Sum of squares

A sum of squares adds squared deviations from a chosen reference, such as a mean or model prediction. It quantifies variation or unexplained error in statistical analyses.

Squared deviation: The square of the difference between an observation and a reference value. Each term in a sum of squares is a squared deviation.
Analysis of variance: A statistical method that compares group means by partitioning variation into components. It separates total variation into between-group and within-group sums of squares.
Pythagorean theorem: A geometric theorem stating that the square of a right triangle's hypotenuse equals the sum of its legs' squares. Orthogonal fitted and residual components yield an analogous sum-of-squares decomposition.
Mean absolute error: The average absolute difference between observed and predicted values. Unlike squared error, it penalizes deviations linearly rather than quadratically.
Arithmetic mean: The sum of numerical values divided by their count. Deviations from the mean define the standard total sum of squares.
Ordinary least squares: A method that estimates model coefficients by minimizing squared residuals. Its objective function is the residual sum of squares.
Orthogonal projection: The projection of a vector onto a subspace along a perpendicular direction. Least-squares fitted values are projections, making residuals orthogonal to the model space.
L1 loss: A loss function that sums or averages absolute errors. Its linear penalty makes extreme errors less influential than in a sum of squares.
Residual: The difference between an observed value and a value predicted by a model. Squaring and summing residuals measures model discrepancy.
Regression sum of squares: The variation in fitted values around the overall mean in a regression model. It measures the fitted model's contribution to total variation.

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