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Concept that permeates much of inferential statistics and descriptive statistics
The partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning
Partition_of_sums_of_squares
Index of articles associated with the same name
elsewhere, sums of squares occur in a number of contexts: For partitioning of variance, see Partition of sums of squares For the "sum of squared deviations"
Sum_of_squares
Statistical measure of the discrepancy between data and an estimation model
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
Residual_sum_of_squares
Statistical testing method
and the error term.[page needed] The main difference between the sum of squares of the within-subject factors and between-subject factors is that within-subject
Mixed-design analysis of variance
Mixed-design_analysis_of_variance
Condition under which an odd prime is a sum 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 , {\displaystyle p=x^{2}+y^{2}
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Calculations in probability theory
each of the T i {\displaystyle T_{i}} will be zero. It is now possible to calculate three sums of squares: Individual I = ∑ x 2 {\displaystyle I=\sum x^{2}}
Squared deviations from the mean
Squared_deviations_from_the_mean
Value 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
Lack-of-fit_sum_of_squares
Decomposition of an integer as a sum of positive integers
combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ
Integer_partition
Number-theoretical function
sum of k {\displaystyle k} squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are
Sum_of_squares_function
Statistical quantity
total sum of squares is the sum of the so-called "within-samples" sum of squares and "between-samples" sum of squares, i.e., partitioning of the sum of squares
Total_sum_of_squares
The sum of the reciprocals of the primes of the form 4n + 1 is divergent. By Fermat's theorem on sums of two squares, it follows that the sum of reciprocals
List_of_sums_of_reciprocals
matrix, and partition of the sum of squares in statistics problems, especially in the analysis of variance, quotition and partition, two ways of viewing the
List_of_partition_topics
squares Partial least squares regression Partial leverage Partial regression plot Partial residual plot Particle filter Partition of sums of squares Parzen
List_of_statistics_articles
Statistical quantity
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
Explained_sum_of_squares
Collection of statistical models
according to Stigler. These include hypothesis testing, the partitioning of sums of squares, experimental techniques and the additive model. Laplace was
Analysis_of_variance
Methods of constructing magic squares
Graeco-Latin squares. Odd squares: For the 3×3 odd square, since α, β, and γ are in arithmetic progression, their sum is equal to the product of the square's order
Constructions of magic squares
Constructions_of_magic_squares
Number of partitions of an integer
In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because
Partition function (number theory)
Partition_function_(number_theory)
expected mean squares (EMS) are the expected values of certain statistics arising in partitions of sums of squares in the analysis of variance (ANOVA)
Expected_mean_squares
Square of numbers with equal row, column and diagonal totals
magic square is a multiple of the order of the smaller squares. Such squares can usually be partitioned into smaller non-overlapping magic sub-squares. Inlaid
Magic_square
multiway number partitioning is the problem of partitioning a multiset of numbers into a fixed number of subsets, such that the sums of the subsets are
Multiway_number_partitioning
Approximation method in statistics
In regression analysis, least squares is a method to determine the best-fit model by minimizing the sum of the squared residuals—the differences between
Least_squares
Magic square with extra constraints
sums to the magic constant, 4 × 4 pandiagonal magic squares are most-perfect magic squares. In addition, the two numbers at the opposite corners of any
Pandiagonal_magic_square
1947 plan to divide British Palestine
The United Nations Partition Plan for Palestine was a proposal developed by the United Nations for the future government of Mandatory Palestine following
United Nations Partition Plan for Palestine
United_Nations_Partition_Plan_for_Palestine
Term in number theory and combinatorics
combinatorics, the rank of an integer partition is a certain number associated with the partition. In fact at least two different definitions of rank appear in
Rank_of_a_partition
Class of binary number
equal multisets of pairwise sums. As 19th-century mathematician Eugène Prouhet showed, the partition into evil and odious numbers of the numbers from
Evil_number
Natural number
exactly 220 different ways of partitioning 64 = 82 into a sum of square numbers. It is a tetrahedral number, the sum of the first ten triangular numbers
220_(number)
Operation in calculus
Al-Haytham invented a way of computing the sums of kth powers, which he applied for the sums of squares and the sums of fourth powers. The next significant
Integral
Set of basic shapes which assemble into a polygon
polygon partition problem is a problem of finding a partition which is minimal in some sense, for example a partition with a smallest number of units or
Polygon_partition
Special functions of several complex variables
such as number theory: "in how many ways can a number be written as a sum of squares?" physics: "how does heat flow on a toroidal ring?", "how do quantum
Theta_function
Numbers parameterizing ways to partition a set
combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Matrix defined using smaller matrices called blocks
{\textstyle \sum _{i=1}^{p}m_{i}=m} , and ∑ j = 1 q n j = n {\textstyle \sum _{j=1}^{q}n_{j}=n} . The elements A i j {\displaystyle A_{ij}} of the partition are
Block_matrix
Figurate number
their partition. In this way they can be used to prove the pentagonal number theorem referenced above. A formula for the sum of the reciprocals of the pentagonal
Pentagonal_number
Divide and conquer sorting algorithm
called partition-exchange sort. The sub-arrays are then sorted recursively. This can be done in-place, requiring small additional amounts of memory to
Quicksort
Metric for fit of statistical models
chi-square test). In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares. In assessing
Goodness_of_fit
Indicator for how well data points fit a line or curve
can be measured with two sums of squares formulas: The sum of squares of residuals, also called the residual sum of squares: S S res = ∑ i ( y i − f i
Coefficient_of_determination
Measure of the error of an estimator
unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the true
Mean_squared_error
Method for estimating the unknown parameters in a linear regression model
least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model by the principle of least
Ordinary_least_squares
Count of the possible partitions of a set
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th
Bell_number
Function whose domain is the positive integers
number of ways n can be represented as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the square
Arithmetic_function
Euler's four-square identity, which shows that the product of two sums of four squares can itself be expressed as the sum of four squares. Euler's identity
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
list of algebraic number theory topics Unimodular lattice Fermat's theorem on sums of two squares Proofs of Fermat's theorem on sums of two squares Riemann
List_of_number_theory_topics
Periodicity computation method
Least-squares spectral analysis (LSSA) is a class of methods for estimating a frequency spectrum by fitting sinusoids to data using a least-squares fit
Least-squares spectral analysis
Least-squares_spectral_analysis
Vector quantization algorithm minimizing the sum of squared deviations
This results in a partitioning of the data space into Voronoi cells. k-means clustering minimizes within-cluster variances (squared Euclidean distances)
K-means_clustering
Square of a triangular number
power sums, namely that odd power sums (sums of odd powers) are a polynomial in triangular numbers. These are called Faulhaber polynomials, of which the
Squared_triangular_number
Computational method in group theory
set of squares filled with the integer i form a border strip, that is, a connected skew-shape with no 2×2-square. The height, ht(T), is the sum of the
Murnaghan–Nakayama_rule
Type of integer in number theory
trapezoid. The problem of representing numbers as sums of consecutive integers and of counting the number of representations of this type has been studied
Polite_number
Mathematical problem
squares are said to be group based. Euler was able to construct Graeco-Latin squares of orders that are multiples of four, and seemed to be aware of the
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Ratio of concentrations in a mixture at equilibrium
physical sciences, a partition coefficient (P) or distribution coefficient (D) is the ratio of concentrations of a compound in a mixture of two immiscible solvents
Partition_coefficient
Logic puzzle
sizes of these products and by computing row and column sums according to this initial setup. Depending on how these sums deviate from the true sums, it
Survo_puzzle
Combinatorial object in representation theory
same set of squares, each filled with the same entries. Young tableaux can be identified with skew tableaux in which μ is the empty partition (0) (the
Young_tableau
Number of values in the final calculation of a statistic that are free to vary
split-plot designs, the sums-of-squares no longer have scaled chi-squared distributions. Comparison of sum-of-squares with degrees-of-freedom is no longer
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
_{i}x_{i}^{2}{\Bigr )}+{\Bigl (}2\sum _{k}X(L_{k})X(R_{k}){\Bigr )}\\&=\sum _{i}x_{i}^{2}.\\\end{aligned}}} But a sum of squares of real variables can only be
Graham–Pollak_theorem
Integer partition attribute, in number theory
Durfee square is an attribute of an integer partition. A partition of n has a Durfee square of size s if s is the largest number such that the partition contains
Durfee_square
Number of points in an octagonal arrangement
number of partitions of 6 n − 5 {\displaystyle 6n-5} into 1, 2, or 3s. For example, there are x 2 = 8 {\displaystyle x_{2}=8} such partitions for 2 ⋅
Octagonal_number
Technique in analytic number theory
theory of theta functions. In the context of Waring's problem, powers of theta functions are the generating functions for the sum of squares function
Hardy–Ramanujan–Littlewood circle method
Hardy–Ramanujan–Littlewood_circle_method
Shape with four equal sides and angles
Several problems of squaring the square involve subdividing squares into unequal squares. Mathematicians have also studied packing squares as tightly as
Square
Expression for sums of powers
18. pp. 124–126. Povolotsky, Alexander R. (2025). "RECURRENCE RELATIONS FOR k-FOLD NESTED POWER SUMS". A visual proof for the sum of squares and cubes.
Faulhaber's_formula
Procedure for comparing multivariate sample means
entries are the same kinds of sums of squares that appear in univariate ANOVA. The off-diagonal entries are corresponding sums of products. Under normality
Multivariate analysis of variance
Multivariate_analysis_of_variance
Number of orderings allowing ties
representation of n {\displaystyle n} as an ordered sum of positive integers. For instance, the ordered partition {a,b},{c},{d,e,f} discussed above corresponds
Ordered_Bell_number
Statistical hypothesis test
tested. The test statistic T in this instance could be set to be the sum of squares about the sample mean, divided by the nominal value for the variance
Chi-squared_test
Statistical hypothesis test
decomposition of the variability in a collection of data in terms of sums of squares. The test statistic in an F-test is the ratio of two scaled sums of squares reflecting
F-test
Natural number
as the singular sum of the digits in its square in the same order. "Facts about the integer". mathworld.wolfram.com. "Partitions of 17". mathworld.wolfram
297_(number)
-1)=(-1)^{n}n!.} Some infinite sums include ∑ n = k ∞ [ n k ] z n n ! = ( − log ( 1 − z ) ) k k ! {\displaystyle \sum _{n=k}^{\infty
Stirling numbers and exponential generating functions in symbolic combinatorics
Stirling_numbers_and_exponential_generating_functions_in_symbolic_combinatorics
Figurate number
of Theon of Smyrna, is visually demonstrated in the following sum, which represents T 4 + T 5 = 5 2 {\displaystyle T_{4}+T_{5}=5^{2}} as digit sums:
Triangular_number
Number of close-packed spheres in an octahedron
of atoms in octahedral clusters; in this context they are called magic numbers. An octahedral packing of spheres may be partitioned into two square pyramids
Octahedral_number
On domino tiling after removing two corners
corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2×1 so as to cover all of these squares? It is an impossible puzzle:
Mutilated_chessboard_problem
Approximation method in statistics
Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters
Non-linear_least_squares
Generating function for quantum correlation functions
In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral
Partition function (quantum field theory)
Partition_function_(quantum_field_theory)
Natural number
since the digit sum of 324 equals to 9 and 324 is a multiple of 9. 325 = 52 × 13. It is the smallest number to be the sum of two squares in 3 different
300_(number)
Natural number
(ed.). "Sequence A036469 (Partial sums of A000009 (partitions into distinct parts))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
700_(number)
Optimization problem in mathematics
and a set of identical squares, the goal is to find the largest number of non-overlapping squares that can be packed in points of S. Suppose that, for each
Rectangle_packing
Physical quantity
the second moment operator is a tensor defined as the sum of the tensor squares of the positions of a single electron. Then, this is a one-electron operator
Total_position_spread
Natural number
totient sum of the first 52 integers, and the sum of four consecutive primes (197 + 199 + 211 + 223). 831 = 3 × 277. There are 831 partitions of 32 into
800_(number)
Statistical measure of how far values spread from their average
uncorrelated. Similar decompositions are possible for the sum of squared deviations (sum of squares, S S {\displaystyle {\mathit {SS}}} ): S S total = S S
Variance
Set of statistical processes for estimating the relationships among variables
For example, the method of ordinary least squares computes the unique line (or hyperplane) that minimizes the sum of squared differences between the true
Regression_analysis
Mathematical concept
terms define different compositions of their sum, while they are considered to define the same integer partition of that number. Every integer has finitely
Composition_(combinatorics)
Integers occurring in the coefficients of the Taylor series of 1/cosh t
unit with i2 = −1. The Euler number E2n can be expressed as a sum over the even partitions of 2n, E 2 n = ( 2 n ) ! ∑ 0 ≤ k 1 , … , k n ≤ n ( K k 1 , …
Euler_number
Natural number
partitions of 1 into reciprocals of positive integers <= 16 Egyptian fraction. 1013 is a prime number, a Sophie Germain prime, and a centered square number
1000_(number)
Natural number
of five remaining Seventeen or Bust numbers in the Sierpiński problem 21209 = number of reduced trees with 23 nodes 21637 = number of partitions of 37
20,000
Natural number
(ed.). "Sequence A036469 (Partial sums of A000009 (partitions into distinct parts))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
400_(number)
Infinite sum
infinity of the finite sums of the n {\displaystyle n} first terms of the series if the limit exists. These finite sums are called the partial sums of the
Series_(mathematics)
Measure of tropical cyclone activity
originated as the Hurricane Destruction Potential (HDP) index, which sums the squares of tropical cyclones' maximum sustained winds while at hurricane strength
Accumulated_cyclone_energy
(Mathematical) decomposition into a product
factorization of x 4 + 1. {\displaystyle x^{4}+1.} If one introduces the non-real square root of –1, commonly denoted i, then one has a difference of squares x 4
Factorization
Concept in combinatorics
t-core partition is a partition which has no hooks of length t. Such partitions have been used in the study of Ramanujan's congruences on the partition function
Core_partition
Number divisible only by 1 and itself
consequence of Fermat's theorem on sums of two squares, which states that an odd prime p {\displaystyle p} is expressible as the sum of two squares, p
Prime_number
Cluster analysis algorithm
K-medians clustering is a partitioning technique used in cluster analysis. It groups data into k clusters by minimizing the sum of distances—typically using
K-medians_clustering
Natural number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A036469 (Partial sums of A000009 (partitions into distinct parts))"
600_(number)
Statistical technique
In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational
Total_least_squares
Even integers as sums of two primes
squares: Lagrange proved that every positive integer is the sum of four squares. See Waring's problem and the related Waring–Goldbach problem on sums
Goldbach's_conjecture
Graph of chess rook moves
two squares sharing a row (rank) or column (file), the squares that a rook can move between. These graphs can be constructed for chessboards of any rectangular
Rook's_graph
Number with odd number of 1s in binary
odious and evil numbers is the unique partition of these numbers into two sets that have equal multisets of pair-wise sums. Sloane, N. J. A. (ed.), "Sequence
Odious_number
Natural number
divisible by the squares of its prime factors (i.e. 4 and 25) but is not a power itself, and a Harshad number, meaning it is divisible by the sum of its digits
500_(number)
Natural number
31721 = start of a prime quadruplet 32043 = smallest number whose square is pandigital. 32045 = can be expressed as a sum of two squares in more ways than
30,000
Kind of probability distribution
combination of different normal variables and squares of normal variables. Equivalently, it is also a linear sum of independent noncentral chi-square variables
Generalized chi-squared distribution
Generalized_chi-squared_distribution
Measure of variation in statistics
Corrected Sums of Squares and Products". Technometrics. 4 (3): 419–420. doi:10.1080/00401706.1962.10490022. Dodge, Yadolah (2003). The Oxford Dictionary of Statistical
Standard_deviation
Measure of goodness of fit for a statistical model
a generalization of the idea of using the sum of squares of residuals (SSR) in ordinary least squares to cases where model-fitting is achieved by maximum
Deviance_(statistics)
Shape with three sides
the vertices of every three squares are tangent to all obtuse triangle's sides. Every acute triangle has three inscribed squares (squares in its interior
Triangle
Specialized form of regression analysis, in statistics
least squares and beyond). Springer Vieweg. ISBN 978-3-658-11455-8. Tofallis, Chris (2008). "Least Squares Percentage Regression". Journal of Modern
Robust_regression
Fundamental theorem in probability theory and statistics
function. Then the distribution of the standardized sums 1 s n ∑ i = 1 n ( X i − μ i ) {\displaystyle {\frac {1}{s_{n}}}\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)}
Central_limit_theorem
Statistics concept
is unknown. The sum of squares of the residuals, on the other hand, is observable. The quotient of that sum by σ2 has a chi-squared distribution with
Errors_and_residuals
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PARTITION OF-SUMS-OF-SQUARES
PARTITION OF-SUMS-OF-SQUARES
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : variant of Chappell.Variant of German Kappel.
Surname or Lastname
Respelling of Kroll.English
Respelling of Kroll.English : variant of Curl.
Surname or Lastname
English (of French origin)
English (of French origin) : variant of Rounsaville.
Biblical
Ecclesiasticus or the Sirach = Joshua, Joshua, saviour, or whose help is Jehovah Jehovah, I am; the eternal living one Jehovah, self-subsisting
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddox.
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddox.
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddox.
Boy/Male
Arabic
Partition; Curtain
Boy/Male
Hindu, Indian, Traditional
Noble Partition
Biblical
peace; abundance
Boy/Male
Shakespearean
King Henry IV, Part 1' Earl of March. Scroop.
Surname or Lastname
English
English : patronymic from a short form of the personal name Simon.Jewish (from Ukraine; Symes, Symis) : metronymic from the Yiddish female personal name Sime (see Sima).Benjamin Syms was a planter and philanthropist, probably the earliest inhabitant of any North American colony to bequeath property for the establishment of a free school. His name was spelled variously as Sims, Simes, Sym, Symms, Syms, and Symes. He was probably born in England, but was reported in the VA census of 1624/25 as age 33 and living at Basse’s Choice in what was later known as Isle of Wight County.
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : variant of Duley.
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddock.
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddox.
Boy/Male
Hindu, Indian, Marathi
Fragrance; Flower; Sum; Total
Girl/Female
Arabic, Muslim
Name of an Early Distinguished Woman
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddox.
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddox.
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddox.
PARTITION OF-SUMS-OF-SQUARES
PARTITION OF-SUMS-OF-SQUARES
PARTITION OF-SUMS-OF-SQUARES
PARTITION OF-SUMS-OF-SQUARES
PARTITION OF-SUMS-OF-SQUARES
PARTITION OF-SUMS-OF-SQUARES
PARTITION OF-SUMS-OF-SQUARES
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