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Quantiles of Student's $t$-distribution
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Numbers
$\nu$
$p$ 
$t_{\nu,p}$
1
1/200:
-63.65674116287158099500242331072784510349629211231565289288838206735678663325145438110134597227962434
1
1/100:
-31.82051595377395803933954943193656540467519466363209450032332029855776574860764573642128833412492341
1
1/40:
-12.70620473617470464602167997884208746766873833004756927928550210985681826776165079308153752964061677
1
1/20:
-6.313751514675043098979464244768186059447320503149318151310053495785384214132895817869008130045887710
1
1/10:
-3.077683537175253402570290576036909824006702143537792427039156250374863288495090918454593721667105435
1
1/4:
-1
comment: The lower quartile of the standard Cauchy distribution is exactly $-1$.
1
1/2:
0
1
3/4:
1
comment: The upper quartile of the standard Cauchy distribution is exactly $1$.
1
9/10:
3.077683537175253402570290576036909824006702143537792427039156250374863288495090918454593721667105435
1
19/20:
6.313751514675043098979464244768186059447320503149318151310053495785384214132895817869008130045887710
1
39/40:
12.70620473617470464602167997884208746766873833004756927928550210985681826776165079308153752964061677
1
99/100:
31.82051595377395803933954943193656540467519466363209450032332029855776574860764573642128833412492341
1
199/200:
63.65674116287158099500242331072784510349629211231565289288838206735678663325145438110134597227962434
1
999/1000:
318.3088389855504459216866954369214201827749367273530577933415421411151519312142567646174547252258650
2
1/200:
-9.924843200918293114742383624018663473435995346454545731573002915155036699933932067451479099611202165
2
1/100:
-6.964556734283274187082299259505419628752217858096215773535739171510127484997234737958292854758048898
2
1/40:
-4.302652729749463852320943892621175008187644875325562530717791191758606636576241383770212185868734842
2
1/20:
-2.919985580353725686960617443846747264585295590052693205466902560638189308019630416668450957515268395
2
1/10:
-1.885618083164126731735584965612930771426229167169264097568906317320976637949476051800516712436855430
2
1/4:
-0.8164965809277260327324280249019637973219824935522233761442308557503201258191050088466198110348800783
2
1/2:
0
2
3/4:
0.8164965809277260327324280249019637973219824935522233761442308557503201258191050088466198110348800783
2
9/10:
1.885618083164126731735584965612930771426229167169264097568906317320976637949476051800516712436855430
2
19/20:
2.919985580353725686960617443846747264585295590052693205466902560638189308019630416668450957515268395
2
39/40:
4.302652729749463852320943892621175008187644875325562530717791191758606636576241383770212185868734842
2
99/100:
6.964556734283274187082299259505419628752217858096215773535739171510127484997234737958292854758048898
2
199/200:
9.924843200918293114742383624018663473435995346454545731573002915155036699933932067451479099611202165
2
999/1000:
22.32712477011987543527500362467222708401509402230763489997190421307068649332258430290663070302304304
3
1/200:
-5.840909309733357260681947467496643198627268624108446322913455116661435388024841804195637464356917407
3
1/100:
-4.540702858568133555255198567550619039575557091570467212865611048821332670803759877094450758077224828
3
1/40:
-3.182446305283709592723225425779868376268944042901145241114653032670843071471503188599891680661311743
3
1/20:
-2.353363434801823877671223978899932607419607950935052747724014469494789935974375752798666394023729783
3
1/10:
-1.637744353696210105458905056896857491490680754930043322476334393119261092944193761433217669734565467
3
1/4:
-0.7648923284043452806575452223324014640295384635668160722302025877701349994112700814576619080682826075
3
1/2:
0
3
3/4:
0.7648923284043452806575452223324014640295384635668160722302025877701349994112700814576619080682826075
3
9/10:
1.637744353696210105458905056896857491490680754930043322476334393119261092944193761433217669734565467
3
19/20:
2.353363434801823877671223978899932607419607950935052747724014469494789935974375752798666394023729783
3
39/40:
3.182446305283709592723225425779868376268944042901145241114653032670843071471503188599891680661311743
3
99/100:
4.540702858568133555255198567550619039575557091570467212865611048821332670803759877094450758077224828
3
199/200:
5.840909309733357260681947467496643198627268624108446322913455116661435388024841804195637464356917407
3
999/1000:
10.21453185240738654988619840065523805041388601824921760381715631389209492497707474191820649322694416
4
1/200:
-4.604094871349993225385464412853251257128286533876668464664198766385360700301271167143363946741618264
4
1/100:
-3.746947387979196836607465640985535611552425663727797567709286611048821885135931279842490953888544367
4
1/40:
-2.776445105197794357803104846748627563933140057058378416698452428342197397640022679202984724223078038
4
1/20:
-2.131846786326650318346866666717486250429642785182236008256882546832783945712047061553273050211861829
4
1/10:
-1.533206274058943910848661059758045153325502721760828927806088344845520074400015727896985155532897210
4
1/4:
-0.7406970841126826329843998899178183373835844382377242536398960599337209463424054266204421116608267764
4
1/2:
0
4
3/4:
0.7406970841126826329843998899178183373835844382377242536398960599337209463424054266204421116608267764
4
9/10:
1.533206274058943910848661059758045153325502721760828927806088344845520074400015727896985155532897210
4
19/20:
2.131846786326650318346866666717486250429642785182236008256882546832783945712047061553273050211861829
4
39/40:
2.776445105197794357803104846748627563933140057058378416698452428342197397640022679202984724223078038
4
99/100:
3.746947387979196836607465640985535611552425663727797567709286611048821885135931279842490953888544367
4
199/200:
4.604094871349993225385464412853251257128286533876668464664198766385360700301271167143363946741618264
4
999/1000:
7.173182219782308478877025190015615636561175416937140132353057494397031836733470695220307819976441760
5
1/200:
-4.032142983555228078368419532691210020588928927633482148624791737368809708297025371393405371878710225
5
1/100:
-3.364929998907218592760012782472535667362857107095476227619208943489950181742553582547893983533902545
5
1/40:
-2.570581835636315514696246217439633513341530557396998188266595518627344140451669857408307413768496721
5
1/20:
-2.015048373333024237840722028225977667527532016972820998137316551733189013692463937728448331610219525
5
1/10:
-1.475884048824481078546458947808863261815926029645273821110692470280284692769760863085083105651294252
5
1/4:
-0.7266868438004226530150011071574763764937072106382390714466401229259195636461384985550032220074286754
5
1/2:
0
5
3/4:
0.7266868438004226530150011071574763764937072106382390714466401229259195636461384985550032220074286754
5
9/10:
1.475884048824481078546458947808863261815926029645273821110692470280284692769760863085083105651294252
5
19/20:
2.015048373333024237840722028225977667527532016972820998137316551733189013692463937728448331610219525
5
39/40:
2.570581835636315514696246217439633513341530557396998188266595518627344140451669857408307413768496721
5
99/100:
3.364929998907218592760012782472535667362857107095476227619208943489950181742553582547893983533902545
5
199/200:
4.032142983555228078368419532691210020588928927633482148624791737368809708297025371393405371878710225
5
999/1000:
5.893429531356010127585150028340387224529070378520506143280068151845084061354529581916406200039836272
6
1/200:
-3.707428021324779824207471062540950151925389966910879841632962164349040342566774994394209226996838389
6
1/100:
-3.142668403290983359120152634587806992257024203083517624395383494399520919785528573020434373647123306
6
1/40:
-2.446911851144969971071296845547715344688605486561931560926956681145466589019775849809029681046789079
6
1/20:
-1.943180280515303206605725457075614476259626300709164075610087912519213012762664706950018709318669282
6
1/10:
-1.439755747265148411790816636441516917528677614195892768084865849557024365581710339180392482420408015
6
1/4:
-0.7175581964914125662114071841071827693213654651578163351767276011760588600004005489046140535256063888
6
1/2:
0
6
3/4:
0.7175581964914125662114071841071827693213654651578163351767276011760588600004005489046140535256063888
6
9/10:
1.439755747265148411790816636441516917528677614195892768084865849557024365581710339180392482420408015
6
19/20:
1.943180280515303206605725457075614476259626300709164075610087912519213012762664706950018709318669282
6
39/40:
2.446911851144969971071296845547715344688605486561931560926956681145466589019775849809029681046789079
6
99/100:
3.142668403290983359120152634587806992257024203083517624395383494399520919785528573020434373647123306
6
199/200:
3.707428021324779824207471062540950151925389966910879841632962164349040342566774994394209226996838389
6
999/1000:
5.207626238725363713314955070330519528202461557904847304937708059913520001810383745618924849909523049
7
1/200:
-3.499483297350493920084204906470941365737311728637312623274880792947381024742369520297035709837976817
7
1/100:
-2.997951566868528433846264535863529580688019699772584215260595671123385636999979113809497168880692742
7
1/40:
-2.364624251592785341680901473780490283130290883489564918705253683970455133718102162516840688688809906
7
1/20:
-1.894578605090007389472209126340128660312015450843298253540106546320560698324795869539011253431668653
7
1/10:
-1.414923927650508477589853609318249009457284021063311609855092466957756201494132756284485420660152436
7
1/4:
-0.7111417780817863056323710558464578522202781054652115484585360781991229991576521146614353302897745515
7
1/2:
0
7
3/4:
0.7111417780817863056323710558464578522202781054652115484585360781991229991576521146614353302897745515
7
9/10:
1.414923927650508477589853609318249009457284021063311609855092466957756201494132756284485420660152436
7
19/20:
1.894578605090007389472209126340128660312015450843298253540106546320560698324795869539011253431668653
7
39/40:
2.364624251592785341680901473780490283130290883489564918705253683970455133718102162516840688688809906
7
99/100:
2.997951566868528433846264535863529580688019699772584215260595671123385636999979113809497168880692742
7
199/200:
3.499483297350493920084204906470941365737311728637312623274880792947381024742369520297035709837976817
7
999/1000:
4.785289628638334063761540675628610375386969462407546129154409524496129842928609233229009626645008763
8
1/200:
-3.355387331333395505165238377981457702150503652107758281761944122951595948396959381152649537965192838
8
1/100:
-2.896459447709622333888721930980613034600260725801541345915381533688076935179115072907415523726207795
8
1/40:
-2.306004135204166683295120954601636980266514487856528910889138000944289002780225479755111771767976640
8
1/20:
-1.859548037530898390006502177535276849411329153852838505479103197710332358654000142958560875172527751
8
1/10:
-1.396815309743864716941073899091600121482233931271515690367250085076719965517782212342999959630541486
8
1/4:
-0.7063866126448386000788233972015281449868151610978175481429593136808216856841339728711964162154804286
8
1/2:
0
8
3/4:
0.7063866126448386000788233972015281449868151610978175481429593136808216856841339728711964162154804286
8
9/10:
1.396815309743864716941073899091600121482233931271515690367250085076719965517782212342999959630541486
8
19/20:
1.859548037530898390006502177535276849411329153852838505479103197710332358654000142958560875172527751
8
39/40:
2.306004135204166683295120954601636980266514487856528910889138000944289002780225479755111771767976640
8
99/100:
2.896459447709622333888721930980613034600260725801541345915381533688076935179115072907415523726207795
8
199/200:
3.355387331333395505165238377981457702150503652107758281761944122951595948396959381152649537965192838
8
999/1000:
4.500790933723724817028557687882093853076342984548680132420688558042020976582464014308702129256156860
9
1/200:
-3.249835541592126275630476767552765520866944378853103537600252477588365090263082678637291298122590422
9
1/100:
-2.821437925025808764325696817122588783373443645156063633388564988297627905569956182198836645415084288
9
1/40:
-2.262157162798205542607769637943216168573499909959832001416316911431477501550196076901297775713001975
9
1/20:
-1.833112932656237168685466510235538248239709564729162288751531418948375521441255110381083158866118012
9
1/10:
-1.383028738396632308695435582143561725196670235219932574904517179497506175825234313414797396410139726
9
1/4:
-0.7027221467513263671826864186165328612446732561180449339735890660165183255091393732389580354491809149
9
1/2:
0
9
3/4:
0.7027221467513263671826864186165328612446732561180449339735890660165183255091393732389580354491809149
9
9/10:
1.383028738396632308695435582143561725196670235219932574904517179497506175825234313414797396410139726
9
19/20:
1.833112932656237168685466510235538248239709564729162288751531418948375521441255110381083158866118012
9
39/40:
2.262157162798205542607769637943216168573499909959832001416316911431477501550196076901297775713001975
9
99/100:
2.821437925025808764325696817122588783373443645156063633388564988297627905569956182198836645415084288
9
199/200:
3.249835541592126275630476767552765520866944378853103537600252477588365090263082678637291298122590422
9
999/1000:
4.296805662729918486055640751536445068692912060591886713842792844017389719933401081580144919243985184
10
1/200:
-3.169272672616951234596992132153615016398630331005879078365653586811818436119563866780642744832922510
10
1/100:
-2.763769458112696198752588877048461669226540611882049688879292946540616766178846642024519569588022570
10
1/40:
-2.228138851986274748395490663201806717856019366416733154719481647360478452035285544279468244028619306
10
1/20:
-1.812461122811676413625523662470264350510870592658365993435574758323287135906070475073358825050476632
10
1/10:
-1.372183641110335627219156967662553919805071883437384543970686004785697538966788569045533098295200211
10
1/4:
-0.6998120613124316273440687660527011240985229455511156138478264602296925974162071758883417199459287281
10
1/2:
0
10
3/4:
0.6998120613124316273440687660527011240985229455511156138478264602296925974162071758883417199459287281
10
9/10:
1.372183641110335627219156967662553919805071883437384543970686004785697538966788569045533098295200211
10
19/20:
1.812461122811676413625523662470264350510870592658365993435574758323287135906070475073358825050476632
10
39/40:
2.228138851986274748395490663201806717856019366416733154719481647360478452035285544279468244028619306
10
99/100:
2.763769458112696198752588877048461669226540611882049688879292946540616766178846642024519569588022570
10
199/200:
3.169272672616951234596992132153615016398630331005879078365653586811818436119563866780642744832922510
10
999/1000:
4.143700494046589666208730019829750495663668993935952115285916254394390743755472224411482327628328506
11
1/200:
-3.105806515539281007098277692803869028190786280133583282989607054588586923085275076668389802025715692
11
1/100:
-2.718079183813862397201515706567058676833972870571918845473942617412685311328685254634333851144147136
11
1/40:
-2.200985160091639867877200361696501825384612427965665511335808235871627373001680580296541267635947955
11
1/20:
-1.795884818704044100921202802018652625121204977475206770553392052403910524210729592571503014665029738
11
1/10:
-1.363430318020540645015295983542677146925180130684110328673062242719298323789713063412225123925475213
11
1/4:
-0.6974453275598815311704866688891382142675489365169496975246799949501466374604417278124204279307698157
11
1/2:
0
11
3/4:
0.6974453275598815311704866688891382142675489365169496975246799949501466374604417278124204279307698157
11
9/10:
1.363430318020540645015295983542677146925180130684110328673062242719298323789713063412225123925475213
11
19/20:
1.795884818704044100921202802018652625121204977475206770553392052403910524210729592571503014665029738
11
39/40:
2.200985160091639867877200361696501825384612427965665511335808235871627373001680580296541267635947955
11
99/100:
2.718079183813862397201515706567058676833972870571918845473942617412685311328685254634333851144147136
11
199/200:
3.105806515539281007098277692803869028190786280133583282989607054588586923085275076668389802025715692
11
999/1000:
4.024701037630738258426955456501944051986042674126238281889249217192363994318475500959450465094113802
12
1/200:
-3.054539589392902130857092986300098542868937170171308032759758658907456434089629485746206066807853097
12
1/100:
-2.680997993120914894865691158557138993896819055937748023644288501868235680345655267710021131285864205
12
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100
1/20:
-1.660234326085339555542109214831187529726097384594476142393471398591090636370374891971810161558078622
100
1/10:
-1.290074761346515969155319959877461623987264647832279137944356384929576705142376322264969525511111654
100
1/4:
-0.6769510430114714779440129744637281019722825607287176291172261444640918811692807881759820311737138410
100
1/2:
0
100
3/4:
0.6769510430114714779440129744637281019722825607287176291172261444640918811692807881759820311737138410
100
9/10:
1.290074761346515969155319959877461623987264647832279137944356384929576705142376322264969525511111654
100
19/20:
1.660234326085339555542109214831187529726097384594476142393471398591090636370374891971810161558078622
100
39/40:
1.983971518523552286595184867990339164772199712781309652030301958770848346874352835964049197651244815
100
99/100:
2.364217366238482136293036437124740549015886818811362986950505494948621560856007376310299229863538884
100
199/200:
2.625890521438017930612365919569391426485789626945088601247105390050591170910173964984652366834213264
100
999/1000:
3.173739493738782938943249609004581530413998679439336875180600708322964682009661600325946198059917273
Definition
For the central Student $t$-distribution with $\nu$ degrees of freedom, location zero, scale one and cumulative distribution function $F_\nu$ [1], this table gives the lower-tail quantile $t_{\nu,p}=\inf\{x:F_\nu(x)\ge p\}$.
Parameters
$\nu$
—   degrees of freedom ($\nu$ is a positive integer)
$p$
—   lower-tail probability ($0<p<1$)
Formulas
(1)
$F_\nu(x)=\int_{-\infty}^x f_\nu(u)\,du$, where $f_\nu(u)=\frac{\Gamma((\nu+1)/2)}{\sqrt{\nu\pi}\,\Gamma(\nu/2)}(1+u^2/\nu)^{-(\nu+1)/2}$ [1].
(2)
For $x\geq0$, $F_\nu(x)=1-\tfrac12 I_{\nu/(\nu+x^2)}(\nu/2,1/2)$; for $x<0$, $F_\nu(x)=1-F_\nu(-x)$ [1]. Here $I_z(a,b)=B(z;a,b)/B(1;a,b)$ is the regularized form of the incomplete beta function.
(3)
$t_{\nu,1-p}=-t_{\nu,p}$, and $t_{\nu,1/2}=0$.
(4)
For a two-sided significance level $0<\alpha<1$, the cutoffs are $t_{\nu,\alpha/2}$ and $t_{\nu,1-\alpha/2}=-t_{\nu,\alpha/2}$, so $\Pr(-t_{\nu,1-\alpha/2}\leq T_\nu\leq t_{\nu,1-\alpha/2})=1-\alpha$ [3].
(5)
$t_{1,p}=-\cot(\pi p)$.
(6)
$t_{2,p}=(2p-1)/\sqrt{2p(1-p)}$.
Comments
(7)
The table uses the lower-tail convention $Q_\nu(p)=t_{\nu,p}$. If $Q_\nu^{\mathrm{upper}}(\alpha)$ is defined by $\Pr(T_\nu>Q_\nu^{\mathrm{upper}}(\alpha))=\alpha$, then $Q_\nu(p)=Q_\nu^{\mathrm{upper}}(1-p)$. In a two-sided table the column labelled $\alpha=0.05$ gives the positive cutoff at $p=39/40=0.975$ [3].
(8)
The probability labels are exact rationals in lowest terms. Thus $p=19/20$ is the familiar probability $0.95$, and $p=199/200$ is $0.995$.
(9)
The parameter $\nu$ counts degrees of freedom. The usual one-sample statistic based on $n$ independent normal observations has $\nu=n-1$ [1]. Although the Student distribution is defined for every positive real $\nu$, this table indexes integer degrees of freedom.
(10)
All median rows $t_{\nu,1/2}=0$ are exact by symmetry. For $\nu=1$ the distribution is the standard Cauchy distribution, and its quartiles $t_{1,1/4}=-1$ and $t_{1,3/4}=1$ are also stored exactly [2].
Programs
(P1)
R
qt(39/40, df = 10, lower.tail = TRUE)
(P2)
Python
import mpmath as mp
mp.mp.dps = 110
nu = 10
p = mp.mpf(39)/40
# Positive quantile, since p > 1/2.
cdf = lambda x: 1 - mp.betainc(mp.mpf(nu)/2, mp.mpf(1)/2,
                             0, nu/(nu+x*x), regularized=True)/2
mp.findroot(lambda x: cdf(x)-p, (2, 3))
Links
Similar tables
Quantiles of the standard normal distribution —   the Student distribution tends to the standard normal distribution as $\nu$ tends to infinity [2]
Quantiles of the chi-squared distribution —   if $Z$ is standard normal and $V$ is an independent chi-squared variable with $\nu$ degrees of freedom, then $Z/\sqrt{V/\nu}$ has the Student distribution [1]
Values of the incomplete beta function $B(x;a,b)$ —   the regularized incomplete beta function gives the Student cumulative distribution function in (2)
Values of the upper incomplete gamma function $\Gamma(a,x)$ —   its regularized values give upper-tail probabilities of the chi-squared variable in the normal-to-chi-squared ratio defining the Student distribution
Values of the lower incomplete gamma function $\gamma(a,x)$ —   its regularized values give lower-tail probabilities of the chi-squared variable in the normal-to-chi-squared ratio defining the Student distribution
Values of the error function $\operatorname{erf}(x)$ —   the error function gives the cumulative distribution function of the limiting standard normal distribution
Differential entropies of continuous probability distributions —   contains differential entropies of the Student and Cauchy distributions
Kullback-Leibler divergences between probability distributions —   contains divergences between normal distributions, the limiting family as the Student degrees of freedom increase
Data properties
Entries are of type: real number
digits: 100 (Unknown key)
How well the digits are known: proven
Table is complete: no (it holds every pair with $\nu\in\{1,2,\ldots,30,40,50,60,100\}$ and $p\in\{1/200,1/100,1/40,1/20,1/10,1/4,1/2,3/4,9/10,19/20,39/40,99/100,199/200,999/1000\}$. The six cumulative probabilities $9/10,19/20,39/40,99/100,199/200,999/1000$ are those tabulated in [3]. The stored selection also includes five complementary probabilities giving negative quantiles, the quartiles and the median. The stored entries include $p=999/1000$ but omit its complement $p=1/1000$. The degrees of freedom cover small samples and selected larger reference rows, matching the chi-squared quantile table.)
How they were obtained:

Rational bisection encloses each positive quantile. Every endpoint comparison is certified by Arb ball evaluations of the regularized incomplete beta function through the hypergeometric representation and complement identity in [4]. The returned ball contains the entire final bracket. Computation uses 493 bits for 100 requested digits, with 160 guard bits; the smallest measured relative accuracy of a nonexact returned ball exceeds 373 bits. Negative values follow by symmetry, and the 36 exact rows are written as integers.

more

An independent check substitutes $x=\sqrt{\nu}\tan\theta$ in the density integral and evaluates $\int_0^\theta\cos^{\nu-1}u\,du$ by integration by parts. This uses neither beta, gamma nor hypergeometric evaluation. Known Cauchy and two-degree-of-freedom values control the method. For every nonexact written value, the independent CDF is strictly less than $p$ at the lower endpoint of its last-place interval and strictly greater at the upper endpoint. Checks also cover the exact rows, symmetry, the two-sided conversion, the closed forms at $\nu=1,2$, ordering in $p$ and $\nu$, and all 204 entries shared with [3] to its printed three decimal places. The generator and independent checks are attached.

parameter selection: The probability selection uses the six cumulative probabilities in [3], five complementary probabilities giving negative quantiles, the quartiles and the median. The stored entries include $p=999/1000$ but omit its complement $p=1/1000$. (Unknown key)


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