统计学外文翻译.docx

统计学外文翻译.docx

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统计学外文翻译

外文翻译原文

名称：

Fundamentals_of_Statistics

MeasuresofCentralTendencyandLocation:

mean,median,mode,percentiles,quartilesanddeciles.

xsortedx

5353

5553

7053

5855

6457

5757

5358

6964

5768

6869

5370

TheMeasuresofCentralTendencyareMean,MedianandMode

Meanx-baror

foragivenvariable,itisthesumofthevaluesdividedbythenumberofvalues（∑xi/n）.Inthiscase,wehaven=11.Soweneedtoaddallofthevaluestogetheranddivideby11.∑=657,

=59.73

Medianthenumberinadistributionofavariable’sresponsewhereonehalfofthevaluesareaboveandonehalfofthevaluesarebelow.Tofindthemedian,wefirstneedtoputourdatainascendingorder（smallesttolargest）.Thenwecandeterminethemedian…ifthevalueofnisodd,itissimplythemiddleobservation,butifthevalueofniseven,itistheaverageofthetwomiddleobservations.

Inthiscase,nisodd,sothemedianwillbethemiddleobservationofoursortedvalues（the6thvalue）...57

Modethevaluethatoccursmostfrequently.Iftherearetwodifferentvaluesmostfrequentlyoccurring,thedataaresaidtobebi-modal.Iftherearemorethantwomodes,andthedistributionissaidtobemulti-modal.Inthiscase,thevaluethatoccursmostoftenis53.So,themodeis53.

ThemeasuresoflocationarePercentile,QuartileandDecile

Percentilethepthpercentileisavaluesuchthatatleastppercentoftheobservationsarelessthanorequaltothisvalueandatleast（100–p）percentoftheobservationsaregreaterthanorequaltothisvalue.Tocalculatepercentiles,weuseindices（i）.

i=（p/100）nforp1,p2,p3,…p99

Iftheanswerisawholenumber（aninteger）,theniistheaverageof（P/100）nand1+（P/100）n.

Iftheindexnumberisnotawholenumber,weALWAYSroundup.Thepositionoftheindexisthenextwholenumber（integer）greaterthanthecomputedindex.

Forexample:

i（p50）=（50/100）11=5.5...thisroundsupto6

So,wewouldcountfromthelowestvalueofthesorteddatatotheindexnumber（6）.Sincethecalculatediwasnotawholenumberwehadtorounduptofindthevaluewhereatleast50%ofthevaluesareequaltoorlowerthanthisvalueandatleast50%areequaltoorhigherthanthisvalue.Inthiscase,thevalueofthe50thpercentileisthe6thvalue...57…Doesthislookfamiliar?

The50thpercentileisthesamethingasthemedian.

Whatdoesittellus?

Inthisdistribution,ATLEAST50%oftheobservationsareLESSTHANOREQUALTO57ANDATLEAST50%oftheobservationsareGREATERTHANOREQUALTO57.

i（p80）=（80/100）11=8.8...thisroundupto9.The9thvalueis68.

Again,sincetheindexnumberisnotawholenumber,weroundup.So,wewouldcountfromthelowestvalueofthesorteddatatotheindexnumber（9）.Inthiscase,thevalueofthe80thpercentileis68.

Sincethisdatasethas11observations,wewon’thaveanyinstanceswhereourcalculatedindexnumberisawholenumber.However,ifwejustremoveourvalueof70andcreateanewdistribution,wewillbeabletoseeanexample...

53535355575758646869

i（p30）=（30/100）10=3...thisisawholenumber,sowemusttakethe3rdand4thvaluesandaveragethemtofindthe30thpercentile.（53+55）/2=54

So,thevalueofthe30thpercentileis54.

Returntoouroriginaldatadistribution...

Quartiles–arespecialcasesofpercentiles…Q1=P25,Q2=P50,Q3=P75,

Thesethreevaluesdividethedistributioninto4equalquarters

i（Q1）=（25/100）11=2.75...thisroundsto3,soQ1isthe3rdvalue...53

i（Q2）=（50/100）11=5.5...thisroundto6,soQ2isthe6thvalue...57

i（Q3）=（75/100）11=8.25...thisroundsto9,soQ3isthe9thvalue...64

MeasuresofDispersionorVariability:

Range,interquartilerange（IQR）,variance,standarddeviationandcoefficientofvariation.

Range=Thistellsushowwidethespanisfromthemaximumvaluetotheminimumvalue.（Max–Min）=Range.Inthisinstance,therangeis69-53=16.

InterquartileRange（IQR）=Thistellsushowwidethespanisinthemiddle50%ofthedata.（Q3–Q1）=IQR.Inthiscase...64–53=11

WewilluseIQRinlaterprocesses,sowewillwanttokeepthis

x

（x-xbar）

（x-xbar）2

5353

-6.73-6.73

45.2945.29

5353

-6.73-6.73

45.2945.29

5353

-6.73-6.73

45.2945.29

5555

-4.73-4.73

22.3722.37

5757

-2.73-2.73

7.457.45

5757

-2.73-2.73

7.457.45

5858

-1.73-1.73

2.992.99

6464

4.274.27

18.2318.23

6868

8.278.27

68.3968.39

6969

9.279.27

85.9385.93

7070

10.2710.27

105.47105.47

657657

-0.03-0.03

454.18454.18

657/11=59.73

454.18/10≈45.2

Weusetheformula:

=s2

Thevarianceforthesedatais454.18.Forourpurposeshere,thecomputationofvarianceisjustasteptowardsthecomputationofthestandarddeviation.

Samplestandarddeviation（s）isthepositivesquarerootofthevariance.

=s

Sotheformulaforsamplestandarddeviationis…

PopulationVariance

（2）usesthesameformulainthenumerator,butNinsteadofn-1inthedenominator.Sincewerarelyhaveinformationabouttheentirepopulation,wealmostalwaysusetheformulaforsamplevariance,s2.

PopulationStandardDeviation:

=

…sincewerarelyhaveinformationfromtheentirepopulation,weusetheformulaforsamplestandarddeviation,s.

CoefficientofVariation:

tellsuswhatpercentthesamplestandarddeviationisofthesamplemean

Thisnumberis“relative”andisonlyofuseincomparingthedistributionoftwoormorevariables.

SupposeIhavetwosamples,andIwanttoknowwhichsamplehasmorevariability…

Ifbothsampleshavethesamemean,theonewiththehigherstandarddeviationwillhavethegreatervariability.However,iftheyhavedifferentmeans,Ineedtocalculatethecoefficientofvariationtodeterminewhichonehasthemostvariability.xbar=458,s=112versusxbar=687,s=192

StandardizedDataandDetectingOutliers

Z-score:

z=

Thez-scoretellsushowmanystandarddeviationsavalueisfromthemean.Wecanlookatapictureofwhataz-scoretellsus.IntheNormalCurve…themeanisatthehighestpointandthecurvetailsoffsymmetricallyinbothdirections.

Thesignofthez-scoretellsuswhichdirectionthevalueisfromthemeanontheNormalCurve.Negativevalueswillbetotheleft,andpositivevalueswillbetotheright.

StandardizingScores:

StandardNormalCurve…themeaniszero,andthestandarddeviationis1.Thedistributionisbell-shapedandsymmetrical.Theareaunderthecurveis1,andthetailsofthecurveextendoutinfinitely.Theyneveractuallytouchthehorizontalaxis.Thehighestpointonthecurveisatthemean

Returntoourdata…let’scalculatethez-scoresforeachofthevalues…

EmpiricalRuleusedwhenthedistributionisassumedtoknowntobeapproximatelynormal.

Approximately68%ofthevalueswillfallwithin1sdofthemean

Approximately95%ofthevalueswillfallwithin2sdofthemean

Approximately99.9%ofthevalueswillfallwithin3sdofthemean

Chebyshev’sTheoremdoesn’trequirethatthedatahaveanormaldistribution

Saysthatatleast（1–1/z2）valueswillfallwithinzstandarddeviationsofthemean.

1-1/12=0,1-1/22=.75,1-1/32=.88889,1-1/42=.9375,1-1/52=.96

Wecan’tmakeanyassumptionsaboutthepercentofvaluesthatarewithin1sdofthemean

But…

Atleast75%ofthevalueswillfallwithin2sdofthemean

Atleast88.9%ofthevalueswillfallwithin3sdofthemean

WeuseChebyshev’sTheoremtoestimatethevariationinadistributionwhen

n<30,or

theshapeofthedistributionisunknown,or

thedistributionisassumedtobenon-normal.

Outliers:

suspectorextremevaluesofdatathatmustbeidentifiedandscrutinized.Iftheyareinstancesofincorrectlyentereddata,theyshouldbecorrected.Ifthevaluewasenteredcorrectlyanditisavalidnumber,itshouldremaininthedatasetaspartoftheinitialanalysis.

Whenweusethez-scoremethodforidentifyingoutliers,weassumethatanyvaluethathasaz-scorewithanabsolutevaluegreaterthan3.0（thatislessthan-3.0orgreaterthan+3.0）isanoutlier.Beforeweproceedwithdataanalysis,weneedtoexaminealloutliersforaccuracy.Ifwedeterminethatthevalueisvalid,weoftenruntwosetsofanalysis.Onewiththeoutlier,andonewithout.

Anotherwaytoidentifyoutliers…

RelatedtoIQRistheFivenumbersummary…minimum,Q1,Q2,Q3,&maximum.Thesevaluesfeedintoupperandlowerlimits,andwegraphtheminaboxplot.

FiveNumberSummary

Minimum

53

Q1

53

Q2

57

Q3

64

Maximum

70

Usetheboxplot…TheadvantageoftheboxplotisthatitisnotinfluencedbyoutliersorextremevaluesasareZ-scores.

BoxPlots–Whiskersshowtherangeofdatawithintheinnerfences

3（IQR）1.5（IQR）Q1MedianQ31.5（IQR）3（IQR）

belowQ1belowQ1（IQR）aboveQ3aboveQ3

（LowerOuter&InnerFences）（UpperInner&OuterFences）Anyvaluesbetweentheinnerandouterfencesare“unusual,”andanyvaluesoutbeyondtheouterfencesare“outliers.”

Advantageofusingtheboxplotmethodaswellasthez-scoremethod...theboxplotmethodisnotinfluencedbyextremevaluesinthesamewaythatthemeanandthestandarddeviationare....itissaidtobeamoreconservativemethodofevaluatingoutliers.

外文翻译原文

课题名称：

统计基础

MeasuresofCentralTendencyandLocation:

趋势和位置的划分：

mean,median,mode,percentiles,quartilesanddeciles.意思是说，中位数，众数，百分位数，四分位数和十分位数。

xx                sortedx排序x

5353                          5353

5555                          5353

7070                          5353

5858