Download Section 3-3 - Gordon State College

Section 3-3
Measures of Variation
WAITING TIMES AT
DIFFERENT BANKS
JeffersonValleyBank
(singlewaitingline)
6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7
BankofProvidence
(multiplewaiting
lines)
4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0
Allthemeasuresofcenterareequalforbothbanks.
Mean=7.15min
Median=7.20min
Mode=7.7min
Midrange=7.10min
RANGE
Therange ofasetofdatavaluesisthe
differencebetweenthehighestvalueandthe
lowestvalue.
range=(maximumdatavalue)−(minimum
datavalue)
EXAMPLE:
JeffersonValleyBankrange=7.7−6.5=1.2min
BankofProvidencerange=10.0−4.2=5.8min
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STANDARD DEVIATION
FOR A SAMPLE
Thestandarddeviation ofasetofsample
values,denotedby ,isameasureofhowmuch
thedatavaluesdeviateawayfromthemean
STANDARD DEVIATION
FORMULAS
SampleStandard
Deviation
ShortcutFormula
forSample
StandardDeviation
∑
̅
1
∑
∑
1
EXAMPLE
Useboththeregularformulaandshortcut
formulatofindthestandarddeviationofthe
following.
3742
2
PROPERTIES OF THE
STANDARD DEVIATION
• Thestandarddeviationisameasureofvariationofall
valuesfromthemean.
• Thevalueofthestandarddeviations isusually
positive.Itiszeroonlywhenallofthedatavaluesare
thesamenumber.(Itisnevernegative.)Also,larger
valuesofs indicategreateramountsofvariation.
• Thevalueofthestandarddeviations canincrease
dramaticallywiththeinclusionofoneormore
outliers.
• Theunitsofthestandarddeviations arethesameas
theunitsoftheoriginaldatavalues.
• Thesamplestandarddeviationisabiasedestimator
ofthepopulationstandarddeviation .
STANDARD DEVIATION
OF A POPULATION
Thestandarddeviationforapopulation is
denotedbyσ andisgivenbytheformula
∑
FINDING THE STANDARD
DEVIATION ON THE TI-83/84
1. PressSTAT;select1:Edit….
2. EnteryourdatavaluesinL1.(Youmayenterthe
valuesinanyofthelists.)
3. Press2ND,MODE (forQUIT).
4. PressSTAT;arrowovertoCALC.Select1:1‐Var
Stats.
5. EnterL1bypressing2ND,1.
6. PressENTER.
7. Thesample standarddeviationisgivenbySx.The
population standarddeviationisgivenbyσx.
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FINDING THE STANDARD DEVIATION
ON THE TI-84 WITH NEW OS
1. PressSTAT;select1:Edit….
2. EnteryourdatavaluesinL1.(Youmayenterthe
valuesinanyofthelists.)
3. Press2ND,MODE (forQUIT).
4. PressSTAT;arrowovertoCALC.Select1:1‐Var
Stats.
5. For“List”,enterL1bypressing2ND,1.
6. Leave“FreqList”blank.
7. Highlight“Calculate”andpressENTER.
8. Thesample standarddeviationisgivenbySx.The
population standarddeviationisgivenbyσx.
SYMBOLS FOR
STANDARD DEVIATION
Sample
s
Sx
Population
σ
σx
textbook
TI‐83/84Calculators
textbook
TI‐83/84Calculators
EXAMPLE
Useyourcalculatortofindthestandard
deviationforwaitingtimesattheJefferson
ValleyBankandtheBankofProvidence.
JeffersonValley
Bank(single
waitingline)
6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7
7.7
BankofProvidence 4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0
(multiplewaiting
lines)
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VARIANCE
Thevariance ofsetofvaluesisameasureof
variationequaltothesquareofthestandard
deviation.
samplevariance=s2
populationvariance=σ2
PROPERTIES OF THE
VARIANCE
• Theunitsofthevariancearethesquares oftheunitsof
theoriginaldatavalues.
• Thevalueofthevariancecanincreasedramatically
withtheinclusionofoneormoreoutliers.
• Thevalueofthevarianceisusuallypositive.Itiszero
onlywhenallofthedatavaluesarethesamenumber.
(Itisnevernegative.)
• Thesamplevarianceisanunbiasedestimator ofthe
populationvariance .
ROUND-OFF RULE FOR
MEASURES OF VARIATON
Carryonemoredecimalplace
thanispresentinthe
originaldataset.
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STANDARD DEVIATION FROM A
FREQUENCY DISTRIBUTION
∑
·
∑
1
·
Usetheclassmidpointsasthex values.
NOTE:Wewillnot usethisformulabut
willusetheTI‐83/84calculator.
FINDING THE STANDARD
DEVIATION FROM A FREQUENCY
TABLE ON THE TI-83/84
1. EntertheclassmidpointsinL1.
2. EnterthefrequenciesinL2.
3. PressSTAT,arrowovertoCALC,andselect
1:1‐VarStats.
4. PressL1,L2 followedbyENTER.
5. ThesamplestandarddeviationwillbeSx.
Thepopulationstandarddeviationwillbe
σx.
FINDING THE STANDARD
DEVIATION FROM A FREQUENCY
TABLE ON THE TI-84 WITH NEW OS
1. EntertheclassmidpointsinL1.
2. EnterthefrequenciesinL2.
3. PressSTAT,arrowovertoCALC,andselect1:1‐
VarStats.
4. For“List”,enterL1.
5. For“FreqList”,enterL2.
6. Highlight“Calculate”andpressENTER.
7. ThesamplestandarddeviationwillbeSx.The
populationstandarddeviationwillbeσx.
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EXAMPLE
Findthesamplestandarddeviationofthe
following.
CLASS
FREQUENCY
1‐3
4
4‐6
9
7‐9
6
RANGE RULE OF THUMB —
PART 1
ForInterpretingaKnownValueofthe
StandardDeviation: Ifthestandard
deviations isknown,useittofindrough
estimatesoftheminimumandmaximum
“usual”samplevaluesbyusingthefollowing
minimum“usual”value
maximum“usual”value
̅
̅
2
2
RANGE RULE OF THUMB —
PART 2
ForEstimatingaValueoftheStandard
Deviation:Toroughly estimatethestandard
deviation,usethe
range
4
where
range=(maximumdatavalue)−(minimum
datavalue)
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EXAMPLES
1. Heightsofmenhaveameanof69.0inand
astandarddeviationof2.8in.Usethe
rangeruleofthumbtoestimatethe
minimumandmaximum“usual”heightsof
men.Inthiscontext,isitunusualfora
mantobe6ft,6intall?
2. Theshortesthome‐runhitbyMark
McGwirewas340ft andthelongestwas
550ft.Usetherangeruleofthumbto
estimatethestandarddeviation.
MORE PROPERTIES OF THE
STANDARD DEVIATION
• Thestandarddeviationmeasuresthevariation amongthedata
values.
• Valuesclosetogetherhaveasmallstandarddeviation,but
valueswithmuchmorevariationhavealargerstandard
deviation.
• Thestandarddeviationhasthesameunitsofmeasurementas
theoriginalvalues.
• Formanydatasets,avalueisunusual ifitdiffersfromthemean
bymorethantwostandarddeviations.
• Whencomparingvariationintwodifferentdatasets,compare
thestandarddeviationsonlyifthedatasetsusethesamescale
andunitsandtheyhavemeansthatareapproximatelythe
same.
THE EMPIRICAL
(OR 68-95-99.7) RULE FOR DATA WITH
A BELL-SHAPED DISTRIBUTION
Theempirical (or68‐95‐99.7)rule statesthatfor
datasetshavingadistributionthatisapproximately
bell‐shaped,thefollowingpropertiesapply.
• About68%ofallvaluesfallwithin1
standarddeviationofthemean.
• About95%ofallvaluesfallwithin2
standarddeviationsofthemean.
• About99.7%ofallvaluesfallwithin3
standarddeviationsofthemean.
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EXAMPLE
Arandomsampleof50gasstationsinCookCounty,
Illinois,resultedinameanpricepergallonof$2.56
andastandarddeviationof$0.07.Ahistogram
indicatedthatthedatafollowabell‐shaped
distribution.
(a) UsetheEmpiricalRuletodeterminethe
percentageofgasstationsthathaveprices
withinthreestandarddeviationsofthe
mean.Whatarethesegasprices?
(b) Determinethepercentageofgasstationswith
pricesbetween$2.42and$2.70,accordingtothe
empiricalrule.
CHEBYSHEV’S THEOREM
Theproportion(orfraction)ofany setofdata
lyingwithinK standarddeviationsofthemean
isalwaysatleast 1−1/K2,whereK isany
positivenumbergreaterthan1.ForK =2and
K =3,wegetthefollowingstatements:
• Atleast3/4(or75%)ofallvalueslie
within2standarddeviationsofthemean.
• Atleast8/9(or89%)ofallvalueslie
within3threestandarddeviationsofthe
mean.
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EXAMPLE
Asurveyoflocalcompaniesfoundthatthe
meanamountoftravelallowancefor
executiveswas$0.25permile.Thestandard
deviationwas$0.02.Findtheminimum
percentageofdatavaluesthatwillfallbetween
$0.19and$0.31.
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