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Hack 1. Know the Big Secret

Hack 1. Know the Big Secret

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however,aretheuseofdescriptivestatisticstodescribeagroup

ofscores,ortheuseofinferentialstatisticstomakeguesses

aboutapopulationofscoresusingonlytheinformation

containedinasampleofscores.Insocialscience,thescores

usuallydescribeeitherpeopleorsomethingthatishappeningto

them.

Itturnsout,then,thatresearchersandmeasurers(thepeople

whoaremostlikelytousestatisticsintherealworld)arecalled

upontodomorethancalculatetheprobabilityofcertain

combinationsandpermutationsofinterest.Theyareableto

applyawidevarietyofstatisticalprocedurestoanswer

questionsofvaryinglevelsofcomplexitywithoutonceneeding

tocomputetheoddsofthrowingapairofsix-sideddiceand

gettingthree7sinarow.



Thoseoddsare.005or1/2of1percentifyoustartfromscratch.Ifyou

havealreadyrolledtwo7s,youhavea16.6percentchanceofrolling

thatthird7.



TheBigSecret

Thekeyreasonthatprobabilityissocrucialtowhatstatisticians

doisbecausetheyliketomakeprobabilitystatementsabout

thescoresinrealortheoreticaldistributions.



Adistributionofscoresisalistofallthedifferentvaluesand,

sometimes,howmanyofeachvaluethereare.



Forexample,ifyouknowthataquizjustadministeredina

classyouaretakingresultedinadistributionofscoresinwhich

25percentoftheclassgot10points,thenImightsay,without

knowingyouoranythingaboutyou,thatthereisa25percent

chancethatyougot10points.Icouldalsosaythatthereisa

75percentchancethatyoudidnotget10points.AllIhave

doneistakenknowninformationaboutthedistributionofsome

valuesandexpressedthatinformationasastatementof

probability.Thisisatrick.Itisthesecrettrickthatall

statisticiansknow.Infact,thisismostlyallthatstatisticians

everdo!

Statisticianstakeknowninformationaboutthedistributionof

somevaluesandexpressthatinformationasastatementof

probability.Thisisworthrepeating(or,technically,

threepeating,asIfirstsaiditfivesentencesago).Statisticians

takeknowninformationaboutthedistributionofsomevalues

andexpressthatinformationasastatementofprobability.

HeavenstoBetsy,wecanalldothat.Howhardcoulditbe?

Imaginethattherearethreemarblesinanotherwiseempty

coffeecan.Furtherimaginethatyouknowthatonlyoneofthe

marblesisblue.Therearethreevaluesinthedistribution:one

bluemarbleandtwomarblesofsomeothercolor,foratotal

samplesizeofthree.Thereisonebluemarbleoutofthree

marbles.Oh,statistician,whatarethechancesthat,without

looking,Iwilldrawthebluemarbleoutfirst?Oneoutofthree.

1/3.33percent.

Tobefair,thevaluesandtheirdistributionsmostcommonly

usedbystatisticiansareabitmoreabstractorcomplexthan

thoseofthemarblesinacoffeecanscenario,andsomuchof

whatstatisticiansdoisnotquitethattransparent.Appliedsocial

scienceresearchersusuallyproducevaluesthatrepresentthe

differencebetweentheaveragescoresofseveralgroupsof

people,forexample,oranindexofthesizeoftherelationship

betweentwoormoresetsofscores.Theunderlyingprocessis

thesameasthatusedwiththecoffeecanexample,though:



referencetheknowndistributionofthevalueofinterestand

makeastatementofprobabilityaboutthatvalue.

Thekey,ofcourse,ishowoneknowsthedistributionofall

theseexotictypesofvaluesthatmightinterestastatistician.

Howcanoneknowthedistributionofaveragedifferencesorthe

distributionofthesizeofarelationshipbetweentwosetsof

variables?Conveniently,pastresearchersandmathematicians

havedevelopedordiscoveredformulasandtheoremsandrules

ofthumbandphilosophiesandassumptionsthatprovideus

withtheknowledgeofthedistributionsofthesecomplexvalues

mostoftensoughtbyresearchers.Theworkhasbeendonefor

us.



ASmaller,DirtierSecret

Mostoftheproceduresthatstatisticiansusetotakeknown

informationaboutadistributionofscoresandexpressthat

informationasastatementofprobabilityhavecertain

requirementsthatmustbemetfortheprobabilitystatementto

beaccurate.Oneoftheseassumptionsthatalmostalwaysmust

bemetisthatthevaluesinasamplehavebeenrandomly

drawnfromthedistribution.

NoticethatinthecoffeecanexampleIslippedinthat"without

looking"business.Ifsomeforceotherthanrandomchanceis

guidingthesamplingprocess,thentheassociatedprobabilities

reportedaresimplywrongandhere'stheworstpartwecan't

possiblyknowhowwrongtheyare.Much,andmaybemost,of

theappliedpsychologicalandeducationalresearchthatoccurs

todayusessamplesofpeoplethatwerenotrandomlydrawn

fromsomepopulationofinterest.

Collegestudentstakinganintroductorypsychologycourse

makeupthesamplesofmuchpsychologicalresearch,for

example,andstudentsatelementaryschoolsconveniently



locatednearwhereaneducationalresearcherlivesareoften

chosenforstudy.Thisisaproblemthatsocialscience

researcherslivewithorignoreorworryabout,but,

nevertheless,itisalimitationofmuchsocialscienceresearch.



Hack2.DescribetheWorldUsingJustTwo

Numbers



Mostofthestatisticalsolutionsandtoolspresentedin

thisbookworkonlybecauseyoucanlookatasample

andmakeaccurateinferencesaboutalargerpopulation.

TheCentralLimitTheoremisthemeta-tool,theprime

directive,thekingofallsecretsthatallowsustopulloff

theseinferentialtricks.

Statisticsprovidesolutionstoproblemswheneveryourgoalis

todescribeagroupofscores.Sometimesthewholegroupof

scoresyouwanttodescribeisinfrontofyou.Thetoolsforthis

taskarecalleddescriptivestatistics.Moreoften,youcansee

onlypartofthegroupofthescoresyouwanttodescribe,but

youstillwanttodescribethewholegroup.Thissummary

approachiscalledinferentialstatistics.Ininferentialstatistics,

thepartofthegroupofscoresyoucanseeiscalledasample,

andthewholegroupofscoresyouwishtomakeinferences

aboutisthepopulation.

Itisquiteatrick,though,whenyouthinkaboutit,tobeableto

describewithanyconfidenceapopulationofvalueswhen,by

definition,youarenotdirectlyobservingthosevalues.Byusing

threepiecesofinformationtwosamplevaluesandan

assumptionabouttheshapeofthedistributionofscoresinthe

populationyoucanconfidentlyandaccuratelydescribethose

invisiblepopulations.Thesetofproceduresforderivingthat

eerilyaccuratedescriptioniscollectivelyknownastheCentral

LimitTheorem.



SomeQuickStatisticsBasics



Inferentialstatisticstendtousetwovaluestodescribe

populations,themeanandthestandarddeviation.



Mean

Ratherthandescribeasampleofvaluesbyshowingthemall,it

issimplymoreefficienttoreportsomefairsummaryofagroup

ofscoresinsteadoflistingeverysinglescore.Thissingle

numberismeanttofairlyrepresentallthescoresandwhat

theyhaveincommon.Consequently,thissinglenumberis

referredtoasthecentraltendencyofagroupofscores.

Typically,thebestmeasureofcentraltendency,foravarietyof

reasons,isthemean[Hack#21].Themeanisthearithmetic

averageofallthescoresandiscalculatedbyaddingtogetherall

thevaluesinagroup,andthendividingthattotalbythe

numberofvalues.Themeanprovidesmoreinformationabout

allthescoresinagroupthanothercentraltendencyoptions

(suchasreportingthemiddlescore,themostcommonscore,

andsoon).

Infact,mathematically,themeanhasaninterestingproperty.A

sideeffectofhowitiscreated(addingupallscoresanddividing

bythenumberofscores)producesanumberthatisascloseas

possibletoalltheotherscores.Themeanwillbeclosetosome

scoresandfarawayfromsomeothers,butifyouaddupthose

distances,yougetatotalthatisassmallaspossible.Noother

number,realorimagined,willproduceasmallertotaldistance

fromallthescoresinagroupthanthemean.



Standarddeviation

Justknowingthemeanofadistributiondoesn'tquitetellus

enough.Wealsoneedtoknowsomethingaboutthevariability

ofthescores.Aretheymostlyclosetothemeanormostlyfar



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