Simple linear regression least squares estimator

Gives the slope and intercept of the straight line that minimises the sum of squared vertical distances to a set of data points. Used to fit a simple linear model y = beta0 + beta1 x to paired observations.

β^_{1} = \frac{\sum _{i = 1}^{n} ( x _{i} - x ˉ ) ( y _{i} - y ˉ )}{\sum _{i = 1}^{n} ( x _{i} - x ˉ ) ^{2}}, β^_{0} = y ˉ - β^_{1} x ˉ

\hat{\beta}_1 = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}, \quad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1\bar{x}

What the symbols mean

$β^_{0}$	estimated intercept of the regression line
$β^_{1}$	estimated slope of the regression line
$x_{i}$	value of the explanatory variable for observation i
$y_{i}$	value of the response variable for observation i
$x ˉ$	sample mean of the x values
$y ˉ$	sample mean of the y values
$n$	number of observations

About this formula

FieldStatistics
TopicRegression analysis
LevelUniversity
Also calledordinary least squares slope and intercept, OLS estimator simple regression, linear regression slope formula, best fit line slope, regression coefficient estimate, least squares line formulas

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