Research any topic before you write.

Find related topics.Discover entities.See connections.Build a topical map.

Student's t-distribution

In probability theory and statistics, Student's t distribution (or simply the t distribution) t ν {\displaystyle t_{\nu }} is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.

History, Applications & Standards

Interactive map loads when it comes into view.
Use the mouse wheel or two fingers (on touchscreens) to zoom in and out of the map.

Topic orientation

Student's t-distribution at a glance

The strongest research directions include Occurrence and applications and History. Use the connected concepts below as starting points, not as a keyword checklist.

ErrorsStandard deviationHypothesis testsQuantilesStandard scoreHelmertLürothStigler's Law of Eponymy

Research this topic

Explore the main themes, entities and connections around Student's t-distribution. Start with the topic map, then use the sections below for research and deeper semantic analysis.

Explore this topic

Start with a few of the strongest sections from the source topic. These are research directions, not a list of keywords you must use.

Key facts & relationships

High-confidence facts extracted from structured source data. Use them as anchors for further research.

CDF
1 2 + x Γ ( ν + 1 2 ) π ν Γ ( ν 2 ) × 2 F 1 ( 1 2 , ν + 1 2 ; 3 2 ; − x 2 ν ) , {\displaystyle {\begin{aligned}&{\frac {1}{2}}+x{\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}…
CF
( ν | t | ) ν / 2 K ν / 2 ( ν | t | ) Γ ( ν / 2 ) 2 ν / 2 − 1 {\displaystyle {\frac {{\big (}{\sqrt {\nu }}\,|t|{\big )}^{\nu /2}\,K_{\nu /2}{\big (}{\sqrt {\nu }}\,|t|{\big )}}…
Entropy
ν + 1 2 [ ψ ( ν + 1 2 ) − ψ ( ν 2 ) ] + ln ⁡ [ ν B ( ν 2 , 1 2 ) ] (nats) , {\displaystyle {\begin{aligned}&{\frac {\nu +1}{2}}\left[\psi {\left({\frac {\nu +1}{2}}\right)}-\psi…
Excess kurtosis
6 ν − 4 {\displaystyle {\frac {6}{\nu -4}}} for ν > 4 , {\displaystyle \nu >4,} ∞ {\displaystyle \infty } for 2 < ν ≤ 4 , {\displaystyle 2<\nu \leq 4,} otherwise undefined
Mean
0 {\displaystyle 0} for ν > 1 , {\displaystyle \nu >1,} otherwise undefined
Median
0 {\displaystyle 0}

Topics to explore

A structured outline of related entities, concepts and subtopics. Open any item to build a new map centered on it.

Browse the full topic structure. Each item opens a new analysis centered on that subject.

Overview

Definitions

Properties

Related distributions

Occurrence and applications

Computational methods

History

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

How this topic connects Entity context

Quick relationship hints grouped by predicate. Useful for spotting recurring semantic connections around the current entity.

See the strongest relationship patterns around the current topic before diving into the raw triples.

Student's t-distribution

Top relations

related to history · 22
Student's t-distribution → Another, As, Biometrika, Dublin, English-language, Eponymy, Gosset's, Guinness, Guinness Brewery, Helmert, In, Ireland, IV, Karl Pearson's, Lüroth, One, Pearson, Stigler's Law, Student, Student's
related to External links · 18
Student's t-distribution → Archived, Distribution, Earliest Known Uses, EMS Press, Encyclopedia, Estimation, First Students, Mathematics, PDF, Probability, Remarks, Rouaud, Some, Statistics, Student, Student's, Wayback Machine, Words
related to Special cases · 2
Student's t-distribution → Certain, Student's
CDF · 1
Student's t-distribution → 1 2 + x Γ ( ν + 1 2 ) π ν Γ ( ν 2 ) × 2 F 1 ( 1 2 , ν + 1 2 ; 3 2 ; − x 2 ν ) , {\displaystyle {\begin{aligned}&{\frac {1}{2}}+x{\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}…
CF · 1
Student's t-distribution → ( ν | t | ) ν / 2 K ν / 2 ( ν | t | ) Γ ( ν / 2 ) 2 ν / 2 − 1 {\displaystyle {\frac {{\big (}{\sqrt {\nu }}\,|t|{\big )}^{\nu /2}\,K_{\nu /2}{\big (}{\sqrt {\nu }}\,|t|{\big )}}…
Entropy · 1
Student's t-distribution → ν + 1 2 [ ψ ( ν + 1 2 ) − ψ ( ν 2 ) ] + ln ⁡ [ ν B ( ν 2 , 1 2 ) ] (nats) , {\displaystyle {\begin{aligned}&{\frac {\nu +1}{2}}\left[\psi {\left({\frac {\nu +1}{2}}\right)}-\psi…
Excess kurtosis · 1
Student's t-distribution → 6 ν − 4 {\displaystyle {\frac {6}{\nu -4}}} for ν 4 , {\displaystyle \nu 4,} ∞ {\displaystyle \infty } for 2 ν ≤ 4 , {\displaystyle 2\nu \leq 4,} otherwise undefined
Mean · 1
Student's t-distribution → 0 {\displaystyle 0} for ν 1 , {\displaystyle \nu 1,} otherwise undefined
Median · 1
Student's t-distribution → 0 {\displaystyle 0}
MGF · 1
Student's t-distribution → undefined

Important terminology Word statistics

Frequent words and multi-word phrases across the lead, headings, infobox and body. Useful for terminology coverage.

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

distribution displaystyle nu student's frac mean variance normal left right probability sqrt mu freedom function degrees value confidence data gamma

Entity relationships Subject–Predicate–Object triples

Extracted RDF-like relationships with confidence and source. The table includes structured facts and lower-confidence contextual relations.
SubjectPredicateObjectConfidenceSrc
Student's t-distributionCDF1 2 + x Γ ( ν + 1 2 ) π ν Γ ( ν 2 ) × 2 F 1 ( 1 2 , ν + 1 2 ; 3 2 ; − x 2 ν ) , {\displaystyle {\begin{aligned}&{\frac {1}{2}}+x{\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}…1.00infobox
Student's t-distributionCF( ν | t | ) ν / 2 K ν / 2 ( ν | t | ) Γ ( ν / 2 ) 2 ν / 2 − 1 {\displaystyle {\frac {{\big (}{\sqrt {\nu }}\,|t|{\big )}^{\nu /2}\,K_{\nu /2}{\big (}{\sqrt {\nu }}\,|t|{\big )}}…1.00infobox
Student's t-distributionEntropyν + 1 2 [ ψ ( ν + 1 2 ) − ψ ( ν 2 ) ] + ln ⁡ [ ν B ( ν 2 , 1 2 ) ] (nats) , {\displaystyle {\begin{aligned}&{\frac {\nu +1}{2}}\left[\psi {\left({\frac {\nu +1}{2}}\right)}-\psi…1.00infobox
Student's t-distributionExcess kurtosis6 ν − 4 {\displaystyle {\frac {6}{\nu -4}}} for ν > 4 , {\displaystyle \nu >4,} ∞ {\displaystyle \infty } for 2 < ν ≤ 4 , {\displaystyle 2<\nu \leq 4,} otherwise undefined1.00infobox
Student's t-distributionMean0 {\displaystyle 0} for ν > 1 , {\displaystyle \nu >1,} otherwise undefined1.00infobox
Student's t-distributionMedian0 {\displaystyle 0}1.00infobox
Student's t-distributionMGFundefined1.00infobox
Student's t-distributionMode0 {\displaystyle 0}1.00infobox
Student's t-distributionParametersν > 0 {\displaystyle \nu >0} degrees of freedom (real, nearly always a positive integer)1.00infobox
Student's t-distributionPDFΓ ( ν + 1 2 ) π ν Γ ( ν 2 ) ( 1 + x 2 ν ) − ν + 1 2 {\displaystyle {\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}}{{\sqrt {\pi \nu }}\,\Gamma {\left({\frac {\nu }{2}}\right)}…1.00infobox
Student's t-distributionSkewness0 {\displaystyle 0} for ν > 3 , {\displaystyle \ \nu >3\ ,} otherwise undefined1.00infobox
Student's t-distributionSupportx ∈ ( − ∞ , ∞ ) {\displaystyle x\in (-\infty ,\infty )}1.00infobox
Student's t-distributionVarianceν ν − 2 {\displaystyle {\frac {\nu }{\nu -2}}} for ν > 2 , {\displaystyle \nu >2,} ∞ {\displaystyle \infty } for 1 < ν ≤ 2 , {\displaystyle 1<\nu \leq 2,} otherwise undefined1.00infobox
Student's t-distributionis aexample of Stigler's Law of Eponymy0.90text

Related concept clusters Concept neighborhoods

Clusters of nearby vocabulary surrounding the topic. Scan them for adjacent concepts and language you may have missed.

These clusters group vocabulary that occurs around closely connected concepts in the source material.

    Connections between topic areas Semantic bridges

    Bridge nodes connect otherwise separate parts of the map. Expand a row to inspect the topic groups on each side.

    Bridges can reveal useful research angles that are easy to miss in a flat list of related terms.

    Map overview Semantic statistics

    Number of nodes, edges, triples, density and central hubs. Use it to gauge the size and connectivity of the map.

    Student's t-distribution

    Nodes109
    Edges108
    Triples56
    Avg. degree1.98
    Density0.018349
    Components1
    For writers, content strategists, SEOs, marketers and creators — from quick topic research to advanced semantic analysis.