Difference Fit vs. Correlation
Claus
in the code below, using Correlation I get 0.501338, using Fit I get
0.514093.
How come?
Thanks for your help,
Claus
In[47]:= GaltonX = {65.04851`, 63.25094`, 64.95532`, 65.7525`,
61.13723`, 63.02254`, 65.37053`, 64.72398`, 66.06509`, 66.96738`,
59.008`, 62.93203`, 63.67063`, 64.07386`, 64.68851`, 65.15466`,
66.37353`, 65.57704`, 67.36765`, 66.75929`, 67.79764`, 69.5348`,
62.54014`, 63.81016`, 64.4565`, 64.9665`, 64.70662`, 65.73829`,
65.50141`, 65.55416`, 66.90294`, 66.57388`, 66.53198`, 67.90613`,
68.29198`, 67.74123`, 68.46917`, 69.38327`, 71.81791`, 62.41615`,
64.49083`, 64.4876`, 63.92303`, 65.44592`, 65.02977`, 65.67038`,
68.66078`, 66.54597`, 64.85954`, 69.60692`, 64.8135`, 65.32703`,
65.83169`, 65.83427`, 65.56135`, 65.8688`, 65.77339`, 67.09116`,
66.61374`, 67.41075`, 67.27716`, 66.61134`, 68.17412`, 67.80555`,
68.32529`, 68.07787`, 68.96795`, 68.9255`, 69.60862`, 70.35646`,
70.00643`, 72.75425`, 62.82147`, 62.86855`, 63.91261`, 64.54118`,
64.51906`, 66.02688`, 65.81011`, 66.01812`, 65.53742`, 65.73346`,
66.77424`, 66.67869`, 66.97585`, 66.94021`, 66.84575`, 67.12008`,
67.74181`, 68.27194`, 67.60691`, 68.44737`, 67.65832`, 68.71193`,
69.32025`, 68.7403`, 69.13395`, 69.34995`, 68.6297`, 69.58258`,
70.39517`, 71.19284`, 70.82429`, 71.65367`, 73.28133`, 63.03753`,
63.66428`, 65.38311`, 64.60205`, 66.40661`, 65.76985`, 65.66729`,
66.05712`, 66.54635`, 66.5723`, 66.64876`, 67.71899`, 68.48726`,
67.82769`, 67.55801`, 68.07763`, 69.16715`, 68.693`, 68.61779`,
68.64797`, 68.61666`, 70.36485`, 70.0099`, 70.13984`, 70.2073`,
71.28063`, 70.90445`, 72.19036`, 74.96203`, 64.88218`, 65.55174`,
65.65889`, 65.92826`, 65.94198`, 67.43646`, 67.48444`, 67.61344`,
68.4518`, 68.49456`, 68.1056`, 69.48308`, 69.07389`, 69.39251`,
69.44294`, 69.52131`, 70.15463`, 69.86911`, 70.44394`, 70.85393`,
71.30074`, 72.13978`, 72.19045`, 75.43393`, 64.51788`, 66.36335`,
65.56855`, 66.78957`, 66.91193`, 68.1714`, 67.50102`, 67.83492`,
68.95255`, 69.26`, 68.74223`, 69.74288`, 70.29942`, 70.04402`,
70.23559`, 70.88075`, 71.19091`, 72.37396`, 73.03345`, 74.8083`,
67.39521`, 67.31523`, 67.77952`, 69.33827`, 68.72404`, 69.98991`,
69.75383`, 70.65942`, 71.10882`, 72.4811`, 72.4801`, 67.52214`,
68.23931`, 68.7997`, 69.89568`, 71.48641`, 70.76482`, 71.9137`,
63.08399`, 68.23241`, 70.05787`, 70.83996`, 72.56944`, 67.93353`,
67.54033`, 72.79855`, 62.67482`, 68.19196`, 65.07215`, 73.42952`,
62.94125`, 62.53667`, 69.90248`, 71.07217`, 71.11059`, 69.48402`,
70.18338`, 65.46124`, 62.04073`, 63.60063`, 65.60613`, 68.02969`,
61.81571`, 63.78886`, 64.89023`, 65.79072`, 66.42371`, 68.80877`,
61.76521`, 64.30302`, 63.69064`, 64.99662`, 65.34899`, 64.81278`,
65.71736`, 66.03148`, 67.04912`, 67.77522`, 69.04964`, 62.1833`,
62.76393`, 63.88391`, 65.41055`, 64.62347`, 65.64621`, 66.16257`,
65.99538`, 65.79786`, 67.42936`, 67.11933`, 67.32907`, 67.86233`,
68.31609`, 67.96359`, 68.68615`, 71.18085`, 62.4296`, 62.88524`,
64.07313`, 63.88597`, 64.8272`, 65.39653`, 64.77344`, 68.66581`,
65.19783`, 64.32621`, 66.08278`, 69.01679`, 64.50194`, 66.07654`,
65.73631`, 66.45485`, 65.63018`, 66.39423`, 67.33687`, 66.83938`,
66.69739`, 67.34708`, 67.35319`, 67.93575`, 67.76608`, 68.11569`,
68.2548`, 68.83427`, 69.40758`, 69.32068`, 70.48`, 70.06429`,
70.63764`, 62.41134`, 63.15274`, 63.51409`, 63.91446`, 64.61683`,
64.86185`, 65.91332`, 66.1992`, 66.45978`, 65.50286`, 66.87795`,
67.0791`, 66.83256`, 67.18231`, 66.51763`, 66.95295`, 66.63586`,
68.18453`, 67.60565`, 68.27918`, 67.97994`, 69.00094`, 69.31139`,
68.88641`, 68.94824`, 69.42186`, 69.49516`, 70.34647`, 69.92358`,
70.69078`, 70.56571`, 72.40997`, 72.47685`, 62.72118`, 63.9001`,
64.50433`, 64.81813`, 65.37582`, 65.82612`, 66.29436`, 65.81042`,
66.80854`, 67.16724`, 67.30525`, 67.54577`, 67.86855`, 68.35003`,
67.60818`, 68.48397`, 68.56724`, 69.07405`, 68.56831`, 69.15373`,
68.60744`, 68.5443`, 69.65536`, 70.41632`, 69.97528`, 70.26613`,
71.31667`, 71.29178`, 72.59442`, 64.40268`, 64.9127`, 66.34756`,
66.04291`, 65.69477`, 66.71512`, 66.64971`, 68.33241`, 67.94335`,
67.96734`, 68.27657`, 68.75008`, 69.05441`, 69.00755`, 69.39488`,
69.7628`, 69.6464`, 70.45427`, 69.89852`, 70.984`, 71.22015`,
71.45442`, 71.60966`, 73.20296`, 63.89132`, 65.99384`, 66.0424`,
67.47658`, 66.68527`, 68.27034`, 67.72898`, 68.28372`, 68.39453`,
69.12925`, 69.10844`, 69.74196`, 69.9987`, 69.56148`, 69.97779`,
70.88691`, 70.64687`, 72.26404`, 71.50569`, 73.01402`, 66.20254`,
67.26312`, 67.8086`, 69.08284`, 69.35107`, 69.59962`, 70.08948`,
70.18385`, 71.26955`, 72.07717`, 72.33894`, 67.06776`, 67.85929`,
69.07274`, 69.59893`, 69.73728`, 70.54066`, 72.40962`, 72.78889`,
68.10144`, 68.6106`, 71.293`, 72.09445`, 74.69236`, 71.16246`,
68.30804`, 60.41857`, 63.94459`, 63.06849`, 63.98036`, 73.89524`,
72.03687`, 61.02471`, 65.39914`, 71.96564`, 71.29643`, 65.15837`,
68.90263`, 64.45192`, 63.99469`, 66.46124`, 67.12556`, 62.27941`,
63.8488`, 65.43612`, 65.62145`, 66.16956`, 67.62675`, 60.21393`,
62.99378`, 63.57239`, 65.15088`, 65.13462`, 64.79732`, 66.24856`,
66.22743`, 66.6056`, 67.99167`, 69.04862`, 62.12902`, 63.16227`,
64.45497`, 63.75483`, 64.63959`, 64.59106`, 66.06003`, 65.98544`,
65.66176`, 67.11501`, 67.31302`, 67.16149`, 68.38679`, 67.64175`,
67.55466`, 68.53614`, 70.02702`, 61.07412`, 62.72387`, 64.28642`,
64.11073`, 63.68318`, 64.68393`, 64.66753`, 67.95848`, 64.60933`,
66.07775`, 65.25267`, 69.67201`, 65.33965`, 65.54375`, 66.1112`,
66.22968`, 66.00879`, 66.03248`, 65.94624`, 67.09734`, 66.98893`,
67.38837`, 67.4075`, 67.31108`, 68.27151`, 68.33586`, 67.56752`,
68.23182`, 69.07604`, 68.77033`, 70.49154`, 70.00767`, 69.52007`,
61.50617`, 62.93875`, 64.10318`, 63.85157`, 65.08731`, 64.62588`,
65.8912`, 66.24386`, 65.66031`, 66.05344`, 66.30711`, 67.15303`,
66.90986`, 66.89436`, 66.56566`, 67.23789`, 67.05684`, 67.94515`,
67.94026`, 67.84234`, 67.91887`, 68.63961`, 68.60477`, 69.04036`,
69.03336`, 69.12497`, 68.53779`, 70.24335`, 69.84308`, 69.90057`,
71.48946`, 72.37013`, 71.94413`, 61.01904`, 63.0108`, 64.5793`,
65.39658`, 64.84995`, 65.54542`, 65.56421`, 66.09747`, 67.40602`,
66.5839`, 67.41035`, 67.21634`, 68.23175`, 67.88053`, 68.0153`,
67.69332`, 68.85462`, 68.73264`, 69.0711`, 68.79964`, 68.79876`,
69.32241`, 69.60923`, 69.76071`, 70.1611`, 70.12109`, 71.10802`,
71.44164`, 71.75006`, 64.35591`, 65.16253`, 66.09529`, 65.78674`,
65.53927`, 66.71703`, 67.17284`, 67.16853`, 68.06061`, 67.85988`,
68.1968`, 69.29537`, 68.84067`, 69.18095`, 68.69659`, 69.45833`,
70.0483`, 69.6916`, 70.20658`, 70.79086`, 70.87977`, 71.30702`,
72.3266`, 72.7824`, 64.40052`, 64.89121`, 65.84506`, 67.35461`,
67.44953`, 67.86439`, 67.92443`, 68.49998`, 67.60602`, 69.37288`,
69.34931`, 69.48306`, 70.21887`, 70.02161`, 69.83698`, 69.6041`,
71.49829`, 72.16894`, 72.41045`, 72.52521`, 64.6702`, 66.78417`,
67.83401`, 68.15907`, 69.19301`, 68.86739`, 70.11126`, 70.12007`,
71.03691`, 71.43196`, 72.04564`, 72.738`, 68.02564`, 69.04939`,
68.86356`, 69.63421`, 71.24085`, 71.95679`, 72.89983`, 65.86028`,
68.48883`, 70.72676`, 72.34582`, 72.59922`, 68.82986`, 73.49987`,
70.0599`, 67.9291`, 64.66998`, 60.7795`, 74.94058`, 70.64925`,
60.91526`, 65.28934`, 71.362`, 71.66893`, 62.68868`, 71.39326`,
66.56302`, 67.18788`, 62.532`, 64.85512`, 66.94161`, 62.3603`,
63.58145`, 65.44633`, 65.00539`, 66.43717`, 66.6676`, 59.4898`,
63.00108`, 64.0892`, 64.0753`, 64.52103`, 65.19231`, 66.30604`,
66.03081`, 66.83096`, 68.19724`, 68.6599`, 61.29369`, 63.26089`,
63.86162`, 64.13176`, 64.81105`, 64.55987`, 66.19437`, 66.5`,
66.43436`, 67.30468`, 67.36559`, 67.29162`, 67.69874`, 68.38221`,
67.76712`, 69.29587`, 69.66477`, 60.03436`, 62.1356`, 63.64047`,
63.82278`, 64.46555`, 64.73798`, 65.43021`, 65.07381`, 69.29517`,
67.67996`, 64.94267`, 69.47933`, 65.05963`, 65.01253`, 65.61963`,
65.94819`, 65.68803`, 65.87689`, 65.91481`, 67.41946`, 66.53833`,
67.41995`, 67.21977`, 67.33721`, 68.11283`, 68.02017`, 68.02829`,
68.35139`, 68.80254`, 69.49307`, 70.2297`, 69.71997`, 70.16651`,
61.07369`, 62.66947`, 64.32254`, 64.27362`, 64.96148`, 64.77222`,
66.2365`, 66.10686`, 66.42772`, 66.22727`, 65.87574`, 67.03017`,
67.48545`, 66.7367`, 66.69181`, 67.31766`, 67.1219`, 67.82592`,
68.4376`, 67.74551`, 68.0934`, 67.71665`, 68.99349`, 69.24616`,
69.2828`, 69.26254`, 68.75605`, 69.62519`, 69.61942`, 69.94715`,
70.58916`, 72.34058`, 71.56142`, 72.94457`, 63.33777`, 64.3445`,
65.14077`, 65.2424`, 65.90397`, 65.98763`, 65.57065`, 65.82733`,
66.88576`, 66.98594`, 67.41537`, 67.93319`, 67.77812`, 68.44999`,
68.32281`, 68.35148`, 68.73678`, 68.97673`, 69.31931`, 68.80442`,
68.90434`, 69.80381`, 69.66712`, 70.38052`, 70.29892`, 71.02634`,
71.28432`, 72.03259`, 63.26539`, 64.87623`, 65.72722`, 66.13831`,
66.48902`, 66.06395`, 66.73532`, 67.12329`, 67.57161`, 68.18286`,
68.33868`, 68.14658`, 69.17022`, 69.17597`, 68.52789`, 68.84874`,
69.69833`, 69.86528`, 70.48646`, 71.16238`, 70.58946`, 70.54353`,
72.38576`, 73.18522`, 64.12344`, 64.62855`, 65.74244`, 67.11563`,
66.81497`, 68.38001`, 68.33835`, 68.25066`, 68.36488`, 68.67873`,
68.70293`, 69.0466`, 70.17375`, 69.86849`, 70.15821`, 70.24204`,
71.41811`, 70.66031`, 72.42614`, 73.04341`, 64.6891`, 67.09928`,
67.79858`, 68.37299`, 69.19364`, 68.5938`, 69.86969`, 70.17952`,
70.51811`, 71.00743`, 71.83086`, 72.81121`, 67.95451`, 67.81317`,
69.32812`, 70.45127`, 71.38916`, 72.31022`, 71.8821`, 64.61601`,
67.75382`, 69.90335`, 70.87039`, 72.71533`, 69.03942`, 72.44463`,
69.07253`, 70.1758`, 72.37219`, 59.63777`, 73.27737`, 68.12621`,
63.15234`, 66.00489`, 71.45303`, 68.1456`, 65.25527`, 70.41584`,
68.38452`, 61.89133`, 64.98391`, 66.21887`, 59.48391`, 62.77645`,
63.86507`, 64.56486`, 65.60265`, 66.28333`, 70.55256`, 62.14539`,
64.37906`, 64.35191`, 64.91304`, 65.33085`, 65.17545`, 66.10285`,
66.05004`, 67.38231`, 67.83999`, 70.27309`, 63.19263`, 63.66834`,
64.12671`, 64.71388`, 65.14662`, 65.94428`, 65.67785`, 65.95134`,
67.30819`, 66.69404`, 66.56885`, 66.80951`, 68.24011`, 67.5344`,
68.4157`, 69.09588`, 70.81036`, 61.68183`, 63.04493`, 64.34945`,
63.89336`, 65.48176`, 64.726`, 64.65128`, 67.47271`, 66.76701`,
64.98147`, 69.38684`, 63.7123`, 64.63358`, 66.00037`, 65.96621`,
66.49751`, 66.27177`, 65.67892`, 66.86814`, 66.73495`, 66.53357`,
67.05051`, 67.04958`, 67.55908`, 67.58076`, 67.51942`, 68.12592`,
68.57465`, 68.52484`, 69.20286`, 69.7551`, 69.91868`, 70.50329`,
62.83528`, 63.24504`, 64.44602`, 64.31819`, 64.73521`, 64.72352`,
65.97741`, 65.63739`, 65.83449`, 65.87064`, 66.61001`, 66.75821`,
66.77012`, 67.21373`, 67.09047`, 66.93359`, 68.03852`, 67.98476`,
68.25046`, 68.26078`, 68.21235`, 68.54029`, 68.50469`, 69.13248`,
69.21981`, 69.34385`, 69.41153`, 70.25056`, 69.68641`, 70.51091`,
71.29598`, 72.20218`, 72.33929`, 62.91628`, 63.93525`, 64.59851`,
65.47401`, 65.864`, 66.30684`, 65.77587`, 66.48386`, 67.31983`,
67.39901`, 67.05939`, 68.45161`, 67.7061`, 68.08389`, 67.58923`,
67.80388`, 69.25919`, 69.05302`, 68.70921`, 69.02788`, 68.94943`,
68.81536`, 69.91801`, 70.45072`, 69.96349`, 69.75787`, 71.38281`,
71.08122`, 72.74936`, 65.26855`, 66.01549`, 65.5142`, 65.62374`,
65.92199`, 67.11129`, 66.89727`, 67.94195`, 67.7274`, 67.54999`,
67.96013`, 68.83038`, 69.43917`, 68.55957`, 69.10638`, 70.4814`,
69.92992`, 70.04238`, 69.56779`, 71.01077`, 70.62603`, 71.46383`,
71.62112`, 73.87758`, 65.00573`, 66.30354`, 65.86743`, 66.86667`,
67.15958`, 68.06904`, 67.57382`, 67.63354`, 69.41356`, 68.87192`,
69.07779`, 69.81177`, 70.01463`, 69.56649`, 70.254`, 70.68296`,
70.87874`, 72.45099`, 73.04359`, 74.40915`, 67.40139`, 66.53122`,
68.04593`, 68.82772`, 69.33069`, 70.34676`, 70.173`, 70.80883`,
70.7029`, 72.35959`, 72.3661`, 67.17534`, 67.84045`, 68.9053`,
70.3752`, 71.18523`, 71.41254`, 71.70673`, 72.60072`, 67.57652`,
68.55814`, 70.95019`, 71.99783`, 75.17469`, 73.07157`, 69.92407`,
65.54323`, 72.56261`, 72.15051`, 63.22006`, 73.2645`, 65.81296`,
67.70657`, 66.99681`, 71.33181`, 71.78314`, 70.73837`, 70.30609`};
In[59]:= GaltonY = {59.77827`, 63.21404`, 63.34242`, 62.79238`,
64.28113`, 64.24221`, 64.08231`, 63.99574`, 64.61338`, 63.97944`,
65.24451`, 65.35102`, 65.67992`, 65.43664`, 65.29391`, 64.79017`,
65.01881`, 65.5464`, 65.08145`, 65.49008`, 65.1352`, 65.51262`,
66.58436`, 66.37842`, 66.14782`, 65.97259`, 65.95887`, 66.45202`,
65.74797`, 65.97164`, 65.90548`, 65.90773`, 66.47103`, 65.75023`,
66.34666`, 66.08577`, 66.28037`, 66.34354`, 66.41711`, 67.22558`,
67.46394`, 66.99362`, 66.76647`, 67.0342`, 66.84316`, 69.2989`,
69.09114`, 61.66358`, 65.7288`, 68.98012`, 67.37397`, 67.17459`,
67.06712`, 67.29446`, 67.62665`, 67.47923`, 66.92162`, 67.58324`,
67.21191`, 66.81629`, 67.24268`, 67.51036`, 67.57026`, 67.17258`,
67.54129`, 67.37113`, 67.59714`, 67.25104`, 66.82533`, 66.72679`,
66.92996`, 67.64847`, 68.04291`, 68.4885`, 68.04777`, 68.34573`,
68.33829`, 68.3138`, 68.18333`, 68.51056`, 68.12014`, 68.31563`,
68.04774`, 68.61273`, 67.9425`, 68.08381`, 68.36754`, 67.85181`,
68.19806`, 68.01897`, 68.23431`, 67.89742`, 68.62409`, 67.97912`,
68.2668`, 67.94192`, 68.62072`, 68.23857`, 68.58554`, 68.06568`,
68.38188`, 67.75943`, 68.59991`, 67.91483`, 67.83861`, 68.77562`,
69.60782`, 69.69143`, 69.41491`, 69.68931`, 69.20775`, 69.45627`,
69.59614`, 69.27994`, 69.52338`, 68.74406`, 69.30987`, 69.22571`,
69.24073`, 69.4912`, 69.11921`, 68.93751`, 69.45774`, 69.27599`,
69.22869`, 68.70547`, 69.34638`, 68.87137`, 69.30262`, 69.1579`,
69.62727`, 69.11014`, 69.09596`, 69.03281`, 69.85677`, 70.06274`,
69.89765`, 70.26185`, 70.52663`, 70.52744`, 69.7865`, 70.36684`,
69.97604`, 69.83308`, 70.69951`, 70.16871`, 70.11632`, 69.95433`,
70.31279`, 69.80903`, 69.96792`, 69.93939`, 69.73486`, 70.08397`,
70.01015`, 70.65025`, 70.01253`, 70.09077`, 71.34319`, 70.81542`,
71.44929`, 71.18676`, 71.66276`, 71.39516`, 70.73612`, 71.2608`,
71.04721`, 71.2592`, 70.89486`, 71.31732`, 71.55712`, 71.17701`,
71.03412`, 71.37275`, 71.64741`, 71.1205`, 70.91756`, 71.66455`,
72.39198`, 71.87928`, 72.34482`, 72.17265`, 72.52078`, 71.98836`,
72.34482`, 72.45347`, 72.26945`, 71.96844`, 72.18169`, 72.7864`,
73.0414`, 72.87858`, 73.17304`, 73.63274`, 72.88245`, 73.18148`,
74.3209`, 74.40727`, 73.76791`, 73.92119`, 73.75503`, 75.56214`,
75.70222`, 77.21338`, 61.28882`, 74.28343`, 70.93102`, 75.34802`,
61.78758`, 64.00009`, 62.4658`, 73.70465`, 66.3495`, 62.20543`,
65.421`, 63.9922`, 62.78458`, 63.52486`, 63.49966`, 62.89658`,
63.72784`, 63.73231`, 64.39579`, 63.70072`, 64.44796`, 64.2795`,
65.18412`, 65.10118`, 65.56277`, 64.94454`, 65.14227`, 65.16843`,
65.58149`, 65.63474`, 64.87191`, 65.4176`, 64.7231`, 65.84559`,
65.77333`, 66.67371`, 66.54726`, 66.43228`, 66.3326`, 66.24935`,
66.39759`, 66.69366`, 66.22114`, 66.43499`, 66.65758`, 65.76873`,
66.20784`, 66.30166`, 65.81894`, 66.34562`, 66.88369`, 66.8162`,
67.36245`, 67.68308`, 67.19885`, 67.26903`, 67.45131`, 68.95907`,
62.013`, 64.10658`, 66.00879`, 65.77796`, 66.94446`, 66.81229`,
67.01431`, 67.43705`, 66.8442`, 67.25391`, 67.52273`, 67.17782`,
67.6068`, 66.8991`, 66.74675`, 67.16928`, 67.23833`, 66.75589`,
66.79828`, 67.488`, 67.1104`, 67.06455`, 66.93899`, 67.54723`,
66.85662`, 67.70971`, 67.88296`, 68.66534`, 68.58164`, 68.44493`,
68.41199`, 68.44665`, 68.38258`, 67.92987`, 68.3033`, 67.97479`,
68.18304`, 68.57984`, 68.56974`, 67.81854`, 67.8796`, 68.22794`,
68.01258`, 67.85023`, 67.80077`, 68.59968`, 68.08671`, 67.87656`,
68.47849`, 68.60006`, 68.07838`, 68.08199`, 67.73211`, 68.4132`,
68.59541`, 68.28458`, 68.06489`, 68.41445`, 69.36998`, 69.00435`,
69.37297`, 68.89496`, 69.37382`, 69.34415`, 68.96686`, 69.59887`,
69.24597`, 69.68301`, 68.90541`, 69.07283`, 69.03297`, 69.18202`,
69.64233`, 68.82818`, 69.24165`, 69.10265`, 68.98512`, 69.31551`,
69.20435`, 68.93765`, 69.36661`, 68.71181`, 68.75662`, 69.35439`,
69.42891`, 69.52003`, 69.2275`, 69.95069`, 69.9199`, 69.98123`,
69.74222`, 69.82326`, 69.90223`, 70.2992`, 69.83745`, 69.94847`,
70.27479`, 70.67752`, 70.38903`, 70.37578`, 70.60668`, 70.16921`,
69.85223`, 70.23353`, 70.28373`, 70.33881`, 69.81443`, 70.54739`,
69.96746`, 69.73702`, 69.8604`, 70.87111`, 71.36657`, 71.1878`,
70.75001`, 70.72998`, 71.32092`, 70.88794`, 70.9085`, 70.87662`,
71.19803`, 71.31925`, 71.22916`, 71.35939`, 71.60205`, 70.97309`,
71.1886`, 71.40319`, 71.39524`, 70.88752`, 71.27449`, 72.10289`,
72.03197`, 71.9969`, 71.72577`, 71.92914`, 72.20518`, 72.29401`,
72.5854`, 72.04673`, 71.80541`, 71.80464`, 73.06559`, 73.44548`,
73.19223`, 73.37471`, 73.22554`, 72.88357`, 72.77826`, 73.26989`,
73.96165`, 74.64342`, 73.92565`, 74.58767`, 74.31029`, 75.10003`,
77.15504`, 60.78204`, 62.36303`, 66.40186`, 58.50708`, 70.00052`,
70.99433`, 64.96417`, 70.47639`, 69.23383`, 64.90184`, 68.7749`,
69.85805`, 62.52373`, 63.28805`, 63.52861`, 62.87456`, 63.91754`,
64.32325`, 64.22405`, 64.59643`, 64.28548`, 63.78835`, 65.66404`,
64.80838`, 65.63528`, 64.74897`, 64.79081`, 65.10988`, 65.55215`,
65.30083`, 65.1426`, 65.09433`, 64.81397`, 65.95058`, 66.37511`,
66.38624`, 66.31904`, 65.80919`, 66.09556`, 66.49077`, 66.32632`,
66.59779`, 66.00909`, 66.45585`, 66.00818`, 66.46605`, 66.37783`,
66.34685`, 66.31558`, 66.62303`, 66.79229`, 67.5336`, 67.15587`,
66.78051`, 67.25544`, 66.79006`, 66.88298`, 68.1121`, 63.61572`,
63.17978`, 65.67211`, 67.23403`, 67.4089`, 67.57913`, 67.20643`,
67.10018`, 66.80557`, 67.09308`, 66.91372`, 66.77528`, 66.97572`,
67.66639`, 67.62089`, 67.66841`, 67.32572`, 67.04153`, 66.96445`,
67.07223`, 67.169`, 67.48077`, 67.67057`, 66.86465`, 67.04437`,
67.72523`, 68.52969`, 67.8741`, 67.84693`, 68.26632`, 68.18364`,
68.55723`, 67.77137`, 67.92976`, 67.72522`, 67.98916`, 68.0679`,
67.772`, 68.11035`, 68.16623`, 68.61054`, 68.2893`, 68.38263`,
68.6212`, 68.27802`, 67.93914`, 67.7105`, 68.351`, 68.14712`,
67.97421`, 68.23682`, 68.05699`, 68.49856`, 68.58894`, 67.74098`,
68.69765`, 68.63531`, 68.37844`, 69.11352`, 69.35385`, 69.5135`,
69.11`, 69.64324`, 69.40655`, 69.18575`, 68.80709`, 69.11679`,
69.02706`, 69.05703`, 69.14581`, 68.7308`, 68.9798`, 68.95303`,
69.1639`, 68.81649`, 68.73746`, 69.00102`, 69.28222`, 69.37676`,
68.90879`, 69.32318`, 69.63872`, 69.43581`, 68.97885`, 69.54252`,
68.73497`, 69.20127`, 70.36167`, 70.10118`, 70.39507`, 70.03877`,
69.76556`, 70.39131`, 70.0372`, 70.25908`, 69.87081`, 70.69845`,
70.25756`, 69.7177`, 70.31993`, 69.95735`, 69.72691`, 70.22698`,
70.02038`, 70.32035`, 70.39947`, 69.78074`, 69.7598`, 70.53827`,
70.03449`, 70.59135`, 71.13378`, 71.15371`, 71.03018`, 70.93936`,
70.78603`, 71.50695`, 71.57023`, 70.83563`, 70.96118`, 71.55428`,
71.08274`, 71.49587`, 71.36997`, 71.49891`, 71.01177`, 71.62784`,
71.36701`, 71.23426`, 70.90129`, 71.53087`, 72.441`, 72.5999`,
71.79167`, 72.00806`, 72.01746`, 72.44981`, 71.92864`, 72.0908`,
72.57398`, 72.53335`, 71.78483`, 72.57594`, 73.06795`, 73.18852`,
73.39371`, 73.3754`, 73.67679`, 72.84789`, 72.99608`, 74.67822`,
73.8507`, 74.35692`, 73.99906`, 73.92525`, 75.67784`, 76.10484`,
78.2476`, 61.89619`, 69.86215`, 62.26801`, 71.40839`, 71.60925`,
65.51351`, 71.86574`, 71.16645`, 64.53321`, 65.25053`, 69.62871`,
65.10223`, 61.03526`, 63.12132`, 63.5968`, 63.47848`, 63.83162`,
64.18388`, 64.66888`, 64.06363`, 64.38072`, 64.60295`, 64.81735`,
65.22864`, 65.62354`, 65.62755`, 64.8584`, 65.6482`, 65.6904`,
65.58218`, 65.08945`, 64.87234`, 65.68794`, 66.54461`, 66.23255`,
65.91034`, 66.29882`, 66.33603`, 66.19024`, 66.12728`, 66.1938`,
65.82389`, 66.61389`, 66.45299`, 66.25633`, 66.53703`, 66.28457`,
65.90354`, 66.61969`, 66.59349`, 67.30641`, 67.41523`, 67.51658`,
66.94071`, 67.11246`, 66.99107`, 66.94124`, 68.06898`, 67.47908`,
63.00731`, 67.40208`, 68.75691`, 66.90827`, 67.28305`, 67.15978`,
67.12405`, 67.07313`, 67.17665`, 66.74724`, 66.81865`, 67.43735`,
67.37322`, 66.96049`, 67.40154`, 66.91834`, 66.93007`, 66.78425`,
66.7842`, 67.15158`, 67.45025`, 67.68985`, 67.13767`, 67.60589`,
67.96334`, 68.65674`, 68.42972`, 68.01111`, 68.61891`, 68.50647`,
67.77439`, 68.09837`, 68.19228`, 68.54918`, 68.59413`, 68.65282`,
68.40294`, 68.54259`, 68.59444`, 67.87885`, 68.0454`, 67.96346`,
68.55092`, 68.07785`, 67.93252`, 68.5916`, 67.90891`, 68.63842`,
68.6514`, 68.40634`, 68.19563`, 68.3391`, 68.13175`, 67.79551`,
68.24471`, 67.96366`, 68.02669`, 68.07695`, 69.23339`, 69.22874`,
68.9062`, 69.33247`, 69.53206`, 69.02418`, 69.26288`, 69.55433`,
69.3664`, 68.96569`, 69.33038`, 69.10425`, 69.29199`, 69.26009`,
68.71424`, 69.48488`, 69.48141`, 68.88311`, 68.83524`, 69.42378`,
68.82387`, 69.06951`, 69.18528`, 69.26989`, 68.83727`, 69.11594`,
69.39569`, 68.70258`, 70.39288`, 70.49606`, 70.11817`, 69.75304`,
70.25238`, 70.28837`, 69.71807`, 70.13835`, 70.41752`, 69.91638`,
69.7803`, 69.90789`, 70.29221`, 70.20136`, 70.42766`, 70.40239`,
70.65713`, 69.7276`, 70.45174`, 70.52536`, 70.45682`, 70.52011`,
70.25621`, 70.32714`, 71.39933`, 71.03487`, 71.61276`, 70.90129`,
71.36831`, 71.46158`, 71.22939`, 71.59368`, 71.6476`, 71.33885`,
71.13133`, 71.52388`, 70.95837`, 71.48199`, 70.85433`, 71.41922`,
71.30872`, 71.22321`, 71.4506`, 70.76122`, 72.65576`, 71.96434`,
72.3071`, 72.2446`, 72.38029`, 72.04211`, 71.77644`, 72.54877`,
72.57381`, 71.90584`, 72.62022`, 72.4218`, 73.13823`, 73.60773`,
73.2649`, 73.0769`, 73.69792`, 73.60551`, 72.87015`, 73.90142`,
74.32431`, 73.87776`, 74.20972`, 74.47833`, 75.07304`, 76.3516`,
78.36479`, 61.22021`, 72.63356`, 64.94291`, 73.02984`, 71.81569`,
65.29123`, 60.90911`, 72.131`, 70.98853`, 61.7019`, 67.51436`,
64.37789`, 63.2825`, 63.47892`, 63.66235`, 63.94637`, 63.93831`,
63.94722`, 64.08187`, 64.68702`, 64.38337`, 63.91132`, 64.84282`,
65.33216`, 65.2375`, 65.47348`, 65.16309`, 65.35727`, 65.59545`,
64.70258`, 64.76966`, 65.55892`, 65.42338`, 66.54624`, 65.70133`,
66.12315`, 66.2678`, 66.55539`, 66.42398`, 66.50633`, 66.27712`,
66.42142`, 66.25827`, 66.1285`, 66.53558`, 66.32575`, 66.11991`,
65.87925`, 65.70978`, 66.10198`, 67.57237`, 67.4361`, 67.00761`,
67.21048`, 67.1921`, 67.39528`, 67.44979`, 68.18883`, 62.74094`,
66.34954`, 65.69966`, 68.36264`, 67.52548`, 67.24632`, 67.21519`,
67.45383`, 66.93472`, 67.6832`, 66.97218`, 66.73049`, 66.75153`,
67.4346`, 66.92896`, 67.49282`, 66.93944`, 67.03135`, 66.91542`,
66.97758`, 67.14932`, 67.53093`, 67.52202`, 67.27928`, 67.56893`,
67.75773`, 68.10565`, 68.49664`, 68.5851`, 67.85579`, 68.60796`,
68.19156`, 68.25828`, 68.15401`, 68.18623`, 68.16064`, 68.6097`,
68.34391`, 68.65678`, 68.51914`, 67.72093`, 67.96986`, 67.75073`,
68.09745`, 68.64602`, 68.2318`, 68.18244`, 68.15011`, 68.13611`,
68.10048`, 68.53037`, 68.37728`, 68.17652`, 67.7452`, 67.93037`,
68.25872`, 68.33639`, 68.27405`, 69.5816`, 68.70027`, 68.93075`,
68.78663`, 69.0649`, 68.99285`, 68.99516`, 68.73929`, 69.47357`,
69.08392`, 69.07281`, 69.62286`, 69.35683`, 69.16149`, 68.81413`,
68.9529`, 69.47561`, 69.5506`, 69.38889`, 69.57241`, 69.37092`,
68.92229`, 68.71081`, 69.2671`, 68.74473`, 68.81509`, 69.47472`,
69.69307`, 69.65334`, 70.46902`, 70.00607`, 70.61711`, 70.51318`,
69.97204`, 69.92313`, 70.46938`, 69.91923`, 69.71908`, 70.61155`,
70.20868`, 70.27586`, 70.18348`, 70.5115`, 70.50015`, 70.24017`,
70.001`, 70.21726`, 70.20554`, 69.97788`, 70.36632`, 70.38931`,
70.18593`, 70.23168`, 70.74197`, 71.47899`, 71.41695`, 71.20207`,
70.87987`, 71.26557`, 71.41653`, 71.24074`, 71.15528`, 71.20588`,
70.98515`, 71.35642`, 71.21331`, 71.59639`, 71.25854`, 71.53325`,
70.92787`, 71.54787`, 71.07293`, 70.86163`, 72.67319`, 71.77245`,
72.64278`, 72.48159`, 71.9078`, 72.18324`, 72.55104`, 72.33861`,
72.07098`, 72.30647`, 72.05185`, 72.75204`, 72.79178`, 73.50606`,
73.68578`, 72.79593`, 73.41049`, 73.02077`, 73.20758`, 74.49237`,
73.81263`, 73.81376`, 73.82642`, 73.81739`, 75.57544`, 77.23474`,
60.05859`, 76.81927`, 66.72684`, 58.79456`, 67.89277`, 61.04946`,
59.81693`, 70.75232`, 68.26774`, 69.30589`, 69.30199`, 67.015`};
In[60]:= lm = LinearModelFit[GaltonDat, {1, x}, x]
Out[60]= FittedModel[\!\(\*
PanelBox[
TagBox[
RowBox[{"33.88660435407788`", " ", "+",
RowBox[{"0.5140930386233082`", " ", "x"}]}],
Short],
FrameMargins->5]\)]
In[62]:= lm["BestFit"]
Out[62]= 33.8866 + 0.514093 x
In[63]:= r2 = Correlation[GaltonX, GaltonY]
Out[63]= 0.501338
Darren Glosemeyer
The correlation and slope are two different quantities. Correlation
gives a measure of how well one variable could predict the other, while
slope is part of the linear prediction. Note, also that correlation is
bounded by -1 and 1, while slope could be outside that range. If we take
the formulas from simple linear regression, we see that the results are
correct and that correlation and slope will only be equal if the x and y
sums of squares, sxx and syy, are equal.
In[3]:= xbar = Mean[GaltonX];
In[4]:= ybar = Mean[GaltonY];
In[5]:= n = Length[GaltonX];
In[6]:= sxy = GaltonX.GaltonY - n*xbar*ybar
Out[6]= 4171.58
In[7]:= sxx = GaltonX.GaltonX - n*xbar^2
Out[7]= 8114.44
In[8]:= syy = GaltonY.GaltonY - n*ybar^2
Out[8]= 8532.58
In[9]:= cor = sxy/Sqrt[sxx*syy]
Out[9]= 0.501338
In[10]:= slope = sxy/sxx
Out[10]= 0.514093
Darren Glosemeyer
Wolfram Research
Ray Koopman
only when the two standard deviations equal one another.
Dimensions[XY = Transpose@{X,Y}]
{1078,2}
f = Fit[XY,{1,x},x]
a = f[[1]]
b = f[[2,1]]
33.8866 + 0.514093 x
33.8866
0.514093
{mx,my} = Mean@XY
{sx,sy} = StandardDeviation@XY
r = Correlation[X,Y]
{67.6871,68.6841}
{2.74487,2.8147}
0.501338
b == r*sy/sx
True
a == my - b*mx
True
Jens-Peer Kuska
can you explain what "using Fit" mean
lm["RSquared"] // Sqrt
gives 0.501338
and this corresponds to the correlation value.
Regards
Jens
Bill Rowe
>Hi, in the code below, using Correlation I get 0.501338, using Fit I
>get 0.514093. How come?
<snip>
>In[60]:= lm = LinearModelFit[GaltonDat, {1, x}, x]
>In[62]:= lm["BestFit"]
>Out[62]= 33.8866 + 0.514093 x
>In[63]:= r2 = Correlation[GaltonX, GaltonY]
>Out[63]= 0.501338
You seem to expect the slope of the best fit line to have the
same value as the correlation between x and y. These are two
separate things and will never have the same value with real data.
The slope of the best fit line is given by
Covariance[x,y]/Variance[y]
The correlation between x and y (Pearson's correlation) is given by
Covariance[x,y]/Sqrt[ Variance[x] Variance[y] ]
The only case where the slope and correlation have the same
value is the trivial case where x = y and there is no error
term, i.e., perfect correlation with a slope of 1.
dh
Hi Claus,
you are confusing two different things.
The correlation coefficient is not the slope of the regression line. The
correlation tells you how much the points are "scattered" around the line.
hope this helps, Daniel
DrMajorBob
GaltonDat.
Hence, the following is only a guess as to what you've actually done:
lm=LinearModelFit[Transpose@{GaltonX,GaltonY},{1,x},x]
FittedModel[33.8866+0.514093 x]
lm["BestFit"]
33.8866+0.514093 x
r2=Correlation[GaltonX,GaltonY]
0.501338
Now the question is, why should the two numbers agree? Here's an example
from Help:
data = {{0, 1}, {1, 0}, {3, 2}, {5, 4}};
lm = LinearModelFit[data, x, x];
lm["BestFit"]
0.186441 + 0.694915 x
The corresponding Correlation is
Correlation @@ Transpose@data // N
0.902244
Fit seeks to reproduce Y from X, but Correlation divides covariance by the
two standard deviations:
"Correlation[Subscript[v, 1],Subscript[v, 2]] is equivalent to
Covariance[Subscript[v, 1],Subscript[v,
2]]/(StandardDeviation[Subscript[v, 1]]StandardDeviation[Subscript[v,
2]]). "
That's a measure of how much a certain model explains (without the
constant term, where your model DOES have a constant term), but it IS NOT
a coefficient in the best fit model.
Bobby
Claus
thanks for your comments, I was writing my original post in a hurry, and
I must have been in a hurry when I was writing the original code.
My problem was that I had calculated the slope of the regression line by
multiplying Correlation[GaltonX,GaltonY] with
StandardDeviation[GaltonX]/StandardDeviation[GaltonY]
However, the correct slope of the regression line is
Correlation[GaltonX,GaltonY] *
StandardDeviation[GaltonY]/StandardDeviation[GaltonX]
Below is some code that should clarify things (use GaltonX and GaltonY
from original post).
Thank you,
Claus
In[98]:= lm = LinearModelFit[Transpose@{GaltonX, GaltonY}, {1, x}, x]
lm["BestFit"]
Out[98]= FittedModel[\!\(\*
PanelBox[
TagBox[
RowBox[{"33.88660435407788`", " ", "+",
RowBox[{"0.5140930386233082`", " ", "x"}]}],
Short],
FrameMargins->5]\)]
Out[99]= 33.8866 + 0.514093 x
should be the same if I use the original pairs
In[100]:= lm2 = LinearModelFit[GaltonDat, {1, x}, x]
lm2["BestFit"]
Out[100]= FittedModel[\!\(\*
PanelBox[
TagBox[
RowBox[{"33.88660435407788`", " ", "+",
RowBox[{"0.5140930386233082`", " ", "x"}]}],
Short],
FrameMargins->5]\)]
Out[101]= 33.8866 + 0.514093 x
In[102]:= CorrelXY = Correlation[GaltonX, GaltonY]
Out[102]= 0.501338
In[103]:= SDevX = StandardDeviation[GaltonX]
SDevY = StandardDeviation[GaltonY]
MY = Mean[GaltonY]
MX = Mean[GaltonX]
CovXY = Covariance[GaltonX, GaltonY]
Out[103]= 2.74487
Out[104]= 2.8147
Out[105]= 68.6841
Out[106]= 67.6871
Out[107]= 3.87333
CorrelXY2 should be the same as CorrelXY
In[108]:= CorrelXY2 = CovXY/(SDevX*SDevY)
Out[108]= 0.501338
Slope of the regression line
In[109]:= slopeRegrL = CorrelXY*SDevY/SDevX
Out[109]= 0.514093
In[110]:= interceptRegrL = MY - slopeRegrL*MX
Out[110]= 33.8866
In[111]:= regrLine = interceptRegrL + slopeRegrL*x
Out[111]= 33.8866 + 0.514093 x
Now this is the same as the result from LinearModelFit
In[112]:= lm["BestFit"]
Out[112]= 33.8866 + 0.514093 x
Sjoerd C. de Vries
NOT the coefficient of a linear fit. The correlation between two
variables can have values between -1 and +1, with 0 indicating no
(linear) relationship between them and 1 perfect linear relationship.
The coefficient of x in the fit is not restricted to this range.
Evidently, there is no reason for them to be equal.
Cheers -- Sjoerd
> 67.70657`, 66.99681`, 71.33181`, 71.78314`, 70.73837`, 70.30609`}=