Example — computing a regression line

Let us apply the least-squares formulae to a small set of data.

Example

Given the points $(1;\,2)$, $(2;\,3{,}5)$, $(3;\,4)$, $(4;\,5{,}5)$, $(5;\,6)$: $\bar{x}=3,\;\bar{y}=4{,}2,\; \sum x_iy_i = 66{,}5,\; \sum x_i^2 = 55.$ $m = \frac{5\cdot 66{,}5 - 15\cdot 21}{5\cdot 55 - 225} = \frac{332{,}5-315}{275-225} = \frac{17{,}5}{50} = \boxed{0{,}95},\qquad q = 4{,}2-0{,}95\cdot 3 = \boxed{1{,}35}.$ Line: $y = 0{,}95\,x+1{,}35$.

The graph shows the data (in red), the regression line (in blue) and the centroid $(\bar{x},\bar{y})=(3;\,4{,}2)$ through which the line passes (in green).

The five experimental points, the least-squares line and the centroid of the data.

Links

Topics: Radicals
Concepts: Least squares · Linear regression
Skills: Calculating · Interpreting a graph · Plotting a graph