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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/Skewness7/24/2019 Statistika Chap 2
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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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