217 - 516.33 \cdot 49947.9957}{ 12 \cdot 22602.3217 - 516.33^2} \approx 105.1 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 12 \cdot 49947.9957 - 516.33 \cdot 1162.55 }{ 12 \cdot 22602.3217 - \left( 516.33 \right)^2} \approx -0.1908\end{aligned}$$

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

$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~105.1 ~-~ 0.1908 \cdot x\end{aligned}$$