Tchebysheff'sandtheEmpiricalRule
Tchebysheff'sandtheEmpiricalRule
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Needs["Statistics`DescriptiveStatistics`"]
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Needs["Statistics`DataManipulation`"]
Seriouslyusefultoolsfordescribingthedistributionofvaluesinthevariable.Ittakesmeanandthestandarddeviationandallowsyoutodescribethedistributioninnon-technicalterms.Youonlyneedtwovaluesandtwofunctionstogobeyondthedescriptionofthemeanandmeasureofdispersion.
Tchebysheff'sTheorem
Tchebysheff'sTheorem
Givenanumberkgreaterthanorequaltoone,andasetofnmeasurements,atleast
1-
1
2
k
ofthedistributionliesintheintervalwithink*(standarddeviation)fromthemean.
Asmentionedinclass,thereisarelationshipbetweenthevalueofkandthepercentageofthedistributionaroundthemeanvalue.Hereistheformulathatshowthis.
k=
1
(1-P)
wherePisthepercentageofvaluesaroundthecenter(Pcannotbeequalto1). Hereisatableofpercentagevaluesaroundthemean(P)(columnone)andkvalues(columntwo).
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TableForm[Table[{P,Sqrt[(1/(1-P))]},{P,0.05,0.95,0.05}]]
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Inclass,wediscussedthestartingwagesofeconundergraduates.Thefollowingtwovalues(meanandstandarddeviation)weregiven.
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MeanWages=32900
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SDWages=3850
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Thefunctionbelowgeneratesthevaluesinthetable.Column1ofthetableisthepercentageofvaluesaroundthemean$32,900,column2isthelowerboundoftherange,column3hastheupperboundoftherange.
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TableForm[Table[{P,(MeanWages-SDWages*Sqrt[(1/(1-P))]),(MeanWages+SDWages*Sqrt[(1/(1-P))])},{P,0.05,0.95,0.05}]]
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Interpretationusing50%values,basedonthemeanandstandarddeviationvalues,atleast50%oftheeconundergradsareexpectedtoobtainstartingwagesfrom$27,455.30to$38,344.70.
EmpiricalRule
EmpiricalRule
Tchebysheff'sTheoremappliestoanydatadistribution(symmetric,skewedtotheleft,skewedtotheright).Ifweknowthatthedistributionissymmetricandunimodal(bellshaped),wecanusetheempiricalrule. TheEmpiricalRuleislimitedinthatittellusthreesetsofranges.
For68%ofthedistribution
μ±σ
For95%ofthedistribution
μ±2*σ
Foralmost100%ofthedistribution
μ±3*σ
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Herearethe68,95and100percentagerangesforthewageexample.
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TableForm[Table[{MeanWages-i*SDWages,MeanWages+i*SDWages},{i,1,3}]]
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