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{"Title": "Quantiles of the studentized range distribution", "Definition": "For independent standard normal variables $Z_1,\\ldots,Z_k$ and an independent $V\\sim\\chi^2_\\nu$, this table gives the lower-tail quantile $q_{k,\\nu,p}=\\inf\\{x:\\Pr(Q_{k,\\nu}\\leq x)\\geq p\\}$ of the studentized range $Q_{k,\\nu}=(\\max_i Z_i-\\min_i Z_i)/\\sqrt{V/\\nu}$ CITE{R}.", "Parameters": {"nu": {"type": "Z", "title": "degrees of freedom", "display": "$\\nu$", "constraints": "$\\nu$ is a positive integer"}, "k": {"type": "Z", "title": "number of means", "display": "$k$", "constraints": "$k$ is an integer with $k\\geq2$"}, "p": {"type": "Q", "title": "lower-tail probability", "display": "$p$", "constraints": "$0<p<1$"}}, "Tags": ["probability theory"], "Links": {"R": {"title": "R: The Studentized Range Distribution", "url": "https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Tukey.html"}, "SciPy": {"title": "SciPy: Studentized Range Distribution", "url": "https://docs.scipy.org/doc/scipy-1.14.1/tutorial/stats/continuous_studentized_range.html"}, "CriticalValues": {"title": "University of Washington: critical values for the studentized range", "url": "https://faculty.washington.edu/heagerty/Books/Biostatistics/TABLES/StudentizedRange/index.html"}, "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 combination of $k\\in\\{2,3\\}$, $\\nu\\in\\{1,2,\\ldots,10,20,30,60,120\\}$ and $p\\in\\{1/200,1/100,1/40,1/20,1/10,1/4,1/2,3/4,9/10,19/20,39/40,99/100,199/200,999/1000\\}$, giving 392 entries. The degrees of freedom select small samples and larger reference rows from CITE{CriticalValues}; the probabilities cover the median, quartiles and conventional significance thresholds. The two- and three-mean cases provide the ordinary absolute Student comparison and the first nontrivial simultaneous comparison, with a rigorous method capable of 100 significant digits", "parameter selection": "The finite degrees of freedom are a subset of the reference rows in CITE{CriticalValues}. The probabilities use the family convention for statistical quantiles. Larger numbers of means and the infinite-degree-of-freedom limit are outside the computed selection.", "rigour details": "Each quantile is enclosed by exact rational endpoints. Numerical quadrature at 135 decimal digits proposes a bracket, and 525-bit Arb arithmetic certifies strict CDF signs at both endpoints. For two means the CDF uses the regularized incomplete beta function. For three means a convergent power series for CITE{formula-three-means} carries an explicit Cauchy remainder bound; the derivation and bound are in the attached mathematical notes. The returned ball contains the entire bracket of width $4\\cdot10^{-115}$. Every returned ball retained at least 374 bits of relative accuracy before rounding to 100 significant digits.\n\nThe independent checker uses finite elementary integration recurrences in 2048-bit ball arithmetic. It checks opposite CDF signs at the two endpoints represented by every stored decimal, rejects 392 deliberately shifted intervals, checks 42 closed-form rows, and checks strict ordering in probability and in the number of means. Four controls precede these comparisons. SciPy evaluates the general distribution on all 392 entries, with maximum CDF discrepancy $1.82\\cdot10^{-13}$. Of 56 shared printed values in CITE{CriticalValues}, 55 agree within half a last-place unit; the remaining printed value is discussed in its entry comment. All 56 agree within one printed unit. The generator, independent checker, source comparisons and numerical evidence are attached."}, "Display properties": {"number-header": "$q_{k,\\nu,p}$"}, "Numbers": {"1": {"2": {"1/200": "0.01110743573382805081769333477531538922109735148400186730462708029154619887401124257647760790141349087", "1/100": "0.02221624193350866958966156001836525086317167300100400016684551185013959091605648152429758550692758527", "1/40": "0.05556460219733658753870201804502076501094660533560100347047199710477618593715789722156458684897871566", "1/20": "0.1113010211732865818742430016108524363780394982337018195657317404981402869330800270795916349953932035", "1/10": "0.2239894235758313586004765366468808518930806315235050055739810989336605151571916411792772681472491377", "1/4": "0.5857864376269049511983112757903019214303281246230519268233202620092675215378929611496124656723584273", "1/2": "1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573", "3/4": "3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573", "9/10": "8.928993021507117403000688000637952372685373397880824794494388082632993319045498879209994017266634598", "19/20": "17.96928706418752138787561900288675718174878629399535140855450790576412692502407778549957966952866240", "39/40": "35.99413873057237936328994002381853512850851919332630382057948780863303003598531346822072392590630352", "99/100": {"number": "90.02422668900665565823192120827206577523932794077255490105580351015123017476435560286879154827785879", "comment": "The exact expression is $\\sqrt{2}\\cot(\\pi/200)$. CITE{CriticalValues} prints $90.03$; the value $90.024226689\\ldots$ rounds to $90.02$ at two decimal places."}, "199/200": "180.0595608137471393672815357513194469396997532330291116694162341005940065484027224483140607044571311", "999/1000": "900.3155756764945723072019273867844424672907129123193406286915550254723039035705836507797460529612129"}, "3": {"1/200": "0.1351887157455562619825423474843505767533723002644449249410214112985242699576031364633942155335310920", "1/100": "0.1919158256623758835094625618829156581557382677066608548569039456532311332360576770504694609860683054", "1/40": "0.3069764531398581995648607437072117545331376416587076414007371747090893264784815212468693676580084826", "1/20": "0.4427866284343222226188797143528043731002160191852926751759843927783342641604813992174708597677859303", "1/10": "0.6526254654544146364707438318344372557321268903843952356090390047135567900092751480800542871777649818", "1/4": "1.189207115002721066717499970560475915292972092463817413019002224719466668226917159870781344538137674", "1/2": "2.337541788960735261544740444279985617686782606732438558824578788789767250433553583096886018068595966", "3/4": "5.229506694755301555039906482252219979213632784171580742702887371650083275078634573930446201974520637", "9/10": "13.43669932118315144456232033978243300006067967718189648154654725831796674068211359831731893932282996", "19/20": "26.97552986950007693001052678053762301132217827088312589890327614278748627866954508736384879397316202", "39/40": "54.00200701157065747772016695905399495644008073540627687672302606664846245540911647178385518192600578", "99/100": "135.0406595185365708777029572844927581667313218133014409102577506443151655068710009506541475652367853", "199/200": "270.0915009573206200550309265503693843020669043532473506693353083536243578526485637950572768280176430", "999/1000": "1350.473795461709686845982986960633591165332729665503172392068953074987713302427522903924045404077804"}}, "2": {"2": {"1/200": "0.01000012500234379882919313926751853314072057380998081535749918987950895728143427303513155461514590764", "1/100": "0.02000100007500625054692422326213773459233281092383859683995255863540896155677177565456295025470381229", "1/40": "0.05001563232803553460233127157290769183960949716688003688632971747997988858187575454744481771483666355", "1/20": "0.1001252348643517740139367535241016902148500055972001831467824713317161687750533559536413291255309508", "1/10": "0.2010075630518424150978747113133749505410356694296577937611322512797869480369117482853844484320176152", "1/4": "0.5163977794943222513572353866376532814443895607055454435450098354817977455915972044692383169144898024", "1/2": "1.154700538379251529018297561003914911295203502540253752037204652967955344605866691387430791171499050", "3/4": "2.267786838055363363287099217405080364894507871213528726030000965029487562768814538080336759835323095", "9/10": "4.129483209670111786329772405761741150761372139693895203369799709835370100402991870621624463438684289", "19/20": "6.084869844593311072363639451172107423609550747786742044133664958029877580055732077540173647805408777", "39/40": "8.775685627222986459583955071922541389012191735327886724957824925029567680645959094808104434324850420", "99/100": "14.03584785916505083515562015474226453583851361599176226379439199414729098670312450398083622168089169", "199/200": "19.92492173800600302877341395129623635275515111384283644620971179501151758623561783477462449598065921", "999/1000": "44.68781154017418497790276095952539924690943132460717145938696902562744123976783897303232630657383863"}, "3": {"1/200": "0.1350179169929992383182312628995254610262909025378508176460859216898999258661642735510952720578697248", "1/100": "0.1914294101292063287940930847062856580059234850877900348733633523935741087398597460038205251290238293", "1/40": "0.3050133701976869904558600592599249044559498422244048731690357392656983459311192754316338150893473444", "1/20": "0.4370346027529441000321494227943727182201226787538639163104361670250522615558977785881783766328281114", "1/10": "0.6351172539476233959389727598444682305765143595110370786545553421212206124963766501600467872861840197", "1/4": "1.100675751682683675400390637218255312683019257085518986308841341084516777603138225926241243079330462", "1/2": "1.908178518404061997919772596120039148009867575619189274777731720536485906342424165994148851549136367", "3/4": "3.308014268186600621541344849995530980780645120192860794656486535524330821383540317147911421951352681", "9/10": "5.732648952563374202107250537461644225732654542744958306256003484438574473682771963695037298703858649", "19/20": "8.330782645637052541315252965229714342138932338699544111398732505589818752967970872089745089571795104", "39/40": "11.93654388951834433329440258912056036180162118968075636477286660451428589896749295155901656869633393", "99/100": "19.01893598731274918884884847845082505808588624850181326162563929928661166733752509750276968453140347", "199/200": "26.96513446059991859890137254132718967436376542320698395996726013141769066696999383745805014604024781", "999/1000": "60.41777661374330036444659822414657270953532873821544139352386602534440222655953642450713555873687913"}}, "3": {"2": {"1/200": "0.009619222620543507790323410513149707345042218968065584739336284803445101063565032866978623741535393800", "1/100": "0.01923903864732939584542052371017827774993500518975223826553841438286288299439873904394401042888117474", "1/40": "0.04810798545747449497908714763584886403681989613988652066493400607862159719303058292394550386399254596", "1/20": "0.09629029799716595961557806031659172024380615432374972017495636897966063096160820220248445427171607667", "1/10": "0.1931790261217447417256873505526935113754725343151802672750548410879857390998713950608251986481995212", "1/4": "0.4938689573245775743484561195213066130049065679288578184987273350427889735258981868215549797402264759", "1/2": "1.081721104584560456230653251862552704432349429603965975423818546852598658853522918183319439656996845", "3/4": "2.011895967218132372351139962045598277737615025805008829615545855286539222408893193070636184408496407", "9/10": "3.328158486689670355693570207875205140742294114849109715668744978745384786615754461107998238970769705", "19/20": "4.500658726456369323277261511985011001986607632247105958524454851901003202337889380422903082809885430", "39/40": "5.906512223084809239319096438534094471276527210253050506911666308695532954185119952595209616157834820", "99/100": "8.260293162416186785808784404225270125995920910749200874096929683550612272609267074200759641936998424", "199/200": "10.54058411467037742180028361924956344754011236579613754025561977318563497528445071820844644964789624", "999/1000": "18.27726586782358174703608166018463577167922185296974549541550690997614284273215725480317001899975059"}, "3": {"1/200": "0.1349610618969742032029665816406853383884153024700133923209288933554624277102989516857147689164379288", "1/100": "0.1912677159112926349913918397051786358801092830689515890119368420214303548251151444607154984919260972", "1/40": "0.3043635237132545453860154971893101019744220234931178293268420158691774072000132521610147435350428375", "1/20": "0.4351440335852931710059485242039659513065993982246306050532594146152537842934328182147608205511958458", "1/10": "0.6294481827860904824972237840041906592578364159402494769408522860862039247530036773686335822471428155", "1/4": "1.073447805942574253556386701240784283224509157689615305151658239409222224498817231901026588335621154", "1/2": "1.790912416418855641028645632104878316732878073542123911016867275863879569868684122945298071187553332", "3/4": "2.883772914151589980897753835060452148268128070849542419693857324903731085822303932995291552110052928", "9/10": "4.467363530649994060857523740357183119138065718554570911156463303981165477734873173755913089071032436", "19/20": "5.909598453393489738361606420126853155061257396408243160856070290642912527555128193144431590610790132", "39/40": "7.660535095309175987890412951357750352921752104293513358009194875778233389968808175548206793554024940", "99/100": "10.61853991287509648800160328177750732947257472598009811972425534627349621157768201502679315115533198", "199/200": "13.49944453559165855218553863129535192722124372096409927992373210681836208783320893998065249982478962", "999/1000": "23.31324084743100844938379107959700480618896426455106326981046790263566969812838060993344114150528011"}}, "4": {"2": {"1/200": "0.009428177714360593791572930289003160793833113025034418406838916275803622361076406030573311713308809523", "1/100": "0.01885687925371901960378029155368174771011287745843537868063467043252454178250650740816204266205982619", "1/40": "0.04715136861802725813165134770038009924510188138421549457410297120454804488126254169231805519013951118", "1/20": "0.09436834220255171407242287130458949159793093328779936785141106429296944039112298760147626101261609871", "1/10": "0.1892647202270805135123978706269268674036546066516410920476880355119298428981577983715810247404308022", "1/4": "0.4827779871747337949025649358865056961350692042247255973602004373386235353381446058872767250225676432", "1/2": "1.047503861962360930512696723503306785955051897576822924046271759345566599275855676061061633439412696", "3/4": "1.901265256222192978134136519353748211703319598944874767820202723157203513039550191433619892981323915", "9/10": "3.014886638124646523020021134821199693382011424820882085585189673849244899419818588307609613865661007", "19/20": "3.926486322955115395311212269622800013283699119295024100637664425329691207688412992247723791451222108", "39/40": "4.943250475764121643738115045013432807714751166713986043158004696324092506030245248285957609025068611", "99/100": "6.511173409515570668662176941103289781582639162504119393132242378753260160758489786570112477722816002", "199/200": "7.916157101028733421628795880468762737035929183484248306510845143136648065272282141674068238120394422", "999/1000": "12.17680527250907839613607697480104031886747558573215361593066994283799660338832085688263221023184857"}, "3": {"1/200": "0.1349326489233642870049912807418924986801060679873328575625789814091402259073217281010113051516027628", "1/100": "0.1911869519114174556737864873504898670199348910296653759568272579390466366309447416702344501285868801", "1/40": "0.3040394419386478472951980207343568223333923754458309170162165629276059822753642331761957989492718285", "1/20": "0.4342037090774445838115097197506280713685681686582341886888706509854870805949641803452466859296735749", "1/10": "0.6266442001042812364218039788428819483935304586068183172937319428429385776541021169287053443487956358", "1/4": "1.060234222117246620895018072896017823934674072101543647584667796661233536602007276678121188661974898", "1/2": "1.736411004924546718135073099497637185479249469972630328690805590949932004818569984277291288301199872", "3/4": "2.701018025498736150469167787268495917525555966398270661826156340294136874707453979546942633705397995", "9/10": "3.975595737246607999487534051492359531651433650993693483665762012387194369096864801292302722830537511", "19/20": "5.040241254983199188178600566209659621382075396016084045746537348025622615744185200436776469467793439", "39/40": "6.243652811697559997082179558943950935572618866097547306723071296427912914464346659454277468477473799", "99/100": "8.119791806342719694201588854520423300688018809804558228176590152783272802448833919006041381042124899", "199/200": "9.813409821168509206062032160934949157115679909740063656768549302425680273973522092905861020097892918", "999/1000": "14.98281000097803266742705167026753320224895158653594592893285903809094898240602457630892278854414389"}}, "5": {"2": {"1/200": "0.009313757291531582262945736522176447059999948834709942989722972113836075724996919886773832409275122413", "1/100": "0.01862799936742775415330871729924208548639030619522429126563024283243962270307085447375149217254448005", "1/40": "0.04657848532562426859927101387813330001762528146183947850673114313197699392624745873704351316509532258", "1/20": 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"5.365187042594885450012534917549673814569650739442472944565611810962488079923007465866195568538363545"}}, "120": {"2": {"1/200": "0.008880810201346296533424845833354321378653509362304212180660325020814677147086410740289830407332425086", "1/100": "0.01776197354785946035792975794024049363430316664821803953366146579652588787990972255417868408761218664", "1/40": "0.04441111596051593368089972580087806283945816855648738450427299129014279915110510384231696307386436389", "1/20": "0.08886644839606633419524588533951005595754687071440935163324016687416861279377324472596325608815135573", "1/10": "0.1780884748989377209918044824996155652248368415434536011446848779330839990766180868983363685798659802", "1/4": "0.4516597429756053010310941232186102297149123638340582782413282332629835180825399557685940979443985461", "1/2": "0.9567716544554489831216214811297462680088779291460385421662965155613776654224725779888006377951171915", "3/4": 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"0.1909530491498492875822752020678053239042975868286686946164656970960103018700946538772885070372639192", "1/40": "0.3031027648277479549814474350155624560182204207646538633877632314415617268713314428710074293395494078", "1/20": "0.4314953211540193168767427538290779239589607961116086669835546493130551835343154089149067942481692550", "1/10": "0.6186259965853900635492189570751700623345091332931694807729958673040200728768720997083176851727025600", "1/4": "1.023360023797536315745337903139275982321106680452625545013466881477720292517397875849913246555073314", "1/2": "1.592406303646905877352431331286081145850239027574682805796330882276331756620697777623311576522535864", "3/4": "2.261421350250744518862945372897408436475900284286863665145125192007059664114085122260887210578958377", "9/10": "2.930409035965660150934220097782785455485881777618644831727140421889851215652670012407646583348318956", "19/20": "3.356138396150708925223803859656912020757561571878452831152014136314748732294753923342084936538140575", "39/40": "3.739250173336110001216059664107378093663462636391154453838370980410719699149669592261042037677750399", "99/100": "4.199944004615294934226611561415246917982570699066067277788008972183642087630945146888284125927272519", "199/200": "4.522681034726935945420790582469064061576546048544090192204872898579301311924704813714155275794097139", "999/1000": "5.210644665307017636970982271070845731446708909960675180407132230435513234064952950300427979200109299"}}}, "Keywords": ["critical value", "percentage point", "percentile", "significance level", "p-value", "table", "Tukey", "studentised range", "q distribution", "multiple comparisons"], "Comments": {"comment-convention": "The table stores lower-tail probabilities. If $Q^{\\mathrm{upper}}_{k,\\nu}(\\alpha)$ denotes an upper-tail critical value, then $q_{k,\\nu,p}=Q^{\\mathrm{upper}}_{k,\\nu}(1-p)$. In particular, $p=19/20$ and $p=99/100$ correspond to the upper-tail significance levels $0.05$ and $0.01$ used in CITE{CriticalValues}.", "comment-two-sided": "The range includes differences in both directions. Its level-$\\alpha$ threshold is $q_{k,\\nu,1-\\alpha}$; the conversion to the two-sided Student threshold when $k=2$ is given in CITE{formula-two-sided}.", "comment-scope": "The stored rows cover two and three means. Two means gives the ordinary absolute Student comparison, while three means is the first nontrivial simultaneous comparison. These cases permit certified 100-digit inversion through one-dimensional formulas. The distribution itself is defined for every integer $k\\geq2$.", "comment-scale": "The denominator is an independent estimate of the normal standard deviation. It is not the sample standard deviation of the same $Z_1,\\ldots,Z_k$. The single-range convention agrees with `nranges=1` in CITE{R}.", "comment-probabilities": "The parameters are exact rational probabilities in lowest terms. Their selection includes the median, the quartiles, conventional significance levels and their complements, and the upper $0.1\\%$ critical value."}, "Formulas": {"formula-two-means": "Write $G_{k,\\nu}(x)=\\Pr(Q_{k,\\nu}\\leq x)$. For $x>0$, $G_{2,\\nu}(x)=I_{x^2/(2\\nu+x^2)}(1/2,\\nu/2)$. Here $B(z;a,b)$ is the HREF{Values_of_the_incomplete_beta_function}[incomplete beta function] and $I_z(a,b)=B(z;a,b)/B(a,b)$ is its regularized form CITE{DLMFBeta}.", "formula-student": "In terms of HREF{T454}[Quantiles of Student's $t$-distribution], $q_{2,\\nu,p}=\\sqrt{2}\\,t_{\\nu,(1+p)/2}$.", "formula-two-sided": "For $0<\\alpha<1$, $q_{2,\\nu,1-\\alpha}=\\sqrt{2}\\,t_{\\nu,1-\\alpha/2}$, and $\\Pr\\{|T_\\nu|>q_{2,\\nu,1-\\alpha}/\\sqrt{2}\\}=\\alpha$. Here $T_\\nu$ has the central Student distribution.", "formula-three-means": "For $x>0$, $G_{3,\\nu}(x)=1-\\frac{6}{\\pi}\\int_0^{\\pi/6}\\left(1+\\frac{x^2}{2\\nu\\cos^2\\theta}\\right)^{-\\nu/2}\\,\\mathrm{d}\\theta$. This follows by projecting three independent standard normals onto the plane perpendicular to $(1,1,1)$ and using polar coordinates; the derivation is in the attached mathematical notes.", "formula-upper-tail": "$G_{k,\\nu}(q_{k,\\nu,p})=p$ and $1-G_{k,\\nu}(q_{k,\\nu,1-\\alpha})=\\alpha$ for $0<p,\\alpha<1$.", "formula-cauchy": "For one degree of freedom, $q_{2,1,p}=\\sqrt{2}\\tan(\\pi p/2)$ and $q_{3,1,p}=\\sqrt{\\frac{1}{2\\sin^2(\\pi(1-p)/6)}-2}$.", "formula-two-df": "For two means and two degrees of freedom, $q_{2,2,p}=2p/\\sqrt{1-p^2}$."}, "Similar tables": [{"table": "HREF{Quantiles_of_the_standard_normal_distribution}[Quantiles of the standard normal distribution]", "relation": "the numerator is the range of independent standard normal variables"}, {"table": "HREF{Quantiles_of_the_chi-squared_distribution}[Quantiles of the chi-squared distribution]", "relation": "the denominator contains an independent chi-squared variable divided by its degrees of freedom"}, {"table": "HREF{T454}[Quantiles of Student's $t$-distribution]", "relation": "for two means, the range is distributed as $\\sqrt{2}$ times the absolute Student variable, as in CITE{formula-student}"}, {"table": "HREF{T455}[Quantiles of the $F$-distribution]", "relation": "for two means, $Q_{2,\\nu}^2/2$ has an F distribution with degrees of freedom $1,\\nu$"}, {"table": "HREF{T456}[Quantiles of the Kolmogorov distribution]", "relation": "gives Kolmogorov critical values using the same lower-tail probability convention"}, {"table": "HREF{Values_of_the_upper_incomplete_gamma_function}[Values of the upper incomplete gamma function $\\Gamma(a,x)$]", "relation": "regularized upper incomplete gamma values give upper-tail probabilities of the chi-squared denominator variable"}, {"table": "HREF{Values_of_the_lower_incomplete_gamma_function}[Values of the lower incomplete gamma function $\\gamma(a,x)$]", "relation": "regularized lower incomplete gamma values give lower-tail probabilities of the chi-squared denominator variable"}, {"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 two-mean cumulative distribution function in CITE{formula-two-means}"}, {"table": "HREF{Values_of_the_error_function}[Values of the error function $\\operatorname{erf}(x)$]", "relation": "the error function gives the normal cumulative distribution function appearing in the general studentized-range integral CITE{SciPy}"}, {"table": "HREF{Differential_entropies_of_continuous_probability_distributions}[Differential entropies of continuous probability distributions]", "relation": "includes differential entropies of the normal, chi-squared and Student distributions used in the construction"}, {"table": "HREF{Kullback-Leibler_divergences_between_probability_distributions}[Kullback-Leibler divergences between probability distributions]", "relation": "includes divergences between normal distributions, while this table gives thresholds for their studentized sample ranges"}], "Programs": {"program-scipy": {"language": "Python", "code": "from scipy.stats import studentized_range\nstudentized_range.ppf(19/20, 3, 10) # machine-precision lower-tail quantile"}, "program-r": {"language": "R", "code": "qtukey(19/20, nmeans=3, df=10, nranges=1, lower.tail=TRUE)"}, "program-mpmath": {"language": "Python", "code": "# Numerical inversion for three means; the attached generator certifies its result.\nimport mpmath as mp\nmp.mp.dps = 60\nnu, p = 10, mp.mpf(19)/20\ndef cdf(q):\n return 1 - 6/mp.pi * mp.quad(\n lambda theta: (1+q*q/(2*nu*mp.cos(theta)**2))**(-mp.mpf(nu)/2),\n [0, mp.pi/6])\nmp.findroot(lambda q: cdf(q)-p, (3, 4))"}}}