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Hack 1. Know the Big Secret

# Hack 1. Know the Big Secret

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however,aretheuseofdescriptivestatisticstodescribeagroup

ofscores,ortheuseofinferentialstatisticstomakeguesses

containedinasampleofscores.Insocialscience,thescores

usuallydescribeeitherpeopleorsomethingthatishappeningto

them.

Itturnsout,then,thatresearchersandmeasurers(thepeople

whoaremostlikelytousestatisticsintherealworld)arecalled

upontodomorethancalculatetheprobabilityofcertain

combinationsandpermutationsofinterest.Theyareableto

questionsofvaryinglevelsofcomplexitywithoutonceneeding

tocomputetheoddsofthrowingapairofsix-sideddiceand

gettingthree7sinarow.

Thoseoddsare.005or1/2of1percentifyoustartfromscratch.Ifyou

thatthird7.

TheBigSecret

Thekeyreasonthatprobabilityissocrucialtowhatstatisticians

thescoresinrealortheoreticaldistributions.

sometimes,howmanyofeachvaluethereare.

25percentoftheclassgot10points,thenImightsay,without

chancethatyougot10points.Icouldalsosaythatthereisa

75percentchancethatyoudidnotget10points.AllIhave

valuesandexpressedthatinformationasastatementof

probability.Thisisatrick.Itisthesecrettrickthatall

statisticiansknow.Infact,thisismostlyallthatstatisticians

everdo!

somevaluesandexpressthatinformationasastatementof

probability.Thisisworthrepeating(or,technically,

threepeating,asIfirstsaiditfivesentencesago).Statisticians

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

Thekey,ofcourse,ishowoneknowsthedistributionofall

theseexotictypesofvaluesthatmightinterestastatistician.

Howcanoneknowthedistributionofaveragedifferencesorthe

distributionofthesizeofarelationshipbetweentwosetsof

variables?Conveniently,pastresearchersandmathematicians

havedevelopedordiscoveredformulasandtheoremsandrules

ofthumbandphilosophiesandassumptionsthatprovideus

withtheknowledgeofthedistributionsofthesecomplexvalues

mostoftensoughtbyresearchers.Theworkhasbeendonefor

us.

ASmaller,DirtierSecret

Mostoftheproceduresthatstatisticiansusetotakeknown

informationasastatementofprobabilityhavecertain

requirementsthatmustbemetfortheprobabilitystatementto

beaccurate.Oneoftheseassumptionsthatalmostalwaysmust

bemetisthatthevaluesinasamplehavebeenrandomly

drawnfromthedistribution.

NoticethatinthecoffeecanexampleIslippedinthat"without

guidingthesamplingprocess,thentheassociatedprobabilities

reportedaresimplywrongandhere'stheworstpartwecan't

possiblyknowhowwrongtheyare.Much,andmaybemost,of

theappliedpsychologicalandeducationalresearchthatoccurs

todayusessamplesofpeoplethatwerenotrandomlydrawn

fromsomepopulationofinterest.

Collegestudentstakinganintroductorypsychologycourse

makeupthesamplesofmuchpsychologicalresearch,for

example,andstudentsatelementaryschoolsconveniently

locatednearwhereaneducationalresearcherlivesareoften

chosenforstudy.Thisisaproblemthatsocialscience

nevertheless,itisalimitationofmuchsocialscienceresearch.

Hack2.DescribetheWorldUsingJustTwo

Numbers

Mostofthestatisticalsolutionsandtoolspresentedin

thisbookworkonlybecauseyoucanlookatasample

TheCentralLimitTheoremisthemeta-tool,theprime

directive,thekingofallsecretsthatallowsustopulloff

theseinferentialtricks.

Statisticsprovidesolutionstoproblemswheneveryourgoalis

todescribeagroupofscores.Sometimesthewholegroupof

scoresyouwanttodescribeisinfrontofyou.Thetoolsforthis

onlypartofthegroupofthescoresyouwanttodescribe,but

youstillwanttodescribethewholegroup.Thissummary

approachiscalledinferentialstatistics.Ininferentialstatistics,

thepartofthegroupofscoresyoucanseeiscalledasample,

andthewholegroupofscoresyouwishtomakeinferences

describewithanyconfidenceapopulationofvalueswhen,by

definition,youarenotdirectlyobservingthosevalues.Byusing

threepiecesofinformationtwosamplevaluesandan

populationyoucanconfidentlyandaccuratelydescribethose

invisiblepopulations.Thesetofproceduresforderivingthat

eerilyaccuratedescriptioniscollectivelyknownastheCentral

LimitTheorem.

SomeQuickStatisticsBasics

Inferentialstatisticstendtousetwovaluestodescribe

populations,themeanandthestandarddeviation.

Mean

Ratherthandescribeasampleofvaluesbyshowingthemall,it

issimplymoreefficienttoreportsomefairsummaryofagroup

numberismeanttofairlyrepresentallthescoresandwhat

theyhaveincommon.Consequently,thissinglenumberis

referredtoasthecentraltendencyofagroupofscores.

Typically,thebestmeasureofcentraltendency,foravarietyof

reasons,isthemean[Hack#21].Themeanisthearithmetic

thevaluesinagroup,andthendividingthattotalbythe

allthescoresinagroupthanothercentraltendencyoptions

(suchasreportingthemiddlescore,themostcommonscore,

andsoon).

Infact,mathematically,themeanhasaninterestingproperty.A

bythenumberofscores)producesanumberthatisascloseas

possibletoalltheotherscores.Themeanwillbeclosetosome

distances,yougetatotalthatisassmallaspossible.Noother

number,realorimagined,willproduceasmallertotaldistance

fromallthescoresinagroupthanthemean.

Standarddeviation