Regression slope

Regression slope is the coefficient on an explanatory variable in a regression model. It gives the predicted change in the outcome for a one-unit rise in that variable, holding all other variables constant.

Also known asB1 · slope coefficient

ByHoang TruongUpdated June 30, 2026

FrameworkOrdinary least squares (OLS)

See it move

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A scatter-line chart plots price in euros on the horizontal axis against profit in thousands of euros on the vertical axis, with the fitted line ŷ = 40.0 − 2.5x overlaid. A slope triangle on the chart marks the change associated with a one-euro rise in price: predicted profit falls by €2,500, which is the slope coefficient of −2.5 expressed in the units of the vertical axis.

Where it fits

SubjectData Analysis & StatisticsCoreTopicSimple Linear Regression & OLSCore

The formula

LaTeX

\hat{\beta}_1 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}

Variables

$\hat{β}_{1}$: Estimated OLS slope coefficient
$X_{i}$: Value of explanatory variable for observation i
$\overset{ˉ}{X}$: Sample mean of X
$Y_{i}$: Value of outcome variable for observation i
$\overset{ˉ}{Y}$: Sample mean of Y

OLS minimises the sum of squared residuals; this formula is the closed-form solution for the slope.

Check yourself

PracticeCORE

A firm fits the regression: monthly profit (€000) = 8.5 + 0.6 × units sold. Which of the following correctly interprets the slope?

Select an answer to check your understanding.