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Statistical proof is the rational demonstration of degree of certainty for a proposition, hypothesis or theory that is used to convince others subsequent
Statistical_proof
Reasoning for mathematical statements
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The
Mathematical_proof
Topics referred to by the same term
a branch of mathematical logic that represents proofs as formal mathematical objects Statistical proof, demonstration of degree of certainty for a hypothesis
Proof
Obligation on a party in a dispute to provide sufficient warrant for their position
The burden of proof (Latin: onus probandi, shortened from Onus probandi incumbit ei qui dicit, non ei qui negat – the burden of proof lies with the one
Burden_of_proof_(philosophy)
Evidence that either supports or counters a scientific theory
infallible proof, in practice theories may be said to be proved according to some standard of proof used in a given inquiry. In this limited sense, proof is the
Scientific_evidence
Realization of a certain method or idea in order to demonstrate its feasibility
A proof of concept (POC or PoC), also known as proof of principle, is an inchoate realization of a certain idea or method in order to demonstrate its feasibility
Proof_of_concept
Proving validity without revealing other data
In cryptography, a zero-knowledge proof (also known as a ZK proof or ZKP) is a protocol in which one party (the prover) can convince another party (the
Zero-knowledge_proof
Statistical proof by contradiction technique
Surrogate data testing (or the method of surrogate data) is a statistical proof by contradiction technique similar to permutation tests and parametric
Surrogate_data_testing
American Professor in [[Classics]] and [[Military History]] (born 1953)
this controversial opinion as undeniable fact without exhaustive statistical proof is undeniably racist." Journalist Kelefa Sanneh observed that "It's
Victor_Davis_Hanson
cholera epidemic in Soho, with the use of a dot distribution map and statistical proof to illustrate the connection between the quality of the water source
History_of_water_filters
Law of statistics
lead to dogmatic or absolute knowledge, see: statistical proof. Graphing calculator at Desmos (graphing) Proof in: Elemér Elad Rosinger, (2016), "Quanta
Law_of_truly_large_numbers
Formula in probability theory
In probability theory, the law of total covariance, covariance decomposition formula, or conditional covariance formula states that if X, Y, and Z are
Law_of_total_covariance
Theorem in probability theory
Statistics. Retrieved 9 July 2014. "Law of total variance". The Book of Statistical Proofs. Billingsley, Patrick (1995). "Problem 34.10(b)". Probability and
Law_of_total_variance
Statistical principle
is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic for a model parameter
Sufficient_statistic
Type of fan mounted to a ceiling horizontally
of visual flicker induced the ceiling fan. The findings revealed statistical proof that one out of three cognitive performances (digit-span task) may
Ceiling_fan
Variety of proofs provided for the different types of convergence of random variables
article is supplemental for “Convergence of random variables” and provides proofs for selected results. Several results will be established using the portmanteau
Proofs of convergence of random variables
Proofs_of_convergence_of_random_variables
Water purification device
cholera epidemic in Soho, with the use of a dot distribution map and statistical proof to illustrate the connection between the quality of the water source
Slow_sand_filter
In mathematics, the Bhatia–Davis inequality, named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance σ2 of any bounded probability
Bhatia–Davis_inequality
Mass exodus of white people from areas becoming more diverse
machines by the US Census Bureau, failed to attain any approved level of statistical proof. It was rigorous reprocessing of the same raw data on a UNIVAC I,
White_flight
Probability theory upper bound
In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ2 of any bounded probability distribution
Popoviciu's inequality on variances
Popoviciu's_inequality_on_variances
Jungian theories
view. Critics cite that Jung's experiments that sought to provide statistical proof for this theory did not yield satisfactory result. His experiment
Analytical_psychology
Physical law in electrochemistry
activities are posited very close to the true concentrations). Fundamental statistical proof of the mentioned linearity goes beyond the scope of this section,
Nernst_equation
Statistic used in statistical hypothesis testing
statistics and their corresponding statistical tests or models. Test statistic is a quantity derived from the sample for statistical hypothesis testing. A hypothesis
Test_statistic
Statistical probability Statistical process control Statistical proof Statistical randomness Statistical range – see range (statistics) Statistical regularity
List_of_statistics_articles
Correction for sample variance bias
In statistics, Bessel's correction is the use of n − 1 instead of n in the formula for the sample variance and sample standard deviation, where n is the
Bessel's_correction
In information theory and statistics, Kullback's inequality is a lower bound on the Kullback–Leibler divergence expressed in terms of the large deviations
Kullback's_inequality
Distance between two statistical objects
probability theory, and information theory, a statistical distance quantifies the distance between two statistical objects, which can be two random variables
Statistical_distance
Process of removing impurities from water
cholera epidemic in Soho, with the use of a dot distribution map and statistical proof to illustrate the connection between the quality of the water source
Water_purification
Erroneous method of proof
In the common discourse, a proof by example can also be used to describe an attempt to establish a claim using statistically insignificant examples. In
Proof_by_example
Study of the properties of codes and their fitness
introduced the qualitative and quantitative model of communication as a statistical process, opening with the assertion that "The fundamental problem of
Coding_theory
Effect of variables' uncertainties on the uncertainty of a function based on them
"Variance of the linear combination of two random variables". The Book of Statistical Proofs. Retrieved 2022-01-29. Lee, S. H.; Chen, W. (2009). "A comparative
Propagation_of_uncertainty
Resistance to a malicious network attack
Proof of personhood (PoP) is a means of resisting malicious attacks on peer-to-peer networks, particularly attacks that use multiple fake identities,
Proof_of_personhood
Vector in statistics
In multivariate statistics, if ε {\displaystyle \varepsilon } is a vector of n {\displaystyle n} random variables, and Λ {\displaystyle \Lambda } is an
Quadratic_form_(statistics)
Processing of natural language by a computer
which include both statistical and neural networks, on the other hand, have many advantages over the symbolic approach: both statistical and neural network
Natural_language_processing
Theorem on magnetism
The Bohr–Van Leeuwen theorem states that when statistical mechanics and classical mechanics are applied consistently, the thermal average of the magnetization
Bohr–Van_Leeuwen_theorem
Probability and computer science concept
{\sigma }}^{2}+{\frac {B\varepsilon }{3}}\right)}}\right).} The following proof of McDiarmid's inequality constructs the Doob martingale tracking the conditional
McDiarmid's_inequality
These early demonstrations and others elsewhere gradually allowed statistical proof that roundabouts were also safe and effective in North America, paving
Leif_Ourston
Field of knowledge
shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas
Mathematics
Digital asset using a distributed ledger
of coin ownership. The two most common consensus mechanisms are proof of work and proof of stake. Despite the name, which has come to describe many of
Cryptocurrency
Probability distribution
"Kullback–Leibler divergence for the Dirichlet distribution". The Book of Statistical Proofs. StatProofBook. Retrieved 2025-06-23. Connor, Robert J.; Mosimann, James
Dirichlet_distribution
Branch of mathematics
studied uniformly accelerated motion, and Nicole Oresme gave a graphical proof of the mean speed theorem, representing displacement by the area under a
Mathematical_analysis
Mathematical Assumption
The rare disease assumption is a mathematical assumption in epidemiologic case-control studies where the hypothesis tests the association between an exposure
Rare_disease_assumption
cholera epidemic in Soho, with the use of a dot distribution map and statistical proof to illustrate the connection between the quality of the water source
History of water supply and sanitation
History_of_water_supply_and_sanitation
Alternative decimal expansion of 1
mathematically rigorous proofs. The intuitive arguments are generally based on properties of finite decimals that are extended without proof to infinite decimals
0.999...
Medical research using human test subjects
Journals (URM) British Doctors Study – in 1956 provided convincing statistical proof that tobacco smoking increased the risk of lung cancer. Framingham
Outline_of_clinical_research
English poet and essayist (1843–1901)
that "the chief aim of [the] book was to produce a cumulative quasi-statistical proof of telepathy." It was enthusiastically praised by psychologist William
Frederic_W._H._Myers
US labor law passed in 1991
plaintiff's burden with respect to statistical proof, in which the court had held: "The mere existence of a statistical imbalance in an employer's workforce
Civil_Rights_Act_of_1991
Branch of applied mathematics
approaches and ideas have been extended to other areas of physics, such as statistical mechanics, continuum mechanics, classical field theory, and quantum field
Mathematical_physics
Method in statistics
{D}}{\mathcal {N}}(0,\sigma ^{2}[g'(\theta )]^{2})}.} This concludes the proof. Alternatively, one can add one more step at the end, to obtain the order
Delta_method
Set of objects whose state must satisfy limits
doi:10.1109/FOCS.2017.37. ISBN 978-1-5386-3464-6. Zhuk, Dmitriy (2020). "A Proof of the CSP Dichotomy Conjecture". Journal of the ACM. 67 (5): 1–78. arXiv:1704
Constraint satisfaction problem
Constraint_satisfaction_problem
Application of mathematical methods to other fields
discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical
Applied_mathematics
Theorem about the power of the likelihood ratio test
the size of the ratio (i.e. whether a large statistic corresponds to a small ratio or to a large one). Proof Given any hypothesis test with rejection set
Neyman–Pearson_lemma
Topics referred to by the same term
hypothesis Statistical hypothesis test, techniques to reach conclusions about probabilistic behavior Metal testing Mechanical testing Proof test, a stress
Test
Statistical error measure
Median ( y | X = x ) {\displaystyle {\hat {f}}(x)={\text{Median}}(y|X=x)} . Proof: The Loss functions for classification is L = E [ | y − a | | X = x ] =
Mean_absolute_error
Lower bound on variance of an estimator
integral converges uniformly for all θ {\displaystyle \theta } . Proof based on. Proof First equation: Let δ {\displaystyle \delta } be an infinitesimal
Cramér–Rao_bound
Study of discrete mathematical structures
it can be easily verified with a truth table. The study of mathematical proof is particularly important in logic, and has accumulated to automated theorem
Discrete_mathematics
Sequence of operations for a task
"The Spirit of Truth" for a history leading to, and a discussion of, his proof. Kleene, Stephen C. (1936). "General Recursive Functions of Natural Numbers"
Algorithm
Theorem of convex functions
convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder
Jensen's_inequality
Mathematics award
Yuri I. Manin, with the first-ever IMU silver plaque in recognition of his proof of Fermat's Last Theorem. Don Zagier referred to the plaque as a "quantized
Fields_Medal
In linear algebra, relation between 3 dimensions
{nullity} (M)=n.} Here we provide two proofs. The first operates in the general case, using linear maps. The second proof looks at the homogeneous system A
Rank–nullity_theorem
Rules in probabilistic logic
justifies the rare-event approximation often used in reliability theory. The proofs are elementary. Recall that P(A ∨ B) = P(A) + P(B) − P(A & B), which implies
Fréchet_inequalities
Unique positive real number which when multiplied by itself gives 2
{\displaystyle {\sqrt {2}}} . One proof of the number's irrationality is the following proof by infinite descent. It is also a proof of a negation by refutation:
Square_root_of_2
American climate researcher
stating that evidence had reached the 5-sigma "gold standard level" of statistical proof of human influence in global climate change using three sets of satellite
Benjamin_D._Santer
Property of a thermodynamic system
introduced the concept of statistical disorder and probability distributions into a new field of thermodynamics, called statistical mechanics, and found the
Entropy
Vladimir Putin, and said that "what has happened to Navalny is yet more proof of Putin’s brutality"; Gladiators on Network 10 and 10Play (with Margaret
List of The Weekly with Charlie Pickering episodes
List_of_The_Weekly_with_Charlie_Pickering_episodes
Branch of mathematics concerning probability
both discrete and continuous distributions as well as others; separate proofs are not required for discrete and continuous distributions. Certain random
Probability_theory
Method of quality control
Statistical process control (SPC) or statistical quality control (SQC) is the application of statistical methods to monitor and control the quality of
Statistical_process_control
Method of logical reasoning
non-random and the sample size is very small. Statistical generalizations are also called statistical projections and sample projections. An anecdotal
Inductive_reasoning
Proposition in probability theory
{G}}_{1}]=\operatorname {E} [X\mid {\mathcal {G}}_{1}]\quad {\text{(a.s.)}}.} Proof. Since a conditional expectation is a Radon–Nikodym derivative, verifying
Law_of_total_expectation
Formulation of classical mechanics using momenta
{d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}.} Proof The Hamiltonian H ( p , q ) {\displaystyle {\mathcal {H}}({\boldsymbol {p}}
Hamiltonian_mechanics
industry. The most probative type of evidence seems to be statistical data showing "gross statistical disparities between the proportion of minorities hired
Strong-basis-in-evidence standard
Strong-basis-in-evidence_standard
Norwegian bridge player
Woolsey published an article on Bridge Winners demonstrating a sound statistical proof of Mevius's findings. The accused pair refused to issue a public statement
Boye_Brogeland
1932 book by John von Neumann
than the statistical one be possible. — pp. 324-325 This proof was rejected as early as 1935 by Grete Hermann who found a flaw in the proof. The additive
Mathematical Foundations of Quantum Mechanics
Mathematical_Foundations_of_Quantum_Mechanics
doi:10.1002/bdm.734. Wasserman, David T. (1991–1992). "The morality of statistical proof and the risk of mistaken liability". Cardozo Law Review. 13: 935–976
Wells_effect
{\frac {4}{9\lambda ^{2}}}.} For a relatively elementary proof see. The rough idea behind the proof is that there are two cases: one where the mode of X {\displaystyle
Vysochanskij–Petunin inequality
Vysochanskij–Petunin_inequality
Document used to identify a person
but they have de facto equivalents since these countries still require proof of identity in many situations. For example, all vehicle drivers must have
Identity_document
Canadian historian (1963–2010)
of ethnic conflict in the twentieth century are here': Erroneous statistical proofs and the search for ethnic violence in revolutionary Ireland, 1917–1923"
Peter_Hart_(historian)
Phenomenon in decision theory and estimation theory
} from that point. For a sketch of the proof of this result, see Proof of Stein's example. An alternative proof is due to Larry Brown: he proved that the
Stein's_example
Harmonic functions as solutions to Laplace's equation
geometry software Optimization software Statistical software Numerical-analysis software Numerical libraries Proof assistants Solvers Discrete Computer algebra
Potential_theory
1960 article by Eugene Wigner
geometry software Optimization software Statistical software Numerical-analysis software Numerical libraries Proof assistants Solvers Discrete Computer algebra
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
Technique used in data-driven research
Statistical disclosure control (SDC), also known as statistical disclosure limitation (SDL) or disclosure avoidance, is a technique used in data-driven
Statistical disclosure control
Statistical_disclosure_control
Wetland in Indiana and Illinois, United States
settlers faced systemic discrimination, including mandatory registration, proof of freedom, and $500 good-behavior bonds. Such was the case of William Greenwood's
Grand_Kankakee_Marsh
History of the evolutionary theory
Essay on the Principle of Population which reminded him of Malthus's statistical proof that human populations breed beyond their means and compete to survive
Inception_of_Darwin's_theory
On distances between points on a circle
1950s by Vera T. Sós, János Surányi [hu], and Stanisław Świerczkowski; more proofs were added by others later. Applications of the three-gap theorem include
Three-gap_theorem
2015 contract bridge cheating scandal
published an article on the Bridge Winners website demonstrating a statistical proof of Mevius's hypothesis. The allegations were reported both by official
Fantoni and Nunes cheating scandal
Fantoni_and_Nunes_cheating_scandal
Kind of differential equation
Essentials of stochastic finance: Facts, models, theory. Advanced Series on Statistical Science & Applied Probability No. 3. River Edge, NJ: World Scientific
Tanaka's_formula
The purpose of Proofs involving ordinary least squares page is to provide supplementary materials for the ordinary least squares article, reducing the
Proofs involving ordinary least squares
Proofs_involving_ordinary_least_squares
Multiple comparisons correction
Means of Multivariate Normal Distributions". Journal of the American Statistical Association. 62 (318): 626–633. doi:10.1080/01621459.1967.10482935. Seidler
Šidák_correction
Randomized controlled trial Sequential analysis Sequential probability ratio test Glossary Category Mathematics portal Statistical outline Statistical topics
Fisher's_inequality
Country in Eastern Europe and North Asia
offered the first ever Clay Millennium Prize Problems Award for his final proof of the Poincaré conjecture in 2002, as well as the Fields Medal in 2006
Russia
Statistics function
simulation and estimation via minimax tilting". Journal of the Royal Statistical Society, Series B. 79: 125–148. arXiv:1603.04166. Bibcode:2016arXiv160304166B
Q-function
Measure of distance to normality
(see Differential entropy § Maximization in the normal distribution for a proof): J ( Y ) = D K L ( Y ‖ Y G ) {\displaystyle J(Y)=D_{KL}(Y\ \Vert \
Negentropy
Political party in the United States
imposition of strict voter ID laws. The 2016 Republican platform advocated proof of citizenship as a prerequisite for registering to vote and photo ID as
Republican Party (United States)
Republican_Party_(United_States)
Framework to describe phase transitions
results of statistical field theory can be applied directly to its quantum equivalent.[citation needed] The correlation functions of a statistical field theory
Statistical_field_theory
Quantum field theory enjoying conformal symmetry
applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are
Conformal_field_theory
AI standards for robustness and data privacy
Differential privacy for exposing statistical data while guaranteeing that no private information is exposed. Zero-knowledge proof - providing proven validity
Trustworthy_AI
Theorem about diagonalizing matrices
Nicacio, F. (2021-12-01). "Williamson theorem in classical, quantum, and statistical physics". American Journal of Physics. 89 (12): 1139–1151. arXiv:2106
Williamson_theorem
Area of mathematics
theorems (for example the four color theorem), and the design and use of proof assistants. Computational mathematics emerged as a distinct part of applied
Computational_mathematics
Response of Hungary to the 2022 Russian invasion of Ukraine
Ukraine. However, she later corrected herself by admitting that he had no proof of the leaks; she understood what he was saying to mean that the Russians
Hungary and the Russo-Ukrainian war
Hungary_and_the_Russo-Ukrainian_war
Intelligence of machines
statements that are given and assumed to be true (the premises). Proofs can be structured as proof trees, in which nodes are labelled by sentences, and children
Artificial_intelligence
STATISTICAL PROOF
STATISTICAL PROOF
Girl/Female
Muslim
Many signs & proofs, Verses in the Quran, Royal
Boy/Male
Muslim
Proof
Girl/Female
Indian
Many signs & proofs, Verses in the Quran, Royal
Boy/Male
Arabic, Muslim
Evidence; Proof
Girl/Female
Muslim
Guide, Proof
Boy/Male
Indian
Proof
Girl/Female
Muslim
Proof
Boy/Male
Muslim
Proof
Boy/Male
Arabic
Evidence; Proof; Distinction Between Truth and Falsehood
Boy/Male
Muslim
Argument, Reasoning, Proof
Boy/Male
Indian
Proof
Girl/Female
Biblical
Flight, proof, temptation, delicate.
Girl/Female
Indian
Many signs & proofs, Verses in the Quran, Royal
Girl/Female
Indian
Witness; Proof
Boy/Male
Arabic, Muslim
Another Name for God; Evidence; Proof
Boy/Male
Indian
Argument, Reasoning, Proof
Girl/Female
Assamese, Bengali, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu, Traditional
Witness; Justice; Proof; Cute Princess; Loved by Everyone; Grace; Purity; Pluck; Witness Truth; Queen; Princess; Real; Truth
Boy/Male
Muslim
Evidence. Proof.
Surname or Lastname
English
English : from Middle English, Old French palmer, paumer (from palme, paume ‘palm tree’, Latin palma), a nickname for someone who had been on a pilgrimage to the Holy Land. Such pilgrims generally brought back a palm branch as proof that they had actually made the journey, but there was a vigorous trade in false souvenirs, and the term also came to be applied to a cleric who sold indulgences.Swedish (Palmér) : ornamental name formed with palm ‘palm tree’ + the suffix -ér, from Latin -erius ‘descendant of’.Irish : when not truly of English origin (see 1 above), a surname adopted by bearers of Gaelic Ó Maolfhoghmhair (see Milford) perhaps because they were from an ecclesiastical family.German : topographic name for someone living among pussy willows (see Palm 2).German : from the personal name Palm (see Palm 3).
Girl/Female
Muslim
Many signs & proofs, Verses in the Quran, Royal
STATISTICAL PROOF
STATISTICAL PROOF
Boy/Male
Muslim
Insist, Never gives up
Boy/Male
Australian, Finnish, German, Turkish
Brother
Male
African
father.
Boy/Male
Hindu, Indian, Sanskrit
Another Name for the Sun; Lord Shiva
Boy/Male
Gaelic Irish
Victorious.
Girl/Female
Tamil
Existence, Real
Girl/Female
Indian
Pour
Boy/Male
Hindu
Servant a devotee of Goddess Durga
Girl/Female
Arabic, Muslim
Female Warrior
Boy/Male
African, American, British, Christian, Dutch, English, Hawaiian, Hebrew, Hindu, Indian, Jamaican
Descends; Descending One who Rules; Descent
STATISTICAL PROOF
STATISTICAL PROOF
STATISTICAL PROOF
STATISTICAL PROOF
STATISTICAL PROOF
n.
The science which has to do with the collection and classification of certain facts respecting the condition of the people in a state.
n.
An official registration of the number of the people, the value of their estates, and other general statistics of a country.
n.
See Statistics, 2.
a.
Proof against proofs; obstinate in the wrong.
n.
Classified facts respecting the condition of the people in a state, their health, their longevity, domestic economy, arts, property, and political strength, their resources, the state of the country, etc., or respecting any particular class or interest; especially, those facts which can be stated in numbers, or in tables of numbers, or in any tabular and classified arrangement.
n.
The act of forming into a table or tables; as, the tabulation of statistics.
n.
An account, or formal report, of an action performed, of a duty discharged, of facts or statistics, and the like; as, election returns; a return of the amount of goods produced or sold; especially, in the plural, a set of tabulated statistics prepared for general information.
n.
The branch of mathematics which studies methods for the calculation of probabilities.
adv.
In the way of statistics.
a.
Of or pertaining to statistics; as, statistical knowledge, statistical tabulation.
v. t.
To arm with proof armor; to arm securely; as, to proof-arm herself.
a.
Highly rectified; very strongly alcoholic; as, high-proof spirits.
a.
Arranged in a schedule; as, tabular statistics.
n.
Vital statistics.
n.
One versed in statistics; one who collects and classifies facts for statistics.
a.
Alt. of Statistical
n.
A book published yearly; any annual report or summary of the statistics or facts of a year, designed to be used as a reference book; as, the Congregational Yearbook.
n.
A book or table, containing a calendar of days, and months, to which astronomical data and various statistics are often added, such as the times of the rising and setting of the sun and moon, eclipses, hours of full tide, stated festivals of churches, terms of courts, etc.
n.
A statistician.