Linear Regression Calculator

The calculator will find the line of best fit for the given set of paired data using the least squares method, with steps shown.

Related calculator: Quadratic Regression Calculator

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Find the line of best fit for $\left\{\left(1, 2\right), \left(2, 5\right), \left(3, 7\right), \left(4, 11\right), \left(5, 15\right)\right\}$ .

Solution

The number of observations is $n = 5$ .

Generate the following table:

	$x$	$y$	$x y$	$x^{2}$	$y^{2}$
	$1$	$2$	$2$	$1$	$4$
	$2$	$5$	$10$	$4$	$25$
	$3$	$7$	$21$	$9$	$49$
	$4$	$11$	$44$	$16$	$121$
	$5$	$15$	$75$	$25$	$225$
$\sum$	$15$	$40$	$152$	$55$	$424$

The line of best fit is $y = m x + b$ .

$m = \frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^2)-(\sum x)^2} = \frac{5 \cdot 152 - \left(15\right)\cdot \left(40\right)}{5 \cdot 55 - 15^{2}} = \frac{16}{5}$

$b = \frac{(\sum y)(\sum x^2)-(\sum x)(\sum xy)}{n(\sum x^2)-(\sum x)^2} = \frac{\left(40\right)\cdot \left(55\right) - \left(15\right)\cdot \left(152\right)}{5 \cdot 55 - 15^{2}} = - \frac{8}{5}$

Thus, the line of best fit is $y = \frac{16 x}{5} - \frac{8}{5}$ .

Answer

The line of best fit is $y = \frac{16 x}{5} - \frac{8}{5} = 3.2 x - 1.6$ A.

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