NumberDB$^\beta$ Advanced search Tags Tables Help Login
Quantiles of the studentized range distribution
edit · history · discussion · files · short url · probability theory
Numbers
$\nu$
$k$
$p$ 
$q_{k,\nu,p}$
1
2
1/200:
0.01110743573382805081769333477531538922109735148400186730462708029154619887401124257647760790141349087
1
2
1/100:
0.02221624193350866958966156001836525086317167300100400016684551185013959091605648152429758550692758527
1
2
1/40:
0.05556460219733658753870201804502076501094660533560100347047199710477618593715789722156458684897871566
1
2
1/20:
0.1113010211732865818742430016108524363780394982337018195657317404981402869330800270795916349953932035
1
2
1/10:
0.2239894235758313586004765366468808518930806315235050055739810989336605151571916411792772681472491377
1
2
1/4:
0.5857864376269049511983112757903019214303281246230519268233202620092675215378929611496124656723584273
1
2
1/2:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
1
2
3/4:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
1
2
9/10:
8.928993021507117403000688000637952372685373397880824794494388082632993319045498879209994017266634598
1
2
19/20:
17.96928706418752138787561900288675718174878629399535140855450790576412692502407778549957966952866240
1
2
39/40:
35.99413873057237936328994002381853512850851919332630382057948780863303003598531346822072392590630352
1
2
99/100:
90.02422668900665565823192120827206577523932794077255490105580351015123017476435560286879154827785879
comment: The exact expression is $\sqrt{2}\cot(\pi/200)$. [3] prints $90.03$; the value $90.024226689\ldots$ rounds to $90.02$ at two decimal places.
1
2
199/200:
180.0595608137471393672815357513194469396997532330291116694162341005940065484027224483140607044571311
1
2
999/1000:
900.3155756764945723072019273867844424672907129123193406286915550254723039035705836507797460529612129
1
3
1/200:
0.1351887157455562619825423474843505767533723002644449249410214112985242699576031364633942155335310920
1
3
1/100:
0.1919158256623758835094625618829156581557382677066608548569039456532311332360576770504694609860683054
1
3
1/40:
0.3069764531398581995648607437072117545331376416587076414007371747090893264784815212468693676580084826
1
3
1/20:
0.4427866284343222226188797143528043731002160191852926751759843927783342641604813992174708597677859303
1
3
1/10:
0.6526254654544146364707438318344372557321268903843952356090390047135567900092751480800542871777649818
1
3
1/4:
1.189207115002721066717499970560475915292972092463817413019002224719466668226917159870781344538137674
1
3
1/2:
2.337541788960735261544740444279985617686782606732438558824578788789767250433553583096886018068595966
1
3
3/4:
5.229506694755301555039906482252219979213632784171580742702887371650083275078634573930446201974520637
1
3
9/10:
13.43669932118315144456232033978243300006067967718189648154654725831796674068211359831731893932282996
1
3
19/20:
26.97552986950007693001052678053762301132217827088312589890327614278748627866954508736384879397316202
1
3
39/40:
54.00200701157065747772016695905399495644008073540627687672302606664846245540911647178385518192600578
1
3
99/100:
135.0406595185365708777029572844927581667313218133014409102577506443151655068710009506541475652367853
1
3
199/200:
270.0915009573206200550309265503693843020669043532473506693353083536243578526485637950572768280176430
1
3
999/1000:
1350.473795461709686845982986960633591165332729665503172392068953074987713302427522903924045404077804
2
2
1/200:
0.01000012500234379882919313926751853314072057380998081535749918987950895728143427303513155461514590764
2
2
1/100:
0.02000100007500625054692422326213773459233281092383859683995255863540896155677177565456295025470381229
2
2
1/40:
0.05001563232803553460233127157290769183960949716688003688632971747997988858187575454744481771483666355
2
2
1/20:
0.1001252348643517740139367535241016902148500055972001831467824713317161687750533559536413291255309508
2
2
1/10:
0.2010075630518424150978747113133749505410356694296577937611322512797869480369117482853844484320176152
2
2
1/4:
0.5163977794943222513572353866376532814443895607055454435450098354817977455915972044692383169144898024
2
2
1/2:
1.154700538379251529018297561003914911295203502540253752037204652967955344605866691387430791171499050
2
2
3/4:
2.267786838055363363287099217405080364894507871213528726030000965029487562768814538080336759835323095
2
2
9/10:
4.129483209670111786329772405761741150761372139693895203369799709835370100402991870621624463438684289
2
2
19/20:
6.084869844593311072363639451172107423609550747786742044133664958029877580055732077540173647805408777
2
2
39/40:
8.775685627222986459583955071922541389012191735327886724957824925029567680645959094808104434324850420
2
2
99/100:
14.03584785916505083515562015474226453583851361599176226379439199414729098670312450398083622168089169
2
2
199/200:
19.92492173800600302877341395129623635275515111384283644620971179501151758623561783477462449598065921
2
2
999/1000:
44.68781154017418497790276095952539924690943132460717145938696902562744123976783897303232630657383863
2
3
1/200:
0.1350179169929992383182312628995254610262909025378508176460859216898999258661642735510952720578697248
2
3
1/100:
0.1914294101292063287940930847062856580059234850877900348733633523935741087398597460038205251290238293
2
3
1/40:
0.3050133701976869904558600592599249044559498422244048731690357392656983459311192754316338150893473444
2
3
1/20:
0.4370346027529441000321494227943727182201226787538639163104361670250522615558977785881783766328281114
2
3
1/10:
0.6351172539476233959389727598444682305765143595110370786545553421212206124963766501600467872861840197
2
3
1/4:
1.100675751682683675400390637218255312683019257085518986308841341084516777603138225926241243079330462
2
3
1/2:
1.908178518404061997919772596120039148009867575619189274777731720536485906342424165994148851549136367
2
3
3/4:
3.308014268186600621541344849995530980780645120192860794656486535524330821383540317147911421951352681
2
3
9/10:
5.732648952563374202107250537461644225732654542744958306256003484438574473682771963695037298703858649
2
3
19/20:
8.330782645637052541315252965229714342138932338699544111398732505589818752967970872089745089571795104
2
3
39/40:
11.93654388951834433329440258912056036180162118968075636477286660451428589896749295155901656869633393
2
3
99/100:
19.01893598731274918884884847845082505808588624850181326162563929928661166733752509750276968453140347
2
3
199/200:
26.96513446059991859890137254132718967436376542320698395996726013141769066696999383745805014604024781
2
3
999/1000:
60.41777661374330036444659822414657270953532873821544139352386602534440222655953642450713555873687913
3
2
1/200:
0.009619222620543507790323410513149707345042218968065584739336284803445101063565032866978623741535393800
3
2
1/100:
0.01923903864732939584542052371017827774993500518975223826553841438286288299439873904394401042888117474
3
2
1/40:
0.04810798545747449497908714763584886403681989613988652066493400607862159719303058292394550386399254596
3
2
1/20:
0.09629029799716595961557806031659172024380615432374972017495636897966063096160820220248445427171607667
3
2
1/10:
0.1931790261217447417256873505526935113754725343151802672750548410879857390998713950608251986481995212
3
2
1/4:
0.4938689573245775743484561195213066130049065679288578184987273350427889735258981868215549797402264759
3
2
1/2:
1.081721104584560456230653251862552704432349429603965975423818546852598658853522918183319439656996845
3
2
3/4:
2.011895967218132372351139962045598277737615025805008829615545855286539222408893193070636184408496407
3
2
9/10:
3.328158486689670355693570207875205140742294114849109715668744978745384786615754461107998238970769705
3
2
19/20:
4.500658726456369323277261511985011001986607632247105958524454851901003202337889380422903082809885430
3
2
39/40:
5.906512223084809239319096438534094471276527210253050506911666308695532954185119952595209616157834820
3
2
99/100:
8.260293162416186785808784404225270125995920910749200874096929683550612272609267074200759641936998424
3
2
199/200:
10.54058411467037742180028361924956344754011236579613754025561977318563497528445071820844644964789624
3
2
999/1000:
18.27726586782358174703608166018463577167922185296974549541550690997614284273215725480317001899975059
3
3
1/200:
0.1349610618969742032029665816406853383884153024700133923209288933554624277102989516857147689164379288
3
3
1/100:
0.1912677159112926349913918397051786358801092830689515890119368420214303548251151444607154984919260972
3
3
1/40:
0.3043635237132545453860154971893101019744220234931178293268420158691774072000132521610147435350428375
3
3
1/20:
0.4351440335852931710059485242039659513065993982246306050532594146152537842934328182147608205511958458
3
3
1/10:
0.6294481827860904824972237840041906592578364159402494769408522860862039247530036773686335822471428155
3
3
1/4:
1.073447805942574253556386701240784283224509157689615305151658239409222224498817231901026588335621154
3
3
1/2:
1.790912416418855641028645632104878316732878073542123911016867275863879569868684122945298071187553332
3
3
3/4:
2.883772914151589980897753835060452148268128070849542419693857324903731085822303932995291552110052928
3
3
9/10:
4.467363530649994060857523740357183119138065718554570911156463303981165477734873173755913089071032436
3
3
19/20:
5.909598453393489738361606420126853155061257396408243160856070290642912527555128193144431590610790132
3
3
39/40:
7.660535095309175987890412951357750352921752104293513358009194875778233389968808175548206793554024940
3
3
99/100:
10.61853991287509648800160328177750732947257472598009811972425534627349621157768201502679315115533198
3
3
199/200:
13.49944453559165855218553863129535192722124372096409927992373210681836208783320893998065249982478962
3
3
999/1000:
23.31324084743100844938379107959700480618896426455106326981046790263566969812838060993344114150528011
4
2
1/200:
0.009428177714360593791572930289003160793833113025034418406838916275803622361076406030573311713308809523
4
2
1/100:
0.01885687925371901960378029155368174771011287745843537868063467043252454178250650740816204266205982619
4
2
1/40:
0.04715136861802725813165134770038009924510188138421549457410297120454804488126254169231805519013951118
4
2
1/20:
0.09436834220255171407242287130458949159793093328779936785141106429296944039112298760147626101261609871
4
2
1/10:
0.1892647202270805135123978706269268674036546066516410920476880355119298428981577983715810247404308022
4
2
1/4:
0.4827779871747337949025649358865056961350692042247255973602004373386235353381446058872767250225676432
4
2
1/2:
1.047503861962360930512696723503306785955051897576822924046271759345566599275855676061061633439412696
4
2
3/4:
1.901265256222192978134136519353748211703319598944874767820202723157203513039550191433619892981323915
4
2
9/10:
3.014886638124646523020021134821199693382011424820882085585189673849244899419818588307609613865661007
4
2
19/20:
3.926486322955115395311212269622800013283699119295024100637664425329691207688412992247723791451222108
4
2
39/40:
4.943250475764121643738115045013432807714751166713986043158004696324092506030245248285957609025068611
4
2
99/100:
6.511173409515570668662176941103289781582639162504119393132242378753260160758489786570112477722816002
4
2
199/200:
7.916157101028733421628795880468762737035929183484248306510845143136648065272282141674068238120394422
4
2
999/1000:
12.17680527250907839613607697480104031886747558573215361593066994283799660338832085688263221023184857
4
3
1/200:
0.1349326489233642870049912807418924986801060679873328575625789814091402259073217281010113051516027628
4
3
1/100:
0.1911869519114174556737864873504898670199348910296653759568272579390466366309447416702344501285868801
4
3
1/40:
0.3040394419386478472951980207343568223333923754458309170162165629276059822753642331761957989492718285
4
3
1/20:
0.4342037090774445838115097197506280713685681686582341886888706509854870805949641803452466859296735749
4
3
1/10:
0.6266442001042812364218039788428819483935304586068183172937319428429385776541021169287053443487956358
4
3
1/4:
1.060234222117246620895018072896017823934674072101543647584667796661233536602007276678121188661974898
4
3
1/2:
1.736411004924546718135073099497637185479249469972630328690805590949932004818569984277291288301199872
4
3
3/4:
2.701018025498736150469167787268495917525555966398270661826156340294136874707453979546942633705397995
4
3
9/10:
3.975595737246607999487534051492359531651433650993693483665762012387194369096864801292302722830537511
4
3
19/20:
5.040241254983199188178600566209659621382075396016084045746537348025622615744185200436776469467793439
4
3
39/40:
6.243652811697559997082179558943950935572618866097547306723071296427912914464346659454277468477473799
4
3
99/100:
8.119791806342719694201588854520423300688018809804558228176590152783272802448833919006041381042124899
4
3
199/200:
9.813409821168509206062032160934949157115679909740063656768549302425680273973522092905861020097892918
4
3
999/1000:
14.98281000097803266742705167026753320224895158653594592893285903809094898240602457630892278854414389
5
2
1/200:
0.009313757291531582262945736522176447059999948834709942989722972113836075724996919886773832409275122413
5
2
1/100:
0.01862799936742775415330871729924208548639030619522429126563024283243962270307085447375149217254448005
5
2
1/40:
0.04657848532562426859927101387813330001762528146183947850673114313197699392624745873704351316509532258
5
2
1/20:
0.09321768245867664511644237758742800938394662163183131485173662780083549210127030226939281788117985707
5
2
1/10:
0.1869239254216925237486020374354064600440898732221575433044691678117035254161897176350999513331757055
5
2
1/4:
0.4761960632923131077965023804497030752924446366276932101062753785362348009586286510249347139444442579
5
2
1/2:
1.027690390100656600677098596040370526350089430688687378868266146024344890040829505196003171206676382
5
2
3/4:
1.839819771972766206916427111699506973760119611539832379551596188552218443529968357404313942176472274
5
2
9/10:
2.849708738405406590815510845829851620010670259966973058218930276379914098779708604593917625050967863
5
2
19/20:
3.635351695146803656009907430980972903570031278070068710826500511096374034483724138409510873776030782
5
2
39/40:
4.473696949193941217106670853046705071618931779661748758411495029016839216846743407543259439551240254
5
2
99/100:
5.702311292771319107656742330504901234386592735498387319599836117460838328949276232190192564950725716
5
2
199/200:
6.750523021212872082082732876049148681492137005604469426679397270733803570542126640471437816906752567
5
2
999/1000:
9.713987771910491516633961190329269536185480069194000409384700701881747590253257808213002829755300239
5
3
1/200:
0.1349156058008165682806836568876221996969863054963974226724819476283787915683774041267049856500808840
5
3
1/100:
0.1911385200813650075807885251327732721470625223041923737078034938824340328171658205247647377619208659
5
3
1/40:
0.3038452615060091073460422171874170029386891958376303700588576676364967535528653483686861062944394653
5
3
1/20:
0.4336410940033437779182269479911896612167931015536695633272260820512633333343002575684730392557947260
5
3
1/10:
0.6249714918398489507688360582090522949468257375264801661530514712024094452020804861124546453588965328
5
3
1/4:
1.052430830472440826007751417893998054386953380133656304426547632728120164347822784326226453185141180
5
3
1/2:
1.704951022467423969358202589859812815712233313941547107083328848300606243395768465270346466234472456
5
3
3/4:
2.599650877952047756856786740430739303219346257089393116851039818080502003558763398535140442901659544
5
3
9/10:
3.717111449623848051201973280930909727435943902505635875658692386211661047756083999585456071442267580
5
3
19/20:
4.601726054362567298083624111754734772655032051214312066792489800557079906827755573140252604600879145
5
3
39/40:
5.558022769659694390313987133685487348145927487062554527211516758253979984468764113105568325859778646
5
3
99/100:
6.975736631249441568805549815228330823815621144289244230477329271727522602872629274203660851353471955
5
3
199/200:
8.195503319462242471817593688654975984880139042999148327481672259268623430581213832297872262644747515
5
3
999/1000:
11.67198293404628501107347864697369258018105690974096870433637822954805497196194669511270761680082052
6
2
1/200:
0.009237680946108141023427565936196882886930167763245616228595270397895526239577583896410167256975325155
6
2
1/100:
0.01847582175478461432343925654079672942517006907321268026258307848366475526376657769268860665700451565
6
2
1/40:
0.04619760493622502133121583131951711015571641324190100186832333521697328118617475607961511106210471614
6
2
1/20:
0.09245279835719323364287447012351552557157102722116587252518003786724985658466741757892152649238049685
6
2
1/10:
0.1853689663331716465725882653476147996849600478083469402202037959013710246019762235996032082012546577
6
2
1/4:
0.4718436363566480367806933340360105001348137819865865579444832957623526714296998268614652017616340225
6
2
1/2:
1.014780533270133878011410316938469267718763591447613966467149150921687964361906226727401230920023870
6
2
3/4:
1.800787862633593313632283993941461568661409829207182099709550161415186971467450273800785410007952425
6
2
9/10:
2.748071906840697084913241162843856569607726369550750668144823293238000522072504117778643316892834107
6
2
19/20:
3.460455925820672457688138734414804095326928035337998368479226167162383491682839749071241114600164797
6
2
39/40:
4.198356971166239273283396152436205114264613077131463326861606670748067070549853387872231572150531364
6
2
99/100:
5.243094989279551872603098470985227922613172306280306415111254016736708078418586966400594108869147840
6
2
199/200:
6.104915436378082249359240306264340218338597768221409776991313363329614214457337180761919720217340568
6
2
999/1000:
8.427038655773711758896913688639653654992235758822863039510724605448224448765516074290024623055004981
6
3
1/200:
0.1349042456609521008612064008040871542579468366039908237204390288752800829653667789098849967081551956
6
3
1/100:
0.1911062432596526666980919714777829078399189634992818756219691717532704648971044335475877669040976992
6
3
1/40:
0.3037159196669748753684099002666369304034064879412153010686655247711195242636821965518534442467660699
6
3
1/20:
0.4332666737148142960592331742120162357232957577771795929273517761549918140426147255193019538188526614
6
3
1/10:
0.6238603649188846141371545405661798915016519491952866696755246350500216138826317099378510125407349293
6
3
1/4:
1.047279788099950813568566537318641434363960084014244593122435004894485357713311475391901736140706940
6
3
1/2:
1.684476814069700103312298189753973488176299468426751637761764741551991967804711526813476748548257803
6
3
3/4:
2.535292667255733043203918815834749071829696537284867928490006702401737632926073601883998843637029608
6
3
9/10:
3.558371594081961311668744074331814378984887232645016336939647900756071810952240258507332811601454316
6
3
19/20:
4.339195476520309919302957931840532733122745197537856613085536649934730043472969046018211272454001511
6
3
39/40:
5.157934346758494552256663849749862861599733481716904760670169124969411896712106690126255179862620922
6
3
99/100:
6.330507598274360255300138913157271679046845200533442446057605880926287605032460953244833020596354855
6
3
199/200:
7.306370852878161700998095805008927283632823129599591642194532408788728383178840004856232199301777371
6
3
999/1000:
9.959519175662708537693375209461609987715256623978972660214870846500077940788072170422942889267651235
7
2
1/200:
0.009183482635233619911341799881727173851688556743284643433591592607077271788652757460248590532369516384
7
2
1/100:
0.01836740786537940163657895207860847670439196802814728434004056796927341820145384851593625650681549046
7
2
1/40:
0.04592626787518220670605333423357039756212929800715378970383640452333500195133554753645680897678377520
7
2
1/20:
0.09190796028491679394517486482179903297325772220811353713638728629345460371407875829213734172746228375
7
2
1/10:
0.1842618400196105562904336659688186809367007831335019471456530998220159880571846947769599487933826440
7
2
1/4:
0.4687540490986774372193870760615143488569139131174333823061725207583653220142362904875271143557730126
7
2
1/2:
1.005706347333380015009519887822575729163614131497327008903345484883373648046967062558516998186837141
7
2
3/4:
1.773817922499836915656921707100835502505862229285888080499363668497458181955602144442605512924413562
7
2
9/10:
2.679338758300188577996505824695467357299181850526982858460337016071126004870060427919835358155532319
7
2
19/20:
3.344083686518846732105601663609021315731801318552467417970780304494614987462751110144033247329126768
7
2
39/40:
4.018126150367998043570999267142232400398825918037914610352895012971860666710832277403284788432789480
7
2
99/100:
4.949016740411187060787303862701736406957242239783032749158634973064826163542911205822378111450506577
7
2
199/200:
5.698343283996131188425188531277849578953360688860857214280767956842629615261791296545240060411298556
7
2
999/1000:
7.647900804723053954604388111259253842432031981266495454484511647071253071256060255093190949285060072
7
3
1/200:
0.1348961322262774144288620979927929533637813464450395931258089904396727329892114235787270142574409335
7
3
1/100:
0.1910831938047648696384642761867475349313408821793071131256610938090789628017519742205371022757784876
7
3
1/40:
0.3036235873434708166694011165713271092523821084906381396924821653550005781129813621956158003266495799
7
3
1/20:
0.4329995516165549195893120070639539300824364244967714843599879735067360511998831513201350398403816797
7
3
1/10:
0.6230686609648020369438684491033597517804796061916231062663160064360761584177719472580417398928933836
7
3
1/4:
1.043625320920787236227356516477137840868455791818025685372241580063988743506160756916663171148235298
7
3
1/2:
1.670091356278506390152844321529557788936960100581159349427175785201328045299334037654047512648236186
7
3
3/4:
2.490830443630051285958222571232724645051078662091645320768642673263480344204341363951391044570606587
7
3
9/10:
3.451165134043074062246481815243136620537453078820529139101736999403198185788567788545168986663405587
7
3
19/20:
4.164941474325877384756404337718751848502296515171254518598848799372694873289879695885963513351743787
7
3
39/40:
4.896970411644711950080378036218403874504613323105290267450562435385375796450254915647627753772152645
7
3
99/100:
5.919293567994987051191663395692786769773144451586998621851905312324975658453904968153004806564900540
7
3
199/200:
6.749607736220656086421079024961603625587763158707395982173608271367903293352313017404995277560911504
7
3
999/1000:
8.930392844751151493658897713462012193778071706799357550018399611506183046111378821222853504252728929
8
2
1/200:
0.009142928794069790575516961399848166485906730012405086562635844568017593385499413331802849395133190003
8
2
1/100:
0.01828628752102230182366169086402850726947461641021033525002414599351611501091565677972922660530883787
8
2
1/40:
0.04572324531792341436265682756964663805237918603603752808870871266299801412950922883779583111516379036
8
2
1/20:
0.09150032845107362850775684105887412847725860706198687731373687679368603072299753442672466956064506722
8
2
1/10:
0.1834337825710916431336984176704296437916233464575752373010017281517356711135925618926847235831381941
8
2
1/4:
0.4664481869317052256489385913444827972046137801147883453127561842390763256089294419386226613365985110
8
2
1/2:
0.9989815278811207852501519640640262068095834870860883183149888388479196838467955408831773222125653934
8
2
3/4:
1.754074906057865643686435392718399535675491617284013114989073205440635004416071163002235923343814955
8
2
9/10:
2.629798054560469663002835677770312595437962601951963171564387813745223967609418246812266286274163790
8
2
19/20:
3.261182322894172887785413474106131779756743958154922579430528252471398506702882282080683151229668794
8
2
39/40:
3.891241986741957837623584075590669405053248630575916470653955275675922756266903098264322325028825510
8
2
99/100:
4.745234270986553867046922453992921864835920307138656048816054982448539518603210317971682630160547196
8
2
199/200:
5.419999902862274011677326565261401851418040975850058098417076652814300599168958847862127819107892344
8
2
999/1000:
7.129482515941789703427705231615928056725445259531078426538008396884707869494716076711441112274826181
8
3
1/200:
0.1348900476702571552482298692059745651390047835699524847154492990764526587403824172772411080707518757
8
3
1/100:
0.1910659096757410539050925806424025142663790935672482569239860885159724609892837480191026678411019584
8
3
1/40:
0.3035543679995688491696301340753548981447966730116260451788410430745538814342640976542322332603584440
8
3
1/20:
0.4327993853600888745201128458666684817504264581794587633955245139977808756919332738384505072347715741
8
3
1/10:
0.6224759512218768444926787436733286522461475004385576115856517834447780464020473258676331493415199780
8
3
1/4:
1.040897969303393322926963347589569864859374301738920117057866473454169622378371564539239998961813321
8
3
1/2:
1.659430890402975367560104935698122106391573422288564442254693144384935749070228775040055570256700927
8
3
3/4:
2.458282912454533213224136544257318450705205006995642936781007953431732534578381432325122272374905537
8
3
9/10:
3.373969209840156393317237732115775603154884867498430171138421364554064807542502815581176193436739763
8
3
19/20:
4.041036471985930567944058095832560855829740943612605922215976552335123541647815106296342267575813896
8
3
39/40:
4.713744679411483370772710811937076470291139946138032774419331858120758499991994818758178310149087223
8
3
99/100:
5.635392684839257886425782812969801219290439240512263754230126866831076837481186302998984707518724358
8
3
199/200:
6.370100172044532029276032172963968307144082902307882158855305045961697952306163963688181039331960842
8
3
999/1000:
8.249500262078329699703004482182570505519075330327814068958886211506764792923034021261192011450839688
9
2
1/200:
0.009111451063769105741863863907933806893170742296951082663084756907452408196056096175509115950207465128
9
2
1/100:
0.01822332238184220031024373127365351112550606877308078040480570361655221634346362655471281425262618403
9
2
1/40:
0.04556566301112057391729209314732674478031103780294158199812805256892475873277570227047107671662999678
9
2
1/20:
0.09118395101994286933795099563540501062898520550401789724259273721791806723027393826456241355229283071
9
2
1/10:
0.1827912496256088304369113765939604898363402480801612600814606562320300939465195108036407029107803534
9
2
1/4:
0.4646618093820637111029376166011055740357430929825606603846975209832782410582918306884175328741570678
9
2
1/2:
0.9937991905156621435941604411313711284699160167109900280088158785057845479134679367110616406560716138
9
2
3/4:
1.739000679957259217797455570486213291892545138965624574216983683893362991156357900852533227635785877
9
2
9/10:
2.592413170723968646911853993335475195753094182458909978834071674048785938069833673113547119806501191
9
2
19/20:
3.199173339848663784670762869589761631140043215150364469494692850771051653807407448339586152203690376
9
2
39/40:
3.797178754686699871883370859497312840373320254236944675513711376057894911419592799905222264572992682
9
2
99/100:
4.595961498401697601400947203657781421165377695483799358999890494800452481145288829084405612155459507
9
2
199/200:
5.217970595774601922763278192062165230740578611355995450271758619543154856239923492178606139361267368
9
2
999/1000:
6.761231419544042080214611546158914665583617036967914889389364688829565067938680477212578029526472495
9
3
1/200:
0.1348853155459098502974447722370126466572406856793736560405975611203726813288297073090370559787772312
9
3
1/100:
0.1910524682193153550079076837655533219207506510537292477366924450666385443113587184742728535953415700
9
3
1/40:
0.3035005484420736894912269035173217591035529210738515437053786837483369485020186959730708388148276684
9
3
1/20:
0.4326438042933250089076488491028366124771626685966372507907955561428828302005966101792717677558516972
9
3
1/10:
0.6220155866232289191335133334862426163729862319825530938832280900210397321485716764639699834288198673
9
3
1/4:
1.038784655604998998025120764957520142683322305067718327138656987897818855690877394481494848245711791
9
3
1/2:
1.651214746548405687838784049758914354395100609091656417679945972085804235659975602594738936482526816
9
3
3/4:
2.433430820588731345747334568352049899904445064183755372963639994634087789517134500844205824370476087
9
3
9/10:
3.315758018702452194741600381807919372711087210984957299052397797273959271884283696446821952846757135
9
3
19/20:
3.948492203457505531511296231157937755764964082530395675582473881504091405408616287359641366245537371
9
3
39/40:
4.578203543702534794789123695842223670463314200105225090079460636803070202660392264214008535299761354
9
3
99/100:
5.428042816591381447234596866764202430778397953763915968560625436410798225490311663752673026996273214
9
3
199/200:
6.095598305637575665459861857049142920598196762780781970494427308024081662427473132339751670293530791
9
3
999/1000:
7.768013407455343114565965862818460879466267411718706641781944935818446930381898056588912897397947016
10
2
1/200:
0.009086313245570783564365546492236137052837413997276738882788906214885497450611279401643906810873140958
10
2
1/100:
0.01817303910875837055120087549861427588491460480696204814997533862612313801230913513645704718172466099
10
2
1/40:
0.04543982112094620515218281721568796763796064510930194603727284267245884206291345504255729609271058672
10
2
1/20:
0.09093131031269235910457989500841200291230015076291844447193375558650683043326765506356684237380536654
10
2
1/10:
0.1822782537553832776938304409740990763177853490907694277503334771934051488345152085838988185106415356
10
2
1/4:
0.4632373407633542974048964273212732046259309496709953693567320491096986181674679173729722397653834528
10
2
1/2:
0.9896837082203127417550841488331109064983177873565362803619581222100985420294221496175553705215793757
10
2
3/4:
1.727115942735858075794515763477864603253380578223294580452104906603174640308370283569851901300792327
10
2
9/10:
2.563207101154240627171323104546010642368254802563715850227968599388286082065395506515555257698148492
10
2
19/20:
3.151064183329407960757798501749110819868078355072304789024438157197674619432446507808272657515752965
10
2
39/40:
3.724708892328898728044893917603689291380727090119091096352935754081256679323500431536846756758673596
10
2
99/100:
4.482028396473318409515589749767506429664701438019429407926153813310435543808408850109735934104057478
10
2
199/200:
5.064873223363723967436015323481552910319696396601696517509573753756831620046224756309549588247429932
10
2
999/1000:
6.486847504143126802391586395930531692788009822027526770686779665903520987189097244128759724490570321
10
3
1/200:
0.1348815300405301022070925100406307541101953126224549805205370720848884463590184433956844041967695487
10
3
1/100:
0.1910417161596995022587827051613290984122064253820826846900369236374825257994614440503863528272382693
10
3
1/40:
0.3034575039495880122012564998750223142895092956969843892037882590657944908449446493276504099176120152
10
3
1/20:
0.4325194047886863086134224669102158538315395868102580201615016422540726023449117973184062899691678180
10
3
1/10:
0.6216476924617359913967699641408066124345062708366857923408098282606217377957274344213237257442196404
10
3
1/4:
1.037099000191590060338244564931250116404756443471434475765346805075472717994475447756302423455490144
10
3
1/2:
1.644688858713789424896793870210619015227282364333849466978070647530228510046310734533890006049044870
10
3
3/4:
2.413835421437801688612077329381553995431906329840817000695612944880371133070825197008953545884832794
10
3
9/10:
3.270308407950086232752076576974916572538496222820507609751436072058011445655166715173848901238868043
10
3
19/20:
3.876776750013181841515897897641392102395414718589428242766837851235372951228658545044384044321566500
10
3
39/40:
4.473958672960888764159773561858080556071341991154290607075478314243864721380060226904292691491173906
10
3
99/100:
5.270161537068830933306951757056611191441738974534575062071729026661501561257980103647412605048194319
10
3
199/200:
5.888171716092056036523477598248066436898605300029927292459183115147815750382208653997013068101330073
10
3
999/1000:
7.410580838024006093702116189642009266539744493880821213862295304638542991417140767787811380921435789
20
2
1/200:
0.008973759510675472372596036853961155223065459341651162740107724416280543198077817398370679882374249258
20
2
1/100:
0.01794789842737623328176867431197158398865876531567282689190552219332730553206312880183157363639386582
20
2
1/40:
0.04487638793654843733809619580364315730162661914637367487806046303087923320759524465137054045899398496
20
2
1/20:
0.08980028254682723499474716624825183766619155327254762279388057779353873885398481939144894998974629364
20
2
1/10:
0.1799826561853200807843451383392616689145872075556072458098562912501380894475213313533364802050612627
20
2
1/4:
0.4568817435503913808868465076880103517787943540224913480602735684097190243813454901714120233055954868
20
2
1/2:
0.9715003656110779565587171985143624383839428916135762583654094730555938768006795525748565806162438028
20
2
3/4:
1.675505688644137031833335687039246068617942886781881816412235500102220792676094630277290453593553009
20
2
9/10:
2.439119930410871689967789795999887496307364741727953875475223936717374658523865402884392877724053130
20
2
19/20:
2.949997797737920514386085357092326775967975388699033991600521293633480845761368320942240477524742166
20
2
39/40:
3.426804273899539620827855226905587704109537957403009839269127655848023741376046998848390579559598380
20
2
99/100:
4.023918007138240873551034887880723336752162120012331223484979664668858023335591125159385737197119992
20
2
199/200:
4.459581801230851029555088282896809529926602870234530559406220408694785502884557592424919902791758880
20
2
999/1000:
5.444038124583131910597985029739490081456068374720725555161627136077203181077261565416251569174083954
20
3
1/200:
0.1348644974011495047847236683585013324071785039474846364090955955901488684115946959496822988275708759
20
3
1/100:
0.1909933440493365771048586833614862315429545342401860769180146681695850647937765560428332441162653168
20
3
1/40:
0.3032639263492943469139449301778538096200625133751987906922763157892328524361668768641861644206721073
20
3
1/20:
0.4319603249920468856803749608987691335977492034953077245840152298812197403561963546064702698866761801
20
3
1/10:
0.6199965306778648359431072135555047668304745999709635416784919707335375544631583481111424510494877436
20
3
1/4:
1.029568135784233450245429139644982590232559750565447953414769546294567447140705659072347801615405421
20
3
1/2:
1.615831486627128708104213865641769924837382468676513785280180904933652066382253154529956706951540593
20
3
3/4:
2.328707761970865812479400942157469277252358580550917264174620769797034695547976505278516403521438664
20
3
9/10:
3.077480014934808363810068203436856598978347181015840287629946471082780953371956633744837212961261177
20
3
19/20:
3.577934725220115373544368525410792272757778210260321423826644216593843958350835757094371589683271219
20
3
39/40:
4.047321852853385348090650518488414482427897286660469699312214149088837661676971659350688785904586472
20
3
99/100:
4.639220082834274209306993944934902674321421585702766572796147163995951476589591824916667462089728374
20
3
199/200:
5.074017840219844607212890018162939979880174888717894559795544838139588619670974708755587069931136145
20
3
999/1000:
6.064884378814879525935264377598805099967104398468845958610285030794394256614080140858992710780138081
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 $Q_{k,\nu}=(\max_i Z_i-\min_i Z_i)/\sqrt{V/\nu}$ [1].
Parameters
$\nu$
—   degrees of freedom ($\nu$ is a positive integer)
$k$
—   number of means ($k$ is an integer with $k\geq2$)
$p$
—   lower-tail probability ($0<p<1$)
Formulas
(1)
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 incomplete beta function and $I_z(a,b)=B(z;a,b)/B(a,b)$ is its regularized form [4].
(2)
In terms of Quantiles of Student's $t$-distribution, $q_{2,\nu,p}=\sqrt{2}\,t_{\nu,(1+p)/2}$.
(3)
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.
(4)
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.
(5)
$G_{k,\nu}(q_{k,\nu,p})=p$ and $1-G_{k,\nu}(q_{k,\nu,1-\alpha})=\alpha$ for $0<p,\alpha<1$. If $Q^{\mathrm{upper}}_{k,\nu}(\alpha)$ denotes the upper-tail critical value satisfying $\Pr(Q_{k,\nu}>Q^{\mathrm{upper}}_{k,\nu}(\alpha))=\alpha$, then $q_{k,\nu,p}=Q^{\mathrm{upper}}_{k,\nu}(1-p)$.
(6)
For one degree of freedom, $q_{2,1,p}=\sqrt{2}\tan(\pi p/2)$ and $q_{3,1,p}=\sqrt{\frac{1}{2\sin^2(\pi(1-p)/6)}-2}$.
(7)
For two means and two degrees of freedom, $q_{2,2,p}=2p/\sqrt{1-p^2}$.
Comments
(8)
The quantiles use the lower-tail probability convention. In particular, $p=19/20$ and $p=99/100$ correspond to the upper-tail significance levels $0.05$ and $0.01$ used in [3].
(9)
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 (3).
(10)
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. The distribution itself is defined for every integer $k\geq2$.
(11)
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 [1].
(12)
The probabilities $p$ are exact rationals in lowest terms. Their selection includes the median, the quartiles, conventional significance levels and their complements, and the upper $0.1\%$ critical value.
Programs
(P1)
Python
from scipy.stats import studentized_range
studentized_range.ppf(19/20, 3, 10)  # machine-precision lower-tail quantile
(P2)
R
qtukey(19/20, nmeans=3, df=10, nranges=1, lower.tail=TRUE)
(P3)
Python
# Numerical inversion for three means; the attached generator certifies its result.
import mpmath as mp
mp.mp.dps = 60
nu, p = 10, mp.mpf(19)/20
def cdf(q):
    return 1 - 6/mp.pi * mp.quad(
        lambda theta: (1+q*q/(2*nu*mp.cos(theta)**2))**(-mp.mpf(nu)/2),
        [0, mp.pi/6])
mp.findroot(lambda q: cdf(q)-p, (3, 4))
Links
Similar tables
Quantiles of the standard normal distribution —   the numerator is the range of independent standard normal variables
Quantiles of the chi-squared distribution —   the denominator contains an independent chi-squared variable divided by its degrees of freedom
Quantiles of Student's $t$-distribution —   for two means, the range is distributed as $\sqrt{2}$ times the absolute Student variable, as in (2)
Quantiles of the $F$-distribution —   for two means, $Q_{2,\nu}^2/2$ has an F distribution with degrees of freedom $1,\nu$
Quantiles of the Kolmogorov distribution —   gives Kolmogorov critical values using the same lower-tail probability convention
Values of the upper incomplete gamma function $\Gamma(a,x)$ —   regularized upper incomplete gamma values give upper-tail probabilities of the chi-squared denominator variable
Values of the lower incomplete gamma function $\gamma(a,x)$ —   regularized lower incomplete gamma values give lower-tail probabilities of the chi-squared denominator variable
Values of the incomplete beta function $B(x;a,b)$ —   the regularized incomplete beta function gives the two-mean cumulative distribution function in (1)
Values of the error function $\operatorname{erf}(x)$ —   the error function gives the normal cumulative distribution function appearing in the general studentized-range integral [2]
Differential entropies of continuous probability distributions —   includes differential entropies of the normal, chi-squared and Student distributions used in the construction
Kullback-Leibler divergences between probability distributions —   includes divergences between normal distributions, while this table gives thresholds for their studentized sample ranges
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 combination of $k\in\{2,3\}$, $\nu\in\{1,2,\ldots,10,20\}$ 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 308 entries. The degrees of freedom select small samples and a reference row at $\nu=20$ from [3]; 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)
How they were obtained:

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 (4) 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.

more

The 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 308 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 308 entries, with maximum CDF discrepancy $1.82\cdot10^{-13}$. Of 44 shared printed values in [3], 43 agree within half a last-place unit; the remaining printed value is discussed in its entry comment. All 44 agree within one printed unit. The generator, independent checker, source comparisons and numerical evidence are attached.

parameter selection: The degrees of freedom are $\nu=1,2,\ldots,10,20$, a small-sample subset of the reference rows in [3]. The probabilities use the family convention for statistical quantiles. Degrees of freedom $\nu\geq30$, larger numbers of means and the infinite-degree-of-freedom limit are outside the computed selection. (Unknown key)


API Python package Skill for AI Privacy Legal notice Source
Data licensed CC BY-SA 4.0. No tracking, no analytics, essential cookies only.