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Line of Best Fit (Least Square Method)

Aline of best fitis a straight line that is the best approximation of the given set of data.

It is used to study the nature of the relation between two variables. (We're only considering the two-dimensional case, here.)

A line of best fit can be roughly determined using an eyeball method by drawing a straight line on ascatter plotso that the number of points above the line and below the line is about equal (and the line passes through as many points as possible).

A more accurate way of finding the line of best fit is theleast square method.

使用following steps to find the equation of line of best fit for a set ofordered pairs $\left(x_1,y_1\right),\left(x_2,y_2\right),\mathrm{...}\left(x_n,y_n\right)$.

Step 1: Calculate the mean of the$x$-values and the mean of the$y$-values.

$\begin{array}{l}\bar{X}=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i}}{n}\\ \bar{Y}=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{y}_i}}{n}\end{array}$

Step 2: The following formula gives the slope of the line of best fit:

$m=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(x_i-\bar{X}\right)\left(y_i-\bar{Y}\right)}}{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(x_i-\bar{X}\right)}^2}}$

Step 3: Compute the $y$-interceptof the line by using the formula:

$b=\bar{Y}-m\bar{X}$

Step 4: Use the slope$m$and the$y$-intercept$b$to form the equation of the line.

Solution:

Plot the points on acoordinate plane.

计算means of the$x$-values and the$y$-values.

$\begin{array}{l}\bar{X}=\frac{8+2+11+6+5+4+12+9+6+1}{10}=6.4\\ \bar{Y}=\frac{3+10+3+6+8+12+1+4+9+14}{10}=7\end{array}$

Now calculate$x_i-\bar{X}$,$y_i-\bar{Y}$,$\left(x_i-\bar{X}\right)\left(y_i-\bar{Y}\right)$, and${\left(x_i-\bar{X}\right)}^2$for each$i$.

$i$	$x_i$	$y_i$	$x_i-\bar{X}$	$y_i-\bar{Y}$	$\left(x_i-\bar{X}\right)\left(y_i-\bar{Y}\right)$	${\left(x_i-\bar{X}\right)}^2$
$1$	$8$	$3$	$1.6$	$-4$	$-6.4$	$2.56$
$2$	$2$	$10$	$-4.4$	$3$	$-13.2$	$19.36$
$3$	$11$	$3$	$4.6$	$-4$	$-18.4$	$21.16$
$4$	$6$	$6$	$-0.4$	$-1$	$0.4$	$0.16$
$5$	$5$	$8$	$-1.4$	$1$	$-1.4$	$1.96$
$6$	$4$	$12$	$-2.4$	$5$	$-12$	$5.76$
$7$	$12$	$1$	$5.6$	$-6$	$-33.6$	$31.36$
$8$	$9$	$4$	$2.6$	$-3$	$-7.8$	$6.76$
$9$	$6$	$9$	$-0.4$	$2$	$-0.8$	$0.16$
$10$	$1$	$14$	$-5.4$	$7$	$-37.8$	$29.16$
					$\begin{array}{l}{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(x_i-\bar{X}\right)\left(y_i-\bar{Y}\right)}\\ =-131\end{array}$	$\begin{array}{l}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(x_i-\bar{X}\right)}^2}\\ =118.4\end{array}$

计算slope.

$m=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(x_i-\bar{X}\right)\left(y_i-\bar{Y}\right)}}{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(x_i-\bar{X}\right)}^2}}=\frac{-131}{118.4}\approx -1.1$

计算$y$-intercept.

使用formula to compute the$y$-intercept.

$\begin{array}{l}b=\bar{Y}-m\bar{X}\\ =7-\left(-1.1\times 6.4\right)\\ =7+7.04\\ \approx 14.0\end{array}$

使用slope and$y$-intercept to form the equation of the line of best fit.

The slope of the line is$-1.1$and the$y$-intercept is$14.0$.

Therefore, the equation is$y=-1.1x+14.0$.

Draw the line on the scatter plot.