Download Sections 3-1 and 3-2 - Gordon State College

5/5/2015
Sections 3-1 and 3-2
Overview and
Measures of Center
TWO TYPES OF STATISTICS
• Descriptivestatistics summarizeor
describe theimportantcharacteristicsof
data.
• Inferentialstatistics usesampledatato
makeinferences (orgeneralizations)about
apopulation.
MEASURE OF CENTER
Ameasureofcenter isavalueatthecenter
ormiddleofadataset.
Therearefourmeasuresofcenterthatwe
willdiscussinthisclass
• (Arithmetic)Mean
• Median
• Mode
• Midrange
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THE MEAN
Thearithmeticmean,orthemean,ofasetof
dataisthemeasureofcenterfoundbyadding
allthedatavaluesanddividingbythetotal
numberofdatavalues.Thisiswhatis
commonlyreferredtoastheaverage.
PROPERTIES OF THE MEAN
• Samplemeansdrawnfromthesame
populationtendtovarylessthanother
measuresofcenter.
• Themeanofthedatasetuseseverydata
value.
• Adisadvantageofthemeanisthatjustone
extremevalue(outlier)canchangethevalue
ofthemeansubstantially.(Sincethemean
cannotresistsubstantialchangescausedby
extremevalues,wesaythatthemeanisnota
resistant measureofcenter.
NOTATION
• Σ – denotesthesummation (addition)ofa
setofvalues.
• – isthevariable usuallyusedtorepresent
theindividualdatavalues.
• – representsthenumberofvalues ina
sample.
• N – representsthenumberofvalues ina
population.
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MEAN OF A SAMPLE AND
MEAN OF A POPULATION
•
•
̅
∑
isthemeanofsample values.
∑
isthemeanofpopulation values.
EXAMPLE
ThegivenvaluesarethenumbersofDutchess
Countycarcrashesforeachmonthinarecent
year.Findthemean.
27817111525161414141318
FINDING THE MEAN
ON THE TI-83/84
1. PressSTAT;select1:Edit….
2. EnteryourdatavaluesinL1.(Youmay
enterthevaluesinanyofthelists.)
3. Press2ND,MODE (forQUIT).
4. PressSTAT;arrowovertoCALC.Select
1:1‐VarStats.
5. EnterL1bypressing2ND,1.
6. PressENTER.
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FINDING THE MEAN ON THE
TI-84 WITH THE NEW OS
1. PressSTAT;select1:Edit….
2. EnteryourdatavaluesinL1.(Youmay
enterthevaluesinanyofthelists.)
3. Press2ND,MODE (forQUIT).
4. PressSTAT;arrowovertoCALC.Select
1:1‐VarStats.
5. For“List”enterL1bypressing2ND,1.
6. Leave“FreqList”blank.
7. Highlight“Calculate”andpressENTER.
MEDIAN
Themedian,denotedby ,ofadatasetisthe
middlevaluewhentheoriginaldatavalues
arearrangedinorderofincreasing(or
decreasing)magnitude.
PROPERTIES OF THE
MEDIAN
• Themediandoesnotchangebylarge
amountswhenweincludejustafew
extremevalues(sothemedianisaresistant
measureofcenter).
• Themediandoesnotuseeverydatavalue.
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6.72
3.46
3.60
6.44
3.46
3.60
6.44
6.72
(even number of values)
no exact middle -- shared by two numbers
3.60 + 6.44
MEDIAN is 5.02
2
6.72
3.46
3.60
6.44
26.70
3.46
3.60
6.44
6.72
26.70
(in order -
exact middle
odd number of values)
MEDIAN is 6.44
EXAMPLE
ThegivenvaluesarethenumbersofDutchess
Countycarcrashesforeachmonthinarecent
year.Findthemedian.
27817111525161414141318
FINDING THE MEDIAN
ON THE TI-83/84
Thisisdoneinexactlythesamewayasfinding
themean.Afteryouhavefinishedusethe
downarrowtoscrolldownandyouwillsee
Med=.
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MODE
• Themode,denotedbyM,ofadatasetisthenumberthat
occursmostfrequently.
• Whentwovaluesoccurwiththesamegreatestfrequency,
eachoneisamodeandthedatasetissaidtobebimodal.
• Whenmorethantwovaluesoccurwiththesamegreatest
frequency,eachisamodeandthedatasetissaidtobe
multimodal.
• Whennodatavalueisrepeated,wesaythatthereisno
mode.
• Thisistheonly measureofcenterthatcanbeusedwith
nominal data.
EXAMPLE
a. 5 5 5 3 1 5 1 4 3 5
Mode is 5
b. 1 2 2 2 3 4 5 6 6 6 7 9
Bimodal -
c. 1 2 3 6 7 8 9 10
No Mode
2 and 6
EXAMPLE
ThegivenvaluesarethenumbersofDutchess
Countycarcrashesforeachmonthinarecent
year.Findthemode.
27817111525161414141318
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FINDING THE MODE
ON THE TI-83/84
TheTI‐83/84willNOT calculatethemodeofa
dataset.However,thedatainalistcanbe
easilysortedtohelpinfindingthemode.
TosortL1inascending order:STAT,2:SortA,
L1,), andENTER.
Tosortindescending order,use3:SortD.
MIDRANGE
Themidrange ofadatasetisthemeasureofcenterthat
isthevaluemidwaybetweenthehighestandlowest
valuesoftheoriginaldataset.Itisfoundbyaddingthe
highestdatavalueandthelowestdatavalueandthen
dividingby2;thatis,
midrange maximumdatavalue minimumdatavalue
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PROPERTIES OF THE
MIDRANGE
•
•
Becausethemidrangeusesonlythemaximumand
minimumvalues,itisverysensitivetothoseextremes.
Inpractice,themidrangeisrarelyused,butithasthree
redeemingfeatures:
1. Themidrangeiseasytocompute.
2. Themidrangehelpsreinforcetheveryimportant
pointthatthereareseveraldifferentwaystodefine
thecenterofadataset.
3. Thevalueofthemidrangeissometimesused
incorrectlyforthemedian,soconfusioncanbe
reducedbyclearlydefiningthemidrangealongwith
themedian.
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EXAMPLE
ThegivenvaluesarethenumbersofDutchess
Countycarcrashesforeachmonthinarecent
year.Findthemidrange.
27817111525161414141318
ROUND-OFF RULES FOR MEAN,
MEDIAN, AND MIDRANGE
Forthemean,median,andmidrange,carry
onemoredecimalplacethanispresentin
theoriginalsetofvalues.
Forthemode,leavethevaluewithout
rounding(becausevaluesofthemodearethe
sameassomeoftheoriginaldatavalues).
MEAN FROM A FREQUENCY
DISTRIBUTION
Tocomputethemeanfromafrequency
distribution,weassumethatallsamplevalues
areequaltotheclassmidpoint.
̅
∑
·
∑
=classmidpoint
=frequency
Σf =sumoffrequencies=
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EXAMPLE
Thefollowingdatarepresentthenumberof
peopleaged25–64coveredbyhealthinsurance
in2007.Approximatethemeanage.
Age
25–34
Number
(inbillions)
39.8
35–44
41.9
45–54
43.8
55–64
33.3
Source:ebri.orgNotes•
August2009•Vol.30,No.8
FINDING THE MEAN FROM A
FREQUENCY TABLE ON TI-83/84
1. EntertheclassmidpointsinL1.
2. EnterthefrequenciesinL2.
3. PressSTAT,arrowovertoCALC,and
select1:1‐VarStats.
4. PressL1,L2 followedbyENTER.
5. Themeanwillbethefirstitem.
FINDING THE MEAN FROM A
FREQUENCY TABLE ON TI-84
WITH NEW OS
1. EntertheclassmidpointsinL1.
2. EnterthefrequenciesinL2.
3. PressSTAT,arrowovertoCALC,and
select1:1‐VarStats.
4. For“List”,enterL1
5. For“FreqList”,enter L2.
6. Highlight“Calculate”andpressENTER.
7. Themeanwillbethefirstitem.
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WEIGHTED MEAN
Insomecases,thevaluesvaryintheirdegreeof
importance,sowemaywanttoweightthem
accordingly.Wecancomputetheweighted
mean forsuchvalues.
̅
·
∑
∑
=value
=weightofvalue
∑ =sumofweights
EXAMPLE
Marissajustcompletedherfirstsemesterin
college.Sheearnedan“A”inherfour‐hour
statisticscourse,a“B”inherthree‐hour
sociologycourse,an“A”inherthree‐hour
psychologycourse,a“C”inherfive‐hour
computerprogrammingcourse,andan“A”in
herone‐hourdramacourse.Determine
Marissa’sgradepointaverage.
BEST MEASURE OF CENTER
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