mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

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 The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...

* For the "sum of squared differences", see

Mean squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...

* For the "sum of squared error", see

Residual sum of squares In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared estimate of errors (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of dat ...

* For the "sum of squares due to lack of fit", see

Lack-of-fit sum of squares In statistics, a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null ...

* For sums of squares relating to model predictions, see

Explained sum of squares In statistics, the explained sum of squares (ESS), alternatively known as the model sum of squares or sum of squares due to regression (SSR – not to be confused with the residual sum of squares (RSS) or sum of squares of errors), is a quantity ...

* For sums of squares relating to observations, see

Total sum of squares In statistical data analysis the total sum of squares (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. For a set of observations, y_i, i\leq n, it is defined as the sum over all squared dif ...

* For sums of squared deviations, see

Squared deviations from the mean Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics, the definition of ''variance'' is either the expected value of the SDM (when considering a theoretical distribution) or its average valu ...

* For modelling involving sums of squares, see

Analysis of variance Analysis of variance (ANOVA) is a family of statistical methods used to compare the Mean, means of two or more groups by analyzing variance. Specifically, ANOVA compares the amount of variation ''between'' the group means to the amount of variati ...

* For modelling involving the multivariate generalisation of sums of squares, see

Multivariate analysis of variance In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate random variable, multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often fo ...

Number theory

* For the sum of squares of consecutive integers, see

Square pyramidal number In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the stacked spheres in a pyramid (geometry), pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part ...

* For representing an integer as a sum of squares of 4 integers, see

Lagrange's four-square theorem Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number, nonnegative integer can be represented as a sum of four non-negative integer square number, squares. That is, the squares form an additive basi ...

* Legendre's three-square theorem states which numbers can be expressed as the sum of three squares *

Jacobi's four-square theorem In number theory, Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer can be represented as the sum of four squares (of integers). History The theorem was proved in 1834 by Carl Gustav Jakob Jacob ...

gives the number of ways that a number can be represented as the sum of four squares. * For the number of representations of a positive integer as a sum of squares of ''k'' integers, see

Sum of squares function In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer as the sum of squares, where representations that differ only in the order of the summands or in the si ...

. *

Fermat's theorem on sums of two squares In additive number theory, Pierre de Fermat, Fermat's theorem on sums of two squares states that an Even and odd numbers, odd prime number, prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv ...

says which primes are sums of two squares. ** The

sum of two squares theorem In number theory, the sum of two squares theorem relates the prime decomposition of any integer to whether it can be written as a sum of two Square number, squares, such that for some integers , . An integer greater than one can be written as a ...

generalizes Fermat's theorem to specify which composite numbers are the sums of two squares. *

Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Py ...

s are sets of three integers such that the sum of the squares of the first two equals the square of the third. *A

Pythagorean prime A Pythagorean prime is a prime number of the Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they ...

is a prime that is the sum of two squares;

states which primes are Pythagorean primes. * Pythagorean triangles with integer altitude from the hypotenuse have the sum of squares of inverses of the integer legs equal to the square of the inverse of the integer altitude from the hypotenuse. * Pythagorean quadruples are sets of four integers such that the sum of the squares of the first three equals the square of the fourth. * The

Basel problem The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...

, solved by Euler in terms of

\pi

, asked for an exact expression for the sum of the squares of the reciprocals of all positive integers. * Rational trigonometry's triple-quad rule and triple-spread rule contain sums of squares, similar to Heron's formula. *

Squaring the square Squaring the square is the problem of tessellation, tiling an integral square using only other integral squares. (An integral square is a square (geometry), square whose sides have integer length.) The name was coined in a humorous analogy with sq ...

is a combinatorial problem of dividing a two-dimensional square with integer side length into smaller such squares.

Algebra, algebraic geometry, and optimization

Polynomial SOS In mathematics, a form (i.e. a homogeneous polynomial) ''h''(''x'') of degree 2''m'' in the real ''n''-dimensional vector ''x'' is sum of squares of forms (SOS) if and only if there exist forms g_1(x),\ldots,g_k(x) of degree ''m'' such that h(x) ...

, polynomials that are sums of squares of other polynomials * The Brahmagupta–Fibonacci identity, representing the product of sums of two squares of polynomials as another sum of squares *

Hilbert's seventeenth problem Hilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It concerns the expression of positive definite rational functions as sums of quotients of squares. The original que ...

on characterizing the polynomials with non-negative values as sums of squares *

Sum-of-squares optimization A sum-of-squares optimization program is an optimization problem with a linear cost function and a particular type of constraint on the decision variables. These constraints are of the form that when the decision variables are used as coefficien ...

, nonlinear programming with polynomial SOS constraints * The sum of squared dimensions of a finite group's pairwise nonequivalent complex representations is equal to cardinality of that group.

Euclidean geometry and other inner-product spaces

* The

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

says that the square on the hypotenuse of a right triangle is equal in area to the sum of the squares on the legs. The sum of squares is not factorable. * The squared Euclidean distance between two points, equal to the sum of squares of the differences between their coordinates *

Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is A = \sqrt. It is named after first-century ...

for the area of a triangle can be re-written as using the sums of squares of a triangle's sides (and the sums of the squares of squares) * The British flag theorem for rectangles equates two sums of two squares * The

parallelogram law In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the s ...

equates the sum of the squares of the four sides to the sum of the squares of the diagonals *

Descartes' theorem In geometry, Descartes' theorem states that for every four kissing, or mutually tangent circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given ...

for four kissing circles involves sums of squares * The sum of the squares of the edges of a rectangular

cuboid In geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six Face (geometry), faces; it has eight Vertex (geometry), vertices and twelve Edge (geometry), edges. A ''rectangular cuboid'' (sometimes also calle ...

equals the square of any space diagonal

Statistics

Number theory

Algebra, algebraic geometry, and optimization

Euclidean geometry and other inner-product spaces

See also