Correlation and regression calculator

solution

You entered the following data:

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

The equation of the regression line is:

$$y~=~189.5 ~-~ 1.512 \cdot x$$

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.

$X$	$Y$	$X\cdot Y$	$X \cdot X$
95.42	48.91	4666.9922	9104.9764
95.88	49.10	4707.708	9192.9744
93.05	45.92	4272.856	8658.3025
94.58	38.34	3626.1972	8945.3764
98.22	33.75	3314.925	9647.1684
99.65	33.62	3350.233	9930.1225
98.75	37.04	3657.7	9751.5625
100.21	41.65	4173.7465	10042.0441
97.02	46.59	4520.1618	9412.8804
96.48	45.09	4350.2832	9308.3904
95.85	49.20	4715.82	9187.2225
97.44	47.12	4591.3728	9494.5536

Step 2: Find the sum of every column:

$$ \sum{X} = 1162.55 ~,~ \sum{Y} = 516.33 ~,~ \sum{X \cdot Y} = 49947.9957 ~,~ \sum{X^2} = 112675.5741 $$

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

$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 516.33 \cdot 112675.5741 - 1162.55 \cdot 49947.9957}{ 12 \cdot 112675.5741 - 1162.55^2} \approx 189.5 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 12 \cdot 49947.9957 - 1162.55 \cdot 516.33 }{ 12 \cdot 112675.5741 - \left( 1162.55 \right)^2} \approx -1.512\end{aligned}$$

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

$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~189.5 ~-~ 1.512 \cdot x\end{aligned}$$