The Normal Distribution and Z score正态分布和Z评分.ppt

examperformance gradedistribution homeworkandexamperformance examrules reviewtodayinclassreturnattheendofclassyoumaycomeseetheminmyofficeatyourleisure ourfriend thenormaldistribution frequencydistributions howdowefindthenumberofstudentswhoscorebetween4and6 inclusive add score4 5studentsscore5 3studentsscore6 2students 10students score freq whatdowedowithcontinuousscores frequenciesgetmessydistributionisnotclearweneedsomethingelse probabilitydistributions howdowefindtheprobabilityofastudentscoringbetween4and6 inclusive add p 4 2p 5 1p 6 05 35students score freq continuousprobabilitydistributions whatistheprobabilityofscoring3 141592654 virtuallyzero whenitiscontinuous weneedtofindtheprobabilityofscoringinarange normaldistributioncurve thenormaldistributioncanalsobeaprobabilitydistribution familyofnormalcurves allinfamilyare frequencydistributions whichconformtothe68 95 99 7rule themeansandstandarddeviationsofdifferentdistributionsdifferbutthesymmetryholds changingstandarddeviations whateverthemeanandstddev ifthedistributionisnormallydistributed the68 95 99 7ruleapplies at68 two thirdsofallthecasesfallwithin 1standarddeviationofthemean 95 ofthecaseswithin 2standarddeviationofthemean and99 7 ofthecaseswithin 3sdofthemean howdowedoit wehaveour68 95 99 7rulewejusthavetoknowhowmanystandarddeviationsacertainnumberisawayfromthemean example satscoresarenormallydistributedwithameanof500andastd dev of100 whatpercentageofstudentsscorebetween400and600 68 practice satscoresarenormallydistributedwithameanof500andastd dev of100 whatpercentageofstudentsscorebetween400and500 34 practice satscoresarenormallydistributedwithameanof500andastd dev of100 whatpercentageofstudentsscorelessthan300 2 5 iqscoresarenormallydistributedwithameanof100andastd dev of15 whatpercentageofpeoplehaveaniqbetween85and115 itworkswithdifferentand 68 practice iqscoresarenormallydistributedwithameanof100andastd dev of15 whatpercentageofpeoplehaveaniqbetween85and100 34 practice iqscoresarenormallydistributedwithameanof100andastd dev of15 whatpercentageofpeoplehaveaniqlessthan70 2 5 problem bymanipulatingprobabilities wecanonlyhandlesituationswhereweare1 2 or3std devs awayfromthemean whathappenswhenwewanttoknowtheprobabilityofscoringiqbetween100and105weneedtoconverttheiqscore orsatscoreorwhatever intounitsofthestandarddeviation example distancebetween100and105is 333standarddeviations thinkofthestd dev asaunit howmanyinchesareinafoot 12howmanycupsareinapint 2howmanyiqpointsarethereinastandarddeviationforiq 15howmanysatpointsarethereinastandarddeviationforsatscores 100 howdoyouconvertinchestofeet distanceinfeet distanceininches12distanceiniqstddevs distanceiniqpoints100distanceiniqstd devs considerthisproblem party timeemployeesalariesinacompanyarenormallydistributedwithmean 20 000andstandarddev 1 000howmanystd devs is 18 500awayfromthemean intuitively weseethat1 500is1 5std devs fromusingtheformula weget 1 5 negativespecifiesdirection considerthisproblem howmanystd devs is 19 371awayfromthemean intuitively wecan tdothisusingtheformula weget 269std devs away x 19 371 zscores wecallthesestandarddeviationvalues z scores zscoreisdefinedasthenumberofstandardunitsanyscoreorvalueisfromthemean zscorestateshowmanystandarddeviationstheobservationxfallsawayfromthemeanandinwhichdirection plusorminus whatgooddoesthisdo someonefiguredoutthat68 arewithin 1s d andabout95 arewithin 2s d someonedidthistoshowthat74 16 arewithin 1 13s d inthenormaldistribution1 14s d 74 58 1 15s d 74 98 1 16s d 75 4 itgoesonandonandon theseresultsappearina z table youcalculateazscore findthatscoreincolumnaandthez tablewilltellyou theprobabilityofgettingascorebetweenyourz scoreandthemean columnb theprobabilityofgettingascoregreaterthanyourz score thatis fromyourz scoreouttotheendofthenormaldistribution columnc thistablecanbedownloadedfrommywebsite itlookslikethis supposeyoufindaz scoreof 12columnbsaysthat4 78 ofcasesliebetweenthemeanandyourz score itlookslikethis supposeyoufindaz scoreof 12columncsaysthat45 22 ofcasesliebeyondyourz score columnc iqisnormallydistributedwithameanof100andsdof15 howdoyouinterpretascoreof109 usezscore whatdoesthisz score 60mean doesnotmean60percentofcasesbelowthisscorebutratherthatthiszscoreis 60standardunitsabovethemean weneedthez tabletointerpretthis usingtheztable lookatcolumncfor 60only27 43 ofpeoplehaveaniqhigherthanthis ifyouriqis109 6s d abovethemean youaresmarterthanalmost75 ofpeopleintheworld 72 57 ofpeoplehaveaniqlessthanthis usefulnessofzscore describescoresrelativetootherscoresinasingledistributionwhenwedividethedeviationbythestandarddeviation thezscoreistheprobabilityofgettingaparticularvalu