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Simple linear regression

Simple linear regression is a model that expresses one outcome variable as a straight-line function of a single explanatory variable, plus a random error term. It estimates an intercept and a slope from the data.

Also known asSLR · simple regression

ByHoang TruongUpdated

FrameworkOrdinary least squares (OLS)

See it move

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A scatter chart plots individual observations with Sales calls (X) on the horizontal axis and Profit in €'000 (Y) on the vertical axis; an OLS fitted line passes through the data cloud. The estimated equation is ŷ = 8.2 + 1.5x: the intercept of 8.2 is predicted profit when no calls are made, and the slope of 1.5 means each additional sales call is associated with €1,500 more profit on average. Simple linear regression summarises the linear relationship between one explanatory variable and one outcome using a single straight line.

Where it fits
SubjectData Analysis & StatisticsCoreTopicSimple Linear Regression & OLSCore

The formula

LaTeX
Yi=β0+β1Xi+εiY_i = \beta_0 + \beta_1 X_i + \varepsilon_i

Variables

Outcome (dependent) variable
Intercept — predicted value of Y when X = 0
Slope — change in Y for a one-unit rise in X
Explanatory (independent) variable
Random error term (captures all other influences on Y)

Population regression model.

LaTeX
β^1=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2\hat{\beta}_1 = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum(x_i - \bar{x})^2}

Variables

OLS estimate of the slope
Sample mean of X
Sample mean of Y

OLS slope estimator — minimises the sum of squared residuals.

LaTeX
β^0=yˉ−β^1xˉ\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}

Variables

OLS estimate of the intercept

OLS intercept — ensures the fitted line passes through (x̄, ȳ).

Check yourself

PracticeCORE

A firm regresses quarterly profit (Y, in €000) on the number of sales calls made (X). The estimated equation is Ŷ = 12 + 0.8X. How should the slope coefficient 0.8 be interpreted?

Select an answer to check your understanding.