René's URL Explorer Experiment


Title: Normal Distribution Function -- from Wolfram MathWorld

Open Graph Title: Normal Distribution Function -- from Wolfram MathWorld

X Title: Normal Distribution Function -- from Wolfram MathWorld

Description: A normalized form of the cumulative normal distribution function giving the probability that a variate assumes a value in the range [0,x], Phi(x)=Q(x)=1/(sqrt(2pi))int_0^xe^(-t^2/2)dt. (1) It is related to the probability integral alpha(x)=1/(sqrt(2pi))int_(-x)^xe^(-t^2/2)dt (2) by Phi(x)=1/2alpha(x). (3) Let u=t/sqrt(2) so du=dt/sqrt(2). Then Phi(x)=1/(sqrt(pi))int_0^(x/sqrt(2))e^(-u^2)du=1/2erf(x/(sqrt(2))). (4) Here, erf is a function sometimes called the error function....

Open Graph Description: A normalized form of the cumulative normal distribution function giving the probability that a variate assumes a value in the range [0,x], Phi(x)=Q(x)=1/(sqrt(2pi))int_0^xe^(-t^2/2)dt. (1) It is related to the probability integral alpha(x)=1/(sqrt(2pi))int_(-x)^xe^(-t^2/2)dt (2) by Phi(x)=1/2alpha(x). (3) Let u=t/sqrt(2) so du=dt/sqrt(2). Then Phi(x)=1/(sqrt(pi))int_0^(x/sqrt(2))e^(-u^2)du=1/2erf(x/(sqrt(2))). (4) Here, erf is a function sometimes called the error function....

X Description: A normalized form of the cumulative normal distribution function giving the probability that a variate assumes a value in the range [0,x], Phi(x)=Q(x)=1/(sqrt(2pi))int_0^xe^(-t^2/2)dt. (1) It is related to the probability integral alpha(x)=1/(sqrt(2pi))int_(-x)^xe^(-t^2/2)dt (2) by Phi(x)=1/2alpha(x). (3) Let u=t/sqrt(2) so du=dt/sqrt(2). Then Phi(x)=1/(sqrt(pi))int_0^(x/sqrt(2))e^(-u^2)du=1/2erf(x/(sqrt(2))). (4) Here, erf is a function sometimes called the error function....

Opengraph URL: https://mathworld.wolfram.com/NormalDistributionFunction.html

X: @WolframResearch

direct link

Domain: mathworld.wolfram.com

DC.TitleNormal Distribution Function
DC.CreatorWeisstein, Eric W.
DC.DescriptionA normalized form of the cumulative normal distribution function giving the probability that a variate assumes a value in the range [0,x], Phi(x)=Q(x)=1/(sqrt(2pi))int_0^xe^(-t^2/2)dt. (1) It is related to the probability integral alpha(x)=1/(sqrt(2pi))int_(-x)^xe^(-t^2/2)dt (2) by Phi(x)=1/2alpha(x). (3) Let u=t/sqrt(2) so du=dt/sqrt(2). Then Phi(x)=1/(sqrt(pi))int_0^(x/sqrt(2))e^(-u^2)du=1/2erf(x/(sqrt(2))). (4) Here, erf is a function sometimes called the error function....
DC.Date.Modified2007-09-22
DC.Subject62E
DC.RightsCopyright 1999-2026 Wolfram Research, Inc. See https://mathworld.wolfram.com/about/terms.html for a full terms of use statement.
DC.Formattext/html
DC.Identifierhttps://mathworld.wolfram.com/NormalDistributionFunction.html
DC.Languageen
DC.PublisherWolfram Research, Inc.
DC.Relation.IsPartOfhttps://mathworld.wolfram.com/
DC.TypeText
Last-Modified2007-09-22
og:imagehttps://mathworld.wolfram.com/images/socialmedia/share/ogimage_NormalDistributionFunction.png
og:typewebsite
twitter:cardsummary_large_image
twitter:image:srchttps://mathworld.wolfram.com/images/socialmedia/share/ogimage_NormalDistributionFunction.png
Noneie=edge

Links:

https://www.wolfram.com/mathematica/
https://wolframalpha.com/
https://mathworld.wolfram.com/
https://www.wolfram.com/mathematica/
https://wolframalpha.com/
https://mathworld.wolfram.com/
Algebra https://mathworld.wolfram.com/topics/Algebra.html
Applied Mathematics https://mathworld.wolfram.com/topics/AppliedMathematics.html
Calculus and Analysis https://mathworld.wolfram.com/topics/CalculusandAnalysis.html
Discrete Mathematics https://mathworld.wolfram.com/topics/DiscreteMathematics.html
Foundations of Mathematics https://mathworld.wolfram.com/topics/FoundationsofMathematics.html
Geometry https://mathworld.wolfram.com/topics/Geometry.html
History and Terminology https://mathworld.wolfram.com/topics/HistoryandTerminology.html
Number Theory https://mathworld.wolfram.com/topics/NumberTheory.html
Probability and Statistics https://mathworld.wolfram.com/topics/ProbabilityandStatistics.html
Recreational Mathematics https://mathworld.wolfram.com/topics/RecreationalMathematics.html
Topology https://mathworld.wolfram.com/topics/Topology.html
Alphabetical Index https://mathworld.wolfram.com/letters/
New in MathWorld https://mathworld.wolfram.com/whatsnew/
Probability and Statisticshttps://mathworld.wolfram.com/topics/ProbabilityandStatistics.html
Statistical Distributionshttps://mathworld.wolfram.com/topics/StatisticalDistributions.html
Continuous Distributionshttps://mathworld.wolfram.com/topics/ContinuousDistributions.html
History and Terminologyhttps://mathworld.wolfram.com/topics/HistoryandTerminology.html
Database Collectionshttps://mathworld.wolfram.com/topics/DatabaseCollections.html
Integer Sequence Databaseshttps://mathworld.wolfram.com/topics/IntegerSequenceDatabases.html
Online Encyclopedia of Integer Sequenceshttps://mathworld.wolfram.com/topics/OnlineEncyclopediaofIntegerSequences.html
Download Wolfram Notebookhttps://mathworld.wolfram.com/notebooks/Statistics/NormalDistributionFunction.nb
normal distributionhttps://mathworld.wolfram.com/NormalDistribution.html
probability integralhttps://mathworld.wolfram.com/ProbabilityIntegral.html
erfhttps://mathworld.wolfram.com/Erf.html
erfhttps://mathworld.wolfram.com/Erf.html
root extractionshttps://mathworld.wolfram.com/RootExtraction.html
confidence intervalhttps://mathworld.wolfram.com/ConfidenceInterval.html
Maclaurin serieshttps://mathworld.wolfram.com/MaclaurinSeries.html
erfhttps://mathworld.wolfram.com/Erf.html
A014481http://oeis.org/A014481
erfhttps://mathworld.wolfram.com/Erf.html
A001147http://oeis.org/A001147
continued fractionhttps://mathworld.wolfram.com/ContinuedFraction.html
probable errorhttps://mathworld.wolfram.com/ProbableError.html
Berry-Esséen Theoremhttps://mathworld.wolfram.com/Berry-EsseenTheorem.html
Confidence Intervalhttps://mathworld.wolfram.com/ConfidenceInterval.html
Erfhttps://mathworld.wolfram.com/Erf.html
Erfchttps://mathworld.wolfram.com/Erfc.html
Fisher-Behrens Problemhttps://mathworld.wolfram.com/Fisher-BehrensProblem.html
Gaussian Integralhttps://mathworld.wolfram.com/GaussianIntegral.html
Hh Functionhttps://mathworld.wolfram.com/HhFunction.html
Normal Distributionhttps://mathworld.wolfram.com/NormalDistribution.html
Owen T-Functionhttps://mathworld.wolfram.com/OwenT-Function.html
Probability Integralhttps://mathworld.wolfram.com/ProbabilityIntegral.html
Tetrachoric Functionhttps://mathworld.wolfram.com/TetrachoricFunction.html
normal distribution function https://www.wolframalpha.com/input/?i=normal+distribution+function
erf(x), erf'(x), erf''(x) https://www.wolframalpha.com/input/?i=erf%28x%29%2C+erf%27%28x%29%2C+erf%27%27%28x%29
plot Re(erf(x + I y)), Im(erf(x + I y)) https://www.wolframalpha.com/input/?i=plot+Re%28erf%28x+%2B+I+y%29%29%2C+Im%28erf%28x+%2B+I+y%29%29
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.http://www.amazon.com/exec/obidos/ASIN/0486612724/ref=nosim/ericstreasuretro
CRC Standard Mathematical Tables, 28th ed.http://www.amazon.com/exec/obidos/ASIN/1584882913/ref=nosim/ericstreasuretro
An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed.http://www.amazon.com/exec/obidos/ASIN/0471257087/ref=nosim/ericstreasuretro
An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed.http://www.amazon.com/exec/obidos/ASIN/0471257095/ref=nosim/ericstreasuretro
Approximations for Digital Computers.http://www.amazon.com/exec/obidos/ASIN/0691079145/ref=nosim/ericstreasuretro
Continuous Univariate Distributions, Vol. 1, 2nd ed.http://www.amazon.com/exec/obidos/ASIN/0471584940/ref=nosim/ericstreasuretro
Handbook of the Normal Distribution.http://www.amazon.com/exec/obidos/ASIN/0824793420/ref=nosim/ericstreasuretro
A001147http://oeis.org/A001147
A014481http://oeis.org/A014481
The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed.http://www.amazon.com/exec/obidos/ASIN/B0006BP2CE/ref=nosim/ericstreasuretro
Normal Distribution Functionhttps://www.wolframalpha.com/input/?i=normal+distribution+function
Weisstein, Eric W.https://mathworld.wolfram.com/about/author.html
MathWorldhttps://mathworld.wolfram.com/
https://mathworld.wolfram.com/NormalDistributionFunction.htmlhttps://mathworld.wolfram.com/NormalDistributionFunction.html
Probability and Statisticshttps://mathworld.wolfram.com/topics/ProbabilityandStatistics.html
Statistical Distributionshttps://mathworld.wolfram.com/topics/StatisticalDistributions.html
Continuous Distributionshttps://mathworld.wolfram.com/topics/ContinuousDistributions.html
History and Terminologyhttps://mathworld.wolfram.com/topics/HistoryandTerminology.html
Database Collectionshttps://mathworld.wolfram.com/topics/DatabaseCollections.html
Integer Sequence Databaseshttps://mathworld.wolfram.com/topics/IntegerSequenceDatabases.html
Online Encyclopedia of Integer Sequenceshttps://mathworld.wolfram.com/topics/OnlineEncyclopediaofIntegerSequences.html
About MathWorldhttps://mathworld.wolfram.com/about/
MathWorld Classroomhttps://mathworld.wolfram.com/classroom/
Contributehttps://mathworld.wolfram.com/contact/
MathWorld Bookhttps://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20
wolfram.comhttps://www.wolfram.com
13,439 Entrieshttps://mathworld.wolfram.com/whatsnew/
Last Updated: Mon Jul 6 2026https://mathworld.wolfram.com/whatsnew/
©1999–2026 Wolfram Research, Inc.https://www.wolfram.com
Terms of Usehttps://www.wolfram.com/legal/terms/mathworld.html
https://www.wolfram.com
wolfram.comhttps://www.wolfram.com
Wolfram for Educationhttps://www.wolfram.com/education/

Viewport: width=device-width, initial-scale=1

URLs of crawlers that visited me.