Draw a line through the middle of a cloud of data points that is a "best fit" to the data.

Linear Regression

This explanation looks at regression solely as a descriptive statistic: what is the line which lies "closest" to a given set of points. "Closest" means minimizing the sum of the squared y (vertical) distance of the points from the least squares regression line. I won't derive the formula, merely present it and then use it. Data is given as a set of points in the plane, i.e., as ordered pairs of x and y values.

Statistical Formulae

X-bar, written as an X with a line over it, is the mean (average) of the x-values.

Y-bar, a Y with a line over it, is the mean of the y-values.

SS_xx is the sum of the squares of the x-deviations.  SUM (x_i-(X-bar))²

SS_yy is the sum of the squares of the y-deviations.  SUM (y_i-(Y-bar))²

SS_xy is SUM (x_i-(X-bar))(y_i-(Y-bar))

b₁ = SS_xy/SS_xx 

b₀ = (Y-bar) - b₁(X-bar)

The least squares regression line is y-hat = b₀ + b₁x

(y-hat is written as a y with a circumflex over it.)

Example:

Statistical measures of this data:

The formula for the least squares regression line is

y-hat = b₀ + b₁x

So in our example, where b₀=-1.622 and b₁=1.48, the least squares regression line is

y-hat = -1.622 + (1.48)x

Internet references

Related pages in this website

The webmaster and author of this Math Help site is Graeme McRae.