Sum of squares

Sum of squares decomposes total variation in a dataset into an explained part (SSR) and an unexplained part (SSE); it is the building block behind R-squared, ANOVA and the F-test.

ByHoang TruongUpdated July 5, 2026

See it move

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Four months of advertising spend and sales give total sum of squares SST = 13.0. The fitted regression line explains SSR = 12.8 of that variation, leaving SSE = 0.2 unexplained — and 12.8 + 0.2 reproduces the 13.0 total exactly, since SST always equals SSR plus SSE.

Where it fits

SubjectData Analysis & StatisticsCoreTopicCommon Significance TestsCoreTopicSimple Linear Regression & OLSCore

The formula

LaTeX

SST = SSR + SSE

Variables

$S S T$: Total sum of squares (squared units of y)
$S S R$: Regression (explained) sum of squares (squared units of y)
$S S E$: Error (unexplained) sum of squares (squared units of y)

The core decomposition: total variation in the outcome splits exactly into an explained part and an unexplained part.

LaTeX

SST=\sum (y_i-\bar y)^2,\quad SSR=\sum(\hat y_i - \bar y)^2,\quad SSE=\sum(y_i-\hat y_i)^2

Variables

$y_{i}$: Actual observed value
$\overset{y}{^}_{i}$: Fitted (predicted) value from the regression
$\overset{y}{ˉ}$: Mean of the observed y values

Defines each sum of squares from the actual, fitted and mean values of the outcome variable.

Check yourself

PracticeCORE

A simple regression of exam score on study hours for four students produces SST = 725 and SSE = 5. What is SSR, and what does it represent?

Select an answer to check your understanding.