Median

Definition

Median,alsoknownasmidpoint,median.Themedianisthenumberinthemiddleofagroupofdataarrangedinorder,thatis,inthisgroupofdata,halfofthedataislargerthanhim,andhalfofthedataissmallerthanhim.HereweuseTorepresentthemedian.(Note:Themedianisdifferentfromthemode.Themodereferstothelargestnumber.Sometimesthereismorethanonemode,andtherecanonlybeonemedian.)

Thereisasetofdata:

Sortitfromsmallesttolargestasfollows:

WhenNisanoddnumber,

Atmosthalfofthevalues​​inanumbersetarelessthanthemedian,andatmosthalfofthevalues​​aregreaterthanthemedian.Ifthenumberofvalues​​greaterthanandlessthanthemedianislessthanhalf,thentheremustbeseveralvalues​​inthenumbersetequaltothemedian.

SupposethedistributionfunctionofthecontinuousrandomvariableXis,thenthenumberthatmeetstheconditioniscalledthemedianofXordistributionF.

Features

1)Themedianistherepresentativevalueofthetotalunitmarkvaluedeterminedbyitspositioninallthemarkvalues,andisnotsubjecttothemaximumormaximumvalueofthedistributionsequence.Theinfluenceoftheminimumvalueincreasestherepresentativenessofthemediantothedistributionsequencetoacertainextent.

2)Forsomediscretevariablemonomialseries,whenthefrequencydistributionisskewed,therepresentativenessofthemedianwillbeaffected.

3)Tendtothemiddlepositionofasetofordereddata

Calculationexample

Forasetoffinitenumberofdata,theirmedianAnumberisanumber:halfofthedatainthisgroupofdataislargerthanit,andtheotherhalfofthedataissmallerthanit.Themethodofcalculatingthemedianofafinitenumberofdataistoarrangeallsimilardatainorderofsize.Ifthenumberofdataisodd,themiddledataisthemedianofthisgroupofdata;ifthenumberofdataiseven,thearithmeticaverageofthemiddletwodataisthemedianofthisgroupofdata.

Median:Thatis,thenumberinthemiddleisselected,whichisameasureofcentraltendency.

Example1

Findthemedianofthissetofdata:23,29,20,32,23,21,33,25.

Solution:

Firstarrangethesetofdata(hereinascendingorder),get:

20,21,23,23,25,29,32,33

Becausethissetofdataconsistsof8dataintotal,thatis,nisanevennumber,soaccordingtothemediancalculationmethod,themedian,Whichistheaverageofthefourthandfifthnumbers.

Example2

Findthemedianofthissetofdata:10,20,20,20,30.

Solution:

Firstarrangethesetofdata(hereinascendingorder),get:

10,20,20,20,30

Becausethissetofdataconsistsof5data,thatis,nisanoddnumber,soaccordingtothemediancalculationmethod,themedianis20,whichisthethirdnumber.

Matlabfunction

InMatlab,themedian()functionisabuilt-infunctionforobtainingthemedianofasetofdata.Theusagerulesareasfollows:

M=median(A)

M=median(A,dim)

M=median(___,nanflag)

Example:

Foldall

Themedianofthematrixcolumns

Definea4×3matrix.

A=[011;232;132;422]

A=4×3

011

232

132

422

Calculatethemedianvalueofeachcolumn.

M=median(A)

M=1×3

1.50002.50002.0000

Foreachcolumn,themedianvalueisThemeanvalueofthetwovalues​​inthemiddleofthearrangementorder.

Themedianoftherowsofthematrix

Definea2×3matrix.

A=[011;232]

A=2×3

011

232

Calculatethemedianvalueofeachrow.

M=median(A,2)

M=2×1

1

2

Foreachrow,themedianvalueisthevalueinthemiddleofthesortorder.

Differentiationandconnection

1)Theaverageisobtainedbycalculation,soitwillvarywitheachdatachange.

2)Themedianisobtainedbysorting,anditisnotaffectedbythemaximumandminimumextremevalues.Changesinpartofthedatahavenoeffectonthemedian.Whenindividualdatainasetofdatachangesgreatly,itisoftenusedtodescribethecentraltendencyofthissetofdata.

3)Modeisalsoarepresentativenumberofdata,reflectingthedegreeofconcentrationofasetofdata.Indailylife,suchas"best","mostpopular","mostsatisfied",etc.,arerelatedtothemode,whichreflectsamostcommontendency.

Prosandcons:

Average:Itneedsallthedataofthewholegrouptocalculate;itiseasilyaffectedbyextremevalues​​inthedata.Median:Itcanbedeterminedonlybyarrangingthedatainorder;itisnoteasilyaffectedbytheextremevalues​​inthedata.Mode:Obtainedbycounting;noteasilyaffectedbyextremevalues​​inthedata.