LastLecture

• ImportanceofStatisticalMeasurements

• DistinguishCategoricalfromNumericalvariables

• Knowingdifferentwaysofstatisticallydescribingdata

LectureGoals

• VariabilityMeasurement– Percentiles

• Shape– Skewness,Kurtosis

• NormaldistributionoftheMean,SD

• MoreontheCentralLimitTheorem

• ConfidenceIntervals

Notations

Characteristic Population(Parameter)

Sample(Statistic)

Mean

Variance

StandardDeviation(SD)

1  1  

2 ∑ ̅ 21  2 ∑ 21  

∑ ̅ 21  ∑ 2

1  

StatisticalDescriptionofData…contd.

• Statisticsdescribesanumericsetofdatabyits- Center- Variability- Shape

• Statisticsdescribesacategoricalsetofdataby- Frequency,percentageorproportionofeachcategory

MethodsofVariability Measurement

• Percentiles:Percentiles:Ifdataisorderedanddividedinto100parts,thencutpointsarecalledPercentiles.25th percentileistheQ1,50thpercentileistheMedian(Q2)andthe75th percentileofthedataisQ3.

Innotations,percentilesofadataisthe((n+1)/100)pth observationofthedata,wherepisthedesiredpercentileandnisthenumberofobservationsofdata.

Percentiles

• Apercentileisavaluebelowwhichacertainpercentageoftheobservationslie

Example‐1:

DataSet:2,2,3,4,5,5,5,6,7,8,8,8,8,8,9,9,10,11,11,12

−Whatisthepercentilerankof‘10’?

Percentiles

2,2,3,4,5,5,5,6,7,8,8,8,8,8,9,9,10,11,11,12

−Whatisthepercentilerankof‘10’?

− Ans.:PercentileRankof‘10’= 16

20 * 100 = 80%

Percentiles

• Example‐2:(exerciseforstudents)

DataSet:2,2,3,4,5,5,5,6,7,8,8,8,8,8,9,9,10,11,11,12

−Whatvalueexistsatthepercentilerankingof25%?(i.e.whatscorehad25ofthescoresbelowit?)

Percentiles

2,2,3,4,5,5,5,6,7,8,8,8,8,8,9,9,10,11,11,12

−Whatvalueexistsatthepercentilerankingof25%?

Value#=Percentile

100 * (n+1)

StatisticalDescriptionofData...Contd.

• Statisticsdescribesanumericsetofdatabyits- Center- Variability- Shape

• Statisticsdescribesacategoricalsetofdataby- Frequency,percentageorproportionofeachcategory

ShapeofData• Shapeofdataismeasuredby

- Skewness- Kurtosis

Skewness

• Measuresasymmetryofdata- Positiveorrightskewed:Longerrighttail- Negativeorleftskewed:Longerlefttail

2/3

1

2

1

3

21

)(

)(Skewness

Then, ns.observatio be ,...,Let

n

ii

n

ii

n

xx

xxn

nxxx

Kurtosis

• Measurespeakednessofthedistributionofdata.Thekurtosisofnormaldistributionis0.

3)(

)(Kurtosis

Then, ns.observatio be ,...,Let

2

1

2

1

4

21

n

ii

n

ii

n

xx

xxn

nxxx

SummaryoftheVariable‘Age’inthegivendataset

Mean 90.41666667

Standard Error 3.902649518

Median 84

Mode 84

Standard Deviation 30.22979318

Sample Variance 913.8403955

Kurtosis -1.183899591

Skewness 0.389872725

Range 95

Minimum 48

Maximum 143

Sum 5425

Count 60

Histogram of Age

Age in Month

Num

ber o

f Sub

ject

s

40 60 80 100 120 140 160

02

46

810

manystatisticaltestsassumevaluesarenormallydistributed

notalwaysthecase!examinedatapriortoprocessing

TheNormalDistribution

TheNormalDistribution

TheStandardNormalCurve

• Standardizethe‘mean’tomakeit0

CumulativeProbability

TheEmpiricalRule

• Instatistics,the68–95–99.7rule,alsoknownasthethree‐sigmaruleorempiricalrule,statesthatnearlyallvaluesliewithinthreestandarddeviationsofthemeaninanormaldistribution.

• 68.27%ofthevaluesliewithinonestandarddeviationofthemean.

Exercise‐ 1

• Thetestscoresforaclassarenormallydistributed.Theaveragetestscoreis75andthestandarddeviationis10.Drawthedataonthenormalcurve.

Exercise‐ 2

• Whatistheprobabilityofastudentscoringbelow95onthistest?

• P(X<95)=?

Exercise‐ 3

• Whatistheprobabilityofastudentscoringbetween55and65onthistest?

• P(55<X<95)=?

Z‐test:Exercise‐ 4

• Thetestscoresforaclassarenormallydistributed.Given, =75and =10Whatistheprobabilitythestudentscoredabovea60?

• Z‐distribution:

Exercise‐ 4

• Z

Z‐values

• http://medicalnights.com/table‐of‐z‐values/

T‐test:Conditions

• ThepopulationSDisunknown• Thesamplesizeislessthan30

• T‐distribution:

T‐scores:Exercise‐ 5

• Theavg.testscoresforapopulationis75.Asampleof9studentsarerandomlyselected.TheSDforthesampleis10.

Whatistheprobabilitytheaveragescoreforthesampleisabove80?

CentralLimitTheorem

• Thesamplemeanwillbeapproximatelynormallydistributedforlargesamplesizes,regardlessofthedistributionfromwhichwearesampling.

CriticalValues

ConfidenceInterval

EstimatingthePopulationmean

ConfidenceIntervals

• Confidenceinterval=

ConfidenceLevel

• Themoreconfidentwewishtobe,thelargerourconfidenceintervalwillbe

• Note:ahighconfidenceisnotalwaysveryuseful• i.e.– “Icanguaranteethatthevalueisbetween1~1million.”

GumballGuessingGame

ComputingConfidenceIntervals

• Firstcomputesamplemeanandsamplevariance.

• ThenThasStudent'st‐distributionwithn− 1degreesoffreedom,where

T‐table

• http://en.wikipedia.org/wiki/Student's_t‐distribution

− What%ofthepopulationshouldbeincludedinthesample?dependsonaccuracy

• 2measuresaffecttheaccuratenessofthedata:

• marginoferror(orconfidenceintervals):‐ Positiveandnegativedeviationonresultsofthesample

• 5%marginerror:result=90%non‐congested• for‘sure’85%(90%‐5)and95%(90%+5)oftheentirenetworkactually isnon‐congested

• confidencelevel:‐ Howoftenthepercentageofthepopulationactuallylieswithintheboundariesofthemarginoferror• 95%confidencelevel=95%ofthetime85%‐ 95%non‐congested

Whereapplicable:SampleSize(recap)

t‐tablevalues‐ intuition

NextLecture

• HypothesisTesting

• Correlation,Regression