Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
What is Standard Deviation? Student Worksheet 789101112 TI-Nspire Investigation Student 30min Introduction JamalandRudiplayforalocalcricketclub.Theirfirstfourlotsofscoresfortheseasonare: Jamal:29,35,29and27 Rudi:7,12,18,and83 Thisexplorationcomparesthescoresforeachcricketerandlooksatthewaystheirscoresaresimilarand thewaysinwhichtheyaredifferent.Italsoexplorestheusefulnessofanewstatisticalmeasure,the standarddeviation. Part 1: Exploring deviation Question1. Copyandcompletethefollowingtocalculatethemeanscoreforeachboy. !!!!!!! + !!!!!!! + !!!!!!! + !!!!!!! a) Jamal’smeanbattingscore= ! Rudi’smeanbattingscore= !!!!!!! !!!!!!! + !!!!!!! + !!!!!!! + !!!!!!! !!!!!!! = = !!!!!!! = !!!!!!! !!!!!!! !!!!!!! !!!!!!! = !!!!!!! ! b) Whatdothemeanscorestellusabouttheboys’scores?Whatdothemeanscoresnot tellus abouttheboys’scores? Bothhav escoredthesametotalofruns,andthesamenumberofinnings. Themeanprovidesameasureofthecentreofastatisticalvariable,butnotinformationabouthow spreadoutthescoresmaybe.Tomeasurethe‘spread’ofthedata,wewilllookathowmucheach individualscoredeviates(isdifferent)fromthemeanscore. OntheTI-NspireCAS: • • • PressHOME-1tocreateaNewDocument,andthenpress therelevanticontoaddaLists&Spreadsheetpage. TypescoreasthevariablenameforcolumnA,thenenter Jamal’sscoresintothiscolumn. TypedevasthevariablenameforcolumnB,theninthe cellbelow,typeintheformula=score-mean(score). © TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan 2 WhatisStandardDeviation?–StudentWorksheet Nowuseyourspreadsheettoanswerthefollowingquestions. Question2. a) Whichscoredeviatesthemostfromthemeanscore? b) Whichscoreisclosesttothemeanscore? 29 Question3. a) Apossiblemeasureofthe‘average’deviationisthe‘mean’deviation,whichiscalculated as follows(copyandcomplete). Jamal's!mean!deviation!= = Sum!of!deviations Number!of!scores !!!! + !!!!! + !!!!! + !!!!! = !!!!! !!!!! !!!!! = !!!!! ! b) Isthisausefulmeasureofthe‘average’deviation?Why/whynot? Thepositiveandnegativedeviationscancelout.Themeanscorewillalwaysbe thevaluefor whichthiscancellingoccurs.Hencethemeandeviationisnotauseful measureoftheaverage amountofdeviation. Anotherpossiblewayofmeasuringthe‘average’deviationistosquareeachdeviationfirst,findthemean ofthesquareddeviations,andtofindthesquarerootofthis. OntheTI-NspireCAS • • TypesqdevasthevariablenameforcolumnC Inthecellbelow,typeintheformula=dev2. Notethatallsquareddeviationsarepositive. Themean‘squared’deviationofJamal’sscorescanbecalculatedinthefollowingmanner. Question4. Copyandcompletethefollowing,givingyouranswerinsimplestfractionform. Jamal's!mean!squared!deviation!= = ! © Sum!of!squared!deviations Number!of!scores !!!!! + !!!!! + !!!!! + !!!!! !!!!! = !!!!! !!!!! = !!!!! TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan 3 WhatisStandardDeviation?–StudentWorksheet Tocalculatethemean‘squared’deviationofJamal’sscores, dothefollowing.ontheTI-NspireCAS • MovetocellD1,andtypethelabelmeansqdev. • MovetocellD2,andtypetheformula =approx(mean(sqdev)). Confirmthatitgivesthesameanswerforthemean‘squared’ deviationastheoneyoufoundabove. Finally,tofindthe‘average’deviationbythismethod,findthesquarerootofthevalueoftheaverage squareddeviation. Nowwehaveastandardisedmeasureofthemeandeviation(referredtoasthestandarddeviation), whichignoreswhethertheindividualdeviationsarepositiveornegative. TocalculatethestandarddeviationofJamal’sscoresonthe TI-NspireCAS • • MovetocellD3,andtypethelabelstdev. MovetocellD4,andtypetheformula=sqrt(D2) NowletslookatthespreadofRudi’sbattingscores(7,12,18and83runs). Onthespreadsheet,changethevaluesinColumnAtoRudi’sscores. Question5. Forthesenewscores,find(correctto2decimalplaces) a) themeansquareddeviation 951.5 b) thestandarddeviation 30.8 Question6. ComparethestandarddeviationofRudi’sscoreswiththestandarddeviationofJamal’sscoresfound previously.Whatdoyounotice?Whatdoesthissuggest? ThestandarddeviationofRudi’sscoresismuchgreater,indicatingthathisscoresare,onaverage, furtherfromthemeanscore. © TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan 4 WhatisStandardDeviation?–StudentWorksheet Now,supposeRudiisabouttobatagainforthefifthtime.TrydifferentscoresforRudi’sfifthscore. Question7. a) Whatfifthinningsscorewillmakehisstandarddeviationaslowaspossible? 30 b) Whatisthevalueofthisstandarddeviation(correctto2decimalplaces)? 27.59 c) Explainwhythisscorewillachievethesmallestpossiblevalueforthestandarddeviation. Thenumeratorintheexpressionformeansquareddeviationisunchangedbut thedenominator decreasesbyone.Anyotherchangewouldincreasethe numeratoralso,sothestandard deviationwouldbelarger. Part 2: Exploring the standard deviation Wenowlookatsomeotherdatatoexploretheideaofstandarddeviation,andhowitisaffectedbythe spreadofthedatavalues. UsingtheTI-NspireCASspreadsheettemplateyouhavedeveloped,answerthefollowingquestions. Question8. a) b) c) d) e) f) g) © Find4battingscoresthathaveameanscoreof30andastandarddeviationof0. 30,30,30,30 Find4battingscoresthathaveameanscoreof30andastandarddeviationof10. 20,20,40,40 Find4battingscoresthathaveameanscoreof30andastandarddeviationof20. 10,10,50,50 Explainthemethodyouusedtoarriveatyouranswersabove. Thesumofthescoresmustbe120(forsamemeanscore).Also,thesizeofeach deviationmustbethesame. Find4battingscores(between0and100inclusive)thathavethelargestpossible standarddeviation. 0,0,100,100;SD=50 Find4battingscores(between0and100inclusive)thathavethesmallestpossible standarddeviation. Any4identicalscores(e.g.23,23,23,23)SD=0 Thinkaboutyouranswerstoquestionseandf.Inyourownwordsexplainhowthe standarddeviationisrelatedtothespreadofadataset. Thelargerthestandarddeviation,thegreaterthevariationinthedataset;the smallerstandard deviation,thesmallerthevariationinthedataset. TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan 5 WhatisStandardDeviation?–StudentWorksheet Part 3: Changing the data Considertheoriginal4scoresforJamal.Ifeachscorehadbeen20runsmore,itisclearthathismean score(his‘battingaverage’)wouldbehigher,butbyhowmuch?Also,howwouldsuchanincreaseineach scoreaffectthevalueofthestandarddeviation? Todothiswiththeaidofthespreadsheettemplate,wewill addaformulaincellD5thatcalculatesthemeanscore. • • MovetocellD5. Typetheformula=approx(mean(score). Nowuseyourspreadsheettemplatetohelpanswerthesequestions. Question9. Whateffectdoesadding20runstoeachscorehaveupon a) themeanscore? Meanscoreincreasesby20. b) thestandarddeviationofthescore? Standarddeviationisunchanged c) Tryrepeatingthisbyaddinganother20runstoeachscore. Whatdoyounoticeaboutthevalueinthemeanandthestandarddeviationnow? Meanscoreincreasesby20againandstandarddeviationisstillunchanged. d) Whydoyouthinkthishappens? Allscores(includingmeanscore)areincreasedbyafixedamount,butthe deviationofeachnew scorefromthenewmeanscoreisunchanged. Finally,weinvestigatewhathappenstothemeanandstandarddeviationifwemultiplyeachscorebya givennumber(e.g.×2ordoubling).Whateffectdoesthishaveonthemeanandstandarddeviation? Useyourspreadsheettemplatetohelpanswerthefollowingquestions. Question10. WhateffectdoesdoublingeachofJamal’soriginalscoreshaveuponthe a) themeanscore? Meanscoreisdoubled. b) thestandarddeviationofthescore? Standarddeviationisdoubled © TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan 6 WhatisStandardDeviation?–StudentWorksheet Experimentwithsomeothervalues(e.g.triplingeachscore),andthencopyandcompletethefollowing table. Originalscores Originalscoresx2 Originalscoresx3 Originalscoresx0.5 Score1 29 58 87 14.5 Score2 35 70 108 27.5 Score3 29 58 87 14.5 Score4 27 54 81 13.5 Mean 30 60 90 15 Standarddeviation 3 6 9 1.5 Question11. Inyourownwords,summarisewhathappenstothevaluesofthemeanandstandarddeviationwhen eachscoreismultipliedbyaconstantfactor.Suggestareasonwhythismighthappen. Boththemeanandthestandarddeviationarealsomultipliedbythatconstantfactor.Thestandard deviationincreasesbythefactor,sincethedifferenceofeachscorefromthemeanincreasesbythat factor. Challenge Whatwouldhappentothevaluesofthemeanandstandarddeviationifyouweretodoubleeachscore andthenadd10?TrytodothisbyusingtheresultsfromPart3,andthenchecktoseeifyouarecorrect. Generaliseyourresult:ifxrepresentsthescores,thenwhatwillhappentothemeanandstandard deviationifeachscoreismultipliedbyaandthenbisadded(i.e.changextoax+b)? © TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan