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{"Title": "Quantiles of Student's $t$-distribution", "Definition": "For the central Student $t$-distribution with $\\nu$ degrees of freedom, location zero, scale one and cumulative distribution function $F_\\nu$ CITE{RStudent}, this table gives the lower-tail quantile $t_{\\nu,p}=\\inf\\{x:F_\\nu(x)\\ge p\\}$.", "Parameters": {"nu": {"type": "Z", "title": "degrees of freedom", "display": "$\\nu$", "constraints": "$\\nu$ is a positive integer"}, "p": {"type": "Q", "title": "lower-tail probability", "display": "$p$", "constraints": "$0<p<1$"}}, "Tags": ["probability theory"], "Links": {"RStudent": {"title": "R documentation: The Student t Distribution", "url": "https://stat.ethz.ch/R-manual/R-devel/library/stats/html/TDist.html"}, "NISTDistribution": {"title": "NIST/SEMATECH e-Handbook: t distribution", "url": "https://www.itl.nist.gov/div898/handbook/eda/section3/eda3664.htm"}, "NISTCritical": {"title": "NIST/SEMATECH e-Handbook: Critical values of the Student's t distribution", "url": "https://www.itl.nist.gov/div898/handbook/eda/section3/eda3672.htm"}, "DLMFBeta": {"title": "DLMF 8.17: Incomplete beta functions", "url": "https://dlmf.nist.gov/8.17"}}, "Data properties": {"type": "R", "digits": "100", "rigour": "proven", "complete": "no", "complete-note": "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 probabilities are conventional significance levels, negative counterparts, quartiles and the median; the degrees of freedom cover small samples and selected larger reference rows, matching HREF{Quantiles_of_the_chi-squared_distribution}[the chi-squared quantile table].", "rigour details": "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 CITE{DLMFBeta}. 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.\n\nAn 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 CITE{NISTCritical} to its printed three decimal places. The generator and independent checks are attached.", "parameter selection": "The probabilities $9/10,19/20,39/40,99/100,199/200,999/1000$ are the six cumulative probabilities printed in CITE{NISTCritical}. The family adds the negative counterparts $1/10,1/20,1/40,1/100,1/200$, the two quartiles and the median. The degree rows $1,\\ldots,30$ cover small samples, and $40,50,60,100$ provide larger reference cases on the same selection as HREF{Quantiles_of_the_chi-squared_distribution}[the chi-squared quantile table]."}, "Display properties": {"number-header": "$t_{\\nu,p}$"}, "Numbers": {"1": {"1/200": "-63.65674116287158099500242331072784510349629211231565289288838206735678663325145438110134597227962434", "1/100": "-31.82051595377395803933954943193656540467519466363209450032332029855776574860764573642128833412492341", "1/40": "-12.70620473617470464602167997884208746766873833004756927928550210985681826776165079308153752964061677", "1/20": "-6.313751514675043098979464244768186059447320503149318151310053495785384214132895817869008130045887710", "1/10": "-3.077683537175253402570290576036909824006702143537792427039156250374863288495090918454593721667105435", "1/4": {"number": "-1", "comment": "The lower quartile of the standard Cauchy distribution is exactly $-1$."}, "1/2": "0", "3/4": {"number": "1", "comment": "The upper quartile of the standard Cauchy distribution is exactly $1$."}, "9/10": "3.077683537175253402570290576036909824006702143537792427039156250374863288495090918454593721667105435", "19/20": "6.313751514675043098979464244768186059447320503149318151310053495785384214132895817869008130045887710", "39/40": "12.70620473617470464602167997884208746766873833004756927928550210985681826776165079308153752964061677", "99/100": "31.82051595377395803933954943193656540467519466363209450032332029855776574860764573642128833412492341", "199/200": "63.65674116287158099500242331072784510349629211231565289288838206735678663325145438110134597227962434", "999/1000": "318.3088389855504459216866954369214201827749367273530577933415421411151519312142567646174547252258650"}, "2": {"1/200": "-9.924843200918293114742383624018663473435995346454545731573002915155036699933932067451479099611202165", "1/100": "-6.964556734283274187082299259505419628752217858096215773535739171510127484997234737958292854758048898", "1/40": "-4.302652729749463852320943892621175008187644875325562530717791191758606636576241383770212185868734842", "1/20": "-2.919985580353725686960617443846747264585295590052693205466902560638189308019630416668450957515268395", "1/10": "-1.885618083164126731735584965612930771426229167169264097568906317320976637949476051800516712436855430", "1/4": "-0.8164965809277260327324280249019637973219824935522233761442308557503201258191050088466198110348800783", "1/2": "0", "3/4": "0.8164965809277260327324280249019637973219824935522233761442308557503201258191050088466198110348800783", "9/10": "1.885618083164126731735584965612930771426229167169264097568906317320976637949476051800516712436855430", "19/20": "2.919985580353725686960617443846747264585295590052693205466902560638189308019630416668450957515268395", "39/40": "4.302652729749463852320943892621175008187644875325562530717791191758606636576241383770212185868734842", "99/100": "6.964556734283274187082299259505419628752217858096215773535739171510127484997234737958292854758048898", "199/200": "9.924843200918293114742383624018663473435995346454545731573002915155036699933932067451479099611202165", "999/1000": "22.32712477011987543527500362467222708401509402230763489997190421307068649332258430290663070302304304"}, "3": {"1/200": "-5.840909309733357260681947467496643198627268624108446322913455116661435388024841804195637464356917407", "1/100": "-4.540702858568133555255198567550619039575557091570467212865611048821332670803759877094450758077224828", "1/40": "-3.182446305283709592723225425779868376268944042901145241114653032670843071471503188599891680661311743", "1/20": "-2.353363434801823877671223978899932607419607950935052747724014469494789935974375752798666394023729783", "1/10": "-1.637744353696210105458905056896857491490680754930043322476334393119261092944193761433217669734565467", "1/4": "-0.7648923284043452806575452223324014640295384635668160722302025877701349994112700814576619080682826075", "1/2": "0", "3/4": "0.7648923284043452806575452223324014640295384635668160722302025877701349994112700814576619080682826075", "9/10": "1.637744353696210105458905056896857491490680754930043322476334393119261092944193761433217669734565467", "19/20": "2.353363434801823877671223978899932607419607950935052747724014469494789935974375752798666394023729783", "39/40": "3.182446305283709592723225425779868376268944042901145241114653032670843071471503188599891680661311743", "99/100": "4.540702858568133555255198567550619039575557091570467212865611048821332670803759877094450758077224828", "199/200": "5.840909309733357260681947467496643198627268624108446322913455116661435388024841804195637464356917407", "999/1000": "10.21453185240738654988619840065523805041388601824921760381715631389209492497707474191820649322694416"}, "4": {"1/200": "-4.604094871349993225385464412853251257128286533876668464664198766385360700301271167143363946741618264", "1/100": "-3.746947387979196836607465640985535611552425663727797567709286611048821885135931279842490953888544367", "1/40": "-2.776445105197794357803104846748627563933140057058378416698452428342197397640022679202984724223078038", "1/20": "-2.131846786326650318346866666717486250429642785182236008256882546832783945712047061553273050211861829", "1/10": "-1.533206274058943910848661059758045153325502721760828927806088344845520074400015727896985155532897210", "1/4": "-0.7406970841126826329843998899178183373835844382377242536398960599337209463424054266204421116608267764", "1/2": "0", "3/4": "0.7406970841126826329843998899178183373835844382377242536398960599337209463424054266204421116608267764", "9/10": "1.533206274058943910848661059758045153325502721760828927806088344845520074400015727896985155532897210", "19/20": "2.131846786326650318346866666717486250429642785182236008256882546832783945712047061553273050211861829", "39/40": "2.776445105197794357803104846748627563933140057058378416698452428342197397640022679202984724223078038", "99/100": "3.746947387979196836607465640985535611552425663727797567709286611048821885135931279842490953888544367", "199/200": "4.604094871349993225385464412853251257128286533876668464664198766385360700301271167143363946741618264", "999/1000": "7.173182219782308478877025190015615636561175416937140132353057494397031836733470695220307819976441760"}, "5": {"1/200": "-4.032142983555228078368419532691210020588928927633482148624791737368809708297025371393405371878710225", "1/100": "-3.364929998907218592760012782472535667362857107095476227619208943489950181742553582547893983533902545", "1/40": "-2.570581835636315514696246217439633513341530557396998188266595518627344140451669857408307413768496721", "1/20": "-2.015048373333024237840722028225977667527532016972820998137316551733189013692463937728448331610219525", "1/10": "-1.475884048824481078546458947808863261815926029645273821110692470280284692769760863085083105651294252", "1/4": "-0.7266868438004226530150011071574763764937072106382390714466401229259195636461384985550032220074286754", "1/2": "0", "3/4": "0.7266868438004226530150011071574763764937072106382390714466401229259195636461384985550032220074286754", "9/10": "1.475884048824481078546458947808863261815926029645273821110692470280284692769760863085083105651294252", "19/20": "2.015048373333024237840722028225977667527532016972820998137316551733189013692463937728448331610219525", "39/40": "2.570581835636315514696246217439633513341530557396998188266595518627344140451669857408307413768496721", "99/100": "3.364929998907218592760012782472535667362857107095476227619208943489950181742553582547893983533902545", "199/200": "4.032142983555228078368419532691210020588928927633482148624791737368809708297025371393405371878710225", "999/1000": "5.893429531356010127585150028340387224529070378520506143280068151845084061354529581916406200039836272"}, "6": {"1/200": "-3.707428021324779824207471062540950151925389966910879841632962164349040342566774994394209226996838389", "1/100": "-3.142668403290983359120152634587806992257024203083517624395383494399520919785528573020434373647123306", "1/40": "-2.446911851144969971071296845547715344688605486561931560926956681145466589019775849809029681046789079", "1/20": "-1.943180280515303206605725457075614476259626300709164075610087912519213012762664706950018709318669282", "1/10": "-1.439755747265148411790816636441516917528677614195892768084865849557024365581710339180392482420408015", "1/4": "-0.7175581964914125662114071841071827693213654651578163351767276011760588600004005489046140535256063888", "1/2": "0", "3/4": "0.7175581964914125662114071841071827693213654651578163351767276011760588600004005489046140535256063888", "9/10": "1.439755747265148411790816636441516917528677614195892768084865849557024365581710339180392482420408015", "19/20": "1.943180280515303206605725457075614476259626300709164075610087912519213012762664706950018709318669282", "39/40": "2.446911851144969971071296845547715344688605486561931560926956681145466589019775849809029681046789079", "99/100": "3.142668403290983359120152634587806992257024203083517624395383494399520919785528573020434373647123306", "199/200": "3.707428021324779824207471062540950151925389966910879841632962164349040342566774994394209226996838389", "999/1000": "5.207626238725363713314955070330519528202461557904847304937708059913520001810383745618924849909523049"}, "7": {"1/200": "-3.499483297350493920084204906470941365737311728637312623274880792947381024742369520297035709837976817", "1/100": "-2.997951566868528433846264535863529580688019699772584215260595671123385636999979113809497168880692742", "1/40": "-2.364624251592785341680901473780490283130290883489564918705253683970455133718102162516840688688809906", "1/20": "-1.894578605090007389472209126340128660312015450843298253540106546320560698324795869539011253431668653", "1/10": "-1.414923927650508477589853609318249009457284021063311609855092466957756201494132756284485420660152436", "1/4": "-0.7111417780817863056323710558464578522202781054652115484585360781991229991576521146614353302897745515", "1/2": "0", "3/4": "0.7111417780817863056323710558464578522202781054652115484585360781991229991576521146614353302897745515", "9/10": "1.414923927650508477589853609318249009457284021063311609855092466957756201494132756284485420660152436", "19/20": "1.894578605090007389472209126340128660312015450843298253540106546320560698324795869539011253431668653", "39/40": "2.364624251592785341680901473780490283130290883489564918705253683970455133718102162516840688688809906", "99/100": "2.997951566868528433846264535863529580688019699772584215260595671123385636999979113809497168880692742", "199/200": "3.499483297350493920084204906470941365737311728637312623274880792947381024742369520297035709837976817", "999/1000": "4.785289628638334063761540675628610375386969462407546129154409524496129842928609233229009626645008763"}, "8": {"1/200": "-3.355387331333395505165238377981457702150503652107758281761944122951595948396959381152649537965192838", "1/100": "-2.896459447709622333888721930980613034600260725801541345915381533688076935179115072907415523726207795", "1/40": "-2.306004135204166683295120954601636980266514487856528910889138000944289002780225479755111771767976640", "1/20": "-1.859548037530898390006502177535276849411329153852838505479103197710332358654000142958560875172527751", "1/10": "-1.396815309743864716941073899091600121482233931271515690367250085076719965517782212342999959630541486", "1/4": "-0.7063866126448386000788233972015281449868151610978175481429593136808216856841339728711964162154804286", "1/2": "0", "3/4": "0.7063866126448386000788233972015281449868151610978175481429593136808216856841339728711964162154804286", "9/10": "1.396815309743864716941073899091600121482233931271515690367250085076719965517782212342999959630541486", "19/20": "1.859548037530898390006502177535276849411329153852838505479103197710332358654000142958560875172527751", "39/40": "2.306004135204166683295120954601636980266514487856528910889138000944289002780225479755111771767976640", "99/100": "2.896459447709622333888721930980613034600260725801541345915381533688076935179115072907415523726207795", "199/200": "3.355387331333395505165238377981457702150503652107758281761944122951595948396959381152649537965192838", "999/1000": "4.500790933723724817028557687882093853076342984548680132420688558042020976582464014308702129256156860"}, "9": {"1/200": "-3.249835541592126275630476767552765520866944378853103537600252477588365090263082678637291298122590422", "1/100": "-2.821437925025808764325696817122588783373443645156063633388564988297627905569956182198836645415084288", "1/40": "-2.262157162798205542607769637943216168573499909959832001416316911431477501550196076901297775713001975", "1/20": "-1.833112932656237168685466510235538248239709564729162288751531418948375521441255110381083158866118012", "1/10": "-1.383028738396632308695435582143561725196670235219932574904517179497506175825234313414797396410139726", "1/4": "-0.7027221467513263671826864186165328612446732561180449339735890660165183255091393732389580354491809149", "1/2": "0", "3/4": "0.7027221467513263671826864186165328612446732561180449339735890660165183255091393732389580354491809149", "9/10": "1.383028738396632308695435582143561725196670235219932574904517179497506175825234313414797396410139726", "19/20": "1.833112932656237168685466510235538248239709564729162288751531418948375521441255110381083158866118012", "39/40": "2.262157162798205542607769637943216168573499909959832001416316911431477501550196076901297775713001975", "99/100": "2.821437925025808764325696817122588783373443645156063633388564988297627905569956182198836645415084288", "199/200": "3.249835541592126275630476767552765520866944378853103537600252477588365090263082678637291298122590422", "999/1000": "4.296805662729918486055640751536445068692912060591886713842792844017389719933401081580144919243985184"}, "10": {"1/200": "-3.169272672616951234596992132153615016398630331005879078365653586811818436119563866780642744832922510", "1/100": 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"-1.675905025163097567489344928674723434619153206351903304395212865084549512332970024284985068107400175", "1/10": "-1.298713694194809650223472973498118136790391698528361900690311174135215548768200414957941332981551038", "1/4": "-0.6794282003263460107872713998219854099541451074635437354167089862098310418950952020096864921391579364", "1/2": "0", "3/4": "0.6794282003263460107872713998219854099541451074635437354167089862098310418950952020096864921391579364", "9/10": "1.298713694194809650223472973498118136790391698528361900690311174135215548768200414957941332981551038", "19/20": "1.675905025163097567489344928674723434619153206351903304395212865084549512332970024284985068107400175", "39/40": "2.008559112100761105490532128834518875356242952447527713258444599079173158717495514177566075629339236", "99/100": "2.403271916674171933900724177794967783773235795255242210432502213301557459546274244391192860813947526", "199/200": "2.677793270940844249217028721921012092093229391823036149613183547413391627851973594960809660790030368", "999/1000": "3.261409055798318207210124493668632243629048243418028184432741903930958427417026655422515893741576900"}, "60": {"1/200": "-2.660283028855037187132603529234678899365448513350189162077416680429365302516072900671428227601426976", "1/100": "-2.390119472624913319917346529301895477213911158103395017558506896205330415164270169481962822337790257", "1/40": "-2.000297822014260504503472668184365969341692475472505558641210825341930777978904549702372683058083171", "1/20": "-1.670648864904636514806489395162135099748475925584245446875386181148271801838902738474431564601949530", "1/10": "-1.295821093515730785204560576944862227096825768859624828622682923581799811255937953452291390502753027", "1/4": "-0.6786007206481357816056113745073853912145928839932296484231569678115596463820313993946187693159072964", "1/2": "0", "3/4": "0.6786007206481357816056113745073853912145928839932296484231569678115596463820313993946187693159072964", "9/10": "1.295821093515730785204560576944862227096825768859624828622682923581799811255937953452291390502753027", "19/20": "1.670648864904636514806489395162135099748475925584245446875386181148271801838902738474431564601949530", "39/40": "2.000297822014260504503472668184365969341692475472505558641210825341930777978904549702372683058083171", "99/100": "2.390119472624913319917346529301895477213911158103395017558506896205330415164270169481962822337790257", "199/200": "2.660283028855037187132603529234678899365448513350189162077416680429365302516072900671428227601426976", "999/1000": "3.231709126024359751496586552966052199703765388829884267467686837644022420228609889241672578794235219"}, "100": {"1/200": "-2.625890521438017930612365919569391426485789626945088601247105390050591170910173964984652366834213264", "1/100": "-2.364217366238482136293036437124740549015886818811362986950505494948621560856007376310299229863538884", "1/40": "-1.983971518523552286595184867990339164772199712781309652030301958770848346874352835964049197651244815", "1/20": "-1.660234326085339555542109214831187529726097384594476142393471398591090636370374891971810161558078622", "1/10": "-1.290074761346515969155319959877461623987264647832279137944356384929576705142376322264969525511111654", "1/4": "-0.6769510430114714779440129744637281019722825607287176291172261444640918811692807881759820311737138410", "1/2": "0", "3/4": "0.6769510430114714779440129744637281019722825607287176291172261444640918811692807881759820311737138410", "9/10": "1.290074761346515969155319959877461623987264647832279137944356384929576705142376322264969525511111654", "19/20": "1.660234326085339555542109214831187529726097384594476142393471398591090636370374891971810161558078622", "39/40": "1.983971518523552286595184867990339164772199712781309652030301958770848346874352835964049197651244815", "99/100": "2.364217366238482136293036437124740549015886818811362986950505494948621560856007376310299229863538884", "199/200": "2.625890521438017930612365919569391426485789626945088601247105390050591170910173964984652366834213264", "999/1000": "3.173739493738782938943249609004581530413998679439336875180600708322964682009661600325946198059917273"}}, "Keywords": ["critical value", "percentage point", "percentile", "significance level", "p-value", "table", "Student t", "t-test", "Student distribution"], "Comments": {"comment-convention": "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 $p=39/40=0.975$ CITE{NISTCritical}.", "comment-probabilities": "The probability labels are exact rationals in lowest terms. Thus $p=19/20$ is the familiar probability $0.95$, and $p=199/200$ is $0.995$. The six positive tail levels in CITE{NISTCritical} are supplemented by the five negative counterparts present in the family, the quartiles and the median. The family includes $999/1000$ without its complement $1/1000$.", "comment-integer": "The parameter $\\nu$ counts degrees of freedom. The usual one-sample statistic based on $n$ independent normal observations has $\\nu=n-1$ CITE{RStudent}. Although the Student distribution is defined for every positive real $\\nu$, this table indexes integer degrees of freedom.", "comment-exact": "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 CITE{NISTDistribution}."}, "Formulas": {"formula-density": "$F_\\nu(x)=\\int_{-\\infty}^x f_\\nu(u)\\,du$, where $f_\\nu(u)=\\frac{\\Gamma((\\nu+1)/2)}{\\sqrt{\\nu\\pi}\\,\\Gamma(\\nu/2)}(1+u^2/\\nu)^{-(\\nu+1)/2}$ CITE{RStudent}.", "formula-beta": "For $x\\geq0$, $F_\\nu(x)=1-\\tfrac12 I_{\\nu/(\\nu+x^2)}(\\nu/2,1/2)$; for $x<0$, $F_\\nu(x)=1-F_\\nu(-x)$ CITE{RStudent}. Here $I_z(a,b)=B(z;a,b)/B(1;a,b)$ is the regularized form of HREF{Values_of_the_incomplete_beta_function}[the incomplete beta function].", "formula-symmetry": "$t_{\\nu,1-p}=-t_{\\nu,p}$, and $t_{\\nu,1/2}=0$.", "formula-two-sided": "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 $\\Pr(-t_{\\nu,1-\\alpha/2}\\leq T_\\nu\\leq t_{\\nu,1-\\alpha/2})=1-\\alpha$ CITE{NISTCritical}.", "formula-cauchy": "$t_{1,p}=-\\cot(\\pi p)$.", "formula-nu2": "$t_{2,p}=(2p-1)/\\sqrt{2p(1-p)}$."}, "Programs": {"program-r": {"language": "R", "code": "qt(39/40, df = 10, lower.tail = TRUE)"}, "program-python": {"language": "Python", "code": "import mpmath as mp\nmp.mp.dps = 110\nnu = 10\np = mp.mpf(39)/40\n# Positive quantile, since p > 1/2.\ncdf = lambda x: 1 - mp.betainc(mp.mpf(nu)/2, mp.mpf(1)/2,\n 0, nu/(nu+x*x), regularized=True)/2\nmp.findroot(lambda x: cdf(x)-p, (2, 3))"}}, "Similar tables": [{"table": "HREF{Quantiles_of_the_standard_normal_distribution}[Quantiles of the standard normal distribution]", "relation": "the Student distribution tends to the standard normal distribution as $\\nu$ tends to infinity CITE{NISTDistribution}"}, {"table": "HREF{Quantiles_of_the_chi-squared_distribution}[Quantiles of the chi-squared distribution]", "relation": "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 CITE{RStudent}"}, {"table": "HREF{Values_of_the_incomplete_beta_function}[Values of the incomplete beta function $B(x;a,b)$]", "relation": "the regularized incomplete beta function gives the Student cumulative distribution function in CITE{formula-beta}"}, {"table": "HREF{Values_of_the_upper_incomplete_gamma_function}[Values of the upper incomplete gamma function $\\Gamma(a,x)$]", "relation": "its regularized values give upper-tail probabilities of the chi-squared variable in the normal-to-chi-squared ratio defining the Student distribution"}, {"table": "HREF{Values_of_the_lower_incomplete_gamma_function}[Values of the lower incomplete gamma function $\\gamma(a,x)$]", "relation": "its regularized values give lower-tail probabilities of the chi-squared variable in the normal-to-chi-squared ratio defining the Student distribution"}, {"table": "HREF{Values_of_the_error_function}[Values of the error function $\\operatorname{erf}(x)$]", "relation": "the error function gives the cumulative distribution function of the limiting standard normal distribution"}, {"table": "HREF{Differential_entropies_of_continuous_probability_distributions}[Differential entropies of continuous probability distributions]", "relation": "contains differential entropies of the Student and Cauchy distributions"}, {"table": "HREF{Kullback-Leibler_divergences_between_probability_distributions}[Kullback-Leibler divergences between probability distributions]", "relation": "contains divergences between normal distributions, the limiting family as the Student degrees of freedom increase"}]}