L.O 2-STATISTICS-compeletlyhhhhhhhhh.pdf

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1. Representinganddescribing
quantitativedata
LearningOutcome:Analyze,displayanddescribe
quantitativedatawithafocusonstandarddeviation.
L.O2

3. DescriptiveStatistics:Overview
MeasuresofCenter(Central
tendency)
Mode
Median
Mean
MeasuresofSpread
(dispersions)
Range
Inter-quartileRange
Variance
Standarddeviation
MeasuresofSymmetry
Skewnes
s

4. Findthemode,themedianand
themeanforthefollowing
values
Mean:The"average"number;foundbyaddingalldatapointsanddividingby
thenumberofdatapoints.
Example:Themeanof44,11,and77is(4+1+7)/3=12/3=
4(4+1+7)/3=12/3=4.
Median:Themiddlenumber;foundbyorderingalldatapointsandpicking
outtheoneinthemiddle(oriftherearetwomiddlenumbers,takingthe
meanofthosetwonumbers).
Example:Themedianof44,11,and77is44becausewhenthenumbersare
putinorder(1(1,44,7)7),thenumber44isinthemiddle.
Mode:Themostfrequentnumber—thatis,thenumberthatoccursthe
highestnumberoftimes.

8. Representingdatabydotplot

9. Describingthedot

11. Activity1

12. 212223
24
262728
29
313233
34
1819
48

13. Describethefollowingdistributions:
Bothdistributionsareroughlysymmetric.
Thecenterforbothdistributionsis90goals.
Bothdistributionshavedifferentamountsof
VARIABILITY.

14. MeasuringSpread/Variability
•Range:Thedifferencebetweenthelargestandsmallest
observations.
•Quartiles:(usesmedian)Thequartilesmarkoutthe
middlehalfandimproveourdescriptionofspread.

15. LocateQ1andQ3:
Q1=25
Q3=41
Quartiles

16. Quartiles
1.ArrangeobservationsinAscendingorderandlocateM.
2.Thefirstquartile(Q1)
•liesone-quarterofthewayuplistofordered
observations
•Moflowerhalf
•largerthan25%oforderedobservations.

17. 3.Thethirdquartile(Q3)
liesthree-quartersofthewayuplistoforderedobservations
•largerthan75%oforderedobservations.
•Mofupperhalf
4.The“secondquartile”
•median(M)
•Note:isnotlargerthan50%oforderedobservations
•isat50%mark

18. InterquartileRange(IQR):
Thedistancebetweenthefirstandthird
quartiles.
IQR=Q3–Q1
IQR=41–25=16

19. Note:IfanobservationfallsintheIQRitisnot
unusuallyhighorlow.
WeuseIQRtoidentifysuspectoutliers.

20. Outliers
•Unusuallyhighorunusuallylow
•Basic“ruleofthumb”foridentifyingisiftheobservation
fallsmorethan1.5xIQRabovethethirdquartileorbelow
thefirstquartile.

21. HowtofindOutliers?
3steps
1.FindIQR
2.Q3+1.5xIQR(uppercutoff)
3.Q1–1.5xIQR(lowercutoff)

22. Ex:McDonald’sChickenSandwiches
Problem:DeterminewhetherthePremiumCrispyChicken
ClubSandwichwith28gramsoffatisanoutlier.(2min)
Solution:
Herearethe14amountsoffatinorder:
99101012151616171717202328

23. WhichisresistantrangeorIQR?
•Range:
•IQR:

27. TheShape
•Theshapeofadistributionisdescribedbythefollowing
characteristics.
•Symmetry.
•Numberofpeaks
•Skewness.
•Uniform.

28. Symmetr
y.
•Whenitisgraphed,asymmetricdistributioncanbe
dividedatthecentersothateachhalfisamirrorimage
oftheother.
Numberof
peaks
•Distributionscanhavefewormanypeaks.Distributionswith
oneclearpeakarecalledunimodal,anddistributionswithtwo
clearpeaksarecalledbimodal.Whenasymmetricdistribution
hasasinglepeakatthecenter,itisreferredtoasbell-shaped.

29. Skewness.
•Whentheyaredisplayedgraphically,somedistributions
havemanymoreobservationsononesideofthegraphthan
theother.Distributionswithfewerobservationsontheright
(towardhighervalues)aresaidtobeskewedright;and
distributionswithfewerobservationsontheleft(toward
lowervalues)aresaidtobeskewedleft.

30. Uniform.
•Whentheobservationsinasetofdataareequallyspread
acrosstherangeofthedistribution,thedistributioniscalleda
uniformdistribution.Auniformdistributionhasnoclearpeaks.

31. s

32. InterpretingHistograms
Assesswherea
distributionis
centeredbyfinding
themedian
Assessthespread
ofadistribution
Shapeofa
distribution:roughly
symmetric,skewed
totheright,or
skewedtotheleft
Leftandrightsides
aremirror
images

33. ExamplesofSkewness

34. Shape:TypeofMound

35. ComparingtheMeanandMedian
Meanandmedianofasymmetricdistributionare
close
Meanisoftenpreferredbecauseitusesall
Inaskeweddistribution,themeanisfartheroutinthe
skewedtailthanisthemedian
Medianispreferredbecauseitisbetter
representativeofatypicalobservation

36. ResistantMeasures
Ameasureis
resistantifextreme
observations
(outliers)havelittle,
ifany,influenceon
itsvalue
Medianisresistant
tooutliers
Meanisnot
resistanttooutliers
www.stat.psu.edu

38. NumberofModes
Onewaytodescribetheshapeofadistributionisbyitsnumber
ofpeaks,ormodes.
Uniformdistribution—hasnomodebecausealldatavalueshave
thesamefrequency.

39. Anypeakisconsideredamode,evenifallpeaksdonothavethe
sameheight.
Adistributionwithasinglepeakiscalleda
single-peaked,orunimodal,distribution.
Adistributionwithtwopeaks,eventhoughnotthe
samesize,isabimodaldistribution.
Whatisthefollowingdistribution?

40. SymmetryorSkewness
Adistributionissymmetricifitslefthalf
isamirrorimageofitsrighthalf.
Asymmetricdistributionwithasingle
peakandabellshapeisknownasa
normaldistribution.

41. SymmetryorSkewness
Adistributionisleft-skewed
(ornegativelyskewed)ifthevalues
aremorespreadoutontheleft,
meaningthatsomelowvaluesare
likelytobeoutliers.
Adistributionisrightskewed
orpositivelyskewedifthe
valuesaremorespreadout
ontheright.Ithasatail
pulledtowardtheright.

42. Whatistherelationshipbetweenmean,medianandmode
foranormaldistribution?
Findthemeanmedianandmodeof:
1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,6,6,7
Meanis4.
Medianis4.
Modeis4.

44. Whatistherelationshipbetweenmean,medianandmodeofa
left-skeweddistribution?
Findthemean,medianandmodeof:
0,5,10,20,40,45,45,50,50,50,60,60,60,60,60,60,70,70,70,70,70,
70,70,70
Themeanis51.5.
Themedianis60.
Themodeis70.

46. Whatistherelationshipbetweenmean,medianandmodeofa
right-skeweddistribution?
Findthemean,median,andmodeof:
20,20,20,20,20,20,20,20,30,30,30,30,30,30,45,45,45,50,50,60,
70,90
Themeanis36.1.
Themedianis30.
Themodeis20.

48. Variation
Variationdescribeshowwidelydataare
spreadoutaboutthecenterofadata
set.
Howwouldyouexpectthevariationtodifferbetweentimes
ina5Kcityrunanda5Kruninastatemeet?

49. ResistantMeasure(ofcenter)
Resistsinfluenceofextremeobservations.
•Mean(average)isnotresistant
•Median(midpoint)isresistant
•Themeanandmedianwouldbeexactlythesameifthe
distributionisexactlysymmetric.
•Inaskeweddistribution,themeanisfartheroutinthelong
tailthenthemedian.

50. Howdooutliersaffectthemedian?
Arethereanyoutliers?IfsoremoveandfindM.
•n=15
•Mnew=34vsMold=34
Medianisresistant

51. Howdooutliersaffectthe
mean?
16,19,24,25,25,33,33,34,34,37,37,40,42,46,49,73
Findthemeanoftheoriginaldata.
•EnterdataintoL1
•STAT:CALC:1–VarStats:L1
•,Mean=35.44
Findthemeanwithoutoutlier.
•Remove73fromL1
•STAT:CALC:1–VarStats:L1
•Mean=32.93
Wecanfindmedianalsowith1-VarStats.Scrolldownthelist.

52. IsBarrybonds73anoutlier?
•IQR=
41–25=16
•1.5xIQR=
=24
•Q3+24=
41+24=65(uppercutoff)
•Yesorno?
YesBondsrecordsettingyearof73isanoutlier.

53. StemPlot
s

54. StemPlots

55. •Inastemplot,theentriesontheleftarecalledstems;andtheentriesonthe
rightarecalledleaves.Intheexampleabove,thestemsaretens(8
represents80,9represents90,10represents100,andsoon);andthe
leavesareones.However,thestemsandleavescouldbeotherunits-
millions,thousands,ones,tenths,etc.
•Somestemplotsincludeakeytohelptheuserinterpretthedisplay
correctly.Thekeyinthestemplotaboveindicatesthatastemof11witha
leafof7representsanIQscoreof117.
•Lookingattheexampleabove,youshouldbeabletoquicklydescribethe
distributionofIQscores.Mostofthescoresareclusteredbetween90
and109,withthecenterfallingintheneighborhoodof100.Thescores
rangefromalowof81(twostudentshaveanIQof81)toahighof151.
Thehighscoreof151mightbeclassifiedasanoutlier.
•Note:Intheexampleabove,thestemsandleavesareexplicitlylabeled
foreducationalpurposes.Intherealworld,however,stemplotsusually
donotincludeexplicitlabelsforthestemsandleaves.

56. Stemplots
Anothersimplegraphicaldisplayforsmalldatasetsisastemplot.(Alsocalleda
stem-and-leafplot.)
Stemplotsgiveusaquickpictureofthedistributionwhileincludingtheactualnumerical
values.

57. Stem&LeafPlotsReview
Giventhefollowingvalues,drawastemandleafplot
20,32,45,44,26,37,51,29,34,32,25,41,56
Ages Occurrences
---------------------------------------------------------
---------
2 |0,6,9,5
|
3 |2,7,4,2
|
4 |5,4,1
|
5 |1,6

58. Stem-and-leafplots
•Summarizes
quantitativevariables
•Separateeach
observationintoa
stem(firstpartof#)
andaleaf(lastdigit)
•Writeeachleafto
therightofitsstem;
orderleavesif
desired
Sodiumin
Cereals

59. Example:Who’sTaller?
•Whichgenderistaller,malesorfemales?Asample
of14-year-oldsfromtheUnitedKingdomwas
randomlyselectedusingthe
website.
•Herearetheheightsofthestudents(incm):
•Constructstemplots.Reportback.
•Male:154,157,187,163,167,159,169,162,176,
177,151,175,174,165,165,183,180
•Female:160,169,152,167,164,163,160,163,
169,157,158,153,161,165,165,159,168,153,
166,158,158,166

61. Stem&LeafPlotsReview
Giventhefollowingvalues,drawastemandleafplot
20,32,45,44,26,37,51,29,34,32,25,41,56
Ages Occurrences
-------------------------------------------------------
-----------
2 |0,6,9,5
|
3 |2,3,4,2
|
4 |5,4,1
|
5 |1,6

65. https://www.khanacademy.org/math/ap-statistics/quantitative-data-ap/histograms-ste
m-leaf/e/reading_stem_and_leaf_plots

66. Boxplot

67. •Aboxplotsplitsthedatasetintoquartiles.Thebodyoftheboxplot
consistsofa"box"(hence,thename),whichgoesfromthefirstquartile
(Q1)tothethirdquartile(Q3).
•Withinthebox,averticallineisdrawnattheQ2,themedianofthedata
set.Twohorizontallines,calledwhiskers,extendfromthefrontandback
ofthebox.ThefrontwhiskergoesfromQ1tothesmallestnon-outlierin
thedataset,andthebackwhiskergoesfromQ3tothelargest
non-outlier.

68. HowtoInterpretaBoxplot
•Hereishowtoreadaboxplot.Themedianis
indicatedbytheverticallinethatrunsdown
thecenterofthebox.Intheboxplotabove,
themedianisabout400.
•Additionally,boxplotsdisplaytwocommon
measuresofthevariabilityorspreadinadata
set.

69. •Range.Ifyouareinterestedinthespreadofthedata,itis
representedonaboxplotbythehorizontaldistancebetweenthe
smallestvalueandthelargestvalue,includinganyoutliers.Inthe
boxplotabove,datavaluesrangefromabout-700(thesmallest
outlier)to1700(thelargestoutlier),sotherangeis2400.Ifyou
ignoreoutliers,therangeisillustratedbythedistancebetweenthe
oppositeendsofthewhiskers-about1000intheboxplotabove.
•Interquartilerange(IQR).Themiddlehalfofadatasetfalls
withintheinterquartilerange.Inaboxplot,theinterquartilerangeis
representedbythewidthofthebox(Q3minusQ1).Inthechart
above,theinterquartilerangeisequalto600minus300orabout
300.

76. Stem(andleaf)plot
Graphsquantitativedataandisusedforsmallor
mediumsizeddata.(Verycommongraph).
•Tipfororganizingthedata:Firstenterdatainto
list.Thensort.

77. Howtoconstructastemplot
1.Separateeachobservationintoastemconsistingofallbut
therightmostdigit(1digitleafs).
2.Writestemsverticallyinincreasingorderanddrawa
verticallinetotheirright;writeeachleaftorightofits
stem.
3.Ifdatanotenteredintocalculatorlist,writestemsagain
andrearrangeleavesinincreasingorderoutfromstem
(don’tneedtodoifyousortwithcalculator).

78. Howtoconstructastemplotcont.
4.Titlegraphandaddkeydescribingwhatstemsandleaves
represent
Key:3/5meansthesoft
drinkcontains35mgof
caffeineper8ounce
serving.

79. Split-stemandleafplot
•Whensplittingstems,besureeachstemis
assignedanequalnumberofpossibleleafdigits.
•Noticegraph“a”isvery“skyscraper-ish”.
•Sosplit-stem.

80. Back-to-backstemplots-goodtocompare
twosetsofdata
Remember:5stemsisagoodminimum
•Advantages:easytoconstruct;displayactual
datavalues
•Disadvantages:doesnotworkwellwithlarge
datasets

81. StemPlots
•Astemplotgivesaquickpictureoftheshapeof
adistributionwhileincludingthenumerical
values
–Separateeachobservationintoastemandaleaf
eg.14g->1|4256->25|632.9oz->32|9
–Writestemsinaverticalcolumnanddrawavertical
linetotherightofthecolumn
–Writeeachleaftotherightofitsstem
•Note:
–Stemplotsdonotworkwellforlargedatasets
–Notavailableoncalculator

82. Example1
Theages(measuredbylastbirthday)oftheemployees
ofDewey,CheatumandHowearelistedbelow.
a)Constructastemgraphoftheages
b)Constructaback-to-backcomparingtheoffices
c)Constructahistogramoftheages
22 31 21 49 26 42
42 30 28 31 39 39
20 37 32 36 35 33
45 47 49 38 28 48
OfficeA
OfficeB

83. Histograms
•Likeabarchart,ahistogramismadeupofcolumnsplottedonagraph.
Usually,thereisnospacebetweenadjacentcolumns.Hereishowtoread
ahistogram.
•Thecolumnsarepositionedoveralabelthatrepresentsaquantitative
variable.
•Thecolumnlabelcanbeasinglevalueorarangeofvalues.
•Theheightofthecolumnindicatesthesizeofthegroupdefinedbythe
columnlabel.

84. Histogram

85. Histograms
Histogramsbreaktherangeofdatavaluesinto
classesanddisplaysthecountor%of
observationsthatfallintothatclass
Dividetherangeofdataintoequal-widthclasses
Counttheobservationsineachclass:“frequency”
Drawbarstorepresentclasses:height=frequency
Barsshouldtouch(unlikebargraphs).

86. Histograms
Quantitativevariablesoftentakemanyvalues.Agraphofthe
distributionmaybeclearerifnearbyvaluesaregroupedtogether.
Themostcommongraphofthedistributionofonequantitative
variableisahistogram.

87. HistogramversusBarChart
Histogram BarChart
variables quantitative categorical
barspace nospace spacesbetween

88. Example
Belowaretimesobtainedfromamail-ordercompany's
shippingrecordsconcerningtimefromreceiptoforder
todelivery(indays)foritemsfromtheircatalogue?
a)Constructahistogramofthedeliverytimes
3 7 10 5 14 12
6 2 9 22 25 11
5 7 12 10 22 23
14 8 5 4 7 13
27 31 13 21 6 8
3 10 19 12 11 8

89. Example:Histogram
n=36
k=√36=6
w=(31–2)/6
=29/6≈4.85
K range1 Nr
1 2–6 9
2 7–11 12
3 12–167
4 17–212
5 22–264
6 27–312
2
4
6
8
271217222732
Frequency
DaystoDelivery
10
12

90. DisplayingQuantitativeVariables
HistogramsThemostcommongraphfor
distributionofonequantitativevariable.No
spacesbetweengroups.

91. Howtoconstructahistogram
1.Dividerangeofdataintoclassesofequalwidth
andcountnumberofobservationsineachclass;
besuretospecifyclassespreciselysothateach
observationfallsintoexactlyoneclass.
2.Labelandscaleaxisandtitlegraph!
3.Drawabarthatrepresentsthecountineach
class.

92. Remember:
•Leavenospace
betweenbars.
•Addabreak-in-scale
symbol(//)onanaxis
thatdoesnotstartat
0.
•5classesisagood
minimum.
•HistogramTips(page
39)

93. Histograms
Graphthatusesbars
toportray
frequenciesor
relativefrequencies
ofpossible
outcomesfora
quantitativevariable

94. ConstructingaHistogram
1.Divideintointervalsofequalwidth
2.Count#ofobservationsineach
interval
Sodiumin
Cereals

95. ConstructingaHistogram
3.Labelendpoints
ofintervalson
horizontalaxis
4.Drawabarover
eachvalueor
intervalwith
heightequalto
itsfrequency
(orpercentage)
5.Labelandtitle
SodiuminCereals

96. Histograms
•Histogramsbreaktherangeofdatavaluesinto
classesanddisplaysthecountor%of
observationsthatfallintothatclass
–Dividetherangeofdataintoequal-widthclasses
–Counttheobservationsineachclass:“frequency”
–Drawbarstorepresentclasses:height=frequency
–Barsshouldtouch(unlikebargraphs).

97. Example1
Theages(measuredbylastbirthday)oftheemployees
ofDewey,CheatumandHowearelistedbelow.
a)Constructastemgraphoftheages
b)Constructaback-to-backcomparingtheoffices
c)Constructahistogramoftheages
22 31 21 49 26 42
42 30 28 31 39 39
20 37 32 36 35 33
45 47 49 38 28 48
OfficeA
OfficeB

98. Example1:Histogram
n=24
k=√24≈4.9sopickk=4
w=(49–20)/4
=29/4≈7.38
K range Nr
1 20–274
2 28–358
3 36–437
4 44–515
2
4
6
8
20-27
27-35
36-43
44-51
NumbersofPersonnel
Ages

99. BarChart
•Abarchartismadeupofcolumnsplottedonagraph.
Hereishowtoreadabarchart.
•Thecolumnsarepositionedoveralabelthatrepresents
acategoricalvariable.
•Theheightofthecolumnindicatesthesizeofthegroup
definedbythecolumnlabel.

100. Skewedtotheright
(positivelyskewed)
•Rightsideofhistogramextendsmuchfartherout
thanleftside
•pulledtotheright
•longtailtotheright

101. 154109137115152140154178101
103126126137165165129200148
Makeastemplotofthesedata.
EnterintoL2andsort,thenuse:
•Stemsfrom10-20
•Leaves:ones;so154lookslike15/4

102. Arethereanypotentialoutliers?
Aboutwhereisthecenterofthedistribution?
Potentialoutliers: 10139
Center: 115
Median:meanofthe9
th
and10
th
observ.12 669
1377
14 08
15 244
16 55
17 8
18
19
20 0
200
about138.5
n=18

103. Shape
Describeshape:
Theoverallshapeofthedistributionisirregular,asoften
happenswhenonlyafewobservationsareavailable.

104. Spread
Whatisthespreadofthescores(ignoringanyoutliers)?
178–101=77orfrom101to178

105. Center
b.:Findthemeanscorefromtheformulaforthemean.
Byhand:
Sumofthe18observations/18
=2539/18=
Calculatorkeystrokes:
2ndSTAT(list):MATH:MEAN(L2)or
STAT:CALC:1-VarStat(L2)
=
141.06
141.06

106. c.Findthemedianofthesescores.
Whichislarger:themedianorthemean?
Median=averageofthe9
th
and10
th
scores
=138.5
vs.mean=141.058
Explainwhy?
Themeanislargerthanthemedian
becauseoftheoutlierat200whichpulls
themeantowardthelongrighttailofthe
distribution.

107. DescribingDistributions
Shape
symmetric,skewed(leftorright),multi-modal
Outliers
Outliers-Extremevalues,unliketherestofthesampleand
withnospecialtreatmentcouldleadtoover-estimates.
dotheyexist,howmany,andonwhichends
Center
appropriatemeasure(mean,median,ormode)
Spread
appropriatemeasure(standarddeviationorIQR)

108. DescribingShape
Whenyoudescribeadistribution’sshape,concentrateonthemainfeatures.
Lookforroughsymmetryorclearskewness.

109. Statistics

119. Centraltendency

120. Centraltendency

121. Centraltendency

122. Centraltendency
•Seekstoprovideasinglevaluethatbestrepresentsa
distribution
•Typicalmeasuresare
–mode
–median
–mean

123. Mode
•themostfrequentlyoccurringscorevalue
•correspondstothehighestpointonthefrequency
distribution
Foragivensample
N=16:
333536373838
383939393940
40414145
Themode=39

124. Mode
•Themodeisnotsensitivetoextremescores.
Foragivensample
N=16:
333536373838
383939393940
40414150
Themode=39

125. Mode
•adistributionmayhavemorethanonemode
Foragivensample
N=16:
343435353535
363738383939
39394040
Themodes=35
and39

126. Mode
•theremaybenouniquemode,asinthecaseofa
rectangulardistribution
Foragivensample
N=16:
333334343535
363637373838
39394040
Nouniquemode

127. Median
•thescorevaluethatcutsthedistributionin
half(the“middle”score)
•50thpercentile
ForN=15
themedianis
theeighth
score=37

128. Median
ForN=16
themedianis
theaverage
oftheeighth
andninth
scores=37.5

129. Mean
•thisiswhatpeopleusuallyhaveinmindwhen
theysay“average”
•thesumofthescoresdividedbythenumberof
scores
Changingthevalueofasinglescoremaynotaffectthe
modeormedian,butitwillaffectthemean.
Forapopulation:Forasample:

130. Mean
X=7.07
Inmanycasesthemeanisthe
preferredmeasureofcentral
tendency,bothasadescriptionof
thedataandasanestimateofthe
parameter.
__
Inorderforthemeanto
bemeaningful,the
variableofinterestmust
bemeasuresonan
intervalscale.
0
1
2
3
4
5
Buddhist
Protestant
Catholic
Jewish
Muslim
Score
Frequency
X=2.4
__

131. Mean
Themeanis
sensitivetoextreme
scoresandis
appropriatefor
moresymmetrical
distributions.
X=36.8
__
X=36.5
__
X=93.2
__

132. •asymmetricaldistributionexhibitsnoskewness
•inasymmetricaldistributiontheMean=Median=Mode
Symmetry

133. •Skewnessreferstotheasymmetryofthedistribution
Skeweddistributions
•Apositivelyskewed
distributionisasymmetrical
andpointsinthepositive
direction.
Mode=70,000$
Median=88,700$
Mean=93,600$
modemean
media
n
•mode<median<mean

134. •Anegativelyskeweddistribution
Skeweddistributions
•mode>median>mean
mode
mean
media
n

135. Measuresofcentraltendency
+ -
Mode
•quick&easytocompute
•usefulfornominaldata
•poorsamplingstability
Median
•notaffectedbyextreme
scores
•somewhatpoor
samplingstability
Mean
•samplingstability
•relatedtovariance
•inappropriatefor
discretedata
•affectedbyskewed
distributions

136. Distributions
•Center:mode,median,mean
•Shape:symmetrical,skewed
•Spread

137. MeasuresofSpread
•thedispersionofscoresfromthecenter
•adistributionofscoresishighlyvariableifthe
scoresdifferwildlyfromoneanother
•Threestatisticstomeasurevariability
–range
–interquartilerange
–variance

138. Range
•largestscoreminusthesmallestscore
•thesetwo
havesamerange(80)
butspreadslookdifferent
•saysnothingabouthowscoresvaryaroundthecenter
•greatlyaffectedbyextremescores(definedbythem)

139. Interquartilerange
•Q1isthe"middle"valueinthe halfofthe
rank-ordereddataset.
•Q2isthemedianvalueintheset.
•Q3isthe"middle"valueinthe halfofthe
rank-ordereddataset.

140. Interquartilerange
•thedistancebetweenthe25thpercentileandthe75th
percentile
•Q3-Q1=70-30=40
•Q3-Q1=55-45=10
•effectivelyignoresthetopandbottomquarters,so
extremescoresarenotinfluential
•dismisses50%ofthedistribution

141. Deviationmeasures
•Toseehow
‘deviant’the
distributionis
relativetoanother,
wecouldsumthese
scores
•Butthiswouldleave
uswithabigfat
zero
ScoreDeviation
Amy 10 -40
Theo 20 -30
Max 30 -20
Henry 40 -10
Leticia 50 0
Charlotte 60 10
Pedro 70 20
Tricia 80 30
Lulu 90 40
SUM 0

142. Deviationmeasures
Soweuse
squareddeviations
fromthemean
ScoreDeviation
Sq.
Deviation
Amy 10 -40 1600
Theo 20 -30 900
Max 30 -20 400
Henry 40 -10 100
Leticia 50 0 0
Charlotte 60 10 100
Pedro 70 20 400
Tricia 80 30 900
Lulu 90 40 1600
SUM 0 6000
Thisisthesum
ofsquares
(SS)
SS=∑(X-X)
2
__

143. Variance
Wetakethe
“average”
squareddeviation
fromthemeanand
callitVARIANCE
(tocorrectforthefactthat
samplevariancetendsto
underestimatepop
variance)
Forapopulation:
Forasample:

144. Variance
1.Findthemean.
2.Subtractthemean
fromeveryscore.
3.Squarethe
deviations.
4.Sumthesquared
deviations.
5.DividetheSSbyNor
N-1.
Score
Dev’
n
Sq.Dev.
Amy 10 -40 1600
Theo 20 -30 900
Max 30 -20 400
Henry 40 -10 100
Leticia 50 0 0
Charlotte 60 10 100
Pedro 70 20 400
Tricia 80 30 900
Lulu 90 40 1600
SUM 0 60006000/
8=75
0

145. Thestandarddeviationisthesquarerootofthe
variance
Thestandarddeviationmeasuresspreadintheoriginal
unitsofmeasurement,whilethevariancedoessoin
unitssquared.
Varianceisgoodforinferentialstats.
Standarddeviationisnicefordescriptivestats.
Standarddeviation

146. Example
N=28
X=50
s
2
=555.55
s=23.57
N=28
X=50
s
2
=
140.74
s=11.86

147. DescriptiveStatistics:QuickReview
Forapopulation: Forasample:
Mean
Variance
Standard
Deviation

148. •Treatthislittledistributionasasampleandcalculate:
–Mode,median,mean
–Range,variance,standarddeviation
Exercise

149. UnusualFeatures
•Gaps.
•Outliers

150. Gaps
•Gapsrefertoareasofadistribution
wheretherearenoobservations.The
firstfigurebelowhasagap;thereareno
observationsinthemiddleofthe
distribution.

151. •Sometimes,distributionsarecharacterizedbyextremevaluesthat
differgreatlyfromtheotherobservations.Theseextremevalues
arecalledoutliers.Thesecondfigurebelowillustratesa
distributionwithanoutlier.Exceptforonelonelyobservation(the
outlierontheextremeright),alloftheobservationsfallbetween0
and4.Asa"ruleofthumb",anextremevalueisoftenconsidered
tobeanoutlierifitisatleast1.5interquartilerangesbelowthefirst
quartile(Q1),oratleast1.5interquartilerangesabovethethird
quartile(Q3).
Outliers

152. DotPlotExample
Comparedtoothertypesofgraphicdisplay,dotplotsareused
mostoftentoplotfrequencycountswithinasmallnumberof
categories,usuallywithsmallsetsofdata.
•Hereisanexampletoshowwhatadotplotlookslikeandhowto
interpretit.Suppose30firstgradersareaskedtopicktheir
favoritecolor.Theirchoicescanbesummarizedinadotplot,as
shownbelow

153. DotPlotExample
0
1
2
3
4
5
6
7
8
9
10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

154. •Eachdotrepresentsonestudent,andthenumberof
dotsinacolumnrepresentsthenumberoffirstgraders
whoselectedthecolorassociatedwiththatcolumn.
Forexample,Redwasthemostpopularcolor
(selectedby9students),followedbyBlue(selectedby
7students).Selectedbyonly1student,Indigowasthe
leastpopularcolor.
•Inthisexample,notethatthecategory(color)isa
qualitativevariable;soitisnotappropriatetotalkabout
thesymmetryorskewnessofthisdotplot.Thedotplot
inthenextsectionusesaquantitativevariable,sowe
willillustrateskewnessandsymmetryofdotplotsinthe
nextsection.

155. BarChart
•Abarchartismadeupofcolumnsplottedonagraph.
Hereishowtoreadabarchart.
•Thecolumnsarepositionedoveralabelthatrepresents
acategoricalvariable.
•Theheightofthecolumnindicatesthesizeofthegroup
definedbythecolumnlabel.

156. BarChart
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Column1
Column1

157. Histograms
•Likeabarchart,ahistogramismadeupofcolumnsplottedonagraph.
Usually,thereisnospacebetweenadjacentcolumns.Hereishowtoread
ahistogram.
•Thecolumnsarepositionedoveralabelthatrepresentsaquantitative
variable.
•Thecolumnlabelcanbeasinglevalueorarangeofvalues.
•Theheightofthecolumnindicatesthesizeofthegroupdefinedbythe
columnlabel.

158. Histogram