The Normal Distribution

The Normal Distribution - University of West Georgia

A Normal distribution is described by a Normal density curve Any particular Normal distribution is completely specified by two numbers: its mean ???? and its standard deviation ???? The mean of a Normal distribution is the center of the symmetric Normal curve The standard deviation is the distance from the center to the change-

Normal distribution - University of Notre Dame

have a normal distribution • The normal distribution is easy to work with mathematically In many practical cases, the methods developed using normal theory work quite well even when the distribution is not normal • There is a very strong connection between the size of a sample N and the extent to which a sampling distribution approaches

Normal Distributions - University of Notre Dame

calculated over intervals rather than for speci c values of the random variable Although many types of probability density functions commonly occur, we will restrict our attention to random variables with Normal Distributions and the probabilities will correspond to areas under a Normal Curve (or normal density function)

The Normal Distribution - austinccedu

Normal Distributions37 from the menu to standardize By selecting “Subtract mean and divide by standard deviation” the command calculated the standardized values, z=(x?x)s The results can be stored in C2 The standardized vocabulary scores (or z?score) will tell how far above or below the mean a particular score falls

Searches related to how is normal distribution calculated filetype:pdf

Perhaps the most important ideal distribution used is the 'normal' distribution (Figure 6 1) Once one understands the characteristics of the normal distribution, knowledge of other distributions is easily obtained Figure 6 1 The normal distribution Most people are familiar with the normal distribution described as a “bell-shaped curve,”

134917_6minitab_bps4e_03.pdf

CHAPTERȱ

ȱȱȱȱ3ȱȱȱ

TheȱNormalȱDistributionȱ

NormalȱCalculationsȱ

ȱ Sometimes the Normal density can describe the overall pattern of a distribution. Figure 3

.1 of BPS is a histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vo-

cabulary part of the Iowa Test of Basic Skills. To see how well the Normal density describes these data, we can select Graph Histogram and select the With Fit option to obtain a histo- gram with a Normal curve fitted to the data. ȱ 35

36 Chapter 3

Inȱtheȱfollowingȱdialogȱbox,ȱselectȱtheȱvariableȱtoȱbeȱgraphedȱandȱclickȱOK.ȱȱMinitabȱwillȱ

produceȱaȱhistogramȱwithȱaȱNormalȱdensityȱcurveȱforȱtheȱvariable.ȱȱTheȱNormalȱdensityȱcurveȱ

selectedȱwillȱbeȱbasedȱonȱtheȱdata'sȱmeanȱandȱstandardȱdeviation.ȱ ȱ

TheȱsmoothȱcurveȱdrawnȱthroughȱtheȱhistogramȱisȱaȱgoodȱdescriptionȱofȱtheȱoverallȱpatȬ

ternȱofȱtheȱdata.ȱȱItȱwillȱbeȱeasierȱtoȱuseȱtheȱdensityȱcurveȱinsteadȱofȱtheȱhistogramȱforȱsomeȱcalȬ

culations.ȱ vocabulary score Fr e q u e n c y

12.010.59.07.56.04.53.01.5

140
120
100
80
60
40
20 0

Mean6.843

StDev1.551

N947

Histogram of vocabulary score

Normal

MinitabȱcanȱbeȱusedȱtoȱperformȱNormalȱdistributionȱcalculations.ȱȱIfȱdataȱinȱaȱcolumnȱ

areȱNormallyȱdistributed,ȱthenȱtheȱdataȱcanȱbeȱstandardizedȱtoȱobtainȱdataȱwithȱaȱstandardȱ

Normalȱdistribution,ȱthatȱis,ȱthoseȱwithȱmeanȱequalȱtoȱzeroȱandȱstandardȱdeviationȱequalȱtoȱone.ȱȱ

Selectȱ

CalcȱȱStandardizeȱȱ

Normal Distributions 37

from the menu to standardize. By selecting "Subtract mean and divide by standard deviation" the command calculated the standardized values, sxxz)(. The results can be stored in C2.

Theȱstandardizedȱvocabularyȱscoresȱ(orȱzȬscore)ȱwillȱtellȱhowȱfarȱaboveȱorȱbelowȱtheȱ

meanȱaȱparticularȱscoreȱfalls.ȱȱTheȱmeasureȱisȱinȱunitsȱofȱstandardȱdeviations.ȱȱTheȱfirstȱstudentȱ

hasȱaȱscoreȱofȱ5.8,ȱaȱvalueȱthatȱisȱbelowȱtheȱmeanȱ(zȱ=ȱ0.67213).ȱȱTheȱsecondȱscore,ȱ6.3,ȱisȱalsoȱ

belowȱtheȱmean,ȱbutȱtheȱthird,ȱ7.8,ȱisȱaboveȱtheȱmeanȱ(zȱ=ȱ0.61713).ȱȱȱ

Weȱcouldȱexamineȱtheȱstandardizedȱvaluesȱtoȱseeȱhowȱwellȱtheyȱobeyȱtheȱ68Ȭ95Ȭ99.7ȱ

rule.ȱȱApproximatelyȱ68%ȱofȱtheȱstandardizedȱvaluesȱshouldȱhaveȱvaluesȱbetweenȱ-1ȱandȱ+1,ȱ

95%ȱshouldȱhaveȱvaluesȱbetweenȱ2ȱandȱ+2,ȱandȱ99.7%ȱshouldȱhaveȱvaluesȱbetweenȱ-3ȱandȱ+3.ȱ

38 Chapter 3

Anotherȱwayȱtoȱobtainȱstandardizedȱvaluesȱ(zȬscores)ȱisȱtoȱuseȱMinitab'sȱcalculator.ȱȱ

Fromȱtheȱfollowingȱdescriptiveȱinformation,ȱweȱcanȱstandardizeȱtheȱvocabularyȱscoresȱbyȱsubȬ

tractingȱtheȱmeanȱ(6.8427)ȱandȱdividingȱbyȱtheȱstandardȱdeviationȱ(1.5513).ȱȱȱ

Descriptive Statistics: vocabulary scores

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 vocabulary score 947 0 6.843 0.050 1.551 2.100 5.800 6.800 7 .900

Variable Maximum

vocabulary score 12.100

ToȱstandardizeȱusingȱMinitab'sȱcalculator,ȱselectȱCalcȱȱCalculatorȱfromȱtheȱmenuȱandȱ

enterȱtheȱappropriateȱexpressionȱillustratedȱinȱtheȱfollowingȱdialogȱbox.ȱ

YouȱcanȱuseȱMinitabȱtoȱdoȱprobabilityȱcalculationsȱforȱtheȱNormalȱdistributionȱbyȱselectȬ

ingȱȱ CalcȱȱProbabilityȱDistributionsȱȱNormalȱ

fromȱtheȱmenu.ȱȱBothȱforwardȱandȱbackwardȱprobabilitiesȱcanȱbeȱcalculated.ȱ

Exampleȱ3.4ȱinȱBPSȱconcernsȱtheȱheightsȱofȱyoungȱwomen.ȱȱTheȱheightsȱareȱapproxiȬ

matelyȱNormalȱwithȱaȱmeanȱofȱaboutȱ64ȱinchesȱandȱaȱstandardȱdeviationȱofȱ2.7ȱinches.ȱȱToȱfindȱ

theȱproportionȱofȱwomenȱwhoȱareȱlessȱthanȱ70ȱinchesȱtall,ȱweȱselectȱCalcȱȱProbabilityȱDistriȬ

butionsȱȱNormalȱfromȱtheȱmenu.ȱȱInȱtheȱdialogȱbox,ȱweȱselectȱCumulativeȱprobability,ȱfillȱinȱ

theȱMean,ȱStandardȱdeviation,ȱandȱInputȱconstantȱboxesȱasȱfollows.ȱ

Normal Distributions 39

AfterȱclickingȱonȱOK,ȱweȱobtainȱtheȱfollowingȱinformationȱinȱMinitab'sȱSessionȱwindow.ȱȱTheȱ

resultȱsaysȱthatȱtheȱproportionȱofȱwomenȱwhoȱareȱlessȱthanȱ70ȱinchesȱtallȱisȱ.986866.ȱȱThisȱisȱ

slightlyȱdifferentȱfromȱtheȱresultȱthatȱwouldȱbeȱobtainedȱusingȱTableȱAȱsinceȱitȱisȱnotȱrequiredȱtoȱ

roundȱtheȱstandardizedȱvalue.ȱ

Cumulative Distribution Function

Normal with mean = 64 and standard deviation = 2.7 x P( X <= x )

70 0.986866

WeȱcanȱalsoȱuseȱMinitabȱtoȱdoȱbackwardȱcalculations.ȱȱExampleȱ3.8ȱofȱBPSȱshowsȱscoresȱonȱtheȱ

SATȱverbalȱtestȱinȱ2002.ȱȱTheȱscoresȱfollowȱapproximatelyȱtheȱN(504,ȱ110)ȱdistributionȱandȱwe'dȱ

likeȱtoȱfindȱhowȱhighȱaȱstudentȱmustȱscoreȱȱtoȱplaceȱinȱtheȱtopȱ10%ȱofȱallȱstudentsȱtakingȱtheȱ

SAT.ȱȱAgain,ȱweȱselectȱCalcȱȱProbabilityȱDistributionsȱȱNormalȱfromȱtheȱmenu.ȱȱSinceȱweȱ

areȱdoingȱaȱbackwardȱcalculation,ȱweȱcheckȱInverseȱcumulativeȱprobability.ȱȱWeȱfillȱinȱtheȱ

Mean,ȱStandardȱdeviation,ȱandȱInputȱconstantȱboxesȱasȱfollows.ȱȱNoticeȱthatȱsinceȱweȱwantȱtheȱ

valueȱforȱtheȱtopȱ10%,ȱtheȱinputȱconstantȱisȱ0.9ȱcorrespondingȱtoȱ90%ȱbelowȱtheȱcalculatedȱ

value.ȱ

40 Chapter 3

AfterȱclickingȱonȱOKȱinȱtheȱdialogȱbox,ȱweȱobtainȱtheȱfollowingȱresultsȱinȱtheȱSessionȱwindow.ȱȱ

Weȱseeȱthatȱaȱstudentȱmustȱscoreȱaboveȱ644.971ȱ(i.e.,ȱ645ȱorȱabove)ȱtoȱbeȱinȱtheȱtopȱ10%.ȱ

Inverse Cumulative Distribution Function

Normal with mean = 504 and standard deviation = 110

P( X <= x ) x

0.9 644.971

EXERCISESȱ

3.13ȱSelectȱCalcȱȱProbabilityȱDistributionsȱȱNormalȱfromȱtheȱmenuȱtoȱfindȱtheȱproportionȱ

ofȱobservationsȱfromȱaȱstandardȱNormalȱdistributionȱthatȱsatisfiesȱeachȱofȱtheȱfollowingȱ

statements.ȱInȱeachȱcase,ȱsketchȱaȱstandardȱNormalȱcurveȱwithȱyourȱvalueȱofȱzȱmarkedȱ

onȱtheȱaxis.ȱ

(a)ȱȱTheȱpointȱzȱwithȱ25%ȱofȱtheȱobservationsȱfallingȱbelowȱit.ȱ

(b)ȱȱTheȱpointȱzȱwithȱ40%ȱofȱtheȱobservationsȱfallingȱaboveȱit.ȱ

3.14ȱScoresȱonȱtheȱWechslerȱAdultȱIntelligenceȱScaleȱareȱapproximatelyȱNormallyȱdistributedȱ

withȱȱ=ȱ100ȱandȱȱ=ȱ15.ȱȱSelectȱCalcȱȱProbabilityȱDistributionsȱȱNormalȱfromȱtheȱ

menuȱtoȱanswerȱtheȱfollowingȱquestions.ȱ (a)ȱȱWhatȱscoresȱfallȱinȱtheȱlowestȱ25%ȱofȱtheȱdistribution?ȱ (b)ȱȱHowȱhighȱaȱscoreȱisȱneededȱtoȱbeȱinȱtheȱhighestȱ5%?ȱ ȱ

3.26ȱTableȱ2.3ȱinȱBPSȱandȱEX03Ȭ26.MTWȱgiveȱdataȱonȱtheȱpenetrabilityȱofȱsoilȱatȱeachȱofȱthreeȱ

levelsȱofȱcompression.ȱWeȱmightȱexpectȱtheȱpenetrabilityȱofȱspecimensȱofȱtheȱsameȱsoilȱatȱ

theȱsameȱlevelȱofȱcompressionȱtoȱfollowȱaȱNormalȱdistribution.ȱȱSelectȱGraphȱȱHistoȬ

gramȱfromȱtheȱmenuȱandȱselectȱWithȱFitȱandȱGroupsȱtoȱmakeȱaȱhistogramsȱofȱtheȱdataȱ

forȱlooseȱandȱforȱintermediateȱcompression.ȱDoesȱeitherȱsampleȱseemȱroughlyȱNormal?ȱ

Normal Distributions 41

DoesȱeitherȱappearȱdistinctlyȱnonȬNormal?ȱIfȱso,ȱwhatȱkindȱofȱdepartureȱfromȱNormalityȱ

doesȱyourȱgraphȱshow?ȱ ȱ

3.29ȱHereȱandȱinȱEX03Ȭ29.MTWȱareȱtheȱIQȱtestȱscoresȱofȱ31ȱseventhȬgradeȱgirlsȱinȱaȱMidwestȱ

schoolȱdistrict:ȱ

114ȱ100ȱ104ȱ89ȱ102ȱ91ȱ114ȱ114ȱ103ȱ105ȱȱ

108ȱ130ȱ120ȱ132ȱ111ȱ128ȱ118ȱ119ȱ86ȱ72ȱȱ

111ȱ103ȱ74ȱ112ȱ107ȱ103ȱ98ȱ96ȱ112ȱ112ȱ93ȱ

(a)ȱȱWeȱexpectȱIQȱscoresȱtoȱbeȱapproximatelyȱNormal.ȱSelectȱGraphȱȱHistogramȱ

fromȱtheȱmenuȱandȱselectȱWithȱFitȱȱtoȱmakeȱaȱhistogramȱwithȱtheȱcorrespondingȱ

normalȱdenisityȱcurveȱtoȱseeȱthatȱthereȱareȱnoȱmajorȱdeparturesȱfromȱNormality.ȱ

(b)ȱȱNonetheless,ȱproportionsȱcalculatedȱfromȱaȱNormalȱdistributionȱareȱnotȱalwaysȱ

veryȱaccurateȱforȱsmallȱnumbersȱofȱobservations.ȱȱSelectȱCalcȱȱStandardizeȱtoȱ

findȱtheȱstandizedȱvaluesȱforȱtheseȱIQȱscores.ȱWhatȱproportionsȱofȱtheȱscoresȱareȱ

withinȱoneȱstandardȱdeviationȱandȱwithinȱtwoȱstandardȱdeviationsȱofȱtheȱmean?ȱ

WhatȱwouldȱtheseȱproportionsȱbeȱinȱanȱexactlyȱNormalȱdistribution?ȱ ȱ

3.30ȱSelectȱCalcȱȱProbabilityȱDistributionsȱȱNormalȱfromȱtheȱmenuȱtoȱfindȱtheȱproportionȱ

ofȱobservationsȱfromȱaȱstandardȱNormalȱdistributionȱthatȱfallsȱinȱeachȱofȱtheȱfollowingȱ

regions.ȱInȱeachȱcase,ȱsketchȱaȱstandardȱNormalȱcurveȱandȱshadeȱtheȱareaȱrepresentingȱ

theȱregion.ȱ (a)ȱȱzȱǂȱƺ2.25ȱ (b)ȱȱzȱǃȱƺ2.25ȱ (c)ȱȱzȱ>ȱ1.77ȱ (d)ȱȱƺ2.25ȱ<ȱzȱ<ȱ1.77ȱ ȱ

3.31ȱSelectȱCalcȱȱProbabilityȱDistributionsȱȱNormalȱfromȱtheȱmenuȱtoȱanswerȱtheȱfollowȬ

ingȱquestions.ȱ

(a)ȱȱȱFindȱtheȱnumberȱzȱsuchȱthatȱtheȱproportionȱofȱobservationsȱthatȱareȱlessȱthanȱzȱinȱ

aȱstandardȱNormalȱdistributionȱisȱ0.8.ȱ

(b)ȱȱȱFindȱtheȱnumberȱzȱsuchȱthatȱ35%ȱofȱallȱobservationsȱfromȱaȱstandardȱNormalȱdisȬ

tributionȱareȱgreaterȱthanȱz.ȱ ȱ

3.44ȱTheȱheightsȱofȱwomenȱagedȱ20ȱtoȱ29ȱfollowȱapproximatelyȱtheȱN(64,ȱ2.7)ȱdistribution.ȱ

MenȱtheȱsameȱageȱhaveȱheightsȱdistributedȱasȱN(69.3,ȱ2.8).ȱSelectȱCalcȱȱProbabilityȱ

DistributionsȱȱNormalȱfromȱtheȱmenuȱtoȱfindȱtheȱpercentȱofȱyoungȱwomenȱareȱtallerȱ

thanȱtheȱmeanȱheightȱofȱyoungȱmen?ȱ ȱ

3.45ȱȱTheȱheightsȱofȱwomenȱagedȱ20ȱtoȱ29ȱfollowȱapproximatelyȱtheȱN(64,ȱ2.7)ȱdistribution.ȱ

MenȱtheȱsameȱageȱhaveȱheightsȱdistributedȱasȱN(69.3,ȱ2.8).ȱSelectȱCalcȱȱProbabilityȱ

42 Chapter 3

DistributionsȱȱNormalȱfromȱtheȱmenuȱtoȱfindȱtheȱpercentȱofȱyoungȱmenȱareȱshorterȱ

thanȱtheȱmeanȱheightȱofȱyoungȱwomen.ȱ ȱ

3.46ȱChangingȱtheȱmeanȱofȱaȱNormalȱdistributionȱbyȱaȱmoderateȱamountȱcanȱgreatlyȱchangeȱ

theȱpercentȱofȱobservationsȱinȱtheȱtails.ȱSupposeȱthatȱaȱcollegeȱisȱlookingȱforȱapplicantsȱ

withȱSATȱmathȱscoresȱ750ȱandȱabove.ȱȱSelectȱCalcȱȱProbabilityȱDistributionsȱȱNorȬ

malȱfromȱtheȱmenuȱtoȱanswerȱtheȱfollowingȱquestions.ȱ

(a)ȱȱInȱ2004,ȱtheȱscoresȱofȱmenȱonȱtheȱmathȱSATȱfollowedȱtheȱN(537,ȱ116)ȱdistribution.ȱ

Whatȱpercentȱofȱmenȱscoredȱ750ȱorȱbetter?ȱ

(b)ȱȱWomenȇsȱSATȱmathȱscoresȱthatȱyearȱhadȱtheȱN(501,ȱ110)ȱdistribution.ȱWhatȱperȬ

centȱofȱwomenȱscoredȱ750ȱorȱbetter?ȱYouȱseeȱthatȱtheȱpercentȱofȱmenȱaboveȱ750ȱisȱ

almostȱthreeȱtimesȱtheȱpercentȱofȱwomenȱwithȱsuchȱhighȱscores.ȱWhyȱthisȱisȱtrueȱ

isȱcontroversial.ȱ ȱ

3.48ȱOsteoporosisȱisȱaȱconditionȱinȱwhichȱtheȱbonesȱbecomeȱbrittleȱdueȱtoȱlossȱofȱminerals.ȱToȱ

diagnoseȱosteoporosis,ȱanȱelaborateȱapparatusȱmeasuresȱboneȱmineralȱdensityȱ(BMD).ȱ

BMDȱisȱusuallyȱreportedȱinȱstandardizedȱform.ȱTheȱstandardizationȱisȱbasedȱonȱaȱpopuȬ

lationȱofȱhealthyȱyoungȱadults.ȱTheȱWorldȱHealthȱOrganizationȱ(WHO)ȱcriterionȱforȱosȬ

teoporosisȱisȱaȱBMDȱ2.5ȱstandardȱdeviationsȱbelowȱtheȱmeanȱforȱyoungȱadults.ȱBMDȱ

measurementsȱinȱaȱpopulationȱofȱpeoplesimilarȱinȱageȱandȱsexȱroughlyȱfollowȱaȱNormalȱ

distribution.ȱSelectȱCalcȱȱProbabilityȱDistributionsȱȱNormalȱfromȱtheȱmenuȱtoȱanȬ

swerȱtheȱfollowingȱquestions.ȱ

(a)ȱȱWhatȱpercentȱofȱhealthyȱyoungȱadultsȱhaveȱosteoporosisȱbyȱtheȱWHOȱcriterion?ȱ

(b)ȱȱWomenȱagesȱ70ȱtoȱ79ȱareȱofȱcourseȱnotȱyoungȱadults.ȱTheȱmeanȱBMDȱinȱthisȱageȱ

isȱaboutȱƺ2ȱonȱtheȱstandardȱscaleȱforȱyoungȱadults.ȱSupposeȱthatȱtheȱstandardȱdeȬ

viationȱisȱtheȱsameȱasȱforȱyoungȱadults.ȱWhatȱpercentȱofȱthisȱolderȱpopulationȱhasȱ

osteoporosis?ȱ ȱ

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