intro ba

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Business statistics

Quantitative techniques

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Decision making & Analysisin

Business Management

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DECISION MAKIN

• A metho! "o# !ete#mining o$timalst#ategies %hen "ace! %ith seve#al

!ecision alte#natives an! at times anunce#tain $atte#n o" "utu#e events

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Business & !ecision making

• Business management levels

• Basis o" !ecision

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'y$es o" Decision Making Situations

• (nce#tainty ) –

anything can ha$$en* %e !on+t kno% – Still some analysis is $ossi,le on !i-e#ent outcomes

. $ayo-s/ ,ase! on !i-e#ent a$$#oaches

– 'he !ecision make# is una,le o# un%illing toestimate $#o,a,ilities

– A$$ly a common sense c#ite#ion

0o# e1am$le – Senio# management !ecision making is St#ategic in

natu#e &long te#m – 2a#tici$ating in a com$etition

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Decision making un!e# ce#tainty –

state o" natu#e is kno%n – !ecision is to choose the alte#native %ith the ,est $ayo-

– 3e kno%4assume %ell !e5ne! unchanging system

– Mathematical mo!els an! equations a#e !e#ive!

– 3e t#y to im$lement ,est system* mo!el

0o# e1am$le – 6o%e# management !ecision making involves

Im$lementation o" actiona,le o,7ectives an! $lans & issho#t te#m

– M82* E82* many so"t%a#e ,ase! systems* #igi! #ules an!#egulations

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Decision making un!e# #isk .$#o,a,ility situations/

• E1$ecte! 9alue C#ite#ion

– com$ute e1$ecte! value "o# each !ecision alte#native

– select alte#native %ith :,est; e1$ecte! value

– %e kno% some $#o,a,ilities o" things to ha$$en

0o# E1am$le – Mi!!le management !ecision making involves conve#sion

o" st#ategies into actiona,le o,7ectives an! $lans & isme!ium te#m

– 3he#eve# some $#o,a,ility o" ha$$ening o" event can ,ecalculate! O# some $atte#n can ,e o,se#ve! #ega#!ing

ha$$ening o" events In such cases many time 2#o,a,ilitiesa#e calculate! using $ast !ata

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(nce#tainty vs #isk vsce#tainty

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• Al%ays management st#uggles toconve#t unce#tain to #isk* an! #isk toce#tainty

•

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• 3hole %o#l! o# any %o#l! o$e#atesun!e# ce#tain la%s an! in>uences

– I" one un!e#stan! these la%s* one canmanage ,ette#

– I" one has to "#ee onesel" "#om one la%one has to take hel$ o" anothe# la%

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• I" one is not a,le to kno% an! un!e#stan!la%s o" ho% things ha$$en ? like cause &e-ect* then one look "o# $#o,a,ilistic

outcomes ,ase! on $ast !ata * t#ying to 5n!$atte#ns

• I" one is not a,le to get $ast !ata an!

$#o,a,ility associate!* one goes "o# the$ossi,le choices ,ase! on outcomes .$ayo-s/an! ones o$timistic* $essimistic a$$#oach

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• Kno%ing 4 in"o#mation #e!ucesunce#tainty an! #isks

• 'hat+s %he#e %hole o" ,usiness statisticsan! o$e#ations #esea#ch $lays a #ole

• 3hich is "u#the# !evelo$e! as Businessanalytics

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• In to!ay+s scena#io* %hole o" ,usinessmanagement is to someho% a$$#oachto%a#!s #ight !ecision %ith mo#e an!

mo#e ce#tainty

• But %ith highly changing an! !ynamic

envi#onment an! com$etition mo#ean! mo#e unce#tainty is the#e

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• 2#o,lem is com$oun!e! %hen lo%e#management ten!s to ,ecomest#ategic an! senio# management

ten!s to ,ecome highly #igi! an!!e5ne! . at times to savethemselves "#om !i@cult4ha#sh

lea!e#shi$ !ecisions/ o# at times%himsical

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• 8esulting into .o# #esulte! "#om /"ailu#e o" e1isting mo!els an!systems

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Some a$$#oaches

• Analysis ,ase! !ete#ministic Mo!els likeinvento#y mo!els* $#o!uction $lanning mo!els*m#$ mo!els* 6inea# $#og#amming mo!els

• Ma#ket #esea#ch* quality* s$c* sqc* ,ase! onsam$ling an! $#o,a,ility

• Simulation

• (n!e# unce#tainty ? o$timistic* conse#vative*minima1 #eg#et* game theo#y

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Business statistics

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• Business Statistics is a scienceassisting you to make businessdecisions under

uncertainties ,ase! on somenume#ical an! measu#a,le scalesDecision making $#ocesses must ,e

,ase! on !ata* not on $e#sonalo$inion no# on ,elie"

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Decisions,Decisions,

Decisions,

BUSINESS IS ALL ABOUT

Decisions

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Business Decisions

(nce#tainties

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UNCERTAINITIES

'

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• Just like weater! i" you cannotcontrol someting! you souldlearn ow to measure and

analy#e it! in order to $redict it!e%ecti&ely

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Its allabout'ATA!

IN()R*ATI)N

AN''ECISI)N!dear+

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'he 'he

elementselements

a#e the entities on %hich !ata a#ea#e the entities on %hich !ata a#e

collecte!collecte!

AA va#ia,leva#ia,le is a cha#acte#istic o" inte#est "o# the eleis a cha#acte#istic o" inte#est "o# the ele

'he set o" measu#ements collecte! "o# a $a#ticula# 'he set o" measu#ements collecte! "o# a $a#ticula#

element is calle! anelement is calle! an o,se#vationo,se#vation

'he total num,e# o" !ata values in a !ata set is the 'he total num,e# o" !ata values in a !ata set is the num,e# o" elements multi$lie! ,y the num,e# o" num,e# o" elements multi$lie! ,y the num,e# o" va#ia,lesva#ia,les

Elements* 9a#ia,les* an! O,se#vationsElements* 9a#ia,les* an! O,se#vations

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Stock Annual Ea#n4Stock Annual Ea#n4

E1change Sales.M/ Sha#e./E1change Sales.M/ Sha#e./

Data* Data Sets*Data* Data Sets*Elements* 9a#ia,les* an! O,se#vationsElements* 9a#ia,les* an! O,se#vations

Com$anyCom$any

Data#amData#am

Ene#gySouthEne#gySouth

KeystoneKeystone

6an!Ca#e6an!Ca#e 2sycheme!ics2sycheme!ics

AMEAME F F FGH FGH

O'CO'C FF FF H H

NJSENJSE HFHF FGHFGH

NJSENJSE FF F F AMEAME HF HF F F

9a#ia,les9a#ia,les

ElementElement NamesNames

Data SetData Set

O,se#vationO,se#vation

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Measu#ing DA'A

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Scales o" Measu#ement

'he scale in!icates the !ata summa#iLation an! 'he scale in!icates the !ata summa#iLation an! statistical analyses that a#e most a$$#o$#iatestatistical analyses that a#e most a$$#o$#iate 'he scale in!icates the !ata summa#iLation an! 'he scale in!icates the !ata summa#iLation an! statistical analyses that a#e most a$$#o$#iatestatistical analyses that a#e most a$$#o$#iate

'he scale !ete#mines the amount o" in"o#mation 'he scale !ete#mines the amount o" in"o#mation containe! in the !atacontaine! in the !ata

'he scale !ete#mines the amount o" in"o#mation 'he scale !ete#mines the amount o" in"o#mation containe! in the !atacontaine! in the !ata

Scales o" measu#ement inclu!eScales o" measu#ement inclu!eScales o" measu#ement inclu!eScales o" measu#ement inclu!eNominalNominal

O#!inalO#!inal

Inte#valInte#val

8atio8atio

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Scales o" Measu#ement• Nominal

AA nonnume#ic la,elnonnume#ic la,el o#o# nume#ic co!enume#ic co!e may ,e use!may ,e use!AA nonnume#ic la,elnonnume#ic la,el o#o# nume#ic co!enume#ic co!e may ,e use!may ,e use!

Data a#eData a#e la,els o# namesla,els o# names use! to i!enti"y anuse! to i!enti"y an att#i,ute o" the elementatt#i,ute o" the element

Data a#eData a#e la,els o# namesla,els o# names use! to i!enti"y anuse! to i!enti"y an att#i,ute o" the elementatt#i,ute o" the element

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E1am$leE1am$le Stu!ents o" a unive#sity a#e classi5e! ,y theStu!ents o" a unive#sity a#e classi5e! ,y the school in %hich they a#e en#olle! using aschool in %hich they a#e en#olle! using a

nonnume#ic la,el such as Business*

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Scales o" Measu#ement• O#!inal

AA nonnume#ic la,elnonnume#ic la,el o#o# nume#ic co!enume#ic co!e may ,e use!may ,e use!AA nonnume#ic la,elnonnume#ic la,el o#o# nume#ic co!enume#ic co!e may ,e use!may ,e use!

'he !ata have the $#o$e#ties o" nominal !ata an! 'he !ata have the $#o$e#ties o" nominal !ata an! thethe o#!e# o# #ank o" the !ata is meaning"ulo#!e# o# #ank o" the !ata is meaning"ul

'he !ata have the $#o$e#ties o" nominal !ata an! 'he !ata have the $#o$e#ties o" nominal !ata an! thethe o#!e# o# #ank o" the !ata is meaning"ulo#!e# o# #ank o" the !ata is meaning"ul

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• Inte#val

Inte#val !ata a#eInte#val !ata a#e al%ays nume#ical%ays nume#icInte#val !ata a#eInte#val !ata a#e al%ays nume#ical%ays nume#ic

'he !ata have the $#o$e#ties o" o#!inal !ata* an! 'he !ata have the $#o$e#ties o" o#!inal !ata* an!

the inte#val ,et%een o,se#vations is e1$#esse! inthe inte#val ,et%een o,se#vations is e1$#esse! in te#ms o" a 51e! unit o" measu#ete#ms o" a 51e! unit o" measu#e

'he !ata have the $#o$e#ties o" o#!inal !ata* an! 'he !ata have the $#o$e#ties o" o#!inal !ata* an!

the inte#val ,et%een o,se#vations is e1$#esse! inthe inte#val ,et%een o,se#vations is e1$#esse! in te#ms o" a 51e! unit o" measu#ete#ms o" a 51e! unit o" measu#e

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• Inte#val

E1am$leE1am$le Melissa has an SA' sco#e o"Melissa has an SA' sco#e o" FF* %hile Kevin* %hile Kevin

has an SA' sco#e o"has an SA' sco#e o" FPFFPF Melissa sco#e! Melissa sco#e! $oints mo#e than Kevin$oints mo#e than Kevin

E1am$leE1am$le Melissa has an SA' sco#e o"Melissa has an SA' sco#e o" FF* %hile Kevin* %hile Kevin

has an SA' sco#e o"has an SA' sco#e o" FPFFPF Melissa sco#e! Melissa sco#e! $oints mo#e than Kevin$oints mo#e than Kevin

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Scales o" Measu#ement• 8atio

'he !ata have all the $#o$e#ties o" inte#val !ata 'he !ata have all the $#o$e#ties o" inte#val !ata an! thean! the #atio o" t%o values is meaning"ul#atio o" t%o values is meaning"ul

'he !ata have all the $#o$e#ties o" inte#val !ata 'he !ata have all the $#o$e#ties o" inte#val !ata an! thean! the #atio o" t%o values is meaning"ul#atio o" t%o values is meaning"ul

9a#ia,les such as !istance* height* %eight* an! tim9a#ia,les such as !istance* height* %eight* an! tim use the #atio scaleuse the #atio scale 9a#ia,les such as !istance* height* %eight* an! tim9a#ia,les such as !istance* height* %eight* an! tim use the #atio scaleuse the #atio scale

'his 'his scale must contain a Le#o valuescale must contain a Le#o value that in!icatesthat in!icates

that nothing e1ists "o# the va#ia,le at the Le#o $ointhat nothing e1ists "o# the va#ia,le at the Le#o $oin

'his 'his scale must contain a Le#o valuescale must contain a Le#o value that in!icatesthat in!icates

that nothing e1ists "o# the va#ia,le at the Le#o $ointhat nothing e1ists "o# the va#ia,le at the Le#o $oin

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Scales o" Measu#ement• 8atio

E1am$leE1am$le Melissa+s college #eco#! sho%sMelissa+s college #eco#! sho%s HH c#e!it hou#sc#e!it hou#s ea#ne!* %hile Kevin+s #eco#! sho%sea#ne!* %hile Kevin+s #eco#! sho%s c#e!itc#e!it

hou#s ea#ne! Kevin has t%ice as many c#e!ithou#s ea#ne! Kevin has t%ice as many c#e!it hou#s ea#ne! as Melissahou#s ea#ne! as Melissa

E1am$leE1am$le Melissa+s college #eco#! sho%sMelissa+s college #eco#! sho%s HH c#e!it hou#sc#e!it hou#s ea#ne!* %hile Kevin+s #eco#! sho%sea#ne!* %hile Kevin+s #eco#! sho%s c#e!itc#e!it

hou#s ea#ne! Kevin has t%ice as many c#e!ithou#s ea#ne! Kevin has t%ice as many c#e!it hou#s ea#ne! as Melissahou#s ea#ne! as Melissa

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B(SINESSS'A'IS'ICS

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Al%ays 8emem,e# itsall a,out DA'A

An! %e %ant to take!ecision ,ase! on ou#

collection an!inte#$#etation o" !ata

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'y$es o" Data

Data

Numerical

(Quantitative)

Categorical

(Qualitative)

Discrete Continuous

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Qualitative Data

6a,els o# names6a,els o# names use! to i!enti"y an att#i,ute o" eachuse! to i!enti"y an att#i,ute o" each elementelement 6a,els o# names6a,els o# names use! to i!enti"y an att#i,ute o" eachuse! to i!enti"y an att#i,ute o" each elementelement

O"ten #e"e##e! to asO"ten #e"e##e! to as catego#ical !atacatego#ical !ataO"ten #e"e##e! to asO"ten #e"e##e! to as catego#ical !atacatego#ical !ata

(se eithe# the nominal o# o#!inal scale o" (se eithe# the nominal o# o#!inal scale o" measu#ementmeasu#ement (se eithe# the nominal o# o#!inal scale o" (se eithe# the nominal o# o#!inal scale o" measu#ementmeasu#ement

Can ,e eithe# nume#ic o# nonnume#icCan ,e eithe# nume#ic o# nonnume#icCan ,e eithe# nume#ic o# nonnume#icCan ,e eithe# nume#ic o# nonnume#ic

A$$#o$#iateA$$#o$#iate statistical analysesstatistical analyses a#e #athe#a#e #athe# limite!limite!A$$#o$#iateA$$#o$#iate statistical analysesstatistical analyses a#e #athe#a#e #athe# limite!limite!

i i

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Quantitative DataQuantitative Data

Quantitative !ata a#eQuantitative !ata a#e al%ays nume#ical%ays nume#icQuantitative !ata a#eQuantitative !ata a#e al%ays nume#ical%ays nume#ic

O#!ina#y a#ithmetic o$e#ations a#e meaning"ul "o#O#!ina#y a#ithmetic o$e#ations a#e meaning"ul "o#

quantitative !ataquantitative !ata

O#!ina#y a#ithmetic o$e#ations a#e meaning"ul "o#O#!ina#y a#ithmetic o$e#ations a#e meaning"ul "o#

quantitative !ataquantitative !ata

Q i i

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Quantitative !ata in!icateQuantitative !ata in!icate ho% many o# ho% muchho% many o# ho% muchQuantitative !ata in!icateQuantitative !ata in!icate ho% many o# ho% muchho% many o# ho% much discretediscrete!! i" measu#ing ho% manyi" measu#ing ho% manydiscretediscrete!! i" measu#ing ho% manyi" measu#ing ho% many

continuouscontinuous!! i" measu#ing ho% muchi" measu#ing ho% muchcontinuouscontinuous!! i" measu#ing ho% muchi" measu#ing ho% much

i iQ i i D

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Quantitative !ata in!icateQuantitative !ata in!icate ho% many o# ho% muchho% many o# ho% muchQuantitative !ata in!icateQuantitative !ata in!icate ho% many o# ho% muchho% many o# ho% much discretediscrete!! i" measu#ing ho% manyi" measu#ing ho% manydiscretediscrete!! i" measu#ing ho% manyi" measu#ing ho% many

continuouscontinuous!! i" measu#ing ho% muchi" measu#ing ho% muchcontinuouscontinuous!! i" measu#ing ho% muchi" measu#ing ho% much

I9E EAM26ES O0 BO'<

Cl i" i bl di t

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Te time it ta!es to "rive to #or!$

Te num%er o& cre"it car"s a 'erson as$

Te num%er o& em'loees #or!ing in a large "e'artment

store$

Te num%er o& cars stolen eac #ee! in a large cit$

Classi"y eac &ariable as discrete orcontinuous+

S l " M tS l " M t

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Scales o" Measu#ementScales o" Measu#ement

QualitativeQualitativeQualitativeQualitative QuantitativQuantitativeeQuantitativQuantitativee

Nume#icalNume#icalNume#icalNume#ical Nume#icalNume#icalNume#icalNume#icalNon)Non)nume#icalnume#icalNon)Non)nume#icalnume#ical

DataDataDataData

NominaNominallNominaNominall O#!inaO#!inallO#!inaO#!inall NominalNominalNominalNominal O#!inalO#!inalO#!inalO#!inal Inte#valInte#valInte#valInte#val 8atio8atio8atio8atio

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For each of the following variables, determine whether the

variable is categorical or numerical.

If the variable is numerical, determine whether the variable

is discrete or continuous.

a$ Num%er o& tele'ones 'er ouseol"

%$ Lengt (in minutes) o& te longest long"istance call

made per month

c+ ,eter someone in te ouseold owns acell $one

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I66 NO3 3E

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I66 NO3 3E

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Data

Sources

*rimar Secon"ar

E+'eriment *u%lise"

(, OnLine)Surve O%servation

Data Sou#ces

C S ti l D tC#oss Sectional Data

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C#oss)Sectional DataC#oss)Sectional Data

C#oss)sectional !ataC#oss)sectional !ata a#e collecte! at the same o#a#e collecte! at the same o# a$$#o1imately the same $oint in timea$$#o1imately the same $oint in time

C#oss)sectional !ataC#oss)sectional !ata a#e collecte! at the same o#a#e collecte! at the same o# a$$#o1imately the same $oint in timea$$#o1imately the same $oint in time

E1am$leE1am$le !ata !etailing the num,e# o" ,uil!ing !ata !etailing the num,e# o" ,uil!ing $e#mits issue! in une FF in each o" the counties$e#mits issue! in une FF in each o" the counties o" Ohioo" Ohio

E1am$leE1am$le !ata !etailing the num,e# o" ,uil!ing !ata !etailing the num,e# o" ,uil!ing $e#mits issue! in une FF in each o" the counties$e#mits issue! in une FF in each o" the counties o" Ohioo" Ohio

'ime Se#ies Data'ime Se#ies Data

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'ime Se#ies Data 'ime Se#ies Data

'ime se#ies !ata 'ime se#ies !ata a#e collecte! ove# seve#al timea#e collecte! ove# seve#al time $e#io!s$e#io!s 'ime se#ies !ata 'ime se#ies !ata a#e collecte! ove# seve#al timea#e collecte! ove# seve#al time $e#io!s$e#io!s

E1am$leE1am$le !ata !etailing the num,e# o" ,uil!ing !ata !etailing the num,e# o" ,uil!ing $e#mits issue! in 6ucas County* Ohio in each o" $e#mits issue! in 6ucas County* Ohio in each o" the last H monthsthe last H months

E1am$leE1am$le !ata !etailing the num,e# o" ,uil!ing !ata !etailing the num,e# o" ,uil!ing $e#mits issue! in 6ucas County* Ohio in each o" $e#mits issue! in 6ucas County* Ohio in each o" the last H monthsthe last H months

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3E CAN

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Key De5nitions

• A $o$ulation .unive#se/ is the

collection o" things un!e#

consi!e#ation

• A sam$le is a $o#tion o" the $o$ulation

selecte! "o# analysis

• A $a#amete# is a summa#y measu#e

com$ute! to !esc#i,e a cha#acte#istic

o" the $o$ulation

• A statistic is a summa#y measu#e

com ute! to !esc#i,e a cha#acte#istic

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2o$ulation an! Sam$le

-o$ulation Sam$le

(se $a#amete#s to

summa#iLe "eatu#es

(se statistics tosumma#iLe "eatu#es

In"e#ence on the $o$ulation "#om the sam$

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Statistical Metho!s

Statistical-eto"s

Descri'tive

Statistics

In&erential

Statistics

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Desc#i$tive Statistics

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• Desc#i$tive statistics a#e theta,ula#* g#a$hical* an! nume#icalmetho!s use! to summa#iLe !ata

Descri'tive Statistics9Descri'tive Statistics9

Tese are statisticalTese are statistical

meto"s use" tometo"s use" to"escri%e "ata tat"escri%e "ata tat

ave %een collecte"$ave %een collecte"$


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Desc#i$tive Statistics 'a,ula# an! #a$hical

2#esentations• Summa#iLing Qualitative Data• Summa#iLing Quantitative Data

'y$es o" Data 'y$es o" Data 'y$es o" Data

Data

Numerical(Quantitative)

Categorical(Qualitative)

Discrete Continuous

Data

Numerical(Quantitative)

Categorical(Qualitative)

Discrete Continuous

E1am$le Ma#a!a Inn

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E1am$le Ma#a!a Inn

uests staying at Ma#a!a Inn %e#e

aske! to #ate the quality o" thei#accommo!ations as ,eing excellent *above average* average* below

average* o# poor 'he #atings $#ovi!e!,y a sam$le o" F guests a#e

Belo% Ave#ageBelo% Ave#age

A,ove Ave#ageA,ove Ave#age


Ave#ageAve#age


Ave#ageAve#age


Ave#ageAve#ageA,ove Ave#ageA,ove Ave#age


2oo#2oo#

E1cellentE1cellent


Ave#ageAve#age

A,ove Ave#ageA,ove Ave#age A,ove Ave#ageA,ove Ave#age


2oo#2oo#


Ave#ageAve#age

0#equency Dist#i,ution0#equency Dist#i,ution

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2oo#2oo#


Ave#ageAve#ageA,ove Ave#ageA,ove Ave#age

E1cellentE1cellent

PP

'otal 'otal F F

8ating8ating 0#equency0#equency

8elative 0#equency an!8elative 0#equency an!

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8elative 0#equency an!q y2e#cent 0#equency Dist#i,utions2e#cent 0#equency Dist#i,utions

2oo#2oo#


Ave#ageAve#age


E1cellentE1cellent

FF

FF

'otal 'otal FF FF

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8elative8elative

0#equency0#equency

2e#cent2e#cent

0#equency0#equency8ating8ating

F.FF/ F.FF/ FF

4F 4F FF

Ba# #a$hBa# #a$h

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2oo# Belo%Ave#age

Ave#age A,oveAve#age

E1cellent

( r e .

u en c y

Rating

Ba# #a$hBa# #a$h

H

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FMa#a!a Inn Quality 8atingsMa#a!a Inn Quality 8atings

2ie Cha#t2ie Cha#t

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Belo%Ave#age R

Ave#age

R

A,oveAve#age R

2oo#FR

E1cellent R

Ma#a!a InnMa#a!a Inn QualityQuality

8atings8atings

Ma#a!a InnMa#a!a Inn QualityQuality

8atings8atings

2ie Cha#t2ie Cha#t


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AA "#equency !ist#i,ution"#equency !ist#i,ution is a ta,ula# summa#y o" is a ta,ula# summa#y o" !ata sho%ing the "#equency .o# num,e#/ o" items!ata sho%ing the "#equency .o# num,e#/ o" items in each o" seve#al non)ove#la$$ing classesin each o" seve#al non)ove#la$$ing classes

AA "#equency !ist#i,ution"#equency !ist#i,ution is a ta,ula# summa#y o" is a ta,ula# summa#y o" !ata sho%ing the "#equency .o# num,e#/ o" items!ata sho%ing the "#equency .o# num,e#/ o" items in each o" seve#al non)ove#la$$ing classesin each o" seve#al non)ove#la$$ing classes

'he o,7ective is to 'he o,7ective is to $#ovi!e insights$#ovi!e insights a,out the !ataa,out the !ata that cannot ,e quickly o,taine! ,y looking only atthat cannot ,e quickly o,taine! ,y looking only at the o#iginal !atathe o#iginal !ata

'he o,7ective is to 'he o,7ective is to $#ovi!e insights$#ovi!e insights a,out the !ataa,out the !ata that cannot ,e quickly o,taine! ,y looking only atthat cannot ,e quickly o,taine! ,y looking only at the o#iginal !atathe o#iginal !ata


8elative 0#equency Dist#i,ution8elative 0#equency Dist#i,ution

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'he 'he #elative "#equency#elative "#equency o" a class is the "#action o#o" a class is the "#action o# $#o$o#tion o" the total num,e# o" !ata items$#o$o#tion o" the total num,e# o" !ata items ,elonging to the class,elonging to the class

'he 'he #elative "#equency#elative "#equency o" a class is the "#action o#o" a class is the "#action o# $#o$o#tion o" the total num,e# o" !ata items$#o$o#tion o" the total num,e# o" !ata items ,elonging to the class,elonging to the class

AA #elative "#equency !ist#i,ution#elative "#equency !ist#i,ution is a ta,ula#is a ta,ula# summa#y o" a set o" !ata sho%ing the #elativesumma#y o" a set o" !ata sho%ing the #elative "#equency "o# each class"#equency "o# each class

AA #elative "#equency !ist#i,ution#elative "#equency !ist#i,ution is a ta,ula#is a ta,ula# summa#y o" a set o" !ata sho%ing the #elativesumma#y o" a set o" !ata sho%ing the #elative "#equency "o# each class"#equency "o# each class

8elative 0#equency Dist#i,ution8elative 0#equency Dist#i,ution

e#cen #equency

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e#cen #equencyDist#i,ution

'he 'he $e#cent "#equency$e#cent "#equency o" a class is the #elativeo" a class is the #elative "#equency multi$lie! ,y FF"#equency multi$lie! ,y FF 'he 'he $e#cent "#equency$e#cent "#equency o" a class is the #elativeo" a class is the #elative "#equency multi$lie! ,y FF"#equency multi$lie! ,y FF

AA $e#cent "#equency !ist#i,ution$e#cent "#equency !ist#i,ution is a ta,ula#is a ta,ula#

summa#y o" a set o" !ata sho%ing the $e#centsumma#y o" a set o" !ata sho%ing the $e#cent "#equency "o# each class"#equency "o# each class

AA $e#cent "#equency !ist#i,ution$e#cent "#equency !ist#i,ution is a ta,ula#is a ta,ula#

summa#y o" a set o" !ata sho%ing the $e#centsumma#y o" a set o" !ata sho%ing the $e#cent "#equency "o# each class"#equency "o# each class

Ba# #a$h

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Ba# #a$h

AA ,a# g#a$h,a# g#a$h is a g#a$hical !evice "o# $#esentingis a g#a$hical !evice "o# $#esenting

qualitative !ataqualitative !ata On one a1is .usually the ho#iLontal a1is/* %e s$eci"yOn one a1is .usually the ho#iLontal a1is/* %e s$eci"y the la,els that a#e use! "o# each o" the classesthe la,els that a#e use! "o# each o" the classes

AA "#equency"#equency** #elative "#equency#elative "#equency* o#* o# $e#cent "#equency$e#cent "#equency

scale can ,e use! "o# the othe# a1is .usually thescale can ,e use! "o# the othe# a1is .usually the ve#tical a1is/ve#tical a1is/

'he 'he ,a#s a#e se$a#ate!,a#s a#e se$a#ate! to em$hasiLe the "act that eato em$hasiLe the "act that ea class is a se$a#ate catego#yclass is a se$a#ate catego#y

2ie Cha#t

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2ie Cha#t

'he 'he $ie cha#t$ie cha#t is a commonly use! g#a$hical !eviceis a commonly use! g#a$hical !evice

"o# $#esenting"o# $#esenting #elative "#equency !ist#i,utions "o##elative "#equency !ist#i,utions "o# qualitative !ataqualitative !ata

Since the#e a#e HF !eg#ees in a ci#cle*Since the#e a#e HF !eg#ees in a ci#cle*

a class %ith a #elative "#equency o"a class %ith a #elative "#equency o" %oul!%oul! consumeconsume .HF/ PF.HF/ PF !eg#ees o" the ci#cle!eg#ees o" the ci#cle

E1am$leE1am$le Ma#a!a InnMa#a!a Inn

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Insights aine! "#om the 2#ece!ing 2ie Cha#tInsights aine! "#om the 2#ece!ing 2ie Cha#t

E1am$leE1am$le Ma#a!a Inn Ma#a!a Inn

One)hal" o" the custome#s su#veye! gave Ma#a!One)hal" o" the custome#s su#veye! gave Ma#a! a quality #ating o" :a,ove ave#age; o# :e1cellena quality #ating o" :a,ove ave#age; o# :e1cellen .looking at the le"t si!e o" the $ie/ 'his might.looking at the le"t si!e o" the $ie/ 'his might $lease the manage#$lease the manage#

0o#0o# eacheach custome# %ho gave an :e1cellent; #aticustome# %ho gave an :e1cellent; #ati the#e %e#ethe#e %e#e t%ot%o custome#s %ho gave a :$oo#;custome#s %ho gave a :$oo#; #ating .looking at the to$ o" the $ie/ 'his shoul#ating .looking at the to$ o" the $ie/ 'his shoul !is$lease the manage#!is$lease the manage#

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A categorical variable has three categories with

the following frequencies of occurrence:

a$ Com'ute te 'ercentage o& values in eac categor$

%$ Construct a %ar cart$

c$ Construct a 'ie cart$

"$ Construct a *areto "iagram

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• Jou manage a team that sells com$ute#ha#!%a#e to so"t%a#e !evelo$ment com$aniesAt each com$any* you# #e$#esentatives have a$#ima#y contact Jou have catego#iLe! these

contacts ,y the !e$a#tment o" the com$any in%hich they %o#k .Development * ComputerServices* Finance* Other * Don't Know/

• .contactssav/

• (se 0#equencies to stu!y the !ist#i,ution o"!e$a#tments to see i" it meshes %ith you#goals

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• $#o!uces a "#equency ta,le an! $iecha#t "o# the va#ia,le dept

• 3hatUs the inte#$#etation=

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• At a glance* you see that the$lu#ality o" you# contacts %o#k in thecom$ute# se#vices !e$a#tments o"

thei# #es$ective com$anies* "ollo%e!,y those in the 5nancial an!!evelo$ment !e$a#tments

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Business Statistics

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2A8E'O 28INCI26E

• 'he -areto $rinci$le e/ists wente ma0ority o" items in a set o"data occur in a small num,e# o"

catego#ies an! the "e% #emainingitems a#e s$#ea! out ove# a la#genum,e# o" catego#ies 'hese t%o

g#ou$s a#e o"ten #e"e##e! to as thevital "e% an! the t#ivial many

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'he 2a#eto Diag#am

• In a -areto diagram! tecategori#ed res$onses are $lottedin descending order! according to

thei# "#equencies* an! a#e com,ine!%ith a cumulative $e#centage line onthe same cha#t

• 'he 2a#eto !iag#am can i!enti"ysituations in %hich the 2a#eto $#inci$leoccu#s

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• 'he 2a#eto !iag#am has the a,ility tose$a#ate the vital "e% "#om the t#ivialmany* ena,ling you to "ocus on theim$o#tant catego#ies

• In situations in %hich the !ata involve!consist o" !e"ective o# noncon"o#ming

items* the 2a#eto !iag#am is a $o%e#"ultool "o# $#io#itiLing im$#ovement e-o#ts

V

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V

• 3e have !ata "o# a la#ge in7ection)mol!ingcom$any that manu"actu#es $lasticmol!e! Com$onents use! in com$ute#key,oa#!s* %ashing machines*automo,iles* an! television sets

• 'he !ata $#esente! in 'a,le consist o" all

com$ute# key,oa#!s %ith !e"ects$#o!uce! !u#ing a th#ee)month $e#io!

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Using (re.uencies to Study

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Using (re.uencies to Study)rdinal 'ata

• In a!!ition to the !e$a#tment o" eachcontact* you have #eco#!e! thei#com$any #anks (se 0#equencies to

stu!y the !ist#i,ution o" com$any#anks to see i" it meshes %ith you#goals

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• 'he "#equency ta,le "o# o#!inal !atase#ves much the same $u#$ose asthe ta,le "o# nominal !ata 0o#

e1am$le* you can see "#om the ta,lethat R o" you# contacts a#e

7unio# manage#s

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•

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represent the root causes of pharmac errors at a hospital during a

recent time period:

2eason &or :ailure :re;uenc Additional instructions !"

#ose $%

#rug !&

#uplicate order entr $$

Frequenc &'(mission $!

(rder not discontinued when received !$

(rder not received )$

*atient )

+oute &

(ther

a$ Construct a *areto "iagram$

%$ Discuss te vital &e# an" trivial man reasons &or the root causes

of pharmac errors.

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Summa#iLing QuantitativeData

umma# L ng uan a veData

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gData

• 0#equency Dist#i,ution

• 8elative 0#equency an! 2e#cent0#equency Dist#i,utions

• Dot 2lot

•

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8e$ai# 'he manage# o"

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Sam$le o" 2a#ts Cost "o# F 'une)u$sSam$le o" 2a#ts Cost "o# F 'une)u$s

P E C G P D K C C K K N P P G F P C H N

C E H P C N G P H H C K C P C K C N C H

E F I C I H N H G P C E F K C C H K G F E F P

G K P C G G H G G D H G C E H P H C C IH N G N P G E F E C P E F K C P H P H N C D

P E C G P D K C C K K N P P G F P C H N

C E H P C N G P H H C K C P C K C N C H

E F I C I H N H G P C E F K C C H K G F E F P

G K P C G G H G G D H G C E H P H C C I

H N G N P G E F E C P E F K C P H P H N C D

0#equency Dist#i,ution 'a,le

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• ) Dete#mine #ange• ) Select num,e# o" classes

- Usually ,et%een an! F inclusive

• ) Com$ute class inte#vals .%i!th/

• ) Dete#mine class ,oun!a#ies .limits/

• ) Com$ute class mi!$oints

• H) Count o,se#vations & assign toclasses

Ste$s

0#equency Dist#i,ution

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q y

• ui!elines "o# Selecting Num,e# o"

Classes (se ,et%een(se ,et%een an!an! FF classesclasses Data sets %ith a la#ge# num,e# o" elementsData sets %ith a la#ge# num,e# o" elements usually #equi#e a la#ge# num,e# o" classesusually #equi#e a la#ge# num,e# o" classes

Smalle# !ata sets usually #equi#e "e%e# classesSmalle# !ata sets usually #equi#e "e%e# classes

0#equency Dist#i,ution

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q y.Continue!/• ui!elines "o# Selecting 3i!th o"

Classes

L a r g e s t D a t a V a l u e S m a l l e s t D a t a V a l u e

N u m b e r o f C l a s s e s

−

(se classes o" equal %i!th(se classes o" equal %i!thA$$#o1imate Class 3i!th A$$#o1imate Class 3i!th

E1am$ e 0#equencyDist#i,ution

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Dist#i,ution

0o#

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q yDist#i,utions

F)PF)P

HF)HPHF)HPF)PF)P

GF)GPGF)GP

PF)PPPF)PP

FF)FPFF)FP

2a#ts2a#ts

Cost ./Cost ./

FF

HH

FF

'otal FF 'otal FF

8elative8elative

0#equency0#equency

HH

FF

FFFF

2e#cent2e#cent

0#equency0#equency

4F4F F.FFF.FF//

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Anothe# common g#a$hical $#esentation o" Anothe# common g#a$hical $#esentation o"

quantitative !ataquantitative !ata is ais a histog#amhistog#am 'he va#ia,le o" inte#est is $lace! on the ho#iLontal 'he va#ia,le o" inte#est is $lace! on the ho#iLontal a1isa1is

A #ectangle is !#a%n a,ove each class inte#val %ithA #ectangle is !#a%n a,ove each class inte#val %ith

its height co##es$on!ing to the inte#val+sits height co##es$on!ing to the inte#val+s "#equenc"#equenc #elative "#equency#elative "#equency* o#* o# $e#cent "#equency$e#cent "#equency

(nlike a ,a# g#a$h* a histog#am has(nlike a ,a# g#a$h* a histog#am has no naturalno natural separation between rectanglesse paration between rectangles o" a!7acent classeso" a!7acent classes

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• Symmet#ic

- 6e"t tail is the mi##o# image o" the #ighttail

- E1am$le heights an! %eights o" $eo$le

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• Mo!e#ately Ske%e! 6e"t

- A longe# tail to the le"t

- E1am$le e1am sco#es

8 el at i v e0 # e q

u en c y F

F

F

F

F

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• Mo!e#ately 8ight Ske%e! - A 6onge# tail to the #ight

- E1am$le housing values

8 el at i v e0 # e q u en c y F

F

F

F

F

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•

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Cumulative "#equency !ist#i,utionCumulative "#equency !ist#i,ution −− sho%s thesho%s the num,e# o" items %ith values less than o# equal tonum,e# o" items %ith values less than o# equal to the u$$e# limit o" each classthe u$$e# limit o" each class

Cumulative "#equency !ist#i,utionCumulative "#equency !ist#i,ution −− sho%s thesho%s the num,e# o" items %ith values less than o# equal tonum,e# o" items %ith values less than o# equal to the u$$e# limit o" each classthe u$$e# limit o" each class

CumulativeCumulative

#elative#elative

"#equency !ist#i,ution"#equency !ist#i,ution

? sho%s? sho%s

the $#o$o#tion o" items %ith values less than o#the $#o$o#tion o" items %ith values less than o# equal to the u$$e# limit o" each classequal to the u$$e# limit o" each class

CumulativeCumulative #elative#elative "#equency !ist#i,ution"#equency !ist#i,ution ? sho%s? sho%s the $#o$o#tion o" items %ith values less than o#the $#o$o#tion o" items %ith values less than o# equal to the u$$e# limit o" each classequal to the u$$e# limit o" each class

CumulativeCumulative $e#cent$e#cent "#equency !ist#i,ution"#equency !ist#i,ution ? sho%s? sho%s

the $e#centage o" items %ith values less than o#the $e#centage o" items %ith values less than o# equal to the u$$e# limit o" each classequal to the u$$e# limit o" each class

CumulativeCumulative $e#cent$e#cent "#equency !ist#i,ution"#equency !ist#i,ution ? sho%s? sho%s

the $e#centage o" items %ith values less than o#the $e#centage o" items %ith values less than o# equal to the u$$e# limit o" each classequal to the u$$e# limit o" each class

Cumulative Dist#i,utions

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• E/am$le4

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AnAn ogiveogive is a g#a$h o" ais a g#a$h o" a cumulativecumulative

!ist#i,ution!ist#i,ution 'he !ata values a#e sho%n on the ho#iLontal 'he !ata values a#e sho%n on the ho#iLontala1isa1is Sho%n on the ve#tical a1is a#e theSho%n on the ve#tical a1is a#e the

cumulative "#equencies* o#cumulative "#equencies* o#

cumulative #elative "#equencies* o#cumulative #elative "#equencies* o#

cumulative $e#cent "#equenciescumulative $e#cent "#equencies

'he "#equency . 'he "#equency .one o" the a,oveone o" the a,ove/ o" each class/ o" each classis $lotte! as a $ointis $lotte! as a $oint

'he $lotte! $oints a#e connecte! ,y st#aight 'he $lotte! $oints a#e connecte! ,y st#aightlineslines

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Ogive %ithOgive %ith

C l ti 2 t 0 iCumulative 2e#cent 0#equencies

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2a#ts2a#tsCost ./Cost ./

F

F

HF

GF

FF C um ul at i v

e2 e# c ent 0 # e

q u en c y

C um ul at i v

e2 e# c ent 0 # e

q u en c y

F HF F GF PF FF FF HF F GF PF FF F

.GP* H/.GP* H/

Cumulative 2e#cent 0#equenciesCumulative 2e#cent 0#equencies

'une)u$ 2a#ts Cost 'une)u$ 2a#ts Cost 'une)u$ 2a#ts Cost 'une)u$ 2a#ts Cost

State! an! '#ue .o# 8eal/ Class 6imits

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'#ue Classes '#ue Classes A#e those classes such that theA#e those classes such that theu$$e# t#ue limit o" a class is the same as the lo%e#u$$e# t#ue limit o" a class is the same as the lo%e#t#ue limit o" the ne1t classt#ue limit o" the ne1t class 0o# com$a#ison* the state! class limits an! t#ue0o# com$a#ison* the state! class limits an! t#ueclass limits a#e given in the "ollo%ing ta,leYne1tclass limits a#e given in the "ollo%ing ta,leYne1tsli!esli!e

. /

State! an! '#ue .o# 8eal/ Class 6imits

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State!State! '#ue '#ueHFF)PPHFF)PP PPF u$ to ,ut not inclu!ingPPF u$ to ,ut not inclu!ingPPFPPFGFF)PPPGFF)PPP PPF u$ to ,ut not inclu!ingPPF u$ to ,ut not inclu!ing

PPPFPPPF

In the 5#st column o" the a,ove ta,le the !ata %e#eIn the 5#st column o" the a,ove ta,le the !ata %e#e#oun!e! to the nea#est !olla# 0o# e1am$le*#oun!e! to the nea#est !olla# 0o# e1am$le* PPFPPF

%as #oun!e! u$ to%as #oun!e! u$ to

GFFGFF

an! taile! in the secon!an! taile! in the secon!

class Any amount ove#class Any amount ove# PPPP ,ut un!e#,ut un!e# PPFPPF %as%as#oun!e! !o%n to#oun!e! !o%n to PPPP an! inclu!e! in the 5#st classan! inclu!e! in the 5#st class

'hus* the 'hus* the HFF)PPHFF)PP class actually inclu!es all !ataclass actually inclu!es all !ata"#om"#om PPFPPF inclusive u$ to ,ut not inclu!inginclusive u$ to ,ut not inclu!ing

PPFPPF

. /

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Dot 2lot

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• One o" the sim$lest g#a$hicalsumma#ies o" !ata is a !ot $lot

• A ho#iLontal a1is sho%s the #angeo" !ata values

• 'hen each !ata value is#e$#esente! ,y a !ot $lace! a,ovethe a1is

Dot 2lot

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FF HFHF FF GFGF PFPF FFFF

Cost ./

'une)u$ 2a#ts Cost 'une)u$ 2a#ts Cost

+ + + +++++ ++++++++++ ++ + ++ + + +

++ ++ + + ++ ++ ++ ++ ++ +

Stem an! 6ea" !iag#am

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Stem an! 6ea" !iag#am

• A stem;and;lea" dis$layorgani#es data into grou$s1called stems3 so tat te &alues

witin each g#ou$ .the leaves/,#anch out to the #ight on each #o%

'he #esulting !is$lay allo%s you to

see ho% the !ata a#e !ist#i,ute! an!%he#e concent#ations o" !ata e1ist

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• su$$ose that stu!ents "#om you#class eat lunch at a "ast)"oo!#estau#ant 'he "ollo%ing !ata a#e

the amounts s$ent "o# lunch

• F* F* GF* F* F* GF*

HF* GF* PF* PF* F* F*PF* HF* HHF

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• 'o "o#m the stem)an!)lea" !is$lay*you use the units as the stems an!#oun! the !ecimals

• .the leaves/ to one !ecimal $lace• 0o# e1am$le* the 5#st value is F

Its stem .#o%/ is * an! its lea" is

• 'he secon! value* is F Its stem.#o%/ is * an! its lea" is

V

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C8OSS 'AB(6A'IONS

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C8OSS 'AB(6A'IONS

• 'he stu!y o" $atte#ns that may e1ist,et%een t%o o# mo#e catego#icalva#ia,les is common in ,usiness

'hese $atte#ns a#e e1$laine! ,yc#oss)ta,ulating the !ata

• Jou can $#esent crosstabulations

in tabular "orm 1contingencytables3 or gra$ical "orm

'he Contingency 'a,le

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'he Contingency 'a,le

• A contingency table $resents teresults o" two categorical&ariables+ Te 0oint res$onses

areclassi5e! so that the catego#ies o"one va#ia,le a#e locate! in the #o%san! the catego#ies o" the othe# va#ia,lea#e locate! in the columns 'he values

locate! at the inte#sections o" the #o%san! columns a#e calle! cells+

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• 'e$ending on te ty$e o"contingency table constructed!te cells "or each #o% column

com,ination contain the "#equency*the $e#centage o" the ove#all total*the $e#centage o" the #o% total* o#

the $e#centage o" the column total

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• Mutual"un!s 1ls• C#oss ta,ulation o" catego#y vs

o,7ective

• C#oss ta,ulation o" o,7ective vs #isk

'he Scatte# 2lot

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'he Scatte# 2lot

• Jou use a scatter $lot to e/amine$ossible relationsi$s betweentwo numerical &ariables

• 0o# each o,se#vation* you $lot oneva#ia,le on the X ais and the other

variable on the vertical ! ais

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• the cost o" a "ast)"oo! ham,u#ge#meal an! the cost o" t%o movietickets in F cities a#oun! the %o#l!

• 'he !ata 5le Cost o" 6iving1lscontains the com$lete !ata set

• a+ Construct si/ se$arate scatter $lots+(or eac! use te ove#all cost in!e1 as the !

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(or eac! use te ove#all cost in!e1 as the !ais" Use the monthly rent "o# a t%o),e!#oom

a$a#tment* the costs o" a cu$ o" co-ee %ithse#vice* a "ast)"oo! ham,u#ge# meal* !#ycleaning a men s ,laLe#* tooth$aste* an! movietickets as the X ais"

• b+ ,at conclusions can you reac aboutte relationsi$ o" the ove#all cost in!e1 tothese si1 va#ia,les=

Statistical Metho!s

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Statistical-eto"s

Descri'tive

Statistics

In&erential

Statistics


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• Desc#i$tive statistics a#e the

ta,ula#* g#a$hical* an! nume#icalmetho!s use! to summa#iLe !ata

Descri'tive Statistics9Descri'tive Statistics9

Tese are statisticalTese are statistical

meto"s use" tometo"s use" to"escri%e "ata tat"escri%e "ata tat

ave %een collecte"$ave %een collecte"$


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Nume#ical Measu#es

• Measu#es o" 6ocation• Measu#es o" 9a#ia,ility

Measu#es o" 6ocation

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I" the measu#es a#e com$ute!I" the measu#es a#e com$ute! "o# !ata "#om a sam$le*"o# !ata "#om a sam$le*they a#e calle!they a#e calle! sam$le statisticssam$le statistics

I" the measu#es a#e com$ute!I" the measu#es a#e com$ute!

"o# !ata "#om a $o$ulation*"o# !ata "#om a $o$ulation*they a#e calle!they a#e calle! $o$ulation $a#amete#$o$ulation $a#amete#

A sam$le statistic is #e"e##e! toA sam$le statistic is #e"e##e! to

as theas the $oint estimato#$oint estimato# o" theo" theco##es$on!ing $o$ulation $a#amete#co##es$on!ing $o$ulation $a#amete#0o# e1am$le*0o# e1am$le* thethe sam$le mean is asam$le mean is a$oint estimato# o" the $o$ulation mean$oint estimato# o" the $o$ulation mean

MeanMean

Me!ianMe!ian Mo!eMo!e

2e#centiles2e#centiles

Qua#tilesQua#tiles

Mean

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• 'he mean o" a !ata set is the

ave#age o" all the !ata values• As %e sai!* the sam$le mean is

the $oint estimato# o" the

$o$ulation mean µ

Sam$le MeanSam$le Mean

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Num,e# o" Num,e# o" o,se#vationso,se#vations

in the sam$lein the sam$le

Num,e# o" Num,e# o" o,se#vationso,se#vations

in the sam$lein the sam$le

Sum o" the valuesSum o" the valueso" theo" the nn o,se#vationso,se#vationsSum o" the valuesSum o" the valueso" theo" the nn o,se#vationso,se#vations

i

n

=∑

2o$ulation Mean2o$ulation Mean µ µ

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Num,e# o" Num,e# o" o,se#vations ino,se#vations in

the $o$ulationthe $o$ulation

Num,e# o" Num,e# o" o,se#vations ino,se#vations in

the $o$ulationthe $o$ulation

Sum o" the valuesSum o" the valueso" theo" the ## o,se#vationso,se#vationsSum o" the valuesSum o" the valueso" theo" the ## o,se#vationso,se#vations

i

#

µ =∑

Sam$le MeanSam$le Mean

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Seventy e@ciency a$a#tmentsSeventy e@ciency a$a#tments%e#e #an!omly sam$le! in%e#e #an!omly sam$le! in

a small college to%n 'hea small college to%n 'he

monthly #ent $#ices "o#monthly #ent $#ices "o#

these a$a#tments a#e liste!these a$a#tments a#e liste!

in ascen!ing o#!e# on the ne1t sli!ein ascen!ing o#!e# on the ne1t sli!e

E/am$le4E/am$le4 A$a#tment 8ents A$a#tment 8ents

Sam$le MeanSam$le MeanE1am$le Continue!E1am$le Continue!

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I N K I D F I D F I D K I D K I D K I D K I D K I I F I I F

I I F I I F I I F I I K I I K I I K I I K I I K I K F I K F

I K F I K F I K F I K F I K F I H F I H F I H F I H K I H K

I H K I C F I C F I C N I C K I C K I C K I G F I G F I G F

I G F I G K I P F I P F I P F K F F K F F K F F K F F K E F

K E F K E K K N K K N K K N K K D K K I P K K F K C F K C F

K C K K C K K G F K P F H F F H F F H F F H F F H E K H E K








*ontly Rent "or :

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D I * D K H I P F & G F

C F

i

n

= = =∑















*ontly Rent "or :

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Data

i" a#e the mi!)values an!

a#e the co##es$on!ing

"#equencies* %he#e the su,sc#i$t Z$ +stan!s "o# the num,e# o" classes*then the mean is

∑∑= iii

f z f z

k z z z z .,,.........,, 321

k f f f f ,........,,, 321

eomet#ic Mean

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eomet#ic mean is !e5ne! as the $ositive #oot o" the$#o!uct o" o,se#vations Sym,olically*

It is also o"ten use! "o# a set o" num,e#s %hose valuesa#e meant to ,e multi$lie! togethe# o# a#e e1$onentialin natu#e* such as !ata on the g#o%th o" the human

$o$ulation o# inte#est #ates o" a 5nancial investment

0in! geomet#ic mean o" #ate o" g#o%th * * * **

n

n x x x xG

/1

321 )(

=

3eighte! Mean

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3eighte! Mean

'he 3eighte! mean o" the $ositive #eal

num,e#s 1E*1N* * 1n %ith thei# %eight %E*%N*

* %n is !e5ne! to ,e

∑

∑

=

=

= n

i

i

n

i

ii

w

xw

x

1

1

2otential 2#o,lem %ithMeans

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Means

Sam$le mean gives equal %eight to allmeasu#ements

Outliers can have a la#ge in>uence on

the com$ute! mean valueDisto#ts ou# intuition a,out the central

tendency o" the measu#e! values

2otential 2#o,lem %ithMeans

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Means

Mean

Mean

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Me!ianMe!ian

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PP HH GG

0o# an0o# an o!! num,e#o!! num,e# o" o,se#vationso" o,se#vations

in ascen!ing o#!e#in ascen!ing o#!e#

HH GG PP o,se#vations o,se#vations

the me!ian is the mi!!le valuethe me!ian is the mi!!le value

Me!ian PMe!ian P

Me!ianMe!ian

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PP HH GG

0o# an0o# an even num,e#even num,e# o" o,se#vationso" o,se#vations

in ascen!ing o#!e#in ascen!ing o#!e#

HH GG FF G o,se#vationsG o,se#vations

the me!ian is the ave#age o" the mi!!le t%o valuesthe me!ian is the ave#age o" the mi!!le t%o values

Me!ian .P X H/4 Me!ian .P X H/4

PP

FF

Me!ian E1am$le

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Ave#aging the th an! Hth !ata valuesAve#aging the th an! Hth !ata values

Me!ian . X /4 Me!ian . X /4

*ontly Rent "or :

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6F 6o%e# class ,oun!a#y o" the me!ianclass

h 3i!th o" the me!ian class

" F 0#equency o" the me!ian class

0 Cumulative "#equency o" the $#e)me!ian class

−+= F n

f

h L M

o

oe2

E1am$le)0in! Me!ian

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E1am$le)0in! Me!ian

Age in ears umber of births /umulative number ofbirths

!&.)0!1.) "'' "''

!1.)0$&.) !12 $))

$&.)0$1.) !'%' &%%$

$1.)0%&.) !2&2 )%"$

%&.)0%1.) $1& )")"

%1.)0&&.) 1! )'&'&&.)0&1.) !" )'"%

All ages )'"% 0

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Mo!e

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'he 'he mo!emo!e o" a !ata set is the value that occu#s %ito" a !ata set is the value that occu#s %it

greatest %re&uency greatest %re&uency 'he g#eatest "#equency can occu# at t%o o# mo#e 'he g#eatest "#equency can occu# at t%o o# mo#e !i-e#ent values!i-e#ent values

I" the !ata have e1actly t%o mo!es* the !ata a#eI" the !ata have e1actly t%o mo!es* the !ata a#e

,imo!al,imo!al I" the !ata have mo#e than t%o mo!es* the !ata a#I" the !ata have mo#e than t%o mo!es* the !ata a# multimo!almultimo!al

Mo!e E1am$le

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INK IDF IDF IDK IDK IDK IDK IDK IIF IIF

IIF IIF IIF IIK IIK IIK IIK IIK IKF IKF

IKF IKF IKF IKF IKF IHF IHF IHF IHK IHK

IHK ICF ICF ICN ICK ICK ICK IGF IGF IGF

IGF IGK IPF IPF IPF KFF KFF KFF KFF KEF

KEF KEK KNK KNK KNK KDK KIP KKF KCF KCF

KCK KCK KGF KPF HFF HFF HFF HFF HEK HEK








F occu##e! most "#equently . times/F occu##e! most "#equently . times/

Mo!e FMo!e F

*ontly Rent "or :

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No Mo!e

8a% Data F P GP H One Mo!e8a% Data HF PHF GP H P

P

Mo#e 'han Mo!e8a% Data G G

(se E1cel to Com$ute(se E1cel to Com$utethe Mean* Me!ian* an! Mo!ethe Mean* Me!ian* an! Mo!e

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o" the 0ollo%ing Data an! E1$lain theo" the 0ollo%ing Data an! E1$lain theAns%e#sAns%e#s















Mean* Me!ian* o# Mo!e=

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MeanI" the sum o" all values is meaning"ul

Inco#$o#ates all availa,le in"o#mation

Me!ianIntuitive sense o" cent#al ten!ency %ith

outlie#s

3hat is :ty$ical; o" a set o" values=

Mo!e3hen !ata can ,e g#ou$e! into !istinct

ty$es* catego#ies .categorical data/

3hich mean to use=

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Mean value must still con"o#m tocha#acte#istics o" a good $e#"o#mancemet#ic

6inea#

8elia,le8e$eata,le

Easy to use

Consistent

In!e$en!entBest measu#e o" $e#"o#mance still is

eecution time

2e#centiles

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A $e#centileA $e#centile $#ovi!es in"o#mation a,out ho% the$#ovi!es in"o#mation a,out ho% the

!ata a#e s$#ea! ove# the inte#val!ata a#e s$#ea! ove# the inte#val "#om the smalles"#om the smalles value to the la#gest valuevalue to the la#gest value

A!mission test sco#es "o# colleges an! unive#sitiesA!mission test sco#es "o# colleges an! unive#sities a#e "#equently #e$o#te! in te#ms o" $e#centilesa#e "#equently #e$o#te! in te#ms o" $e#centiles

Jou a#e "amilia# %ith Jou a#e "amilia# %ith $e#centile$e#centile sco#e o" nationalsco#e o" nationale!ucational tests such as MA'* an! SA'* %hiche!ucational tests such as MA'* an! SA'* %hich telltellyou %he#e you stan! in com$a#ison %ith othe#syou %he#e you stan! in com$a#ison %ith othe#s

0o# e1am$le* i" you a#e in the Gth $e#centile*0o# e1am$le* i" you a#e in the Gth $e#centile*

then GR o" the test)take#s sco#e! ,elo% you an!then GR o" the test)take#s sco#e! ,elo% you an!you a#e in the to$ R o" the test take#syou a#e in the to$ R o" the test take#s

'h th til " ! t t i

e#cent esDe5nition

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• 'he pth $e#centile o" a !ata set is a

value such that at least p $e#cent o"the items take on this value o# less an! at least .FF ) p/ $e#cent o" the

items take on this value o# mo#e

Ste$s "o# 0in!ing 2e#centilesSte$s "o# 0in!ing 2e#centiles

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A##ange the !ata in ascen!ing o#!e#A##ange the !ata in ascen!ing o#!e#

Com$ute in!e1Com$ute in!e1 ii* the* the $osition$osition o" theo" the p pthth $e#centile$e#centile

ii . . p p4FF/4FF/nn

I" I" ii is notis not an intege#*an intege#* #oun! u$#oun! u$ 'he 'he p p thth $e#centile$e#centile is the value in theis the value in the ii thth $osition$osition

I"I" ii is an intege#* theis an intege#* the p p thth $e#centile is the ave#age$e#centile is the ave#age

o" the values in $ositionso" the values in $ositions ii an!an! ii XX

GFth 2e#centile E1am$le

ii ..p4FF/4FF/n .GF4FF/F H.GF4FF/F H

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ii . . p p4FF/4FF/nn .GF4FF/F H .GF4FF/F H

Ave#aging the HAve#aging the Hthth

an! an! thth

!ata values!ata valuesGFth 2e#centile . X P/4 GFth 2e#centile . X P/4

Note Data is in ascen!ing o#!e#Note Data is in ascen!ing o#!e#



IKF IKF IKF IKF IKF IHF IHF IHF IHK IHKIHK ICF ICF ICN ICK ICK ICK IGF IGF IGF




GFGFthth 2e#centile2e#centile E1am$le Continue!E1am$le Continue!

::At least GFRAt least GFR ::At least FRAt least FR

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At least GFRAt least GFR o" the itemso" the items

take on a valuetake on a value o" o# less;o" o# less;

At least FRAt least FR o" the itemso" the items

take on a valuetake on a value o" o# mo#e;o" o# mo#e;

H4F G o# GFRH4F G o# GFR 4F o# FR4F o# FR

INK IDF IDF IDK IDK IDK IDK IDK IIF IIFIIF IIF IIF IIK IIK IIK IIK IIK IKF IKF






(se E1cel to 0in! GF(se E1cel to 0in! GFthth 2e#centile2e#centile

E l 0 l 3 k h tE l 0 l 3 k h t GFGFthth til$e#centile

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A B C D E

7

A'art

ment

-ontl

2ent (

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E1cel 9alue 3o#ksheetE1cel 9alue 3o#ksheet

A B C D E

7

A'art

ment

-ontl

2ent (

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E1cel 0o#mula 3o#ksheetE1cel 0o#mula 3o#ksheet

'hi#! Qua#tile 'hi#! Qua#tile

!! iltil

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A B C D E

7

A'art

ment

-ontl

2ent (

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E1cel 9alue 3o#ksheetE1cel 9alue 3o#ksheet

A B C D E

7

A'art

ment

-ontl

2ent (

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At $resent only about = > o" atlas com$uterdo not make it to warranty $eriod+Currently Su$$lier ? su$$lies ci$ 1maincom$onent o" com$uter3+

Two su$$liers o" ci$ are tere ? @ +

Consultant said to take a sam$le o" 6< ci$s"rom eac sam$le and com$are te mean

li"e+

Results are a&erage li"e o" ? ci$s is 6<years and a&erage li"e o" ci$s is 5< years+Consultant orders to sto$ su$$ly "rom ? and

A"te# s%itching to su$$lie# J

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A"te# s%itching to su$$lie# J

mo#e then FR o" com$ute#ssta#te! "ailing ,e"o#e %a##anty$e#io!

3hat ha$$ene!=

Su$$lie# chi$s* n F

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F* F* F* F* P* * F* * P* F

Mean F yea#s* #ange yea#s*an! stan!a#! !eviation FH yea#s

Su$$lie# J chi$s* nF .li"e in yea#s/F* * * * F* * P* * G* Mean F yea#s* #ange G yea#s*an! stan!a#! !eviation FF yea#s

Te Ne# ?or! Cit Transit Autorit as @ust 'urcase" a %ran"

ne# t'e o& &luorescent ligt %ul%$

T % l% li& & 6 7= it t " "

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Tese %ul%s ave an average li&e o& 6$7= ears #it a stan"ar"

"eviation o& =$==$

Consultant &or te Transit Autorit, recommen"s tat on one

Satur"a evening %et#een mi"nigt an" 6 a$m$, #en te num%er

o& ri"ers is small, all te %ul%s in te su%#a sstem soul" %e

re'lace"$

A #or!er &or te sstem suggest re'lacing 68 o& te %ul%s eac

Satur"a evening &or te ne+t t#ent #ee!s until all te %ul%s are

re'lace"$

Consultant insists tat it is cea'er an" more e&&icient to re'lace allte %ul%s at once$ Te Transit Autorit listens to Consultant$

Set A * * * *

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Set B * * * *Set C * * * *

All th#ee sets have ave#age o"

Although all sets a#e !i-e#ent3e nee! measu#es %hich account "o#such !is$e#sion o" !ata

'he %o#! !is$e#sion is use! to !enote the

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!eg#ee o" hete#ogeneity in the !ata

It is an im$o#tant cha#acte#istic in!icatingthe e1tent to %hich o,se#vations va#y

amongst themselves

A measu#e o" !is$e#sion is !esigne! to statenume#ically the e1tent to %hich in!ivi!ual

o,se#vations va#y on the ave#age

easu#es o a# a y.Dis$e#sion/ It i "t ! i ,l t i! "It is o"ten !esi#a,le to consi!e# measu#es o" i ,ia#ia,il

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It is o"ten !esi#a,le to consi!e# measu#es o"It is o"ten !esi#a,le to consi!e# measu#es o" va#ia,iva#ia,il

.!is$e#sion/*.!is$e#sion/* as %ell as measu#es o" locationas %ell as measu#es o" location 0o# e1am$le* in choosing su$$lie#0o# e1am$le* in choosing su$$lie# AA o# su$$lie#o# su$$lie# BB %% might consi!e# not only themight consi!e# not only the average delivery ti average delivery ti each* ,ut also theeach* ,ut also the variability in delivery timevariability in delivery time "o"o

easu#es o a# a ty.Dis$e#sion/ 8ange8ange

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8ange8ange

Inte#qua#tile 8ange o#Inte#qua#tile 8ange o#Mi!s$#ea!Mi!s$#ea! 9a#iance9a#iance

Stan!a#! DeviationStan!a#! Deviation

Coe@cient o" 9a#iationCoe@cient o" 9a#iation

8ange

'he'he #ange#a

nge o" a !ata set is the !i-e#ence ,et%een to" a !ata set is the !i-e#ence ,et%een

th

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'he 'he #ange#ange o" a !ata set is the !i-e#ence ,et%een to" a !ata set is the !i-e#ence ,et%een th la#gest an! smallest !ata valuesla#gest an! smallest !ata values

It is theIt is the sim$lest measu#esim$lest measu#e o" va#ia,ilityo" va#ia,ility

It isIt is ve#y sensitiveve#y sensitive to the smallest an! la#gest !atato the smallest an! la#gest !ata valuesvalues

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n e#qua# e ange o# Mi!s$#ea! 'he'he inte#qua#tile #angeinte#

qua#tile #ange o" a !ata set is the !i-e#eno" a !ata set is the !i-e#en

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'he 'he inte#qua#tile #angeinte#qua#tile #ange o" a !ata set is the !i-e#eno" a !ata set is the !i-e#en ,et%een the,et%een the third quartilethird quartile an! thean! the frst quartil frst quartil

It is the #ange "o# theIt is the #ange "o# the mi!!le FRmi!!le FR o" the !atao" the !ata It ove#comes the sensitivity to e1t#eme !ata valueIt ove#comes the sensitivity to e1t#eme !ata value not e-ecte! ,y the e1t#eme valuesnot e-ecte! ,y the e1t#eme values

I n t e r ; u a r t i l e 2 a n g e = Q Q5 7

n e#qua# e angeE1am$le

#! Qua#tile .#! Qua#tile .++/ /

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#! Qua#tile .#! Qua#tile .++/ /

st Qua#tile .st Qua#tile .++/ /

Inte#qua#tile 8ange Inte#qua#tile 8ange ++ ) ) ++ ) GF ) GF

*ontly Rent "or :

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An average distance that a score deviates from the mean

Set A * * * *

Set B * * * *

Set C

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9a#iance9a#iance

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'he va#iance is com$ute! as "ollo%s 'he va#iance is com$ute! as "ollo%s

'he va#iance is the 'he va#iance is the ave#age o" the squa#e!ave#age o" the squa#e!

!i-e#ences!i-e#ences ,et%een each !ata value an! the mean,et%een each !ata value an! the mean

"o# a"o# asam$lesam$le

"o# a"o# a$o$ulation$o$ulation

σ µ 2

2

= −∑ ( ) x

N i s

x

i

x

n

2

2

1= −∑ −( )

a# ance o# #ou$eData 0o# sam$le !ata0o# sam$le !ata

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$$

0o# $o$ulation !ata0o# $o$ulation !ata

( )2

2

n

x x f ii s ∑

−=

( ) N

x f ii2

2 ∑ −= µ

σ


'h'h t ! ! ! i tit ! ! ! i ti " ! t t i th iti" ! t t i th iti

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'he 'he stan!a#! !eviationstan!a#! !eviation o" a !ata set is the $ositiveo" a !ata set is the $ositive

squa#e #oot o" the va#iancesqua#e #oot o" the va#iance

It is measu#e! in theIt is measu#e! in the same units as the !atasame units as the !ata** makinmakin it mo#e easily inte#$#ete! than the va#ianceit mo#e easily inte#$#ete! than the va#iance

'h t ! ! ! i ti i t ! " ll'he stan!a#! !e iation is com$ute! as "ollo s


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'he stan!a#! !eviation is com$ute! as "ollo%s 'he stan!a#! !eviation is com$ute! as "ollo%s

"o# a"o# a

sam$lesam$le

"o# a"o# a

$o$ulation$o$ulation

s s= 2 σ σ = 2

Coe@cient o" 9a#iation

'h'he @ i t " i ticoe@cient o" va#iation i !i t h l thin!icates ho% la#ge the

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'he coe@cient o" va#iation is com$ute! as "ollo%s 'he coe@cient o" va#iation is com$ute! as "ollo%s

1 0 0 %

s

x

×

'he 'he coe@cient o" va#iationcoe@cient o" va#iation in!icates ho% la#ge thein!icates ho% la#ge the

stan!a#! !eviation is in #elation to the meanstan!a#! !eviation is in #elation to the mean

"o# a"o# asam$lesam$le

"o# a"o# a$o$ulation$o$ulation

1 0 0 %

σ

µ

×

Measu#e o" relative !is$e#sion

oe c en o a# a on.Continue!/

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• Measu#e o" relative !is$e#sion

• Al%ays a R• C9 is the stan!a#! !eviation

e1$#esse! as $e#cent o" the mean

• (se! to com$a#e t%o o# mo#eg#ou$s

• 3eakness C9 is un!e5ne! i" the

mean is Le#o o# i" !ata a#e negative• 'hus* C9 is use! only "o# va#ia,les

%hose values a#e [F

E1am$le Continue!E1am$le Continue!

iven the "ollo%ing monthly #ent $#ices "o# F a$a#tments 5n!iven the "ollo%ing monthly #ent $#ices "o# F a$a#tments 5n!

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I I F I I F I I F I I K I I K I I K I I K I I K I K F I K FI K F I K F I K F I K F I K F I H F I H F I H F I H K I H K



K E F K E K K N K K N K K N K K D K K I P K K F K C F K C FK C K K C K K G F K P F H F F H F F H F F H F F H E K H E K


I I F I I F I I F I I K I I K I I K I I K I I K I K F I K FI K F I K F I K F I K F I K F I H F I H F I H F I H K I H K



K E F K E K K N K K N K K N K K D K K I P K K F K C F K C FK C K K C K K G F K P F H F F H F F H F F H F F H E K H E K

*ontly Rent "or :< A$artments*ontly Rent "or :< A$artments

iven the "ollo%ing monthly #ent $#ices "o# F a$a#tments* 5n!iven the "ollo%ing monthly #ent $#ices "o# F a$a#tments* 5n!va#iance* stan!a#! !eviation* an! the coe@cient o" va#iationva#iance* stan!a#! !eviation* an! the coe@cient o" va#iation

use equations & E1celuse equations & E1cel

9a#iance9a#iance

SolutionsSolutions

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× = × = K I &C IE F F R E F F R E E & E K R

I P F &G Fs

N

N . / N * P P H & E HE

i sn

−= =−∑

NN P P H & I C K I & C Is s= = =

thethe

stan!a#!stan!a#!!eviation is!eviation isa,out Ra,out Ro" o" o" theo" themeanmean


Coe@cient o" 9a#iationCoe@cient o" 9a#iation

Note thatNote that C9 is the stan!a#! !eviation e1$#esse! asC9 is the stan!a#! !eviation e1$#esse! as$e#cent o" the mean$e#cent o" the mean

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A 2#o,lem (sing E1cel• A private research

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organi=ation studing

families in various

countries reported

the following data for

the amount of time &0

ear old children

spent alone with their

fathers each da.

Coutr! "#me $#t% Da&(m#utes)

'elg#um 30

Caa&a

C%#a

*#la& 0

A 2#o,lem (sing E1cel.Continue!/

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• (se E1cel* ans%e# the "ollo%ing questions an! e1$lainyou# ans%e#s .#oun! all num,e#s into t%o !ecimal$laces/

- A 'he mean

- B 'he mo!e - C 'he me!ian

- D 'he th $e#centile - E 'he 5#st an! the thi#! qua#tile

- 0 'he #ange

- 'he inte#qua#tile #ange o# mi!s$#ea! -

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Statistical analysis ty$ically involves %o#king %ithStatistical analysis ty$ically involves %o#king %ith

la#ge amounts o" !atala#ge amounts o" !ata Com$ute# so"t%a#eCom$ute# so"t%a#e is ty$ically use! to con!uct theis ty$ically use! to con!uct the analysisanalysis

0#equently the !ata that is to ,e analyLe! #esi!es i0#equently the !ata that is to ,e analyLe! #esi!es i

s$#ea!sheets$#ea!sheet Mo!e#n s$#ea!sheet $ackages a#e ca$a,le o" !ataMo!e#n s$#ea!sheet $ackages a#e ca$a,le o" !ata management* analysis* an! $#esentationmanagement* analysis* an! $#esentation

MS E1celMS E1cel is the most %i!ely availa,le s$#ea!sheetis the most %i!ely availa,le s$#ea!sheet

so"t%a#e in ,usiness o#ganiLationsso"t%a#e in ,usiness o#ganiLations

Statistical Analysis (sing Mic#oso"t E1celStatistical Analysis (sing Mic#oso"t E1cel

tasks might ,e tasks might ,enee!e!nee!e! Ente# DataEnte# Data

A

7

*arts

Cost

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nee!e!nee!e! Ente# DataEnte# Data

Ente# 0unctions an! 0o#mulasEnte# 0unctions an! 0o#mulas A$$ly 'oolsA$$ly 'ools

7 Cost

/ 1!5 '!

> !2&

6 )

0 "$

'

4 "1

D E

-ean 3A>4+A;46A$:A'!8

-e"ian 3M4#IA.6A$:A'!8

-o"e 3M(#46A$:A'!8

2ange 3MA?6A$:A'!80MI.6A$:A'!8

E1cel 3o#ksheetE1cel 3o#ksheet .sho%ing !ata.sho%ing !ata//

Statistical Analysis (sing Mic#oso"t E1celStatistical Analysis (sing Mic#oso"t E1cel

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Note 8o%s F) a#e not sho%nNote 8o%s F) a#e not sho%n

A B C D

7 Customer Invoice

*arts

Cost (

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Note Columns A)B an! #o%s F) a#e not sho%nNote Columns A)B an! #o%s F) a#e not sho%n

C D E :

7

*arts

Cost (

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C D E :

7

*arts

Cost ( !2& !1$

6 ) !'

0 "$ $&$

' !&

4 "1 !")

'& !12

Documents

intro ba