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 The partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors. Mathematically, the sum of squared deviations ...

* For the "sum of squared deviations", see Least squares * 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 between ...

* 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 estimate of errors (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). It is a measure of the discrepan ...

* 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 nul ...

* 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 v ...

* For modelling involving sums of squares, see

Analysis of variance Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...

* 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 sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often followed by significance tests ...

Number theory

* For the sum of squares of consecutive integers, see Square pyramidal number * 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 can be represented as the sum of four integer squares. That is, the squares form an additive basis of order four. p = a_0^2 + a_1^2 + a_2^2 + a_ ...

Legendre's three-square theorem In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers :n = x^2 + y^2 + z^2 if and only if is not of the form n = 4^a(8b + 7) for nonnegative integers and . The ...

states which numbers can be expressed as the sum of three squares *

Jacobi's four-square theorem Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer ''n'' can be represented as the sum of four squares. History The theorem was proved in 1834 by Carl Gustav Jakob Jacobi. Theorem Two representati ...

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 sign ...

. *

Fermat's theorem on sums of two squares In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv 1 \pmod. The prime numbers for which this is true ar ...

says which primes are sums of two squares. ** The sum of two squares theorem 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 , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...

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 quadruple A Pythagorean quadruple is a tuple of integers , , , and , such that . They are solutions of a Diophantine equation and often only positive integer values are considered.R. Spira, ''The diophantine equation '', Amer. Math. Monthly Vol. 69 (1962), ...

s 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 tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer length.) The name was coined in a humorous analogy with squaring the circle. Squaring the sq ...

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

Algebra and algebraic geometry

* For representing a polynomial as the sum of squares of ''polynomials'', see

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 ...

. ** 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 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 q ...

. * The

Brahmagupta–Fibonacci identity In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity say ...

says the set of all sums of two squares is closed under multiplication. * 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 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 In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two Point (geometry), points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theo ...

(SED) is defined as the sum of squares of the differences between 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 , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, 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 In Euclidean geometry, the British flag theorem says that if a point ''P'' is chosen inside a rectangle ''ABCD'' then the sum of the squares of the Euclidean distances from ''P'' to two opposite corners of the rectangle equals the sum to the oth ...

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, mu ...

for four kissing circles involves sums of squares * The sum of the squares of the edges of a rectangular cuboid equals the square of any space diagonal

Statistics

Number theory

Algebra and algebraic geometry

Euclidean geometry and other inner-product spaces

See also