Med Bevegelig Gjennomsnitt Ms Utmerker Seg

Excel for statistisk dataanalyse Dette er et webtekstkompisjonssted for Bedriftsstatistikk USA Site. Beslektede lenker: Vis alle de tilgjengelige stedene, og velg det som er tilgjengelig. Excel er den mye brukte statistiske pakken, som fungerer som et verktøy for å forstå statistiske begreper og beregninger for å sjekke håndarbeidet beregning for å løse lekseproblemer. Nettstedet gir en introduksjon til å forstå grunnleggende om og arbeide med Excel. Redoing av de illustrerte talleksemplene på dette nettstedet vil bidra til å forbedre kjennskapen din og som et resultat øke effektiviteten og effektiviteten av prosessen din i statistikk. Å søke på nettstedet. prøv E dit F inn på side Ctrl f. Skriv inn et ord eller en setning i dialogboksen, f. eks. quote variancequot eller quot averagequot Hvis det første uttrykket fra ordfrasen ikke er det du leter etter, kan du prøve F inn Next. Innledning Dette nettstedet gir illustrativ erfaring med bruk av Excel for datasammendrag, presentasjon og for annen grunnleggende statistisk analyse. Jeg tror den populære bruken av Excel er på områdene hvor Excel virkelig kan utmerke seg. Dette inkluderer organiseringsdata, dvs. grunnleggende datahåndtering, tabulering og grafikk. For ekte statistisk analyse må man lære å bruke profesjonelle kommersielle statistiske pakker som SAS og SPSS. Microsoft Excel 2000 (versjon 9) gir et sett med dataanalyseverktøy kalt Analysis ToolPak som du kan bruke til å lagre trinn når du utvikler komplekse statistiske analyser. Du oppgir dataene og parametrene for hver analyse verktøyet bruker de aktuelle statistiske makrofunksjonene og viser deretter resultatene i en utmatningstabell. Noen verktøy genererer diagrammer i tillegg til utdatatabeller. Hvis kommandoen Dataanalyse er valgt i Verktøy-menyen, er Analysis ToolPak installert på systemet. Men hvis kommandoen Dataanalyse ikke er på Verktøy-menyen, må du installere Analysis ToolPak ved å gjøre følgende: Trinn 1: På Verktøy-menyen klikker du Add-Ins. Hvis Analysis ToolPak ikke er oppført i dialogboksen Add-ins, klikker du Bla gjennom og finner stasjonen, mappenavnet og filnavnet for Analysis ToolPak Add-in Analys32.xll som vanligvis ligger i Program FilesMicrosoft OfficeOfficeLibraryAnalysis-mappen. Når du har funnet filen, velg den og klikk OK. Trinn 2: Hvis du ikke finner filen Analys32.xll, må du installere den. Sett inn Microsoft Office 2000 Disk 1 i CD ROM-stasjonen. Velg Kjør fra Windows Start-menyen. Bla gjennom og velg stasjonen for CDen din. Velg Setup. exe, klikk Åpne, og klikk OK. Klikk på knappen Legg til eller fjern funksjoner. Klikk ved siden av Microsoft Excel for Windows. Klikk ved siden av Add-ins. Klikk på nedpilen ved siden av Analysis ToolPak. Velg Kjør fra Min datamaskin. Velg Oppdater nå-knappen. Excel vil nå oppdatere systemet for å inkludere Analysis ToolPak. Start Excel. På Verktøy-menyen, klikk på Add-Ins. - og merk av for Analysis ToolPak. Trinn 3: Analysis ToolPak Add-In er nå installert og Data Analysis. vil nå bli valgbar på Verktøy-menyen. Microsoft Excel er en kraftig regnearkpakke tilgjengelig for Microsoft Windows og Apple Macintosh. Regnearkprogramvare brukes til å lagre informasjon i kolonner og rader som deretter kan organiseres og behandles. Regneark er laget for å fungere godt med tall, men inneholder ofte tekst. Excel organiserer arbeidet ditt i arbeidsbøker hver arbeidsbok kan inneholde mange regneark regneark brukes til å liste og analysere data. Excel er tilgjengelig på alle offentlig tilgang PCer (dvs. de, for eksempel i biblioteket og PC Labs). Den kan åpnes enten ved å velge Start - Programmer - Microsoft Excel eller ved å klikke på Excel Short Cut som er enten på skrivebordet eller på hvilken som helst PC, eller på Office Tool-linjen. Åpne et dokument: Klikk på File-Open (CtrlO) for å åpne en eksisterende arbeidsbok for å endre katalogområdet eller kjøre for å se etter filer på andre steder. Hvis du vil opprette en ny arbeidsbok, klikker du på File-New-Blank Document. Lagre og lukke et dokument: Hvis du vil lagre dokumentet med dets nåværende filnavn, plassering og filformat, klikker du enten på Arkiv - Lagre. Hvis du lagrer for første gang, klikker du Fil-lagre valgtype et navn for dokumentet, og klikker deretter OK. Bruk også File-Save hvis du vil lagre til en annen filamellokasjon. Når du er ferdig med å jobbe på et dokument, bør du lukke det. Gå til Fil-menyen og klikk på Lukk. Hvis du har gjort noen endringer siden filen var sist lagret, blir du spurt om du vil lagre dem. Excel-skjermen Arbeidsbøker og regneark: Når du starter Excel, vises et tomt regneark som består av et flertall av celler med nummererte rader nedover siden og alfabetisk navngitte kolonner på tvers av siden. Hver celle er referert til av koordinatene sine (for eksempel A3 brukes til å referere til cellen i kolonne A og rad 3 B10: B20 brukes til å referere til rekkevidden av celler i kolonne B og rader 10 til 20). Ditt arbeid er lagret i en Excel-fil kalt en arbeidsbok. Hver arbeidsbok kan inneholde flere regneark og orkart - det nåværende regnearket kalles det aktive arket. Hvis du vil vise et annet regneark i en arbeidsbok, klikker du på det aktuelle arkfanen. Du kan få tilgang til og utføre kommandoer direkte fra hovedmenyen, eller du kan peke på en av verktøylinjeknappene (skjermboksen som vises under knappen, når du plasserer markøren over den, angir navnet på knappen) og klikker en gang. Flytte rundt regnearket: Det er viktig å kunne flytte regnearket effektivt fordi du bare kan skrive inn eller endre data i markørens posisjon. Du kan flytte markøren ved hjelp av piltastene eller ved å flytte musen til den nødvendige cellen og klikke. Når cellen er blitt valgt, blir den aktive cellen og identifiseres med en tykk kantlinje, kan kun én celle være aktiv om gangen. Hvis du vil flytte fra ett regneark til et annet, klikker du arkfanene. (Hvis arbeidsboken din inneholder mange ark, høyreklikker du på rulleskjermknappene og klikker på arket du vil ha.) Navnet på det aktive arket vises med fet skrift. Flytting mellom celler: Her er et hurtigtast for å flytte den aktive cellen: Hjem - flyttes til den første kolonnen i den nåværende raden CtrlHome - flyttes til øverste venstre hjørne av dokumentet Slutt deretter Hjem - flyttes til den siste cellen i dokumentet Til Flytt mellom celler på et regneark, klikk på en hvilken som helst celle eller bruk piltastene. For å se et annet område av arket, bruk rullefeltene og klikk på pilene eller området overbeløpe rulleboksen i enten de vertikale eller horisontale rullefeltene. Merk at størrelsen på en rulleboks angir proporsjonal mengde av det brukte området på arket som er synlig i vinduet. Plasseringen av en rulleboks viser den relative plasseringen av det synlige området i regnearket. Oppgi data Et nytt regneark er et rutenett av rader og kolonner. Rynene er merket med tall, og kolonnene er merket med bokstaver. Hvert kryss av en rad og en kolonne er en celle. Hver celle har en adresse. som er kolonnebrevet og radnummeret. Pilen på regnearket til høyre peker til celle A1, som for tiden er uthevet. som indikerer at det er en aktiv celle. En celle må være aktiv for å legge inn informasjon i den. For å markere (velg) en celle, klikk på den. For å velge mer enn en celle: Klikk på en celle (for eksempel A1), og hold deretter skift-tasten mens du klikker på en annen (for eksempel D4) for å velge alle celler mellom og inklusive A1 og D4. Klikk på en celle (f. eks. A1) og dra musen over ønsket område, unclicking på en annen celle (for eksempel D4) for å velge alle celler mellom og inklusive A1 og D4. For å velge flere celler som ikke er tilstøtende, trykk på kontrollen og klikk på cellene du vil velge. Klikk på et nummer eller en bokstav som merker en rad eller kolonne for å velge den hele raden eller kolonnen. Ett regneark kan ha opptil 256 kolonner og 65 536 rader, så det er en stund før du går tom for plass. Hver celle kan inneholde en etikett. verdi. logisk verdi. eller formel. Etiketter kan inneholde en kombinasjon av bokstaver, tall eller symboler. Verdier er tall. Kun verdier (tall) kan brukes i beregninger. En verdi kan også være en dato eller en timeLogiske verdier er sanne eller falske. Formuler gjør automatisk beregninger på verdiene i andre spesifiserte celler og viser resultatet i cellen der formelen er angitt (for eksempel kan du angi at cellen D3 er å inneholde summen av tallene i B3 og C3 vil tallet som vises i D3 da være en funksjon av tallene som er inngått B3 og C3). For å legge inn informasjon i en celle, velg cellen og begynn å skrive. Vær oppmerksom på at når du skriver inn informasjon i cellen, vises informasjonen du oppgir, også i formellelinjen. Du kan også legge inn informasjon i formellelinjen, og informasjonen vil vises i den valgte cellen. Når du er ferdig med å skrive inn etiketten eller verdien: Trykk Enter for å flytte til neste celle under (i dette tilfellet A2) Trykk på Tab for å flytte til neste celle til høyre (i dette tilfellet B1) Klikk i hvilken som helst celle for å velge det skriver inn etiketter Med mindre informasjonen du oppgir, er formatert som en verdi eller en formel, vil Excel tolke den som en etikett, og standard for å justere teksten på venstre side av cellen. Hvis du lager et langt regneark, og du vil gjenta den samme etikettinformasjonen i mange forskjellige celler, kan du bruke AutoComplete-funksjonen. Denne funksjonen vil se på andre oppføringer i samme kolonne og forsøke å matche en tidligere oppføring med din nåværende oppføring. For eksempel, hvis du allerede har skrevet Wesleyan i en annen celle, og du skriver W i en ny celle, vil Excel automatisk skrive Wesleyan. Hvis du hadde tenkt å skrive Wesleyan i cellen, er oppgaven din ferdig, og du kan gå videre til neste celle. Hvis du hadde tenkt å skrive noe annet, f. eks. Williams, inn i cellen, bare fortsett å skrive for å skrive inn termen. For å slå på AutoComplete-funksjonen, klikk på Verktøy i menylinjen, velg deretter Alternativer, velg deretter Rediger, og klikk for å legge inn en boks ved siden av Aktiver AutoComplete for celleverdier. En annen måte å raskt angi gjentatte etiketter på, er å bruke Pick List-funksjonen. Høyreklikk på en celle, velg deretter Velg fra liste. Dette gir deg en meny med alle andre oppføringer i celler i den kolonnen. Klikk på et element i menyen for å skrive det inn i den valgte cellen. En verdi er et tall, dato eller klokkeslett, pluss noen få symboler om nødvendig for ytterligere å definere tallene 91such som. - () 93. Tall antas å være positivt for å angi et negativt tall, bruk et minustegn - eller legg inn tallet i parenteser (). Datoer lagres som MMDDYYYY, men du trenger ikke å skrive det nøyaktig i det formatet. Hvis du går inn 9. januar eller 9. januar, vil Excel gjenkjenne det 9. januar i år, og lagre det som 192002. Skriv inn det firesifrede året for et år annet enn det nåværende året (f. eks. 9. januar 1999). For å angi dagens dato, trykk på kontroll og samtidig. Tider som standard til en 24-timers klokke. Bruk a eller p for å indikere am eller pm hvis du bruker en 12-timers klokke (for eksempel 8:30 p tolkes som 8:30 PM). For å gå inn i gjeldende tid, trykk på kontroll og: (skift-semikolon) samtidig. En oppføring tolket som en verdi (tall, dato eller tid) er justert til høyre side av cellen, for å reformatere en verdi. Avrundingsnumre som oppfyller spesifikke kriterier: Å bruke farger til maksimale andor minimumsverdier: Velg en celle i regionen, og trykk CtrlShift (i Excel 2003, trykk dette eller CtrlA) for å velge gjeldende region. Fra Tilpass-menyen velger du Betinget formatering. I Tilstand 1, velg Formel Is, og skriv MAX (F: F) F1. Klikk Format, velg kategorien Skrift, velg en farge, og klikk deretter OK. I Tilstand 2, velg Formel Is, og skriv MIN (F: F) F1. Gjenta trinn 4, velg en annen farge enn du valgte for Tilstand 1, og klikk deretter OK. Merk: Pass på å skille mellom absolutt referanse og relativ referanse når du legger inn formlene. Rundnummer som møter spesifiserte kriterier Problem: Avrund alle tallene i kolonne A til null desimaler, unntatt de som har 5 i første desimal. Løsning: Bruk IF, MOD og ROUND-funksjonene i følgende formel: IF (MOD (A2,1) 0,5, A2, ROUND (A2,0)) Kopier og lim inn alle celler i et ark Velg cellene i arket ved å trykke CtrlA (i Excel 2003, velg en celle i et tomt område før du trykker CtrlA, eller fra en valgt celle i et Current RegionList-område, trykk CtrlAA). ELLER Klikk på Velg alt øverst til venstre i krysset mellom rader og kolonner. Trykk CtrlC. Trykk CtrlPage Down for å velge et annet ark, og velg deretter celle A1. Trykk enter. Slik kopierer du hele arket Når du kopierer hele arket, betyr det at du kopierer cellene, sideoppsettparametrene og det definerte området Navn. Alternativ 1: Flytt musepekeren til en arkfane. Trykk Ctrl, og hold musen for å dra arket til et annet sted. Slett museknappen og Ctrl-tasten. Alternativ 2: Høyreklikk riktig arkfanen. Fra snarveismenyen, velg Flytt eller Kopier. Dialogboksen Flytt eller Kopier gjør det mulig å kopiere arket enten til en annen plassering i gjeldende arbeidsbok eller til en annen arbeidsbok. Pass på at du merker av for Opprett en kopi. Alternativ 3: Velg Ordne fra Vindu-menyen. Velg Tiled til fliser alle åpne arbeidsbøker i vinduet. Bruk Alternativ 1 (dra arket mens du trykker på Ctrl) for å kopiere eller flytte et ark. Sortering etter kolonner Standardinnstillingen for sortering i stigende eller synkende rekkefølge er etter rad. Slik sorterer du etter kolonner: Velg Sorter, og velg Valg i Data-menyen. Velg alternativet Sorter til venstre til høyre og klikk OK. I sorteringsalternativet i sorteringsdialogen velger du radenummeret som kolonnene skal sorteres til og klikker på OK. Beskrivende statistikk Data Analysis ToolPak har et beskrivende statistikkverktøy som gir deg en enkel måte å beregne oppsummeringsstatistikk for et sett med eksempeldata. Sammendragsstatistikk inkluderer Mean, Standard Error, Median, Mode, Standard Avvik, Varians, Kurtosis, Skewness, Range, Minimum, Maximum, Sum og Count. Dette verktøyet eliminerer behovet for å skrive individuelle funksjoner for å finne hvert av disse resultatene. Excel inneholder utførlige og tilpassbare verktøylinjer, for eksempel standard verktøylinje som vises her: Noen av ikonene er nyttige matematiske beregninger: er Autosum-ikonet, som kommer inn i formelsummeret () for å legge til en rekke celler. er FunctionWizard-ikonet, som gir deg tilgang til alle tilgjengelige funksjoner. er GraphWizard-ikonet, som gir tilgang til alle graftyper som er tilgjengelige, som vist på denne skjermen: Excel kan brukes til å generere målinger av plassering og variabilitet for en variabel. Anta at vi ønsker å finne beskrivende statistikk for en eksempeldata: 2, 4, 6 og 8. Trinn 1. Velg rullegardinmenyen Verktøy, hvis du ser dataanalyse, klikk på dette alternativet, ellers klikk på add-in . alternativ til å installere analyseverktøyet pakke. Trinn 2. Klikk på dataanalysen. Trinn 3. Velg Beskrivende statistikk fra listen Analyseverktøy. Trinn 4. Når dialogboksen vises: Skriv inn A1: A4 i inntaksområdet, A1 er en verdi i kolonne A og rad 1. I dette tilfellet er denne verdien 2. Ved å bruke samme teknikk, skriv inn andre verdier til du kommer til den siste. Hvis en prøve består av 20 tall, kan du for eksempel velge A1, A2, A3, etc. som inngangsområde. Trinn 5. Velg et utdataområde. i dette tilfellet B1. Klikk på sammendragsstatistikk for å se resultatene. Når du klikker på OK. Du vil se resultatet i det valgte området. Som du vil se, er gjennomsnittet av prøven 5, medianen 5, standardavviket er 2,581989, prøven variansen er 6,6666667, intervallet er 6 og så videre. Hver av disse faktorene kan være viktig i beregningen av ulike statistiske prosedyrer. Normal distribusjon Vurder problemet med å finne sannsynligheten for å få mindre enn en viss verdi under normal sannsynlighetsfordeling. Som et illustrativt eksempel, la oss anta at SAT-resultatene landsdekkende er normalt fordelt med henholdsvis en gjennomsnittlig og standardavvik på henholdsvis 500 og 100. Svar på følgende spørsmål basert på den oppgitte informasjonen: A: Hva er sannsynligheten for at en tilfeldig valgt studentpoengsumme er mindre enn 600 poeng B: Hva er sannsynligheten for at en tilfeldig valgt studentpoeng vil overstige 600 poeng C: Hva er sannsynligheten at en tilfeldig valgt student score vil være mellom 400 og 600 Tips: Ved hjelp av Excel kan du finne sannsynligheten for å få en verdi som er omtrent mindre enn eller lik en gitt verdi. I et problem, når gjennomsnittet og standardavviket for befolkningen er gitt, må du bruke sunn fornuft til å finne forskjellige sannsynligheter basert på spørsmålet, siden du vet at området under en normal kurve er 1. I arbeidsarket, velg celle hvor du vil at svaret skal vises. Anta at du valgte celle nummer ett, A1. Fra menyene, velg quotinsert pull-downquot. Trinn 2-3 Fra menyene, velg innsats, og klikk deretter på funksjonsalternativet. Trinn 4. Etter å ha klikket på funksjonen Funksjon, vises dialogboksen Lim inn funksjon fra Funksjonskategori. Velg Statistisk deretter NORMDIST fra funksjonsnavn-boksen Klikk OK trinn 5. Etter å ha klikket på OK, vises NORMDIST-distribusjonsboksen: i. Skriv inn 600 i X (verdien boksen) ii. Skriv inn 500 i Mean-boksen iii. Skriv inn 100 i standardavviksboksen iv. Skriv quottruequot i den kumulative boksen, og klikk deretter OK. Som du ser verdien 0,84134474 vises i A1, indikerer sannsynligheten for at en tilfeldig valgt student score er under 600 poeng. Ved hjelp av sunn fornuft kan vi svare på en del quotbquot ved å subtrahere 0.84134474 fra 1. Så del quotequot svaret er 1- 0.8413474 eller 0.158653. Dette er sannsynligheten for at en tilfeldig valgt student score er større enn 600 poeng. For å svare på delkvotot, bruk samme teknikker for å finne sannsynlighetene eller området i venstre side av verdiene 600 og 400. Siden disse områdene eller sannsynlighetene overlapper hverandre for å svare på spørsmålet, bør du trekke den mindre sannsynligheten ut av større sannsynlighet. Svaret er lik 0,84134474 - 0,155865526 ​​som er 0,68269. Skjermbildet skal se ut som følgende: Beregne verdien av en tilfeldig variabel som ofte kalles quotxquot-verdien. Du kan bruke NORMINV fra funksjonsboksen til å beregne en verdi for tilfeldige variabelen - hvis sannsynligheten for den venstre siden av denne variabelen er gitt. Faktisk bør du bruke denne funksjonen til å beregne forskjellige prosentiler. I dette problemet kan man spørre hva som er poenget til en student hvis prosentil er 90 Dette betyr at omtrent 90 av studentene er mindre enn dette nummeret. På den annen side, hvis vi ble bedt om å gjøre dette problemet for hånd, ville vi måtte beregne x-verdien ved hjelp av normalfordelingsformel x m zd. Nå kan vi bruke Excel til å beregne P90. I Paste-funksjonen klikker dialogboksen på statistisk, og klikker deretter på NORMINV. Skjermbildet vil se ut som følgende: Når du ser NORMINV, vises dialogboksen. Jeg. Skriv inn 0,90 for sannsynligheten (dette betyr at omtrent 90 av studentpoengene er mindre enn verdien vi leter etter) ii. Skriv inn 500 for gjennomsnittet (dette er gjennomsnittet av normalfordelingen i vårt tilfelle) iii. Oppgi 100 for standardavviket (dette er standardavviket for normalfordelingen i vårt tilfelle) På slutten av dette skjermbildet vil du se formelresultatet som er ca 628 poeng. Dette betyr at topp 10 av studentene skårte bedre enn 628. Confidence Interval for Mean Anta at vi ønsker å estimere et konfidensintervall for gjennomsnittet av en befolkning. Avhengig av størrelsen på utvalgsstørrelsen kan du bruke ett av følgende tilfeller: Stor prøvestørrelse (n er større enn, si 30): Den generelle formelen for å utvikle et konfidensintervall for en populasjonsmiddel er: I denne formelen er den gjennomsnittlige av prøven Z er intervallkoeffisienten, som kan bli funnet fra normalfordelingstabellen (for eksempel er intervallkoeffisienten for et 95-konfidensnivå 1,96). S er standardavviket til prøven og n er prøvestørrelsen. Nå vil vi gjerne vise hvordan Excel brukes til å utvikle et bestemt konfidensintervall for et populasjonsmiddel basert på en prøveinformasjon. Som du ser for å evaluere denne formelen trenger du quotthe gjennomsnittet av samplequot og feilmarginen Excel vil automatisk beregne disse mengdene for deg. De eneste tingene du må gjøre er å legge til feilmarginen til gjennomsnittet av prøven, Finn den øvre grensen for intervallet og trekke feilmarginen fra gjennomsnittet til den nedre grensen for intervallet. For å demonstrere hvordan Excel finner disse mengdene, vil vi bruke datasettet, som inneholder den timelønte inntekten på 36 arbeidsstudenter her ved University of Baltimore. Disse tallene vises i celler A1 til A36 på et Excel-arbeidsark. Etter at du har skrevet inn dataene, fulgte vi den beskrivende statistiske prosedyren for å beregne de ukjente mengdene. Det eneste ekstra trinnet er å klikke på konfidensintervallet i den beskrivende statistikkdialogboksen og angi det oppgitte konfidensnivået, i dette tilfellet 95. Her er de ovennevnte prosedyrene i trinn for trinn: Trinn 1. Oppgi data i celler A1 til A36 (på regnearket) Trinn 2. Fra menyene velg Verktøy Trinn 3. Klikk på Data Analysis, velg deretter Beskrivende statistikk og klikk deretter OK. På den beskrivende statistiske dialogboksen klikker du på Sammendragsstatistikk. Når du har gjort det, klikker du på konfidensintervallnivået og skriver 95 - eller i andre problemer uansett konfidensintervall du ønsker. I boksen Output Range skal du angi B1 eller hvilken plassering du ønsker. Klikk nå på OK. Skjermbildet vil se ut som følgende: Som du ser, viser regnearket at gjennomsnittet av prøven er 6,902777778 og absoluttverdien av feilmarginen 0,231678109. Dette mener er basert på denne prøveinformasjonen. Et 95 konfidensintervall for timelønnen til UB-arbeidsstudentene har en øvre grense på 6.902777778 0.231678109 og en nedre grense på 6.902777778 - 0.231678109. På den annen side kan vi si at av alle intervaller som dannes på denne måten, inneholder 95 befolkningenes gjennomsnitt. Eller for praktiske formål kan vi være 95 sikre på at gjennomsnittet av befolkningen er mellom 6,902777778 - 0,231678109 og 6,902777778 0,231678109. Vi kan være minst 95 sikre på at intervall 6,68 og 7,13 inneholder gjennomsnittlig timelønn for en arbeidsstudent. Smal prøvestørrelse (si mindre enn 30) Hvis prøven n er mindre enn 30, eller vi må bruke den lille prøveprosedyren til å utvikle et konfidensintervall for gjennomsnittet av en befolkning. Den generelle formelen for å utvikle konfidensintervaller for befolkningen er basert på liten, en prøve er: I denne formelen er gjennomsnittet av prøven. er intervallkoeffisienten som gir et område i øvre hale av en t-fordeling med n-1 frihetsgrader som kan bli funnet fra et t-fordelingsbord (for eksempel er intervallkoeffisienten for et 90-konfidensnivå 1,833 hvis prøven er 10). S er standardavviket til prøven og n er prøvestørrelsen. Nå vil du gjerne se hvordan Excel brukes til å utvikle et bestemt konfidensintervall for et populasjonsmiddel basert på denne lille prøveinformasjonen. Som du ser, for å evaluere denne formelen trenger du quotthe gjennomsnittet av samplequot og feilmarginen Excel beregner automatisk disse mengdene slik den gjorde for store prøver. Igjen, det eneste du må gjøre er å legge til feilmarginen til gjennomsnittet av prøven, finne den øvre grensen for intervallet og å trekke feilmarginen fra gjennomsnittet for å finne den nedre grensen for intervallet. For å demonstrere hvordan Excel finner disse mengdene, vil vi bruke datasettet, som inneholder timelønnene på 10 arbeidsstudenter her ved University of Baltimore. Disse tallene vises i celler A1 til A10 på et Excel-arbeidsark. Etter å ha skrevet inn dataene følger vi beskrivende statistikkprosedyre for å beregne de ukjente mengdene (akkurat slik vi fant kvantiteter for stor prøve). Her følger prosedyrene i trinn-for-trinn-skjema: Trinn 1. Oppgi data i celler A1 til A10 på regnearket Trinn 2. Fra menyene velg Verktøy Trinn 3. Klikk på Data analyse, og velg deretter Beskrivende statistikk. Klikk OK på den beskrivende statistiske dialogboksen, klikk på Sammendragsstatistikk, klikk på konfidensintervallnivå og skriv inn 90 eller i andre problemer avhengig av hvilket konfidensintervall du ønsker. I boksen Output Range, skriv inn B1 eller hvilken som helst plassering du ønsker. Klikk nå på OK. Skjermbildet vil se ut som følgende: Nå, som beregningen av konfidensintervallet for den store prøven, beregner du konfidensintervallet for befolkningen basert på denne lille prøveinformasjonen. Konfidensintervallet er: 6,8 0,414426102 eller 6,39 7,21. Vi kan være minst 90 fortrolige at intervallet 6,39 og 7,21 inneholder det sanne gjennomsnittet av befolkningen. Test av hypotesen om populasjonen betyr igjen, må vi skille to tilfeller med hensyn til størrelsen på prøven Stor prøveformat (si over 30): I denne delen ønsker du å vite hvordan Excel kan brukes til å utføre en hypotesetest om et populasjonsmiddel. Vi vil bruke timelønnene til ulike arbeidsstudenter enn de som ble introdusert tidligere i konfidensintervallet. Data er oppgitt i celler A1 til A36. Målet er å teste følgende null - og alternativ-hypotesen: nullhypotesen indikerer at gjennomsnittlig timelønn for en arbeidsstudent er lik 7 per time, men den alternative hypotesen indikerer at gjennomsnittlig timelønn ikke er lik 7 per time. Jeg vil gjenta trinnene i beskrivende statistikk og i slutten vil det vise seg hvordan man finner verdien av teststatistikken i dette tilfellet, z, ved hjelp av en celleformel. Trinn 1. Skriv inn data i celler A1 til A36 (på regnearket) Trinn 2. Fra menyene velg Verktøy Trinn 3. Klikk på Data analyse og velg alternativet Beskrivende statistikk, klikk OK. På den beskrivende statistiske dialogboksen klikker du på Sammendragsstatistikk. Velg boksen Output Range, skriv inn B1 eller ønsket sted. Klikk nå OK. (For å beregne verdien av teststatistikken, søk etter gjennomsnittet av prøven, deretter standardfeilen. I denne utgangen er disse verdiene i celler C3 og C4.) Trinn 4. Velg celle D1 og skriv inn celleformelen (C3-7 ) C4. Skjermbildet skal se ut som følgende: Verdien i celle D1 er verdien av teststatistikken. Siden denne verdien faller i akseptområdet fra -1,96 til 1,96 (fra normalfordelingstabellen), mislykkes det i å avvise nullhypotesen. Små prøvestørrelser (si mindre enn 30): Ved hjelp av trinn som er tatt med den store prøvestørrelsen, kan Excel brukes til å utføre en hypotese for små prøvesaker. La oss bruke timelønnen til 10 arbeidsstudenter på UB for å utføre følgende hypotese. Nulhypotesen indikerer at gjennomsnittlig timelønn for en arbeidsstudent er lik 7 per time. Den alternative hypotesen indikerer at gjennomsnittlig timelønn ikke er lik 7 per time. Jeg vil gjenta trinnene i beskrivende statistikk og i slutten vil det vise seg hvordan man finner verdien av teststatistikken i dette tilfellet quottquot ved hjelp av en celleformel. Trinn 1. Oppgi data i celler A1 til A10 (på regnearket) Trinn 2. Fra menyene velg Verktøy Trinn 3. Klikk på Data Analysis og velg deretter Beskrivende statistikk. Klikk på OK. På den beskrivende statistiske dialogboksen klikker du på Sammendragsstatistikk. Velg boksene Output Range, skriv inn B1 eller hvilken som helst plassering du valgte. Igjen, klikk på OK. (For å beregne verdien av teststatistikken, søk etter gjennomsnittet av prøven, så er standardfeilen i disse utgangene disse verdiene i celler C3 og C4.) Trinn 4. Velg celle D1 og skriv inn celleformelen (C3-7) C4. Skjermbildet vil se ut som følgende: Siden verdien av teststatistikken t -0.66896 faller i akseptasjonsområdet -2.262 til 2.262 (fra t-tabell, hvor 0,025 og frihetsgraden er 9), unnlater vi å nekte nullhypotesen. Forskjellen mellom gjennomsnittet av to populasjoner I denne delen vil vi vise hvordan Excel brukes til å utføre en hypotesetest om forskjellen mellom to befolkningsgrupper, forutsatt at populasjoner har like avvik. Dataene i dette tilfellet er hentet fra forskjellige kontorer her ved University of Baltimore. Jeg samlet inntektsinntektene for 36 tilfeldige utvalgte arbeidsstudenter og 36 studentassistenter. Timelønnsintervallet for arbeidsstudenter var 6 - 8, mens timelønnsintervallet for studentassistenter var 6-9. Hovedmålet i denne hypotesetestingen er å se om det er en signifikant forskjell mellom middelene til de to populasjonene. NULL og ALTERNATIV hypotesen er at midlene er like og midlene er ikke like, henholdsvis. Med henvisning til regnearket valgte jeg A1 og A2 som etikettsentre. Arbeidsstudiestudentens timelønn for en prøvestørrelse 36 er vist i celler A2: A37. og studentassistentens timelønn for en prøvestørrelse 36 er vist i celler B2: B37 Data for arbeidsstudent Student: 6, 6, 6, 6, 6, 6, 6,5, 6,5, 6,5, 6,5, 6,5, 6,5, 7, 7, 7, 7, 7, 7, 7, 7,5, 7,5, 7,5, 7,5, 7,5, 7,5, 8, 8, 8, 8, 8, 8, 8, 8. Data for studentassistent: 6 , 6, 6, 6, 6, 6,5, 6,5, 6,5, 6,5, 6,5, 7, 7, 7, 7, 7, 7,5, 7,5, 7,5, 7,5, 7,5, 7,5, 8,8,8,8,8 , 8, 8, 8,5, 8,5, 8,5, 8,5, 8,5, 9, 9, 9, 9. Bruk beskrivende statistikkprosedyre til å beregne avvikene til de to prøvene. Excel-prosedyren for å teste forskjellen mellom de to befolkningsmidlene vil kreve informasjon om avvikene fra de to populasjonene. Siden variasjonene i de to populasjonene er ukjente, bør de erstattes med utvalgsvariasjoner. Den beskrivende for begge prøvene viser at variansen av første prøve er s 1 2 0,55546218. mens variansen av den andre prøven er 2 2 0,969748. For å utføre den ønskede testhypotesen med Excel, kan du følge trinnene: Trinn 1. Fra menyene velg Verktøy og klikk deretter på Data Analyse. Trinn 2. Når dialogboksen Dataanalyse vises: Velg z-Test: To eksempler på midler, og klikk deretter OK Trinn 3. Når dialogboksen z-Test: Two Sample for means vises: Angi A1: A36 i rekkefeltet for variabel 1 (arbeidsstudent elevernes timelønn) Skriv inn B1: B36 i variabel 2-feltboksen (studentassistent timelønn) Skriv 0 i Hypothesis Mean Difference-boksen (hvis du ønsker å teste en annen forskjell enn 0, skriv inn den verdien) Skriv inn variansen til den første prøven i Variabel 1 Variant-boksen Angi variansen til den andre prøven i Variabel 2 Variant-boksen og velg Etiketter Angi 0,05 eller, uansett nivå av betydning du ønsker, i alfa-boksen Velg et passende utgangsområde for resultater, jeg valgte C19. klikk deretter OK. Verdien av teststatistikken z-1.9845824 vises i vårt tilfelle i celle D24. Avvisningsregelen for denne testen er z 1,96 fra normalfordelingstabellen. I Excel-utgangene er disse verdiene for en tohaletest z 1.959961082. Siden verdien av teststatistikken z-1.9845824 er mindre enn -1,959961082 avviser vi nullhypotesen. We can also draw this conclusion by comparing the p-value for a two tail - test and the alpha value. Since p-value 0.047190813 is less than a0.05 we reject the null hypothesis. Overall we can say, based on the sample results, the two populations means are different. Small Samples: n 1 OR n 2 are less than 30 In this section we will show how Excel is used to conduct a hypothesis test about the difference between two population means. - Given that the populations have equal variances when two small independent samples are taken from both populations. Similar to the above case, the data in this case are taken from various offices here at the University of Baltimore. I collected hourly income data of 11 randomly selected work-study students and 11 randomly selected student assistants. The hourly income range for both groups was similar range, 6 - 8 and 6-9. The main objective in this hypothesis testing is similar too, to see whether there is a significant difference between the means of the two populations. The NULL and the ALTERNATIVE hypothesis are that the means are equal and they are not equal, respectively. Referring to the spreadsheet, we chose A1 and A2 as label centers. The work-study students hourly income for a sample size 11 are shown in cells A2:A12 . and the student assistants hourly income for a sample size 11 is shown in cells B2:B12 . Unlike previous case, you do not have to calculate the variances of the two samples, Excel will automatically calculate these quantities and use them in the calculation of the value of the test statistic. Similar to the previous case, but a bit different in step 2, to conduct the desired test hypothesis with Excel the following steps can be taken: Step 1. From the menus select Tools then click on the Data Analysis option. Step 2. When the Data Analysis dialog box appears: Choose t-Test: Two Sample Assuming Equal Variances then click OK Step 3 When the t-Test: Two Sample Assuming Equal Variances dialog box appears : Enter A1:A12 in the variable 1 range box (work-study student hourly income) Enter B1:B12 in the variable 2 range box (student assistant hourly income) Enter 0 in the Hypothesis Mean Difference box(if you desire to test a mean difference other than zero, enter that value) then select Labels Enter 0.05 or, whatever level of significance you desire, in the Alpha box Select a suitable Output Range for the results, I chose C1, then click OK. The value of the test statistic t-1.362229828 appears, in our case, in cell D10. The rejection rule for this test is t 2.086 from the t distribution table where the t value is based on a t distribution with n 1 - n 2 -2 degrees of freedom and where the area of the upper one tail is 0.025 ( that is equal to alpha2). In the Excel output the values for a two-tail test are t 2.085962478. Since the value of the test statistic t-1.362229828, is in an acceptance range of t 2.085962478, we fail to reject the null hypothesis. We can also draw this conclusion by comparing the p-value for a two-tail test and the alpha value. Since the p-value 0.188271278 is greater than a0.05 again . we fail to reject the null hypothesis. Overall we can say, based on sample results, the two populations means are equal. Enter data in an Excel work sheet starting with cell A2 and ending with cell C8. The following steps should be taken to find the proper output for interpretation. Step 1. From the menus select Tools and click on Data Analysis option. Step 2. When data analysis dialog appears, choose Anova single-factor option enter A2:C8 in the input range box. Select labels in first row. Step3. Select any cell as output(in here we selected A11). Click OK. The general form of Anova table looks like following: Source of Variation Suppose the test is done at level of significance a 0.05, we reject the null hypothesis. This means there is a significant difference between means of hourly incomes of student assistants in these departments. The Two-way ANOVA Without Replication In this section, the study involves six students who were offered different hourly wages in three different department services here at the University of Baltimore. The objective is to see whether the hourly incomes are the same. Therefore, we can consider the following: Treatment: Hourly payments in the three departments Blocks: Each student is a block since each student has worked in the three different departments The general form of Anova table would look like: Source of Variation Degrees of freedom To find the Excel output for the above data the following steps can be taken: Step 1. From the menus select Tools and click on Data Analysis option. Step2. When data analysis box appears: select Anova two-factor without replication then Enter A2: D8 in the input range. Select labels in first row. Step3. Select an output range (in here we selected A11) then OK. Source of Variation NOTE: FMSTMSE 0.9805560.497222 1.972067 F 3.33 from table (5 numerator DF and 10 denominator DF) Since 1.972067 Goodness-of-Fit Test for Discrete Random Variables The CHI-SQUARE distribution can be used in a hypothesis test involving a population variance. However, in this section we would like to test and see how close a sample results are to the expected results. Example: The Multinomial Random Variable In this example the objective is to see whether or not based on a randomly selected sample information the standards set for a population is met. There are so many practical examples that can be used in this situation. For example it is assumed the guidelines for hiring people with different ethnic background for the US government is set at 70(WHITE), 20(African American) and 10(others), respectively. A randomly selected sample of 1000 US employees shows the following results that is summarized in a table. EXPECTED NUMBER OF EMPLOYEES OBSERVED FROM SAMPLE As you see the observed sample numbers for groups two and three are lower than their expected values unlike group one which has a higher expected value. Is this a clear sign of discrimination with respect to ethnic background Well depends on how much lower the expected values are. The lower amount might not statistically be significant. To see whether these differences are significant we can use Excel and find the value of the CHI-SQUARE. If this value falls within the acceptance region we can assume that the guidelines are met otherwise they are not. Now lets enter these numbers into Excel spread - sheet. We used cells B7-B9 for the expected proportions, C7-C9 for the observed values and D7-D9 for the expected frequency. To calculate the expected frequency for a category, you can multiply the proportion of that category by the sample size (in here 1000). The formula for the first cell of the expected value column, D7 is 1000B7. To find other entries in the expected value column, use the copy and the paste menu as shown in the following picture. These are important values for the chi-square test. The observed range in this case is C7: C9 while the expected range is D7: D9. The null and the alternative hypothesis for this test are as follows: H A . The population proportions are not P W 0.70, P A 0.20 and P O 0.10 Now lets use Excel to calculate the p-value in a CHI-SQUARE test. Step 1. Select a cell in the work sheet, the location which you like the p value of the CHI-SQUARE to appear. We chose cell D12. Step 2. From the menus, select insert then click on the Function option, Paste Function dialog box appears. Step 3. Refer to function category box and choose statistical . from function name box select CHITEST and click on OK . Step 4. When the CHITEST dialog appears: Enter C7: C9 in the actual-range box then enter D7: D9 in the expected-range box, and finally click on OK . The p-value will appear in the selected cell, D12. As you see the p value is 0.002392 which is less than the value of the level of significance (in this case the level of significance, a 0.10). Hence the null hypothesis should be rejected. This means based on the sample information the guidelines are not met. Notice if you type CHITEST(C7:C9,D7:D9) in the formula bar the p-value will show up in the designated cell. NOTE: Excel can actually find the value of the CHI-SQUARE. To find this value first select an empty cell on the spread sheet then in the formula bar type CHIINV(D12,2). D12 designates the p-Value found previously and 2 is the degrees of freedom (number of rows minus one). The CHI-SQUARE value in this case is 12.07121. If we refer to the CHI-SQUARE table we will see that the cut off is 4.60517 since 12.071214.60517 we reject the null. The following screen shot shows you how to the CHI-SQUARE value. Test of Independence: Contingency Tables The CHI-SQUARE distribution is also used to test and see whether two variables are independent or not. For example based on sample data you might want to see whether smoking and gender are independent events for a certain population. The variables of interest in this case are smoking and the gender of an individual. Another example in this situation could involve the age range of an individual and his or her smoking habit. Similar to case one data may appear in a table but unlike the case one this table may contains several columns in addition to rows. The initial table contains the observed values. To find expected values for this table we set up another table similar to this one. To find the value of each cell in the new table we should multiply the sum of the cell column by the sum of the cell row and divide the results by the grand total. The grand total is the total number of observations in a study. Now based on the following table test whether or not the smoking habit and gender of the population that the following sample taken from are independent. On the other hand is that true that males in this population smoke more than females You could use formula bar to calculate the expected values for the expected range. For example to find the expected value for the cell C5 which is replaced in c11 you could click on the formula bar and enter C6D5D6 then enter in cell C11. Step 1. Observed Range b4:c5 Smoking and gender So the observed range is b4:c5 and the expected range is b10:c11. Step 3. Click on fx (paste function) Step 4. When Paste Function dialog box appears, click on Statistical in function category and CHITEST in the function name then click OK. When the CHITEST box appears, enter b4:c5 for the actual range, then b10:c11 for the expected range. Step 5. Click on OK (the p-value appears). 0.477395 Conclusion: Since p-value is greater than the level of significance (0.05), fails to reject the null. This means smoking and gender are independent events. Based on sample information one can not assure females smoke more than males or the other way around. Step 6. To find the chi-square value, use CHINV function, when Chinv box appears enter 0.477395 for probability part, then 1 for the degrees of freedom. Degrees of freedom(number of columns-1)X(number of rows-1) Test Hypothesis Concerning the Variance of Two Populations In this section we would like to examine whether or not the variances of two populations are equal. Whenever independent simple random samples of equal or different sizes such as n 1 and n 2 are taken from two normal distributions with equal variances, the sampling distribution of s 1 2 s 2 2 has F distribution with n 1 - 1 degrees of freedom for the numerator and n 2 - 1 degrees of freedom for the denominator. In the ratio s 1 2 s 2 2 the numerator s 1 2 and the denominator s 2 2 are variances of the first and the second sample, respectively. The following figure shows the graph of an F distribution with 10 degrees of freedom for both the numerator and the denominator. Unlike the normal distribution as you see the F distribution is not symmetric. The shape of an F distribution is positively skewed and depends on the degrees of freedom for the numerator and the denominator. The value of F is always positive. Now let see whether or not the variances of hourly income of student-assistant and work-study students based on samples taken from populations previously are equal. Assume that the hypothesis test in this case is conducted at a 0.10. The null and the alternative are: Rejection Rule: Reject the null hypothesis if Flt F 0.095 or Fgt F 0.05 where F, the value of the test statistic is equal to s 1 2 s 2 2. with 10 degrees of freedom for both the numerator and the denominator. We can find the value of F .05 from the F distribution table. If s 1 2 s 2 2. we do not need to know the value of F 0.095 otherwise, F 0.95 1 F 0.05 for equal sample sizes. A survey of eleven student-assistant and eleven work-study students shows the following descriptive statistics. Our objective is to find the value of s 1 2 s 2 2. where s 1 2 is the value of the variance of student assistant sample and s 2 2 is the value of the variance of the work study students sample. As you see these values are in cells F8 and D8 of the descriptive statistic output. To calculate the value of s 1 2 s 2 2. select a cell such as A16 and enter cell formula F8D8 and enter. This is the value of F in our problem. Since this value, F1.984615385, falls in acceptance area we fail to reject the null hypothesis. Hence, the sample results do support the conclusion that student assistants hourly income variance is equal to the work study students hourly income variance. The following screen shoot shows how to find the F value. We can follow the same format for one tail test(s). Linear Correlation and Regression Analysis In this section the objective is to see whether there is a correlation between two variables and to find a model that predicts one variable in terms of the other variable. There are so many examples that we could mention but we will mention the popular ones in the world of business. Usually independent variable is presented by the letter x and the dependent variable is presented by the letter y. A business man would like to see whether there is a relationship between the number of cases of sold and the temperature in a hot summer day based on information taken from the past. He also would like to estimate the number cases of soda which will be sold in a particular hot summer day in a ball game. He clearly recorded temperatures and number of cases of soda sold on those particular days. The following table shows the recorded data from June 1 through June 13. The weatherman predicts a 94F degree temperature for June 14. The businessman would like to meet all demands for the cases of sodas ordered by customers on June 14. Now lets use Excel to find the linear correlation coefficient and the regression line equation. The linear correlation coefficient is a quantity between -1 and 1. This quantity is denoted by R . The closer R to 1 the stronger positive (direct) correlation and similarly the closer R to -1 the stronger negative (inverse) correlation exists between the two variables. The general form of the regression line is y mx b. In this formula, m is the slope of the line and b is the y-intercept. You can find these quantities from the Excel output. In this situation the variable y (the dependent variable) is the number of cases of soda and the x (independent variable) is the temperature. To find the Excel output the following steps can be taken: Step 1. From the menus choose Tools and click on Data Analysis. Step 2. When Data Analysis dialog box appears, click on correlation. Step 3. When correlation dialog box appears, enter B1:C14 in the input range box. Click on Labels in first row and enter a16 in the output range box. Click on OK. As you see the correlation between the number of cases of soda demanded and the temperature is a very strong positive correlation. This means as the temperature increases the demand for cases of soda is also increasing. The linear correlation coefficient is 0.966598577 which is very close to 1. Now lets follow same steps but a bit different to find the regression equation. Step 1. From the menus choose Tools and click on Data Analysis Step 2 . When Data Analysis dialog box appears, click on regression . Step 3. When Regression dialog box appears, enter b1:b14 in the y-range box and c1:c14 in the x-range box. Click on labels . Step 4. Enter a19 in the output range box . Note: The regression equation in general should look like Ym X b. In this equation m is the slope of the regression line and b is its y-intercept. Adjusted R Square The relationship between the number of cans of soda and the temperature is: Y 0.879202711 X 9.17800767 The number of cans of soda 0.879202711(Temperature) 9.17800767. Referring to this expression we can approximately predict the number of cases of soda needed on June 14. The weather forecast for this is 94 degrees, hence the number of cans of soda needed is equal to The number of cases of soda0.879202711(94) 9.17800767 91.82 or about 92 cases. Moving Average and Exponential Smoothing Moving Average Models: Use the Add Trendline option to analyze a moving average forecasting model in Excel. You must first create a graph of the time series you want to analyze. Select the range that contains your data and make a scatter plot of the data. Once the chart is created, follow these steps: Click on the chart to select it, and click on any point on the line to select the data series. When you click on the chart to select it, a new option, Chart, s added to the menu bar. From the Chart menu, select Add Trendline. The following is the moving average of order 4 for weekly sales: Exponential Smoothing Models: The simplest way to analyze a timer series using an Exponential Smoothing model in Excel is to use the data analysis tool. This tool works almost exactly like the one for Moving Average, except that you will need to input the value of a instead of the number of periods, k. Once you have entered the data range and the damping factor, 1- a. and indicated what output you want and a location, the analysis is the same as the one for the Moving Average model. Applications and Numerical Examples Descriptive Statistics: Suppose you have the following, n 10, data: 1.2, 1.5, 2.6, 3.8, 2.4, 1.9, 3.5, 2.5, 2.4, 3.0 Type your n data points into the cells A1 through An. Click on the Tools menu. (At the bottom of the Tools menu will be a submenu Data Analysis. , if the Analysis Tool Pack has been properly installed.) Clicking on Data Analysis. will lead to a menu from which Descriptive Statistics is to be selected. Select Descriptive Statistics by pointing at it and clicking twice, or by highlighting it and clicking on the Okay button. Within the Descriptive Statistics submenu, a. for the input range enter A1:Dn, assuming you typed the data into cells A1 to An. b. click on the output range button and enter the output range C1:C16. c. click on the Summary Statistics box d. finally, click on Okay. The Central Tendency: The data can be sorted in ascending order: 1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8 The mean, median and mode are computed as follows: (1.2 1.5 2.6 3.8 2.4 1.9 3.5 2.5 2.4 3.0) 10 2.48 The mode is 2.4, since it is the only value that occurs twice. The midrange is (1.2 3.8) 2 2.5. Note that the mean, median and mode of this set of data are very close to each other. This suggests that the data is very symmetrically distributed. Variance: The variance of a set of data is the average of the cumulative measure of the squares of the difference of all the data values from the mean. The sample variance-based estimation for the population variance are computed differently. The sample variance is simply the arithmetic mean of the squares of the difference between each data value in the sample and the mean of the sample. On the other hand, the formula for an estimate for the variance in the population is similar to the formula for the sample variance, except that the denominator in the fraction is (n-1) instead of n. However, you should not worry about this difference if the sample size is large, say over 30. Compute an estimate for the variance of the population . given the following sorted data: 1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8 mean 2.48 as computed earlier. An estimate for the population variance is: s 2 1 (10-1) (1.2 - 2.48) 2 (1.5 - 2.48) 2 (1.9 - 2.48) 2 (2.4 -2.48) 2 (2.4 - 2.48) 2 (2.5 - 2.48) 2 (2.6 - 2.48) 2 (3.0 - 2.48) 2 (3.5 -2.48) 2 (3.8 - 2.48) 2 (1 9) (1.6384 0.9604 0.3364 0.0064 0.0064 0.0004 0.0144 0.2704 1.0404 1.7424) 0.6684 Therefore, the standard deviation is s ( 0.6684 ) 12 0.8176 Probability and Expected Values: Newsweek reported that average take for bank robberies was 3,244 but 85 percent of the robbers were caught. Assuming 60 percent of those caught lose their entire take and 40 percent lose half, graph the probability mass function using EXCEL. Calculate the expected take from a bank robbery. Does it pay to be a bank robber To construct the probability function for bank robberies, first define the random variable x, bank robbery take. If the robber is not caught, x 3,244. If the robber is caught and manages to keep half, x 1,622. If the robber is caught and loses it all, then x 0. The associated probabilities for these x values are 0.15 (1 - 0.85), 0.34 (0.85)(0.4), and 0.51 (0.85)(0.6). After entering the x values in cells A1, A2 and A3 and after entering the associated probabilities in B1, B2, and B3, the following steps lead to the probability mass function: Click on ChartWizard. The ChartWizard Step 1 of 4 screen will appear. Highlight Column at ChartWizard Step 1 of 4 and click Next. At ChartWizard Step 2 of 4 Chart Source Data, enter B1:B3 for Data range, and click column button for Series in. A graph will appear. Click on series toward the top of the screen to get a new page. At the bottom of the Series page, is a rectangle for Category (X) axis labels: Click on this rectangle and then highlight A1:A3. At Step 3 of 4 move on by clicking on Next, and at Step 4 of 4, click on Finish. The expected value of a robbery is 1,038.08. E(X) (0)(0.51)(1622)(0.34) (3244)(0.15) 0 551.48 486.60 1038.08 The expected return on a bank robbery is positive. On average, bank robbers get 1,038.08 per heist. If criminals make their decisions strictly on this expected value, then it pays to rob banks. A decision rule based only on an expected value, however, ignores the risks or variability in the returns. In addition, our expected value calculations do not include the cost of jail time, which could be viewed by criminals as substantial. Discrete Continuous Random Variables: Binomial Distribution Application: A multiple choice test has four unrelated questions. Each question has five possible choices but only one is correct. Thus, a person who guesses randomly has a probability of 0.2 of guessing correctly. Draw a tree diagram showing the different ways in which a test taker could get 0, 1, 2, 3 and 4 correct answers. Sketch the probability mass function for this test. What is the probability a person who guesses will get two or more correct Solution: Letting Y stand for a correct answer and N a wrong answer, where the probability of Y is 0.2 and the probability of N is 0.8 for each of the four questions, the probability tree diagram is shown in the textbook on page 182. This probability tree diagram shows the branches that must be followed to show the calculations captured in the binomial mass function for n 4 and 0.2. For example, the tree diagram shows the six different branch systems that yield two correct and two wrong answers (which corresponds to 4(22) 6. The binomial mass function shows the probability of two correct answers as P(x 2 n 4, p 0.2) 6(.2)2(.8)2 6(0.0256) 0.1536 P(2) Which is obtained from excel by using the BINOMDIST Command, where the first entry is x, the second is n, and the third is mass (0) or cumulative (1) that is, entering BINOMDIST(2,4,0.2,0) IN ANY EXCEL CELL YIELDS 0.1536 AND BINOMDIST(3,4,0.2,0) YIELDS P(x3n4, p 0.2) 0.0256 BINOMDIST(4,4,0.2,0) YIELDS P(x4n4, p 0.2) 0.0016 1-BINOMDIST(1,4,0.2,1) YIELDS P(x 179 2 n 4, p 0.2) 0.1808 Normal Example: If the time required to complete an examination by those with a certain learning disability is believed to be distributed normally, with mean of 65 minutes and a standard deviation of 15 minutes, then when can the exam be terminated so that 99 percent of those with the disability can finish Solution: Because t he average and standard deviation are known, what needs to be established is the amount of time, above the mean time, such that 99 percent of the distribution is lower. This is a distance that is measured in standard deviations as given by the Z value corresponding to the 0.99 probability found in the body of Appendix B, Table 5,as shown in the textbook OR the commands entered into any cell of Excel to find this Z value is NORMINV(0.99,0,1) for 2.326342. The closest cumulative probability that can be found is 0.9901, in the row labeled 2.3 and column headed by .03, Z 2.33, which is only an approximation for the more exact 2.326342 found in Excel. Using this more exact value the calculation with mean m and standard deviation s in the following formula would be Z ( X - m ) s That is, Z ( x - 65)15 Thus, x 65 15(2.32634) 99.9 minutes. Alternatively, instead of standardizing with the Z distribution using Excel we can simply work directly with the normal distribution with a mean of 65 and standard deviation of 15 and enter NORMINV(0.99,65,15). In general to obtain the x value for which alpha percent of a normal random variables values are lower, the following NORMINV command may be used, where the first entry is a. the second is m. and the third is s. Another Example: In the early 1980s, the Toro Company of Minneapolis, Minnesota, advertised that it would refund the purchase price of a snow blower if the following winters snowfall was less than 21 percent of the local average. If the average snowfall is 45.25 inches, with a standard deviation of 12.2 inches, what is the likelihood that Toro will have to make refunds Solution: Within limits, snowfall is a continuous random variable that can be expected to vary symmetrically around its mean, with values closer to the mean occurring most often. Thus, it seems reasonable to assume that snowfall (x) is approximately normally distributed with a mean of 45.25 inches and standard deviation of 12.2 inches. Nine and one half inches is 21 percent of the mean snowfall of 45.25 inches and, with a standard deviation of 12.2 inches, the number of standard deviations between 45.25 inches and 9.5 inches is Z: Z ( x - m ) s (9.50 - 45.25)12.2 -2.93 Using Appendix B, Table 5, the textbook demonstrates the determination of P(x 163 9.50) P(z 163 -2.93) 0.17, the probability of snowfall less than 9.5 inches. Using Excel, this normal probability is obtained with the NORMDIST command, where the first entry is x, the second is mean m. the third is standard deviation s, and the fourth is CUMULATIVE (1). Entering NORMDIST(9.5,45.25,12.2,1), Gives P( x 163 9.50) 0.001693. Sampling Distribution and the Central Limit Theorem : A bakery sells an average of 24 loaves of bread per day. Sales (x) are normally distributed with a standard deviation of 4. If a random sample of size n 1 (day) is selected, what is the probability this x value will exceed 28 If a random sample of size n 4 (days) is selected, what is theprobability that xbar 179 28 Why does the answer in part 1 differ from that in part 2 1. The sampling distribution of the sample mean xbar is normal with a mean of 24 and a standard error of the mean of 4. Thus, using Excel, 0.15866 1-NORMDIST(28,24,4,1). 2. The sampling distribution of the sample mean xbar is normal with a mean of 24 and a standard error of the mean of 2 using Excel, 0.02275 1-NORMDIST(28,24,2,1). Regression Analysis: The highway deaths per 100 million vehicle miles and highway speed limits for 10 countries, are given below: (Death, Speed) (3.0, 55), (3.3, 55), (3.4, 55), (3.5, 70), (4.1, 55), (4.3, 60), (4.7, 55), (4.9, 60), (5.1, 60), and (6.1, 75). From this we can see that five countries with the same speed limit have very different positions on the safety list. For example, Britain. with a speed limit of 70 is demonstrably safer than Japan, at 55. Can we argue that, speed has little to do with safety. Use regression analysis to answer this question. Solution: Enter the ten paired y and x data into cells A2 to A11 and B2 to B11, with the death rate label in A1 and speed limits label in B1, the following steps produce the regression output. Choose Regression from Data Analysis in the Tools menu. The Regression dialog box will will appear. Note: Use the mouse to move between the boxes and buttons. Click on the desired box or button. The large rectangular boxes require a range from the worksheet. A range may be typed in or selected by highlighting the cells with the mouse after clicking on the box. If the dialog box blocks the data, it can be moved on the screen by clicking on the title bar and dragging. For the Input Y Range, enter A1 to A11, and for the Input X Range enter B1 to B11. Because the Y and X ranges include the Death and Speed labels in A1 and B1, select the Labels box with a click. Click the Output Range button and type reference cell, which in this demonstration is A13. To get the predicted values of Y (Death rates) and residuals select the Residuals box with a click. Your screen display should show a Table, clicking OK will give the SUMMARY OUTPUT, ANOVA AND RESIDUAL OUTPUT The first section of the EXCEL printout gives SUMMARY OUTPUT. The Multiple R is the square root of the R Square the computation and interpretation of which we have already discussed. The Standard Error of estimate (which will be discussed in the next chapter) is s 0.86423, which is the square root of Residual SS 5.97511 divided by its degrees of freedom, df 8, as given in the ANOVA section. We will also discuss the adjusted R-square of 0.21325 in the following chapters. Under the ANOVA section are the estimated regression coefficients and related statistics that will be discussed in detail in the next chapter. For now it is sufficient to recognize that the calculated coefficient values for the slope and y intercept are provided (b 0.07556 and a -0.29333). Next to these coefficient estimates is information on the variability in the distribution of the least-squares estimators from which these specific estimates were drawn: the column titled Std. Error contains the standard deviations (standard errors) of the intercept and slope distributions the t-ratio and p columns give the calculated values of the t statistics and associated p-values. As shown in Chapter 13, the t statistic of 1.85458 and p-value of 0.10077, for example, indicates that the sample slope (0.07556) is sufficiently different from zero, at even the 0.10 two-tail Type I error level, to conclude that there is a significant relationship between deaths and speed limits in the population. This conclusion is contrary to assertion that speed has little to do with safety. SUMMARY OUTPUT: Multiple R 0.54833, R Square 0.30067, Adjusted R Square 0.21325, Standard Error 0.86423, Observations 10 ANOVA df SS MS F P-value Regression 1 2.56889 2.56889 3.43945 0.10077 Residual 8 5.97511 0.74689 Total 9 8.54400 Coeffs. Estimate Std. Error T Stat P-value Lower 95 Upper 95 Intercept -0.29333 2.45963 -0.11926 0.90801 -5.96526 5.37860 Speed 0.07556 0.04074 1.85458 0.10077 -0.01839 0.16950 Predicted Residuals 3.86222 -0.86222 3.86222 -0.56222 3.86222 -0.46222 4.99556 -1.49556 3.86222 0.23778 4.24000 0.06000 3.86222 0.83778 4.24000 0.66000 4.24000 0.86000 5.37333 0.72667 Microsoft Excel Add-Ins Forecasting with regression requires the Excel add-in called Analysis ToolPak , and linear programming requires the Excel add-in called Solver . How you check to see if these are activated on your computer, and how to activate them if they are not active, varies with Excel version. Here are instructions for the most common versions. If Excel will not let you activate Data Analysis and Solver, you must use a different computer. Excel 20022003: Start Excel, then click Tools and look for Data Analysis and for Solver. If both are there, press Esc (escape) and continue with the respective assignment. Otherwise click Tools, Add-Ins, and check the boxes for Analysis ToolPak and for Solver, then click OK. Click Tools again, and both tools should be there. Excel 2007: Start Excel 2007 and click the Data tab at the top. Look to see if Data Analysis and Solver show in the Analysis section at the far right. If both are there, continue with the respective assignment. Otherwise, do the following steps exactly as indicated: - click the 8220Office Button8221 at top left - click the Excel Options button near the bottom of the resulting window - click the Add-ins button on the left of the next screen - near the bottom at Manage Excel Add-ins, click Go - check the boxes for Analysis ToolPak and Solver Add-in if they are not already checked, then click OK - click the Data tab as above and verify that the add-ins show. Excel 2010: Start Excel 2010 and click the Data tab at the top. Look to see if Data Analysis and Solver show in the Analysis section at the far right. If both are there, continue with the respective assignment. Otherwise, do the following steps exactly as indicated: - click the File tab at top left - click the Options button near the bottom of the left side - click the Add-ins button near the bottom left of the next screen - near the bottom at Manage Excel Add-ins, click Go - check the boxes for Analysis ToolPak and Solver Add-in if they are not already checked, then click OK - click the Data tab as above and verify that the add-ins show. Solving Linear Programs by Excel Some of these examples can be modified for other types problems Computer-assisted Learning: E-Labs and Computational Tools My teaching style deprecates the plug the numbers into the software and let the magic box work it out approach. Personal computers, spreadsheets, e. g. Excel. professional statistical packages (e. g. such as SPSS), and other information technologies are now ubiquitous in statistical data analysis. Without using these tools, one cannot perform any realistic statistical data analysis on large data sets. The appearance of other computer software, JavaScript Applets. Statistical Demonstrations Applets. and Online Computation are the most important events in the process of teaching and learning concepts in model-based statistical decision making courses. These tools allow you to construct numerical examples to understand the concepts, and to find their significance for yourself. Use any or online interactive tools available on the WWW to perform statistical experiments (with the same purpose, as you used to do experiments in physics labs to learn physics) to understand statistical concepts such as Central Limit Theorem are entertaining and educating. Computer-assisted learning is similar to the experiential model of learning. The adherents of experiential learning are fairly adamant about how we learn. Learning seldom takes place by rote. Learning occurs because we immerse ourselves in a situation in which we are forced to perform and think. You get feedback from the computer output and then adjust your thinking-process if needed. A SPSS-Example . SPSS-Examples . SPSS-More Examples . (Statistical Package for the Social Sciences) is a data management and analysis product. It can perform a variety of data analysis and presentation functions, including statistical analyses and graphical presentation of data. SAS (Statistical Analysis System) is a system of software packages some of its basic functions and uses are: database management inputting, cleaning and manipulating data, statistical analysis, calculating simple statistics such as means, variances, correlations running standard routines such as regressions. Available at: SPSSSAS Packages on Citrix (Installing and Accessing ) Use your email ID and Password: Technical Difficulties OTS Call Center (401) 837-6262 Excel Examples. Excel More Examples It is Excellent for Descriptive Statistics, and getting acceptance is improving, as computational tool for Inferential Statistics. The Value of Performing Experiment: If the learning environment is focused on background information, knowledge of terms and new concepts, the learner is likely to learn that basic information successfully. However, this basic knowledge may not be sufficient to enable the learner to carry out successfully the on-the-job tasks that require more than basic knowledge. Thus, the probability of making real errors in the business environment is high. On the other hand, if the learning environment allows the learner to experience and learn from failures within a variety of situations similar to what they would experience in the real world of their job, the probability of having similar failures in their business environment is low. This is the realm of simulations-a safe place to fail. The appearance of statistical software is one of the most important events in the process of decision making under uncertainty. Statistical software systems are used to construct examples, to understand the existing concepts, and to find new statistical properties. On the other hand, new developments in the process of decision making under uncertainty often motivate developments of new approaches and revision of the existing software systems. Statistical software systems rely on a cooperation of statisticians, and software developers. Beside the professional statistical software Online statistical computation . and the use of a scientific calculator is required for the course. A Scientific Calculator is the one, which has capability to give you, say, the result of square root of 5. Any calculator that goes beyond the 4 operations is fine for this course. These calculators allow you to perform simple calculations you need in this course, for example, enabling you to take square root, to raise e to the power of say, 0.36. and so on. These types of calculators are called general Scientific Calculators. There are also more specific and advanced calculators for mathematical computations in other areas such as Finance, Accounting, and even Statistics. The last one, for example, computes mean, variance, skewness, and kurtosis of a sample by simply entering all data one-by-one and then pressing any of the mean, variance, skewness, and kurtosis keys. Without a computer one cannot perform any realistic statistical data analysis. Students who are signing up for the course are expected to know the basics of Excel. As a starting point, you need visiting the Excel Web site created for this course. If you are challenged by or unfamiliar with Excel, you may seek tutorial help from the Academic Resource Center at 410-837-5385, E-mail. What and How to Hand-in My Computer Assignment For the computer assignment I do recommend in checking your hand computation homework, and checking some of the numerical examples from your textbook. As part of your homework assignment you don not have to hand in the printout of the computer assisted learning, however, you must include within your handing homework a paragraph entitled Computer Implementation describing your (positive or negative) experience. Interesting and Useful Sites The Copyright Statement: The fair use, according to the 1996 Fair Use Guidelines for Educational Multimedia. of materials presented on this Web site is permitted for non-commercial and classroom purposes only. This site may be mirrored intact (including these notices), on any server with public access. All files are available at home. ubalt. eduntsbarshBusiness-stat for mirroring. Kindly e-mail me your comments, suggestions, and concerns. Takk skal du ha. EOF: CopyRights 1994-2015.Simple Moving Average (SMA) Explained A simple moving average (SMA) is the simplest type of moving average in forex analysis (DUH). I utgangspunktet beregnes et enkelt glidende gjennomsnitt ved å legge opp de siste 8220X8221 period8217s sluttkursene og deretter dele det tallet med X. Don8217t bekymre deg, we8217ll gjør det krystallklart. Beregning av det enkle flytende gjennomsnittet (SMA) Hvis du plottet et 5-års simpelt glidende gjennomsnitt på et 1-timers diagram, vil du legge opp sluttkursene de siste 5 timene, og deretter dele det nummeret med 5. Voila Du har gjennomsnittet sluttkurs i løpet av de siste fem timene String de gjennomsnittlige prisene sammen, og du får et glidende gjennomsnitt. Hvis du skulle plotte et 5-års simpel glidende gjennomsnitt på et 10-minutters valutakart, ville du legge til sluttkursene de siste 50 minuttene og divider deretter det tallet med 5. Hvis du skulle plotte et 5-årig enkelt glidende gjennomsnitt på et 30-minutters diagram, ville du legge opp sluttkursene de siste 150 minuttene og deretter dele det nummeret med 5. Hvis du skulle plotte 5-timers enkelt glidende gjennomsnitt på 4 timer. chart8230 Okay, vi vet det, vi vet. Du får bildet De fleste kartleggingspakker vil gjøre alle beregningene for deg. Grunnen til at vi bare kjedde deg (gjes) med en 8220how til8221 ved beregning av enkle bevegelige gjennomsnitt er fordi it8217 er viktig å forstå slik at du vet hvordan du redigerer og justerer indikatoren. Å forstå hvordan en indikator fungerer betyr at du kan justere og opprette forskjellige strategier ettersom markedsmiljøet endres. Nå, som med nesten hvilken som helst annen forexindikator der ute, beveger gjennomsnittet seg med en forsinkelse. Fordi du tar gjennomsnittet av tidligere prishistorie, ser du egentlig bare den generelle veien i den siste tiden og den generelle retningen for kortvarig prishandling på 8220future8221. Ansvarsfraskrivelse: Flytte gjennomsnitt vil ikke gjøre deg til Ms Cleo den psykiske Her er et eksempel på hvordan glidende gjennomsnitt utjevner prishandlingen. På kartet over, plottet we8217ve tre forskjellige SMAer på 1-timers diagrammet til USDCHF. Som du kan se, jo lengre SMA-perioden er, jo mer ligger det bak prisen. Legg merke til hvordan 62 SMA er lenger unna den nåværende prisen enn de 30 og 5 SMAene. Dette skyldes at 62 SMA legger til sluttkursene for de siste 62 periodene og deler den med 62. Jo lengre periode du bruker til SMA, desto langsommere er det å reagere på prisbevegelsen. SMAene i dette diagrammet viser deg den generelle følelsen av markedet på dette tidspunktet. Her kan vi se at paret er trending. I stedet for bare å se på dagens markedspris, gir de bevegelige gjennomsnittene oss et bredere syn, og vi kan nå måle den generelle retningen til fremtidig pris. Ved bruk av SMAer kan vi fortelle om et par trender opp, trender ned, eller bare strekker seg. Det er ett problem med det enkle glidende gjennomsnittet: de er mottakelige for pigger. Når dette skjer, kan dette gi oss falske signaler. Vi tror kanskje at en ny valutatrend kan utvikle seg, men i virkeligheten er ingenting endret. I neste leksjon vil vi vise deg hva vi mener, og også introdusere deg til en annen type glidende gjennomsnitt for å unngå dette problemet. Save your progress by signing in and marking the lesson completeEva Goldwater Biostatistics Consulting Center University of Massachusetts School of Public Health updated February 2007 At A Glance We used Excel to do some basic data analysis tasks to see whether it is a reasonable alternative to using a statistical package for the same tasks. We concluded that Excel is a poor choice for statistical analysis beyond textbook examples, the simplest descriptive statistics, or for more than a very few columns. The problems we encountered that led to this conclusion are in four general areas : Missing values are handled inconsistently, and sometimes incorrectly. Data organization differs according to analysis, forcing you to reorganize your data in many ways if you want to do many different analyses. Many analyses can only be done on one column at a time, making it inconvenient to do the same analysis on many columns. Output is poorly organized, sometimes inadequately labeled, and there is no record of how an analysis was accomplished. Excel is convenient for data entry, and for quickly manipulating rows and columns prior to statistical analysis. However when you are ready to do the statistical analysis, we recommend the use of a statistical package such as SAS, SPSS, Stata, Systat or Minitab. Introduction Excel is probably the most commonly used spreadsheet for PCs. Newly purchased computers often arrive with Excel already loaded. It is easily used to do a variety of calculations, includes a collection of statistical functions, and a Data Analysis ToolPak. As a result, if you suddenly find you need to do some statistical analysis, you may turn to it as the obvious choice. We decided to do some testing to see how well Excel would serve as a Data Analysis application. To present the results, we will use a small example. The data for this example is fictitious. It was chosen to have two categorical and two continuous variables, so that we could test a variety of basic statistical techniques. Since almost all real data sets have at least a few missing data points, and since the ability to deal with missing data correctly is one of the features that we take for granted in a statistical analysis package, we introduced two empty cells in the data: Each row of the spreadsheet represents a subject. The first subject received Treatment 1, and had Outcome 1. X and Y are the values of two measurements on each subject. We were unable to get a measurement for Y on the second subject, or on X for the last subject, so these cells are blank. The subjects are entered in the order that the data became available, so the data is not ordered in any particular way. We used this data to do some simple analyses and compared the results with a standard statistical package. The comparison considered the accuracy of the results as well as the ease with which the interface could be used for bigger data sets - i. e. more columns. We used SPSS as the standard, though any of the statistical packages OIT supports would do equally well for this purpose. In this article when we say quota statistical package, quot we mean SPSS, SAS, STATA, SYSTAT, or Minitab. Most of Excels statistical procedures are part of the Data Analysis tool pack, which is in the Tools menu. It includes a variety of choices including simple descriptive statistics, t-tests, correlations, 1 or 2-way analysis of variance, regression, etc. If you do not have a Data Analysis item on the Tools menu, you need to install the Data Analysis ToolPak. Search in Help for quotData Analysis Toolsquot for instructions on loading the ToolPak. Two other Excel features are useful for certain analyses, but the Data Analysis tool pack is the only one that provides reasonably complete tests of statistical significance. Pivot Table in the Data menu can be used to generate summary tables of means, standard deviations, counts, etc. Also, you could use functions to generate some statistical measures, such as a correlation coefficient. Functions generate a single number, so using functions you will likely have to combine bits and pieces to get what you want. Even so, you may not be able to generate all the parts you need for a complete analysis. Unless otherwise stated, all statistical tests using Excel were done with the Data Analysis ToolPak. In order to check a variety of statistical tests, we chose the following tasks: Get means and standard deviations of X and Y for the entire group, and for each treatment group. Get the correlation between X and Y. Do a two sample t-test to test whether the two treatment groups differ on X and Y. Do a paired t-test to test whether X and Y are statistically different from each other. Compare the number of subjects with each outcome by treatment group, using a chi-squared test. All of these tasks are routine for a data set of this nature, and all of them could be easily done using any of the aobve listed statistical packages. General Issues Enable the Analysis ToolPak The Data Analysis ToolPak is not installed with the standard Excel setup. Look in the Tools menu. If you do not have a Data Analysis item, you will need to install the Data Analysis tools. Search Help for quotData Analysis Toolsquot for instructions. Missing Values A blank cell is the only way for Excel to deal with missing data. If you have any other missing value codes, you will need to change them to blanks. Data Arrangement Different analyses require the data to be arranged in various ways. If you plan on a variety of different tests, there may not be a single arrangement that will work. You will probably need to rearrange the data several ways to get everything you need. Dialog Boxes Choose ToolsData Analysis, and select the kind of analysis you want to do. The typical dialog box will have the following items: Input Range: Type the upper left and lower right corner cells. f. eks A1:B100. You can only choose adjacent rows and columns. Unless there is a checkbox for grouping data by rows or columns (and there usually is not), all the data is considered as one glop. Labels - There is sometimes a box you can check off to indicate that the first row of your sheet contains labels. If you have labels in the first row, check this box, and your output MAY be labeled with your label. Then again, it may not. Output location - New Sheet is the default. Or, type in the cell address of the upper left corner of where you want to place the output in the current sheet. New Worksheet is another option, which I have not tried. Ramifications of this choice are discussed below. Other items, depending on the analysis. Output location The output from each analysis can go to a new sheet within your current Excel file (this is the default), or you can place it within the current sheet by specifying the upper left corner cell where you want it placed. Either way is a bit of a nuisance. If each output is in a new sheet, you end up with lots of sheets, each with a small bit of output. If you place them in the current sheet, you need to place them appropriately leave room for adding comments and labels changes you need to make to format one output properly may affect another output adversely. Example: Output from Descriptives has a column of labels such as Standard Deviation, Standard Error, etc. You will want to make this column wide in order to be able to read the labels. But if a simple Frequency output is right underneath, then the column displaying the values being counted, which may just contain small integers, will also be wide. Results of Analyses Descriptive Statistics The quickest way to get means and standard deviations for a entire group is using Descriptives in the Data Analysis tools. You can choose several adjacent columns for the Input Range (in this case the X and Y columns), and each column is analyzed separately. The labels in the first row are used to label the output, and the empty cells are ignored. If you have more, non-adjacent columns you need to analyze, you will have to repeat the process for each group of contiguous columns. The procedure is straightforward, can manage many columns reasonably efficiently, and empty cells are treated properly. To get the means and standard deviations of X and Y for each treatment group requires the use of Pivot Tables (unless you want to rearrange the data sheet to separate the two groups). After selecting the (contiguous) data range, in the Pivot Table Wizards Layout option, drag Treatment to the Row variable area, and X to the Data area. Double click on ldquoCount of Xrdquo in the Data area, and change it to Average. Drag X into the Data box again, and this time change Count to StdDev. Finally, drag X in one more time, leaving it as Count of X. This will give us the Average, standard deviation and number of observations in each treatment group for X. Do the same for Y, so we will get the average, standard deviation and number of observations for Y also. This will put a total of six items in the Data box (three for X and three for Y). As you can see, if you want to get a variety of descriptive statistics for several variables, the process will get tedious. A statistical package lets you choose as many variables as you wish for descriptive statistics, whether or not they are contiguous. You can get the descriptive statistics for all the subjects together, or broken down by a categorical variable such as treatment. You can select the statistics you want to see once, and it will apply to all variables chosen. Correlations Using the Data Analysis tools, the dialog for correlations is much like the one for descriptives - you can choose several contiguous columns, and get an output matrix of all pairs of correlations. Empty cells are ignored appropriately. The output does NOT include the number of pairs of data points used to compute each correlation (which can vary, depending on where you have missing data), and does not indicate whether any of the correlations are statistically significant. If you want correlations on non-contiguous columns, you would either have to include the intervening columns, or copy the desired columns to a contiguous location. A statistical package would permit you to choose non-contiguous columns for your correlations. The output would tell you how many pairs of data points were used to compute each correlation, and which correlations are statistically significant. Two-Sample T-test This test can be used to check whether the two treatment groups differ on the values of either X or Y. In order to do the test you need to enter a cell range for each group. Since the data were not entered by treatment group, we first need to sort the rows by treatment. Be sure to take all the other columns along with treatment, so that the data for each subject remains intact . After the data is sorted, you can enter the range of cells containing the X measurements for each treatment. Do not include the row with the labels, because the second group does not have a label row. Therefore your output will not be labeled to indicate that this output is for X. If you want the output labeled, you have to copy the cells corresponding to the second group to a separate column, and enter a row with a label for the second group. If you also want to do the t-test for the Y measurements, youll need to repeat the process. The empty cells are ignored, and other than the problems with labeling the output, the results are correct. A statistical package would do this task without any need to sort the data or copy it to another column, and the output would always be properly labeled to the extent that you provide labels for your variables and treatment groups. It would also allow you to choose more than one variable at a time for the t-test (e. g. X and Y). Paired t-test The paired t-test is a method for testing whether the difference between two measurements on the same subject is significantly different from 0. In this example, we wish to test the difference between X and Y measured on the same subject. The important feature of this test is that it compares the measurements within each subject. If you scan the X and Y columns separately, they do not look obviously different. But if you look at each X-Y pair, you will notice that in every case, X is greater than Y. The paired t-test should be sensitive to this difference. In the two cases where either X or Y is missing, it is not possible to compare the two measures on a subject. Hence, only 8 rows are usable for the paired t-test. When you run the paired t-test on this data, you get a t-statistic of 0.09, with a 2-tail probability of 0.93. The test does not find any significant difference between X and Y. Looking at the output more carefully, we notice that it says there are 9 observations. As noted above, there should only be 8. It appears that Excel has failed to exclude the observations that did not have both X and Y measurements. To get the correct results copy X and Y to two new columns and remove the data in the cells that have no value for the other measure. Now re-run the paired t-test. This time the t-statistic is 6.14817 with a 2-tail probability of 0.000468. The conclusion is completely different Of course, this is an extreme example. But the point is that Excel does not calculate the paired t-test correctly when some observations have one of the measurements but not the other. Although it is possible to get the correct result, you would have no reason to suspect the results you get unless you are sufficiently alert to notice that the number of observations is wrong. There is nothing in online help that would warn you about this issue. Interestingly, there is also a TTEST function, which gives the correct results for this example. Apparently the functions and the Data Analysis tools are not consistent in how they deal with missing cells. Nevertheless, I cannot recommend the use of functions in preference to the Data Analysis tools, because the result of using a function is a single number - in this case, the 2-tail probability of the t-statistic. The function does not give you the t-statistic itself, the degrees of freedom, or any number of other items that you would want to see if you were doing a statistical test. A statistical packages will correctly exclude the cases with one of the measurements missing, and will provide all the supporting statistics you need to interpret the output. Crosstabulation and Chi-Squared Test of Independence Our final task is to count the two outcomes in each treatment group, and use a chi-square test of independence to test for a relationship between treatment and outcome. In order to count the outcomes by treatment group, you need to use Pivot Tables. In the Pivot Table Wizards Layout option, drag Treatment to Row, Outcome to Column and also to Data. The Data area should say quotCount of Outcomequot ndash if not, double-click on it and select quotCountquot. If you want percents, double-click quotCount of Outcomequot, and click Options in the ldquoShow Data Asrdquo box which appears, select quot of rowquot. If you want both counts and percents, you can drag the same variable into the Data area twice, and use it once for counts and once for percents. Getting the chi-square test is not so simple, however. It is only available as a function, and the input needed for the function is the observed counts in each combination of treatment and outcome (which you have in your pivot table), and the expected counts in each combination. Expected counts What are they How do you get them If you have sufficient statistical background to know how to calculate the expected counts, and can do Excel calculations using relative and absolute cell addresses, you should be able to navigate through this. If not, youre out of luck. Assuming that you surmounted the problem of expected counts, you can use the Chitest function to get the probability of observing a chi-square value bigger than the one for this table. Again, since we are using functions, you do not get many other necessary pieces of the calculation, notably the value of the chi-square statistic or its degrees of freedom. No statistical package would require you to provide the expected values before computing a chi-square test of indepencence. Further, the results would always include the chi-square statistic and its degrees of freedom, as well as its probability. Often you will get some additional statistics as well. Additional Analyses The remaining analyses were not done on this data set, but some comments about them are included for completeness. Simple Frequencies You can use Pivot Tables to get simple frequencies. (see Crosstabulations for more about how to get Pivot Tables.) Using Pivot Tables, each column is considered a separate variable, and labels in row 1 will appear on the output. You can only do one variable at a time. Another possibility is to use the Frequencies function. The main advantage of this method is that once you have defined the frequencies function for one column, you can use CopyPaste to get it for other columns. First, you will need to enter a column with the values you want counted (bins). If you intend to do the frequencies for many columns, be sure to enter values for the column with the most categories. f. eks if 3 columns have values of 1 or 2, and the fourth has values of 1,2,3,4, you will need to enter the bin values as 1,2,3,4. Now select enough empty cells in one column to store the results - 4 in this example, even if the current column only has 2 values. Next choose InsertFunctionStatisticalFrequencies on the menu. Fill in the input range for the first column you want to count using relative addresses (e. g. A1:A100). Fill in the Bin Range using the absolute addresses of the locations where you entered the values to be counted (e. g. M1:M4). Click Finish. Note the box above the column headings of the sheet, where the formula is displayed. It start with quot FREQUENCIES(quot. Place the cursor to the left of the sign in the formula, and press Ctrl-Shift-Enter. The frequency counts now appear in the cells you selected. To get the frequency counts of other columns, select the cells with the frequencies in them, and choose EditCopy on the menu. If the next column you want to count is one column to the right of the previous one, select the cell to the right of the first frequency cell, and choose EditPaste (ctrl-V). Continue moving to the right and pasting for each column you want to count. Each time you move one column to the right of the original frequency cells, the column to be counted is shifted right from the first column you counted. If you want percents as well, yoursquoll have to use the Sum function to compute the sum of the frequencies, and define the formula to get the percent for one cell. Select the cell to store the first percent, and type the formula into the formula box at the top of the sheet - e. g. N1100N 5 - where N1 is the cell with the frequency for the first category, and N5 is the cell with the sum of the frequencies. Use CopyPaste to get the formula for the remaining cells of the first column. Once you have the percents for one column, you can CopyPaste them to the other columns. Yoursquoll need to be careful about the use of relative and absolute addresses In the example above, we used N5 for the denominator, so when we copy the formula down to the next frequency on the same column, it will still look for the sum in row 5 but when we copy the formula right to another column, it will shift to the frequencies in the next column. Finally, you can use Histogram on the Data Analysis menu. You can only do one variable at a time. As with the Frequencies function, you must enter a column with quotbinquot boundaries. To count the number of occurrences of 1 and 2, you need to enter 0,1,2 in three adjacent cells, and give the range of these three cells as the Bins on the dialog box. The output is not labeled with any labels you may have in row 1, nor even with the column letter. If you do frequencies on lots of variables, you will have difficulty knowing which frequency belongs to which column of data. Linear Regression Since regression is one of the more frequently used statistical analyses, we tried it out even though we did not do a regression analysis for this example. The Regression procedure in the Data Analysis tools lets you choose one column as the dependent variable, and a set of contiguous columns for the independents. However, it does not tolerate any empty cells anywhere in the input ranges, and you are limited to 16 independent variables. Therefore, if you have any empty cells, you will need to copy all the columns involved in the regression to new columns, and delete any rows that contain any empty cells. Large models, with more than 16 predictors, cannot be done at all. Analysis of Variance In general, the Excels ANOVA features are limited to a few special cases rarely found outside textbooks, and require lots of data re-arrangements. One-way ANOVA Data must be arranged in separate and adjacent columns (or rows) for each group. Clearly, this is not conducive to doing 1-ways on more than one grouping. If you have labels in row 1, the output will use the labels. Two-Factor ANOVA Without Replication This only does the case with one observation per cell (i. e. no Within Cell error term). The input range is a rectangular arrangement of cells, with rows representing levels of one factor, columns the levels of the other factor, and the cell contents the one value in that cell. Two-Factor ANOVA with Replicates This does a two-way ANOVA with equal cell sizes . Input must be a rectangular region with columns representing the levels of one factor, and rows representing replicates within levels of the other factor. The input range MUST also include an additional row at the top, and column on the left, with labels indicating the factors. However, these labels are not used to label the resulting ANOVA table. Click Help on the ANOVA dialog for a picture of what the input range must look like. Requesting Many Analyses If you had a variety of different statistical procedures that you wanted to perform on your data, you would almost certainly find yourself doing a lot of sorting, rearranging, copying and pasting of your data. This is because each procedure requires that the data be arranged in a particular way, often different from the way another procedure wants the data arranged. In our small test, we had to sort the rows in order to do the t-test, and copy some cells in order to get labels for the output. We had to clear the contents of some cells in order to get the correct paired t-test, but did not want those cells cleared for some other test. And we were only doing five tasks. It does not get better when you try to do more. There is no single arrangement of the data that would allow you to do many different analyses without making many different copies of the data. The need to manipulate the data in many ways greatly increases the chance of introducing errors. Using a statistical program, the data would normally be arranged with the rows representing the subjects, and the columns representing variables (as they are in our sample data). With this arrangement you can do any of the analyses discussed here, and many others as well, without having to sort or rearrange your data in any way. Only much more complex analyses, beyond the capabilities of Excel and the scope of this article would require data rearrangement. Working with Many Columns What if your data had not 4, but 40 columns, with a mix of categorical and continuous measures How easily do the above procedures scale to a larger problem At best, some of the statistical procedures can accept multiple contiguous columns for input, and interpret each column as a different measure. The descriptives and correlations procedures are of this type, so you can request descriptive statistics or correlations for a large number of continuous variables, as long as they are entered in adjacent columns. If they are not adjacent, you need to rearrange columns or use copy and paste to make them adjacent. Many procedures, however, can only be applied to one column at a time. T-tests (either independent or paired), simple frequency counts, the chi-square test of independence, and many other procedures are in this class. This would become a serious drawback if you had more than a handful of columns, even if you use cut and paste or macros to reduce the work. In addition to having to repeat the request many times, you have to decide where to store the results of each, and make sure it is properly labeled so you can easily locate and identify each output. Finally, Excel does not give you a log or other record to track what you have done. This can be a serious drawback if you want to be able to repeat the same (or similar) analysis in the future, or even if youve simply forgotten what youve already done. Using a statistical package, you can request a test for as many variables as you need at once. Each one will be properly labeled and arranged in the output, so there is no confusion as to whats what. You can also expect to get a log, and often a set of commands as well, which can be used to document your work or to repeat an analysis without having to go through all the steps again. Although Excel is a fine spreadsheet, it is not a statistical data analysis package. In all fairness, it was never intended to be one. Keep in mind that the Data Analysis ToolPak is an quotadd-inquot - an extra feature that enables you to do a few quick calculations. So it should not be surprising that that is just what it is good for - a few quick calculations. If you attempt to use it for more extensive analyses, you will encounter difficulties due to any or all of the following limitations: Potential problems with analyses involving missing data. These can be insidious, in that the unwary user is unlikely to realize that anything is wrong. Lack of flexibility in analyses that can be done due to its expectations regarding the arrangement of data. This results in the need to cutpastesort and otherwise rearrange the data sheet in various ways, increasing the likelyhood of errors. Output scattered in many different worksheets, or all over one worksheet, which you must take responsibility for arranging in a sensible way. Output may be incomplete or may not be properly labeled, increasing possibility of misidentifying output. Need to repeat requests for the some analyses multiple times in order to run it for multiple variables, or to request multiple options. Need to do some things by defining your own functionsformulae, with its attendant risk of errors. No record of what you did to generate your results, making it difficult to document your analysis, or to repeat it at a later time, should that be necessary. If you have more than about 10 or 12 columns, andor want to do anything beyond descriptive statistics and perhaps correlations, you should be using a statistical package. There are several suitable ones available by site license through OIT, or you can use them in any of the OIT PC labs. If you have Excel on your own PC, and dont want to pay for a statistical program, by all means use Excel to enter the data (with rows representing the subjects, and columns for the variables). All the mentioned statistical packages can read Excel files, so you can do the (time-consuming) data entry at home, and go to the labs to do the analysis. A much more extensive discussion of the pitfalls of using Excel, with many additional links, is available at burns-stat Click on Tutorials, then Spreadsheet Addiction. For assistance or more information about statistical software, contact the Biostatistics Consulting Center. Telephone 545-2949