Correlation and regression calculator

solution

You entered the following data:

$$\begin{array}{c|cccccc}X&8.5&7.1&7.2&5.6&5.6&4.0\\Y&1.3&2.3&3.4&4.8&6.1&8.3\end{array}$$

The equation of the regression line is:

$$y~=~14.37 ~-~ 1.58 \cdot x$$

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.

$X$	$Y$	$X\cdot Y$	$X \cdot X$
8.5	1.3	11.05	72.25
7.1	2.3	16.33	50.41
7.2	3.4	24.48	51.84
5.6	4.8	26.88	31.36
5.6	6.1	34.16	31.36
4.0	8.3	33.2	16

Step 2: Find the sum of every column:

$$ \sum{X} = 38 ~,~ \sum{Y} = 26.2 ~,~ \sum{X \cdot Y} = 146.1 ~,~ \sum{X^2} = 253.22 $$

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

$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 26.2 \cdot 253.22 - 38 \cdot 146.1}{ 6 \cdot 253.22 - 38^2} \approx 14.37 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 6 \cdot 146.1 - 38 \cdot 26.2 }{ 6 \cdot 253.22 - \left( 38 \right)^2} \approx -1.58\end{aligned}$$

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

$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~14.37 ~-~ 1.58 \cdot x\end{aligned}$$