Ordinary least squares

Ordinary least squares is the standard method for fitting a regression line, choosing coefficients that minimise the sum of squared residuals.

Also known asOLS

ByHoang TruongUpdated June 30, 2026

FrameworkOrdinary least squares (OLS)

See it move

Loading infographic...

The infographic is a scatter plot with a fitted regression line, showing advertising spend on the horizontal axis and profit on the vertical axis. OLS selects the line by squaring each vertical gap between every observed profit value Y and its predicted value ŷ, then choosing the line that makes the total Σ(Y − ŷ)² as small as possible.

Where it fits

SubjectData Analysis & StatisticsCoreTopicSimple Linear Regression & OLSCore

The formula

LaTeX

\min_{\hat{\beta}_0,\,\hat{\beta}_1} \sum_{i=1}^n (y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i)^2

Variables

$S S R$: Sum of squared residuals — the quantity OLS minimises
$y_{i}$: Observed outcome for observation i
$\hat{β}_{0}$: OLS estimate of the intercept
$\hat{β}_{1}$: OLS estimate of the slope
$x_{i}$: Observed explanatory variable value for observation i

Squaring residuals penalises large errors more heavily and stops positive and negative gaps from cancelling each other.

LaTeX

\hat{\beta}_1 = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum(x_i - \bar{x})^2}

Variables

$\hat{β}_{1}$: OLS slope estimate
$x_{i}$: Explanatory variable value for observation i
$\overset{x}{ˉ}$: Sample mean of the explanatory variable
$y_{i}$: Outcome variable value for observation i
$\overset{y}{ˉ}$: Sample mean of the outcome variable

The OLS slope equals the sample covariance of x and y divided by the sample variance of x.

LaTeX

\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}

Variables

$\hat{β}_{0}$: OLS intercept estimate
$\overset{y}{ˉ}$: Sample mean of the outcome variable
$\hat{β}_{1}$: OLS slope estimate
$\overset{x}{ˉ}$: Sample mean of the explanatory variable

The fitted line always passes through the point (x̄, ȳ).