Sum of squares

In mathematics, statistics and elsewhere, sums of squares occur in a number of contexts:

Statistics

• For partitioning of variance, see Partition of sums of squares
• For the "sum of squared deviations", see Least squares
• For the "sum of squared differences", see Mean squared error
• For the "sum of squared error", see Residual sum of squares
• For the "sum of squares due to lack of fit", see Lack-of-fit sum of squares
• For sums of squares relating to model predictions, see Explained sum of squares
• For sums of squares relating to observations, see Total sum of squares
• For sums of squared deviations, see Squared deviations
• For modelling involving sums of squares, see Analysis of variance
• For modelling involving the multivariate generalisation of sums of squares, see Multivariate analysis of variance

Number theory

• For the sum of squares of consecutive integers, see Square pyramidal number
• For representing an integer as a sum of squares of integers, see Lagrange's four-square theorem
• Fermat's theorem on sums of two squares says which integers are sums of two squares.
  • A separate article discusses Proofs of Fermat's theorem on sums of two squares

Algebra and algebraic geometry

• For representing a polynomial as the sum of squares of polynomials, see Polynomial SOS.
  • For computational optimization, see Sum-of-squares optimization.
• For representing a multivariate polynomial that takes only non-negative values over the reals as a sum of squares of rational functions, see Hilbert's seventeenth problem.
• The Brahmagupta–Fibonacci identity says the set of all sums of two squares is closed under multiplication.

Euclidean geometry and other inner-product spaces

• The Pythagorean theorem says that the square on the hypotenuse of a right triangle is equal in area to the sum of the squares on the legs