Statistika Chap 2

7/24/2019 Statistika Chap 2

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Central Tendency and Variability

The two most essential features of a

distribution


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Numerical DataProperties & Measures

Numerical DataProperties

MeanMean

MedianMedianModeMode

CentralTendency

RangeRangeVarianceVariance

Standard DeviationStandard Deviation

Variation

SkewSkew

Shape


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Variables have distributions

A variable is somethin that chan es orhas different values !e" "# an er$"

A distribution is a collection ofmeasures# usually across people"

Distributions of numbers can besummari%ed with numbers !calledstatistics or parameters$"


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Central Tendency refers to the

Middle of the Distribution


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Variability is about the pread


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Mean

um of scores divided by the number of people" Population mean is !mu$and sample mean is !'(bar$"

)e calculate the sample mean by*

Arit

+eo

X

N

X X

=

n X X = n FX X =


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,n rouped Data

31 36 40

46 33 33

31 17 20

46 39 29

38 34 37

No of Child Frequency

0 3

1 20

2 15

3 8

4 3

5 1

The hei ht !to the nearest mm$ ofeach of a number of seedlin s

Number of a familychildren in leman


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+rouped Data

-.ampleThe hei hts !in cm$ of a roup ofstudents are summari%ed below" Draw ahisto ram and poly on to illustratethese data


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Mean

/" Measure of Central Tendency 0" Most Common Measure 1" Acts as 23alance Point4 5" Affected by -.treme Values

!26utliers4$

7" 8ormula ! ample Mean$

X X

X X

nn

X X X X X X

nn

i i

i i

nn

nn== ==++ ++ ++

==

11 11


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Deviation from the mean

. 9 ' : " Deviations sum to %ero" Deviation score : deviation from the

mean ;aw scores

Deviation scores

X

? = < /> //

>

(/ > /(0 (/ > / 0


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Median

core that separates top 7>@ from bottom 7>@,n rouped Data -ven number of scores# median is half way between two

middle scores"

etaB Med/9 n 0etaB Med0 9 !n 0$ 0

Med 9 !Med/ Med0$ 0 : / 5 E 8 9 /> /? /=: Median is !=


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Median

/" Measure of Central Tendency 0" Middle Value Fn 6rdered eGuence

: Ff 6dd n# Middle Value of eGuence :

Ff -ven n# Avera e of 0 Middle Values 1" Position of Median in eGuence

5" Not Affected by -.treme Values

PositioninPositionin g Pointg Point==

++nn 11


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Median -.ample

6dd( i%ed ample ;aw Data* 05"/ 00"E 0/"7 01"?00"E

PositioningPositioning PointPoint

Median ! "#Median ! "#

== == ==

n +1n +1 $ %1$ %1&&

'rdered('rdered( 1"$1"$ "#"# "#"# &")&") *"1*"1

Position(Position( 11 && ** $$


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Median -.ample

-ven( i%ed ample ;aw Data* />"1 5"< ="< //"? E"1?"?

PositioningPositioning PointPoint

MedianMedian

== == ==

== ==

n +1n +1 # %1# %1&& $$

)") % +",)") % +",+"&+"&

""

'rdered('rdered( *",*", #""& )"))") +",+", 1-"&1-"& 11")11")

Position(Position( 1 1 && * * $ $ # #


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Mode

/" Measure of Central Tendency

0" Value That 6ccurs Most 6ften

1" Not Affected by -.treme Values 5" May 3e No Mode or everal Modes

7" May 3e ,sed for Numerical &Cate orical Data


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The mode : the most freGuentlyoccurrin score" Midpoint of most

populous class interval" Can have

bimodal and multimodal distributions"


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+rouped Classified Data


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Mode -.ample

No Mode;aw Data* />"1 5"< ="< //"? E"1 ?"?

One Mode;aw Data* E"1 4.9 ="< E"1 4.9 4.9

More Than 1 Mode;aw Data* 0/ 28 28 5/ 43 43


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ThinBin Challen e

Hou4re a financial analyst"Hou have collected thefollowin closin stocB

prices of new stocB issues*17 1! 21 18 13 1! 1211.

Describe the stocB pricesin terms of cen"ral"endency "


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6DD & -V-N DATA


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Classified Data


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Comparison of mean# median

and mode Mode : +ood for nominal variables : +ood if you need to Bnow most freGuent

observation : IuicB and easy

Median

: +ood for JbadK distributions : +ood for distributions with arbitrary

ceilin or floor


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Comparison of mean# median

& mode Mean : ,sed for inference as well as descriptionL

best estimator of the parameter

: 3ased on all data in the distribution : +enerally preferred e.cept for JbadK

distribution" Most commonly usedstatistic for central tendency"


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3est +uess interpretations

Mean : avera e of si ned error will be%ero"

Mode : will be absolutely ri ht withreatest freGuency

Median : smallest absolute error


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tatistics for 3usinessand -conomics# Ee Chap 1(0E

hape of a Distribution

Describes how data are distributed Measures of shape

: ymmetric or sBewed

Mean 9 MedianMean Median Median Mean

;i ht( Bewedeft( Bewed ymmetric


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Fnfluence of Distribution

hape


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;eview

)hat is central tendencyO Mode Median Mean


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;eview

;an e Avera e deviation Variance tandard Deviation score


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Variation


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Numerical DataProperties & Measures

Numerical DataProperties

MeanMean

MedianMedian

ModeMode

CentralTendency

RangeRange

VarianceVarianceStandard DeviationStandard Deviation

Variation

SkewSkew

Shape


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5 tatistics* ;an e# Avera e Deviation#

Variance# & tandard Deviation ;an e 9 hi h score minus low score"

: /0 /5 /5 /E /E /= 0> : ran e90>(/09=

Avera e Deviation : mean of absolutedeviations from the median*

N Md X AD = QQ

Note difference between Rays & under rad te.t(

deviation from Median vs" Mean


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Variance

Population Variance* )here means population variance# means population mean# and the other

terms have their usual meanin " The variance is eGual to the avera e sGuared

deviation from the mean" To compute# taBe each score and subtract the

mean" Guare the result" 8ind the avera eover scores" Ta daS The variance"

N

X =

00 $!

0


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Computin the Variance!N97$

7 /7 (/> />>

/> /7 (7 07

/7 /7 > >

0> /7 7 07

07 /7 /> />>Total* ?7 > 07>

Mean* Variance Fs 7>

X X X X 0

$! X X


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tandard Deviation

Variance is avera e #quared deviationfrom the mean"

To return to ori inal# un#quared units#we ust taBe the sGuare root of thevariance" This is the standarddeviation"

Population formula* N

X = 0$!


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tandard Deviation

ometimes called the root(mean(sGuaredeviation from the mean" This namesays how to compute it from the inside

out" 8ind the deviation !difference betweenthe score and the mean$"

8ind the deviations sGuared" 8ind their mean" TaBe the sGuare root"


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Computin the tandard

Deviation!N97$7 /7 (/> />>

/> /7 (7 07

/7 /7 > >0> /7 7 07

07 /7 /> />>

Total* ?7 > 07>Mean* Variance Fs 7>

Grt D Fs

X X X X 0

$! X X

>?"?7> ==


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-.ample* A e Distribution

$-*-&--1-

age

1#

1

+

*

-

. r e / u e n c y

$-*-&--1-

age

Distri0ution o 2ge

Mean! $")&

$-*-&--1-

age

SD ! #"*) 2verage Distrance rom Mean

$-*-&--1-

age

Central Tendency3 Varia0ility3 and Shape

Median ! &

Mode ! 1


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tandard or % score

A % score indicates distance from themean in standard deviation units"8ormula*

Convertin to standard or % scores doesnot chan e the shape of the distribution"

(scores are not normali%ed"

S X X z

=

= X

z


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$%e&ne## and 'ur"o#i#$%e&ne## and %ur"o#i# describe the shape of your

data setUs distribution" Bewness indicates howsymmetrical the data set is# while Burtosis indicateshow heavy your data set is about its mean comparedto its tails"

Perfectly symmetrical data sets will have a sBewnessof %ero !sBewness 9 >$# and a nor(ally di#"ri)u"ed data set will have a Burtosis of appro.imately three!Burtosis91$"
http://en.wikipedia.org/wiki/Skewnesshttp://en.wikipedia.org/wiki/Kurtosishttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Kurtosishttp://en.wikipedia.org/wiki/Skewness


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-)N-


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,;T6 F


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-I,ATF6N

sBewness* / 9 m 1 m01 0

Burtosis* a5 9 m 5 m00


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-.ample


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Calculation of Bewness 6NC A F8F-D DATA

Finally "he #%e&ne## i#*1 + ( 3 , ( 23,2 + -2.!933 , 8.5275 3,2 + -0.1082


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FnterpretationFf sBewness 9 ># the data are perfectly symmetrical" 3ut a sBewness of e.actly%ero is Guite unliBely for real(world data# so ho& can you in"er re" "he#%e&ne## nu()er O

3ulmer# M" +"# Principles of Statistics !Dover# /


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Calculation of urtosis


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Fnfluence of Distributionhape

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Statistika Chap 2

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