Post on 25-Feb-2018
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Descriptive stats – part BDescriptive stats – part B
Understanding Relative Location andUnderstanding Relative Location andDetecting OutliersDetecting Outliers
Five number summaryFive number summary
Box PlotBox Plot
σ
µ
x
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Understanding Relative LocationUnderstanding Relative Locationand Detecting Outliersand Detecting Outliers
"#Scores"#Scores
$mpirical Rule$mpirical Rule
Detecting OutliersDetecting Outliers
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"#Scores"#Scores
&'e &'e "#score"#score is o(ten called t'e standardi"edis o(ten called t'e standardi"ed
value)value) *t denotes t'e number o( standard deviations a*t denotes t'e number o( standard deviations a
data valuedata value x x ii is (rom t'e mean)is (rom t'e mean)
+ data value less t'an t'e sample mean ,ill+ data value less t'an t'e sample mean ,ill
'ave a "#score less t'an "ero)'ave a "#score less t'an "ero)
+ data value greater t'an t'e sample mean ,ill+ data value greater t'an t'e sample mean ,ill
'ave a "#score greater t'an "ero)'ave a "#score greater t'an "ero)
+ data value e-ual to t'e sample mean ,ill 'ave+ data value e-ual to t'e sample mean ,ill 'avea "#score o( "ero)a "#score o( "ero)
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Ra, dataRa, data
.!/ .%0 .%0 .%/ .%/ .%/ .%/ .%/ ..0 ..0
..0 ..0 ..0 ../ ../ ../ ../ ../ ./0 ./0
./0 ./0 ./0 ./0 ./0 .0 .0 .0 ./ ./
./ .20 .20 .2! .2/ .2/ .2/ .30 .30 .30
.30 .3/ .40 .40 .40 /00 /00 /00 /00 /10
/10 /1/ /!/ /!/ /!/ /%/ /.4 //0 /20 /20
/2/ /2/ /30 /40 00 00 00 00 1/ 1/
.!/ .%0 .%0 .%/ .%/ .%/ .%/ .%/ ..0 ..0
..0 ..0 ..0 ../ ../ ../ ../ ../ ./0 ./0
./0 ./0 ./0 ./0 ./0 .0 .0 .0 ./ ./
./ .20 .20 .2! .2/ .2/ .2/ .30 .30 .30
.30 .3/ .40 .40 .40 /00 /00 /00 /00 /10
/10 /1/ /!/ /!/ /!/ /%/ /.4 //0 /20 /20
/2/ /2/ /30 /40 00 00 00 00 1/ 1/
Ra, data on +partment RentsRa, data on +partment Rents
5ean 6 .40)35ean 6 .40)3
Standard deviation 6 /.)2.Standard deviation 6 /.)2.
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"#Score o( Smallest 7alue 8.!/9"#Score o( Smallest 7alue 8.!/9
Standardi"ed 7alues (or +partment RentsStandardi"ed 7alues (or +partment Rents
#1)!0 #1)11 #1)11 #1)0! #1)0! #1)0! #1)0! #1)0! #0)4% #0)4%
#0)4% #0)4% #0)4% #0)3. #0)3. #0)3. #0)3. #0)3. #0)2/ #0)2/
#0)2/ #0)2/ #0)2/ #0)2/ #0)2/ #0)/ #0)/ #0)/ #0).2 #0).2
#0).2 #0)%3 #0)%3 #0)%. #0)!4 #0)!4 #0)!4 #0)!0 #0)!0 #0)!0
#0)!0 #0)11 #0)01 #0)01 #0)01 0)12 0)12 0)12 0)12 0)%/
0)%/ 0).. 0)! 0)! 0)! 0)31 1)0 1)03 1)./ 1)./
1)/. 1)/. 1)% 1)31 1)44 1)44 1)44 1)44 !)!2 !)!2
#1)!0 #1)11 #1)11 #1)0! #1)0! #1)0! #1)0! #1)0! #0)4% #0)4%
#0)4% #0)4% #0)4% #0)3. #0)3. #0)3. #0)3. #0)3. #0)2/ #0)2/
#0)2/ #0)2/ #0)2/ #0)2/ #0)2/ #0)/ #0)/ #0)/ #0).2 #0).2
#0).2 #0)%3 #0)%3 #0)%. #0)!4 #0)!4 #0)!4 #0)!0 #0)!0 #0)!0
#0)!0 #0)11 #0)01 #0)01 #0)01 0)12 0)12 0)12 0)12 0)%/
0)%/ 0).. 0)! 0)! 0)! 0)31 1)0 1)03 1)./ 1)./
1)/. 1)/. 1)% 1)31 1)44 1)44 1)44 1)44 !)!2 !)!2
$xample: +partment Rents$xample: +partment Rents
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$mpirical Rule$mpirical Rule
For data 'aving a bell#s'aped distributionFor data 'aving a bell#s'aped distribution::
; +pproximately+pproximately 68%68% o( t'e data values ,illo( t'e data values ,illbe ,it'inbe ,it'in oneone standard deviationstandard deviation o( t'eo( t'e
mean)mean)
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$mpirical Rule$mpirical Rule
For data 'aving a bell#s'aped distribution:For data 'aving a bell#s'aped distribution:
; +pproximately+pproximately 95%95% o( t'e data values ,illo( t'e data values ,illbe ,it'inbe ,it'in twotwo standard deviationsstandard deviations o( t'eo( t'emean)mean)
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$mpirical Rule$mpirical Rule
For data 'aving a bell#s'aped distribution:For data 'aving a bell#s'aped distribution:
; Almost allAlmost all 844)29 o( t'e items ,ill be844)29 o( t'e items ,ill be
,it'in,it'in threethree standard deviationsstandard deviations o( t'eo( t'emean)mean)
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$xample: +partment Rents$xample: +partment Rents
$mpirical Rule$mpirical Rule
*nterval*nterval in *nterval in *nterval
<it'in =># 1<it'in =># 1ss .%)0 to /./)/..%)0 to /./)/. .3>20 6 4.3>20 6 4
<it'in =># !<it'in =># !ss
%31)%! to 00)!3%31)%! to 00)!3
3>20 6 423>20 6 42
<it'in =># %<it'in =># %ss %!)/3 to //)0!%!)/3 to //)0! 20>20 620>20 6100100
.!/ .%0 .%0 .%/ .%/ .%/ .%/ .%/ ..0 ..0
..0 ..0 ..0 ../ ../ ../ ../ ../ ./0 ./0
./0 ./0 ./0 ./0 ./0 .0 .0 .0 ./ ./
./ .20 .20 .2! .2/ .2/ .2/ .30 .30 .30
.30 .3/ .40 .40 .40 /00 /00 /00 /00 /10
/10 /1/ /!/ /!/ /!/ /%/ /.4 //0 /20 /20
/2/ /2/ /30 /40 00 00 00 00 1/ 1/
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Detecting OutliersDetecting Outliers
+n+n outlieroutlier is an unusually small or unusuallyis an unusually small or unusually
large value in a data set)large value in a data set) + data value ,it' a "#score less t'an #% or+ data value ,it' a "#score less t'an #% or
greater t'an =% mig't be considered angreater t'an =% mig't be considered anoutlier)outlier)
*t mig't be an incorrectly recorded data value)*t mig't be an incorrectly recorded data value)
*t mig't be a data value t'at ,as incorrectly*t mig't be a data value t'at ,as incorrectlyincluded in t'e data set)included in t'e data set)
*t mig't be a correctly recorded data value*t mig't be a correctly recorded data value
t'at belongs to t'e data set ?t'at belongs to t'e data set ?
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$xample: +partment Rents$xample: +partment Rents
Detecting OutliersDetecting Outliers
&'e most extreme "#scores are #1)!0 and &'e most extreme "#scores are #1)!0 and!)!2)!)!2)
Using @Using @ z z @@ AA % as t'e criterion (or an outlier% as t'e criterion (or an outlier
t'ere are no outliers in t'is data set)t'ere are no outliers in t'is data set)
Standardi"ed 7alues (or +partment RentsStandardi"ed 7alues (or +partment Rents#1)!0 #1)11 #1)11 #1)0! #1)0! #1)0! #1)0! #1)0! #0)4% #0)4%
#0)4% #0)4% #0)4% #0)3. #0)3. #0)3. #0)3. #0)3. #0)2/ #0)2/
#0)2/ #0)2/ #0)2/ #0)2/ #0)2/ #0)/ #0)/ #0)/ #0).2 #0).2
#0).2 #0)%3 #0)%3 #0)%. #0)!4 #0)!4 #0)!4 #0)!0 #0)!0 #0)!0#0)!0 #0)11 #0)01 #0)01 #0)01 0)12 0)12 0)12 0)12 0)%/
0)%/ 0).. 0)! 0)! 0)! 0)31 1)0 1)03 1)./ 1)./
1)/. 1)/. 1)% 1)31 1)44 1)44 1)44 1)44 !)!2 !)!2
#1)!0 #1)11 #1)11 #1)0! #1)0! #1)0! #1)0! #1)0! #0)4% #0)4%
#0)4% #0)4% #0)4% #0)3. #0)3. #0)3. #0)3. #0)3. #0)2/ #0)2/
#0)2/ #0)2/ #0)2/ #0)2/ #0)2/ #0)/ #0)/ #0)/ #0).2 #0).2
#0).2 #0)%3 #0)%3 #0)%. #0)!4 #0)!4 #0)!4 #0)!0 #0)!0 #0)!0#0)!0 #0)11 #0)01 #0)01 #0)01 0)12 0)12 0)12 0)12 0)%/
0)%/ 0).. 0)! 0)! 0)! 0)31 1)0 1)03 1)./ 1)./
1)/. 1)/. 1)% 1)31 1)44 1)44 1)44 1)44 !)!2 !)!2
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Five#Cumber SummaryFive#Cumber Summary
Smallest 7alueSmallest 7alue
First uartileFirst uartile
5edian5edian
&'ird uartile &'ird uartile
Largest 7alueLargest 7alue
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$xample: +partment Rents$xample: +partment Rents
Five#Cumber SummaryFive#Cumber Summary
Lo,est 7alue 6 .!/Lo,est 7alue 6 .!/ First uartile 6First uartile 6./0./0
5edian 6 .2/5edian 6 .2/
&'ird uartile 6 /!/ Largest 7alue 6 &'ird uartile 6 /!/ Largest 7alue 61/1/.!/ .%0 .%0 .%/ .%/ .%/ .%/ .%/ ..0 ..0
..0 ..0 ..0 ../ ../ ../ ../ ../ ./0 ./0
./0 ./0 ./0 ./0 ./0 .0 .0 .0 ./ ./
./ .20 .20 .2! .2/ .2/ .2/ .30 .30 .30
.30 .3/ .40 .40 .40 /00 /00 /00 /00 /10
/10 /1/ /!/ /!/ /!/ /%/ /.4 //0 /20 /20
/2/ /2/ /30 /40 00 00 00 00 1/ 1/
.!/ .%0 .%0 .%/ .%/ .%/ .%/ .%/ ..0 ..0
..0 ..0 ..0 ../ ../ ../ ../ ../ ./0 ./0
./0 ./0 ./0 ./0 ./0 .0 .0 .0 ./ ./
./ .20 .20 .2! .2/ .2/ .2/ .30 .30 .30
.30 .3/ .40 .40 .40 /00 /00 /00 /00 /10
/10 /1/ /!/ /!/ /!/ /%/ /.4 //0 /20 /20
/2/ /2/ /30 /40 00 00 00 00 1/ 1/
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Box PlotBox Plot
+ box is dra,n ,it' its ends located at t'e Erst+ box is dra,n ,it' its ends located at t'e Erst
and t'ird -uartiles)and t'ird -uartiles) + vertical line is dra,n in t'e box at t'e+ vertical line is dra,n in t'e box at t'e
location o( t'e median)location o( t'e median)
Limits are located 8not dra,n9 using t'eLimits are located 8not dra,n9 using t'e
inter-uartile range 8*R9)inter-uartile range 8*R9)
; &'e lo,er limit is located 1)/8*R9 belo, &'e lo,er limit is located 1)/8*R9 belo, QQ1)1)
; &'e upper limit is located 1)/8*R9 above &'e upper limit is located 1)/8*R9 aboveQQ%)%)
; Data outside t'ese limits are consideredData outside t'ese limits are consideredoutliersoutliers))
continuedcontinued
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Box Plot 8Gontinued9Box Plot 8Gontinued9
<'isHers 8das'ed lines9 are dra,n (rom t'e<'isHers 8das'ed lines9 are dra,n (rom t'e
ends o( t'e box to t'e smallest and largestends o( t'e box to t'e smallest and largestdata values inside t'e limits)data values inside t'e limits)
&'e locations o( eac' outlier is s'o,n ,it' t'e &'e locations o( eac' outlier is s'o,n ,it' t'e
symbolsymbol II ))
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$xample: +partment Rents$xample: +partment Rents
Box PlotBox Plot
Lo,er Limit: 1 # 1)/8*R9 6 ./0 # 1)/82/9Lo,er Limit: 1 # 1)/8*R9 6 ./0 # 1)/82/96 %%2)/6 %%2)/
Upper Limit: % = 1)/8*R9 6 /!/ = 1)/82/9Upper Limit: % = 1)/8*R9 6 /!/ = 1)/82/9
6 %2)/6 %2)/
&'ere are no outliers &'ere are no outliers))
%2/
.00
.!/
./0
.2/
/00
/!/
//0 /2/ 00 !/
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