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Hack 25. Predict the Future with the Normal Curve

Hack 25. Predict the Future with the Normal Curve

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WhatpercentofstudentsintheU.S.willqualifyasNational

MeritScholars?

WhatarethechancesthatmyUncleFrankcouldpassthe

Mensaqualifyingexam?

Forthesetypesofquestions,aprecisetoolisneeded.Thishack

providesthattool:atableofareasunderthenormalcurve.



TheTableofAreasUndertheNormalCurve

Thenormalcurveisdefinedbythemeanandstandard

deviationofadistribution,andtheshapeofthecurveisalways

thesame,regardlessofwhatwemeasure,aslongasthe

scoringsystemallowsscorestovary.Theproportionsofscores

fallingwithinvariousareasbeneaththecurve,suchasthe

spacebetweencertainstandarddeviationsanddistancesfrom

themean,havebeenspecified.

Thishackreliesonacomplicated-lookingtable,butitissofull

ofusefulinformationthatitwillquicklybecomeaprimarytool

inyourhacker'stoolbox.Withoutfurtherado,takeadeep

breathandlookatTable3-2.

TableAreasunderthenormalcurve



z

score

.00

.12

.25

.39

.52

.67

.84

1.04



Proportionofscores

betweenthemean

andz

.00

.05

.10

.15

.20

.25

.30

.35



Proportionof

scoresinthe

largerarea

.50

.55

.60

.65

.70

.75

.80

.85



Proportionof

scoresinthe

smallerarea

.50

.45

.40

.35

.30

.25

.20

.15



1.28

1.65

1.96

4.00



.40

.45

.475

.50



.90

.95

.975

1.00



.10

.05

.025

.00



DecipheringtheTable

Beforeweusethisniftytool,weneedtotakeaseconddeep

breathandgetthelayoftheland.Ihavesimplifiedthe

informationonthistableinacoupleofways.First,Ihavelisted

onlyafewofthevaluesthatcouldbecomputed.Indeed,many

tablesinstatisticalbookshaveeveryvaluebetweenazof.00

andazof4.00,increasingattherateof.01.That'salotof

informationthatcouldbepresented,soIhavechosentoshow

onlyaglimpseofthemostcommonlyneededvalues,including

thezscoresnecessaryfor90percentconfidence(1.65)and95

percentconfidenceintervals(1.96);see"MeasurePrecisely"

[Hack#6]formoreonconfidenceintervals.

Ihavealsoroundedtheproportionstotwodecimalplaces.

Finally,Iusedthesymbolzinthetabletoindicatethedistance

fromthemeaninstandarddeviations.Youcanlearnmore

aboutzscoresin"GiveRawScoresaMakeover"[Hack#26].

Afterunderstandingthesimplificationsmadetothetable,the

firststeptowardusingittomakeprobabilitypredictionsabout

performanceoranswerstatisticalquestionsistounderstandthe

fourcolumns.



Thezcolumn

Picturethenormalcurve[Hack#23].Ifyouareinterested



insomescorethatcouldfallalongthebottomhorizontal

line,itissomedistancefromthemean.Itcouldbegreater

thanthemeanscoreorlessthanit.Thedistancetothe

meanexpressedinstandarddeviationsisthezscore.Az

scoreof1.04describesascorethatisalittlemorethanone

standarddeviationawayfromthemean.Becausethe

normalcurveissymmetrical,wedon'tbothertonote

whetherthedistanceisnegativeorpositive,soallofthese

zscoresareshownaspositive.



Proportionofscoresbetweenthemeanandz

Inthatspacebetweenagivenscoreandthemean,there

willbeacertainproportionofscores.Thisistheprobability

thatarandomscorewillfallintheareadefinedbythe

meanandanyz.



Proportionofscoresinthelargerarea

Youcouldalsodescribetheareabetweenanygivenzanda

zof4.00,ortheendofthecurve.

Thecurvedoesn'treallyeverend,theoretically,butaz

scoreof4.00willcomeveryclosetoincluding100percent

ofthescores.

Therearetwoendsofthecurve,though.Unlessyourzis

0.0,thedistancebetweenthezandoneendofthecurve

willbegreaterthanthedistancebetweenthezandthe

otherend.Thiscolumnreferstotheareabetweenthezand

thatfurthestendofthecurve,andthevalueinthiscolumn

istheproportionofscoresthatwillfallinthatspace.In

otherwords,itisthechancethatarandompersonwill

produceascoreinthatarea.



Proportionofscoresinthesmallerarea

Thiscolumnreferstotheareabetweenthezandthat

closestendofthecurve.Itistheproportionofscoresthat

willfallinthatspace.



EstimatingtheChanceofScoringAboveor

BelowAnyScore

Ifyouneedtoknowyourchancesofgettingintoyourcollegeof

choice,identifythenecessaryscoreyouneedtobeat,also

knownasthecutscore,onthatschool'sadmissionstests.Once

youknowthescore,findoutthemeanandstandarddeviation

forthetest.(AllofthisinfoisprobablyontheWeb.)Convert

yourrawscoretoazscore[Hack#26],andthenfindthatz

score,orsomethingclosetoit,inTable3-2.

Determinewhetherthecutscoreisabovethemean:

Ifitis,lookatthe"Proportionofscoresinthesmallerarea"

column.Thatrepresentsyourchancesofscoringator

abovethatcutscore,andyourchancesofgettingin.

Ifthecutscoreisbelowthemean(unlikely,butforthesake

ofcompletelytrainingyouonhowtousethistool),identify

"Proportionofscoresinthelargerarea."That'sthe

proportionofstudentsbeingacceptedand,thus,your

chances,allthingsbeingequal.

Forthechancesofscoringbelowagivenscore,theprocessis

theoppositeoftheoptionsjustmentioned.Thechanceof

gettingbelowaspecificcutscorethatisbelowthemeanis

showninthe"smallerarea"column.Thechanceofscoring



belowagivencutscorethatisabovethemeanisshowninthe

"largerarea"column.



EstimatingtheChanceofScoringBetweenAny

TwoScores

Thechancesofgettingascorewithinanyrangeof

scoringscorescanbedeterminedbylookingattheproportionof

scoresthatwillnormallyfallinthatrange.

Ifyouwanttoknowwhatproportionofscoresfallsbetweenany

twopointsunderthecurve,definethosepointsbytheirzscore

andfigureouttherelevantproportion.Dependingonwhether

bothscoresfallonthesamesideofthemean,oneoftwo

methodswillgiveyouthecorrectproportionbetweenthose

points:

Ifthezscoresareonthesamesideofthecurve,lookup

theproportionofscoresineitherthe"largerarea"or

"smallerarea"columnforbothzscoresandsubtractthe

lowervaluefromthehighervalue.

Ifthezscoresfallonbothsidesofthemeanwiththemean

betweenthem,usethe"Proportionofscoresbetweenthe

meanandz"column.Lookupthevalueforbothscoresand

addthemtogether.



ProducingPercentileRanks

Athirduseofthetableistocomputepercentileranks.Youcan

readmoreaboutsuchnorm-referencedscoresin"Produce

Percentiles"[Hack#24].Forscoresabovethemean,the

percentilerankis"Proportionofscoresbetweenthemeanand



z"plus.50.Forscoresbelowthemean,thepercentilerankis

"Proportionofscoresinthesmallerarea."



DeterminingStatisticalSignificance

Anotheruseforthesesortsoftablesistoassignstatistical

significance[Hack#4]todifferencesinscores.Byknowingthe

proportionofscoresthatwillfallacertaindistancefromeach

otherorfurther,youcanassignastatisticalprobabilitytothat

outcome.

Moreusefully,otherstatisticalvaluessuchascorrelationsand

proportionscanbeconvertedtozscores,andthistablecanbe

usedtocomparethosevaluestozeroortoeachother.



WhyItWorks

"SeetheShapeofEverything"[Hack#23]providesagood

pictureofthenormalcurve.However,justbylookingattheway

thesevalueschangeinTable3-2,youcangetagoodsenseof

thenormaldistribution'sshape.Nearthemean,wheretherows

havesmallerzscores,agoodlyproportionofscoreswillfall.As

youmovefurtherandfurtherawayfromthemean,ittakes

largerandlargerareasofthecurvetocontainthesame

proportionofscores.

Forexample,ittakesajumpfromazof1.65to4justtocover

thatlast5percentofthedistribution.Nearthemean,though,it

requiresonlyajumpfromz=.12toz=.25tocover5percent

ofscores.Thetabledemonstrateshowcommonitistobe

commonandhowrareitistobescarce.



SeeAlso



Youwillbeabletocomputeyourownexactareasunderthe

normalcurvebyusingthiswebsite:

http://www.psychstat.missouristate.edu/introbook/sbk11m.htm

Agooddiscussionandsomeinteractivecalculatorsarepart

ofthissitemaintainedbyDavidStockburger.Whenyou

visit,don'tbeconfusedbywordslikeMuandSigma.That's

statstalkformeanandstandarddeviation,respectively.



Hack26.GiveRawScoresaMakeover



Arawscoreonatesthaslittleornomeaning.Change

thatpitifulrawscoretoa"zscore,"though,andyouwill

scarcelybelievehowmuchinformationiscrammedinto

thatonelittlesupernumber.

Itissurprisinghowlittleinformationisconveyedbythatsingle

rawscoreplasteredatthetopofsomethinglikeahighschool

test.Here'swhatImean.IfIcomehomefromschoolandtell

mymomthatIgota16onthebigexaminschooltoday,she'll

probablysayafewthings,including"Whyareyoustilllivingat

homeatage42?"and"That'snice,dear.Is16good?"

Whenyoujusttellsomeonearawscore,verylittlereal

informationhasbeenshared.Youdon'tknowif16isgood.You

don'tknowif16isrelativelyhighorlow.Didmostpeoplegeta

16orhigher,ordidmostpeoplegetsomethinglessthan16?

Evenifweknowtherangeofscoresonthattestandthepoints

possibleandsoon,westillcan'tcompareperformanceonthat

testtoperformanceonthepasttestorthenexttestoratest

onsomeothersubject.Rawscoresarevirtuallymeaningless.

Don'tfret!Youcanstillunderstandyourperformanceandthe

performancesofothers.Youcanstillmakeselectiondecisions

andcompareperformanceacrosspeopleandacrosstests.

Thereisstillhope!

Rawscorescanbechangedintoanewnumberthatdoesallthe

thingsthatthat97-poundweakling,therawscore,couldnever

do.Rawscorescanbetransformedintoasupernumber:az

score.Unlikearawscore,aztellsyouwhetherthe

performanceisaboveorbelowaverage,andhowfaraboveor



belowaverageitis.Azalsoallowsyoutocompareperformance

acrosstestsandoccasions,andevenbetweenpeople.



CalculatingzScores

Azscoreisarawscorethathasbeentransformedinsucha

waythatthenewnumberindicateshowfaraboveorbelowthe

meantherawscoreis.

Here'stheequation:

Tochangearawscoreintoaz,subtractthemeanfromitand

thendividebythestandarddeviation.Thestandarddeviationof

adistributionistheaveragedistanceofeachscorefromthe

mean[Hack#2].



UnderstandingPerformance

zscorestypicallytakeonarangeofvaluesbetween-3and+3.

Examinethetoppartofthezscoreequationandyoumight

noticethefollowing:

Iftherawscoreisgreaterthanthemean,thezwillbe

positive.

Iftherawscoreisbelowthemean,thezwillbenegative.

Iftherawscoreisexactlythemean,thezwillbe0.



zscorestendtorangebetween-3and+3becausethenormal

distributionofscoresistypicallyjustsixstandarddeviationswide[Hack

#23].



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