The least-squares formulae

The slope $m$ and the intercept $q$ of the regression line are computed directly from the data with the following formulae.

Property — Least-squares formulae

$m = \frac{n\,\displaystyle\sum x_i y_i - \biggl(\sum x_i\biggr)\!\biggl(\sum y_i\biggr)}{n\,\displaystyle\sum x_i^2 - \biggl(\sum x_i\biggr)^{\!2}}, \qquad q = \bar{y} - m\,\bar{x}$ where $\bar{x}=\dfrac{1}{n}\sum x_i$ and $\bar{y}=\dfrac{1}{n}\sum y_i$ are the means.

The line always passes through the point $(\bar{x},\,\bar{y})$ (centroid of the data).

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Topics: Radicals
Concepts: Least squares · Linear regression
Skills: Analytic geometry · Using formulae