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统计学外文翻译
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
Mean→x-baror
→foragivenvariable,itisthesumofthevaluesdividedbythenumberofvalues（∑xi/n）.Inthiscase,wehaven=11.Soweneedtoaddallofthevaluestogetheranddivideby11.∑=657,
=59.73
Median→thenumberinadistributionofavariable’sresponsewhereonehalfofthevaluesareaboveandonehalfofthevaluesarebelow.Tofindthemedian,wefirstneedtoputourdatainascendingorder（smallesttolargest）.Thenwecandeterminethemedian…ifthevalueofnisodd,itissimplythemiddleobservation,butifthevalueofniseven,itistheaverageofthetwomiddleobservations.
Inthiscase,nisodd,sothemedianwillbethemiddleobservationofoursortedvalues（the6thvalue）...57
Mode→thevaluethatoccursmostfrequently.Iftherearetwodifferentvaluesmostfrequentlyoccurring,thedataaresaidtobebi-modal.Iftherearemorethantwomodes,andthedistributionissaidtobemulti-modal.Inthiscase,thevaluethatoccursmostoftenis53.So,themodeis53.
ThemeasuresoflocationarePercentile,QuartileandDecile
Percentile→thepthpercentileisavaluesuchthatatleastppercentoftheobservationsarelessthanorequaltothisvalueandatleast（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（σ2）→usesthesameformulainthenumerator,butNinsteadofn-1inthedenominator.Sincewerarelyhaveinformationabouttheentirepopulation,wealmostalwaysusetheformulaforsamplevariance,s2.
PopulationStandardDeviation:
σ=
…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…
EmpiricalRule→usedwhenthedistributionisassumedtoknowntobeapproximatelynormal.
→Approximately68%ofthevalueswillfallwithin1sdofthemean
→Approximately95%ofthevalueswillfallwithin2sdofthemean
→Approximately99.9%ofthevalueswillfallwithin3sdofthemean
Chebyshev’sTheorem→doesn’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 53
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