finance - statistics and econometrics (schaum's outline)

LL The Nature of Statistics 12 Statstis and Econometrics 13 The Methodology of Econometrics Descriptive Statistics LA Frequency Distributions 22° Measures of Central Tendency 23° Measur

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USE With tHESE GGUFSES ‘A Taiisiies and fcanemeines LÝ Sixisicai Wetnods in économies

unniatveKetnds in Ecommies (Mathematical Ecoamies (W! Were-Ecnanis

A wacre-Eeounies (2 Wath fr Economists Math fr Socal Sciences

Theory and Problems of

STATISTICS AND ECONOMETRICS

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‘TERMS OF USE

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[AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RISULTS TO BE OSTAINED FROM USING THE (Ont, AND EAPRESSLY DSCIAIM ANY WARRANTY EXPRESS OR DIPLIED INCLUDING BUT NOT

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‘This book presents clear and concise introduction to statistics and econometrics course in statistics

‘or esonometncs softs one ofthe most useful but also one ofthe most dificult {im colleges and univeraies, The purpose of this book is to help overcome this dificult by vung x of the required courses problem-solving approach ach chapter begins with a statement of theory, principles, or background information, fully ile strated with examples This followed by numerous theoretical and practical problems with detailed Stepby-step solutions While primarily intended asa supplement to al current standard textbooks of

‘Setiis endfor cooncmettien, the nok fam ean be wed ev on tadependca eros well as toeapyiment sass ctu

"The book i aimod at college students in economics, business administration, and the social sciences taking a one-tomester or a one-year course in statistics and/or cconometic It also provides a very sseful source of reference for MLA and MLBLA students and forall those who use (oF WOUML ike to wie) Statistics and econometrics in thei work No prior statistical background is assumed,

“The book is completely selfcontainad in that it covers the statistics (Chaps 1 0 5) required for

‘conometrics (Chaps 610 11) Itis applied in nature, and all proofs appear inthe problems section

‘ather than inthe txt ial Real-world socioeconomic and busines data are wie, whenever possible, {to demonstrate the more advanood econometric techniques and models Several sourcss of online data fare sod, and Web addestes are given forthe studeat’s and rescarcher’s farther use (App 12) Topics froqucntiy encountred in econometrics, such as multcallncarity and autocorrelation, are cleafly and concisely discussed as tothe problems they create, the methods to tet for thee presence, and possible jon techniques In ths second edition, we have expanded the computcr applications to provide a

‘feneral introduction to data handling, and specie programming instruction to perform all estimations {this Book by computer (Chap 12) using Nocrosalt Excel, views, oF SAS statistical packages We

‘ave also added sections on nonparametric testing matrix notation, binary choice model, and an entire chapter on time soies analysis (Chap 11), afield of econometrics which has expandod as of late A

‘Sample statistics and econometrics examination is abo included

"The methodology of this book and much ofits content has been tstod in undergraduate and

‘graduate classes in statistics and econometrics at Fordham University Students found the approach

‘and content ofthe book extremely useful and made many valuable sugpestion fr improvement We Ihave alto received very useful advice from Professors Mary Beth Combs, Edward Downg and Damo- dar Gujarat The following students carefully read through the cnlre maneeerpt and made may useful comments: Luca Bonardi, Kevin Coughiin, San Hennessy, and James Santangelo To all of them we are deeply grateful We owe a great intellectual debt to our former professors of statistics and

‘conometrics: JS Butler, Jack Johnston, Lawrence Klein, and Bernard Okt

‘Weare indebtd tothe Literary Exccutor ofthe late Si Ronald A Fisher, F RS, to Dr Frank

‘Yatss, FR and the Longman Group Ltd Londo, for permission to adapt and reprint Tables TT and FV fiom their book, Static Tables for Biological, Agricultural and Medical Research

Microeconomic Theory Macroeconomie Theos, International Economics, Mathematics for Economists and Principles of Economics

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LL The Nature of Statistics

12 Statstis and Econometrics

13 The Methodology of Econometrics Descriptive Statistics

LA Frequency Distributions 22° Measures of Central Tendency 23° Measures of Dispersion

Probability and Probability Distributions

AA Probability ofa Single Event

2 Probability of Multiple Events A4 The Poisson Distribution 35> Continuous Probability Distrib

43 Estimation Using the Normal Distribution 44° Confidence Interval for the Mean Using the ¢ Distribution

Statistical Inference: Testing Hypotheses

‘Copyright 2002 The McGrail

‘Simple Regression Analysis

61 The Two-Variable Linear Model

62 The Ordinary Least Squares Method

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63 Tests of Significance of Parameter Estimates

64 Test of Goodness of Fit and Corrlation

65 Properties of Ordinary Least Squares Estimators Multiple Regression Analysis

TA The Throe-Variable Linear Model

12 Tests of Significance of Parameter Estimates 2⁄3 The Coefficient of Multiple Determination

TA Test of the Overall Significance ofthe Regression 1⁄8 Partal-Corrlation Cooscats

Tố Matrix Notation

Further Techniques and Applications in Regression Analysis

81 Functional Form 42° Dummy Vanables

3 Distrbuted Lag Mods

£4 Forecasting

AS Binary Choice Mods’

#6 Interpretation of Binary Choice Models

Problems in Regression Analysis

103 Estimation: Indirect Least Squares

1044 Estimation: Two-Stage Least Squares

Time-Series Methods Art

1233 Eviews

124 SAS, ECONOMETRICS EXAMINATION

‘Binomial Distribution Appeodix 1

Kolmogorov Smirnov Critical Values ADF Critcal Values

Data Sources on the Web

Introduction

LAL THENATURE OF STATISTICS,

Satisics refers tothe collection, presentation, analysis, and utilization of numerical data to make Inferences and reach decsions in the face of uncertainty in economics, busines, and other socal and physical sienes ‘Statistics is subdivided ito descriptive and snferental Descriptive statics is concerned with

‘summarizing and describing a body of data Inferonal strstr is the process of reaching general lzations about the whole (called the population) by examining a portion (called the sample) tn ondce foc this to be vad, the sample mast be represemaive of the population und the probably oferor al

us be specie

Descriptive statistics is discussed in detail im Chap 2 This is followed by (the more cru

‘statistical inference; Chap 3 deal with probability, Chap 4 with estimation, and Chap 5 with hypoth

Su ce eondwons are sabe! to cor, we abo would have lo die The probably of ero This

———

L3 STATISTICS AND ECONOMETRICS

Econometrics refers to the application of economic theory, mathematics, and statistical techniques forthe purpose of testing hypotheses and estimating and forsasting eeonomie phenomena Eeono- metrics has become strongly ilemified with regression analyst” This olates a dependent variable to one

‘of more independent or explanatory varabics Since rlabonships among economic sarables are

‘generally inexact, « disturbance or error term (with welhdetined probabilsie properties) must be Incl (sc Prob 8)

‘Chapters 6 and 7 deal with repression analysis: Chap 8 extnds the basc regression model; Chap 9 deals with methods of testing and correcting for wolations im the assumptions ofthe basic reeresion

‘model; and Chaps 10 and 11 deal with two speciic area of econometrics, specifically simultancous-

‘quitions and time-series methods Thus Chaps 1 to 5 deal with the statistics required for econometrics (Chaps 6to 11) Chapter 12s conccrmed with using the computer to ai in the calculations involved in the previows chapters,

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INTRODUCHON (omar EXAMPLE 2 Consumption theory tlh ws tha

thoi daponbe afer a acorn Y, seri ba

am be stata in expt car equation frm a

vera, poop increase tet consemptin expensiure C38 yas mach asthe crease hee pombe come Th,

whore fy and are unknown constants called pramcier ‘The parame by the slope code reecenting he

"Bang propensity to consume (MPC) Since sơn reole XD kiuel dạpoalie locoe ae tay tò Mave -eeneotui Sấếmem cooramgtioecpcaltareeầe theoctcsly cai sai ddermigk hdgiomidp eprtentd by 1g (11) mut be modied fo ncade a random Ssturbance or ero ew aig i stochaste

1.3 THE METHODOLOGY OF ECONOMETRICS

Econometric research, in general invohses the following thee tags:

1 Specification of the model or maintained hypothesis in explicit stochastic equation form, together with the a priori theorctcal expectations about the sign and size of the parameters

‘ofthe function

Collection of data on the variables ofthe model and estimation ofthe coeiciens ofthe function

‘with appropriate econometric tchniques (presented in Chaps 6 to 8)

3 Evaluation ofthe estimated coefcimts ofthe function on the bass of economic, statistical, and sconomethe entra

EXAMPLE 3 The is sommes rar osama hen p49 sate the any ea

‘stochastic equation for, as Eq, (1), with the eapstation that fy > Oe at Y, = 0, C> Oat people dave Sndjorbortow) and 0 by = I The second sage inves th colton of ata on consumption especie and

‘iponble income abd etination o (11 The tiedtay im economic esearch involve (1) cocking to ei

‘he otinatal value of hy > Dandi <by «1: (2}dcerminig i's "atalactory” poportion othe variations CH

“explained” by change iY andi by and are "statistical significant at acceptable ve see rob 1.) and

‘Sex $2 and (3) esting tose if the assumptions of the base represion model are satisfied o,f 0, how to coet {or wolations Wf the estimated relationship des not pass these tet, the Rypothezed relationship mast Be

‘odie! and estate unl 3 salnfactory estimate coupon rclatoeship ached

Solved Problems

‘THE NATURE OF STATISTICS

(LH What isthe purpose and function of (a) The fick of study of statisti? (6) Descriptive star Listes?” (c) Inferential statistics?

(Statistics is the Body of procedures and techniques used to collet, present and analyze dat on which

to ase decsions in the fae of emery or incomplete information Satis analysed today

in pracy techn; tác koderupcmoe may te to to the product Sega evry profenuon The econoent tac fo tent the eisency of aemative production or package that asinine les the scoot to analy the result ofa dug rehabilitation program: the intial mọchaloghl tà canine worker enc 10 plant eirment: he pial nto force ots pater he yen ots the eesti of 4 mew drag: the chemist o proce cheaper feliz: and 80 on ()Doscptive statics summary of data wth on o to paces of oration tht characterize

‘he whe dara Talo refers tothe presentation bay of Sata the frm of tas, chars raph, Sand other Forme of pape ea

‘Seductive reasoning which ase properties tothe specie tring withthe whole)

(a) Are descriptive or inferential statistics more important today? (4) What i the importance

‘of a representative sample in statistical inference? (c) Why is probability theory required? (4) Staitics tated a a purely descriptive sinc, but it grew into a powerful tool of econ making 28 is iferential branch was developed Modern statistical analyse refers paar to tretal or inductive tativcs Howcver, dative tow to generate comple frm papulaons Stor ce ven or to pews fm sgh: to pope and infctive satis re compknentary, We mas sty (8) tn order for statistical inference 1 be Yb it must be based on 3 sample that lly sete the

‘uractriss and propertis ofthe population fom which ti Gruen A represetaive sample

‘nsired by random sampling, wheres each clement ofthe popation bas an egal chanse of Bein fected in the sample (ee Sex 4.1), {0 Sine the possibilty of err exis in tail infrece, estimates of tests ofa poration property oF cSuratrsie ae given tother withthe chance ot protubkty of bong wrong Thus proba

How can the manager ofa frm producing lightbulbs summarize and describe to aboard mecting the results of testing the life ofa sample of 100 lightbulbs produced by the fin?

Providing the (a) data onthe ie cach ithe sample of 100 ightbalts pcodeel by the rs would

bề Xe bSglen rf e ro coơheetng for the tours mamars to raha Tasca, the manager ight Summarize the dat by indicating that the average eof the Balls ested is MO and that 98% ofthe babs

‘ested lasted bermcen 539 and 10h, By doing his the manages proving tro pices of information (he sverage if and the spread the average Me) that characterise the Me ofthe 10 bal ested The manage ako might want to describe the data with a tale or chat seating the umber oe proportion o Bl tested hat lated ship acB 10B đauweben Sacha tubular or raphe representation the data ao

‘ery wel for gaining ick overview o the data In simmartng and dsenbing the dat nthe wap

‘ndiated, the manager engaging descriptive tats It soul! be noted that deseiptve statis ca

be ood to summar and dscbe anybody of data whether itis sample as above) or a popslation whee

al the elements ofthe poption are Known abd iy chracterstcs can be cle)

(0) Why may the manager in Prob, 1.3 want to engage in statistical inference? (b) What would {his involve and require?

(2) Quality contrat rues thatthe manazcr hae fait good idea about the averae ie and the seal

S the We o he phils produced bythe frm However testing al the gba produced would

‘estos theatre outpte'he rm Even when testing oes nt destroy the prot esting the entire

‘utp is rally profiitneyeapensve and time conan.” The weal pred o take sample

‘SF theoutp ad ifr the properties and characteristic of the entire output population) fom he

‘Sorponding daratet of a samrie drawn fom the popaltion|

1) Staisicaliteence toques fest of all thatthe sample be representative of the population being Sampled IC the em peduces lightbulbs ia dierent plat, wth more Man ooe Works, ad

‘witha mates from arent applic thee mt he reece the spl nthe proportion ich they contre to th otal opt ofthe em From he average Me and sped te i of

‘he buts inthe sample the fem manager might estimate with 9% probability of beg comet an 5% probability of bing wrons, the average feo lhe lights prsdueed bythe ñn be heneen S20 nd 408 (ce Se 45) Ttead, the manager may te the sample formation Lo et, with 959%

‘roablity of being correc and Ss peat ‘all he lbs prodice bythe fr rete than 20h (ee Sex 53) Tn etnating of beng eons thatthe aera ef the popaaion or tevig the

` "

INTRODUCHON (omar STATISTICS AND ECONOMETRICS

1s

"6

‘What is meant by (a) Econometrics? (6) Regression analysis? (2) Disturbance of error term? (d) Simultancous-oquations models?

(0) Econometric isthe interation of sonomic theory, mathematic, and statistical techno forthe poo puters sheet enon phentnana, tem etc of econo ats

“hips ad forecasting oe pred Ture vals of economic vara or psnomena” Eoonometos issubdivided into theoretical and ape econometrics Thcreica economics for measurement of economic reatoeships i genta, Aplind emacs examines the problems fers othe mets

‘ocountered an the ndings i particular eds economies, sock at demand theory, peodoction investment, consumption, and ter feks of applied economic xa, In any case, economics arty an at and partys see, case often the nation and god jadpment of the esate Pays acral roe

(6) Regrsion analysis stad the cal reationshiptetmeen one economic arable to be explained (he dependent variable) and one more andependent or explanatory varia When thee i ony ome independent or explanatory variable, we have simple cợncrim Inthe moe stl cae of moc han ‘See indepenent or explanatory variable, we hae mip regres (0 A eandom) disturbance or ero must be ince in the exact relationships posted by eeoomic ‘theory and mathematical economic inorder to make them stochastic (eo onder ores the act

‘att the el word, sonoma febbnshis among esonam vào sóc eae and somewhat (d) Semutancouscyantons ‘han one squation and och tha the econowicvarabls i the various equations aract Sou made fer là nhai rong sao va expressed with more scour equations mod ar the mow comple apt of econoctin ant ate data ia Chap 0

‘commodity inerey felted tos pose The scot fncton of economies to prove nama

‘Stinaes of the cocints of ccoomscrlsonships These a estan dcion making For

‘Lampe a government policymaker needs 1 heat accurate estimate ofthe oxic of he eatin

‘hip between consumption abd income ia cedr to deetnae the stilling (the wiper

‘38 proposed tan seduchon A manager needs to know if a pice reduction ncreabn ord he total sas revenes of the firm and, is, by how much The tied fiction of econcmctris Forecasting of events This 00 necessary inorder for poeymaers to tke appropriate coretve the

‘Sion he rate of unemployment or ifation i predic! to nae a the ate

(0) Therear two asc diferences betneen econometrics (and othe soil zene) on one hand ad most pyscalsonces (sh as piss) on the her One i that (as posed ou carer) slatonips

‘Smong economic vanates are maevart and somewtat erate TC sen at mọi eeoeome Thenomena eur contemporanconly 0 that laboratory experinets cannot he conde The Sieences require special methods of analy (sch the insion ofa istaxranceo or te wth the cac relationships postltd by economic theory) and multvaate analy (sich as maple regres anaes) The later inate the elt ofeach independent oe explanatory variable 06

‘he dependent vartable i the face of contemporaneous change i all explaeatry variable,

{In what way and for what purpose ar (a) economic theory (2) mathematics and (statistical analysis combined to form the ieM of study of econometis?

(4) Exonomircs presepposes the existence ofa body of economic theories or hypthess resins ting Whe variable sugcted by economic theory donot prove a satiactory explanation the nhecber

‘nay capemet th altematne lormalations an varables sugested by previo et oppo theories In this way, esoometie research can led the aceptance recon, and reformation of

18

() Mathematics is wd to exes the wera statment of eeaeomic theres in mathematical form, agen an exact of tennis farcoeal tans Reeur the Gperent and one oe ee independent o explanatory arabes

(0) Static analy applic appropiate echnaguc to timate the otal and nonexpermentl eatin: ships among econome vanubles by wing Felevant economic data and evaluating the rests

‘What justifies the inclusion of a disturbance or ereor term in repression analysis?

‘The inctsion of «(tandow) disturbance or ertor tem (vith well deind probabilistic proper) is required ia regression analysis for tree important reasons Firs, siae the purpos of theory ito generale land ump eemecne sateethgs vay me on (be mot portant forces at mark Th meas

‘hat mumerous other variables with sght and repr ffs ane wot iced The ror tm can be

‘sewed as representing the net eft ofthis are nmber of all and irrepaa fores at work Sood he incasion oth ror erm canbe jin oder to ake into consideration the met eet of possible rors semeasting the depen! varable of Yanable being eine Fal ice human tehasor wal

‘ies in arandom way under drtkalcecamstanes, the dntarbanc ero erm can be Wed ape

"hi hen random human barr Thier term ths alls for inal rapdom Jetons Foe

‘the exact ad deterministic relationships postulated by economic theory and mathematical cennpie

‘Consumer demand theory states that the quantity demanded ofa commodity Dy isa function of,

‘or depends on, its price Py consumer's income Y and the price of other (elated) commodities,

sy, commodity 2 (Le, Pz) Assuming that consumers tases remain constant duting the period

‘of analysis, state the preceding theory in (a) specific or explicit linear form or equation and () im stochastic form (c) Which are the cocficents to be estimated? What are they called?

‘THE METHODOLOGY OF ECONOMETRICS

1.40 With reference tothe consumer demand theory in Prob 19, indicate (a) what the ist step isin

‘sonometnic rescarch and (8) what the @ prior! theoretical expectations are of the sign and Possible sizeof the parameters of the demand function given by Eq (L4)

(0) The fast step in esonometic analysis to express the theory of consumer demand in stochastic

‘sation form, as Eq (14), and indicate thea prion theoretical expectations about the sgn and posi the sz ofthe params ofthe fonction

(8) Consumer demand theory postulates that i Ea (14) 6, <0 Gating tat peice and quantity ate trverly cred, 9x the comme ea normal god inating tha onvomesporane More

‘the commodty at higher incomes), by 0 and Z are sabia, ad by 20 and Z are complements,

Indicate the second stage in econometric research (a) in general and (8) with reference to the

‘demand function specified by Eq (1)

Lộ The esod dhgtlaseosonubSerhoanh ‘on each ofthe independent or caplanatory variable ofthe model and wing thew data forthe lông tây eolloion of ata oth epee eile nd Cty estimation of the params ofthe model Ths wally dove with multiple regression

‘nase scsed in Chap 7)

(8) In order to ctimate the demand faction given by Eq (8, dats mast be callted on (1) he

4 ity demanded of commodity ¥ by consumers, (2) the pie of YQ) consumer's incomes (6) the pie of commodity Z ern of time, prday month Se) a ove a ember

In onder to estima the damand function given by Ea (149, mesial wales of the variables are required ona prod of ine For example we want oct he demand function for cle, we need

‘the mamersal value of he quant of colle demanded, a, pr year overa umber of Jeary a, roe 1960

401990 SeMluy, te pel dựa on the average pee cof, conse income andthe pie ofa ea {a sist for ele) per ear rom 190) 4 180 ata tht gre ames ii chế ah si

‘ancton from pred To prio ae alle meses dita.” Mowever, oct the soesumpton fncion {ora group of tamiis tone peti time, we weed ơơ-evicma do (Le, mera Yas fe the -ewusmpiea cpendinre and dnposale incomes of cach amily the rou ata prticla point i tine,

i 198

‘What is meant by (2) The third stage in conometric analysis? (8) A priori theoretical cri feria? (c) Statistical criteria?) Econometric itera? (e) The forecasting ability of the moderr

(6) The tied tape econometric research involves the ealaton ofthe eximat model on the has of

‘he a piece static and son erteiu, nd the forecasting abit of the model, 1) The gyitecemc or tư tò the sign and eof the params ofthe model postulated by cxonomi theory the etimatd overs Jot conform to thon poste, the mol mst BS

©) The stat ertera weer 19) the proportion of variation in the dependent variable “eplined™

by changes im the independent or explanatory variables and”) vercation that the persion oF spread of cach estimated coef fence a the etna aoa the rue parte fiery matow fo Be Us To (0) The economeric ceria ee 10 tts that the assumptions ofthe base repression model, and parton lan those about the distarbance or ror tm, are sais The forecasting aby ofthe model refers tothe ability ofthe model to accurately pot are Vales

‘tthe Soenaent variable tased ot now oe expestd Ue walt) of he tú hức o explana tory van)

How can the estimated demand function given by Eq, (1.4) be evaluated in tems of (a) Thea ior criteria? (0) The statistical itera? (e) The esonometric eter? (d) The forecasting bility of the mod?

(9 The stated demand function given by Ea (14) canbe evaluate in ems ofthe a prion theoretical tera by checking tht the eimated cosine conform to the theoretical expectations with reper

10 sig ad pose si, as postited im Prob 110) The demand theory pen by Eg (18) 8

‘Sone oa if by <i by > 0 GE ta somal good) and > 00 Z isa subst foe X38 snlote by demand theory

() The stati criteria are satis oly if a “high” proportion ofthe variation ia Dy over tine i “explains” by change ie Py, Yan Py, and ithe duperson of extimate andy around the tre rat “safiety narrow *"There eno generally accepted answer ato what 2 "high roporton ofthe variation a Dy “explained” by Py and Py However, because of common tends

In time-series data, we would expect moe than 300 70% ofthe variation nthe depend variable to

te eapaind by the independent of explanatory varabics for the ml to he judged stator

‘Semi monde foreach estimated count o be “statinicaly grant,” we woul exes he -đapgrden of cach estimate cocfldert abou the trae prance (eased By its andar vation: 2S SG 399v reo Icc thạc bạ the wt wae of the coset

Present in schematic form the various stages of econometric research,

Stage 1: Eennonk thewy

: AMaterateel modgl

¡ esnsmate bietbade) mod Stage 2 Calton of appropriate data

t Estimation ofthe parameters ofthe model Stage 3: EAahaddo of the model o the bass of ono,

(4) To whch el of sady i sa0igiel aalrderpoftàf (9) What are the most important Feton of

‘Ssrpive sarics?"(c) What Is the most portant funcom of trea sak?

“ans a) To economic, busicss, and othe socal and physical ences (8) Summarizing and describing {boy of data fo) Drang inference aboot the characteristics of popsation fom the coresponsing

‘taracteritis ofa sample drawn from the pop

(4) Is static inference associat with dive oe inductive reasoning? (8) What ate the eoaitons required in ode or statisti inference to be vals?

"Asa Tete reasoning (b) A repesetaine sample and probability theory

STATISTICS AND ECONOMETRICS

Expres the form of an ext nar equation the statement thatthe level of svete spending 1s wYeney felted 1 ate of interest

án by +8\R_ with by postulated 1o be nepative us,

vai

What i the ansner Ans Am economi to Prob 1.18 an example heory expres in (tat oF deterministic) mathematical orm of?

Express Bg, (1.) in stochastic form,

Wy isa stochastic form required in esnometi analysis?

Ane" ocssne the atone noes economic arse are tact and women cratic as oppose 19 T "

What are ager (0) ome, (H) to, and (2) the somone resect

Ansa) Speciation ofthe theory i chat egltion form and ideation ofthe expt signs ant posses of estimated parameters () Colton of data onthe variables of he mods and xtmation [the coeicients ofthe fnction ( Eeosomic tata, and econometi evalation othe estate

—

‘What he Sit sage of economic anal forthe investment theory i Prob 118?

‘Ans Stating the theory nthe form of (16) and pricing lị = 0

[What is the sco stage in ononctiamaly for the investment theory ia Prob 1.187

‘An Colicton of tine-sets dats on! aed Rand eximation of Ea (1.8)

What is the hed stage of econometric analysis forthe investment theory in Prob 18?

4s Determination thatthe estimates eect of «0, that an "adeuae™ proportion of he aration (8T sa tine “elaine” by changes that by “satay igicant at customary kvl,” and

“hạt con assumptions ofthe model ae sae

Descriptive Statistics

24 FREQUENCY DISTRIBUTIONS

tis often ustul to organize or arrange a body of data into a frequency darbuton This breaks up {he data into groups or clases and shows the number of observations in each class The number of

‘asses is usually between Sand 15 relative frequency dsrturion obtained by dividing the mambe

‘f observations in cach clas by the total numberof observations in the ata asa whole The sum ofthe felative frequencies cquals I "A histogram isa bar graph of afequency distribution, where clases are tmensured along the horizontal axis and frequencies along the vertical axis A frequency polygon isa ine {raph of #froquency dstbution resulting fom joning the frequency of cach class plotted atthe sass {idpoint, A cumulative frequency dtrdbtion shows, To cach cas, th total amber 0

allasss up to and including that class When plotod, this gives a dissbuion curve, oie

EXAMPLE 1 studce cst he folloming grads (eased from 1010) onthe 10 ics he tok daring 2 femestr 6,716.8, 5.7.6.9, 10, and 6 Thnepades cn be arranged ino frequen dtbutions asin Table 2 1 Jad town gapncaly as we Fie 2

Tatle 21 Feqaeny Dhiba of Gras

Grates Abate Fegaeesy [Rene Pegaemy

" DESCRIPTIVE STATISTICS louar > {EXAMPLE 2 ‘The ans ina sample of 2 cans of it contain et weighs of rat ranging fom 19.310 20.907 08 fen in Table 22 If we want to group thew data ito 6 cles, we pt class inves of O02 [210192376 = 0.300, The weight given in Table 22 canbe arranged ito the frguency đồtnolom ghen {Table 23 and shown graphically a Fe

2.2 MEASURES OF CENTRAL TENDENCY

Cental tendency refers to the location ofa distribution The most important measures of central tendency are (1) the mean, (2) the median, and (3) the mode We will be measuring these for

‘populations (ie the colketion of all the clments that we are desribing) and for samples drawn

om populations, as wll as for grouped and ungrouped data

1 The arithmetic mean oF average, of a population is represented by y (the Greck letter mu); and fora sample by Y (read "X bar”) For ungrouped data, and Y are calulated by the Following Temas

we EX nf te um of he eer, an cs whe nebo Tere hon a Sigh oh Pr ae ny and ng

Ty

where 3> J refers to the sum of the frequency ofeach clas f times the lass midpoint

2 The median for ungrouped data isthe vale of the middle tem when ll he items are arranged in cither ascending oF descending order in terms of vals:

Median = the *P)u item i the data array G3)

where WV refers to the number of i

_grouped data is given by the formula sin the population ( fora sample) The median for

5/2 +

Median = £4! Fe G4) where

lower limit of the median class (ic the class that contains the middle itm of the distribution

‘n= the numberof observations in the dataset

sum of the frequencies upto but not including the median class frequency ofthe median class

fs

(¢= width ofthe clas interval

3 The mode isthe value that ocous most fFequontly inthe dataset For groped data, we obtain

here Z = lower limit of the modal clas (ie, the class with the greatest frequency)

‘d, = frequency of the modal class minus the frequency ofthe previous class

«4; = frequency of the modal class minus the frequency of the following lass

width of the class interval

‘The mean isthe most commonly used measure of central tendency The mean, however, is aeted

‘by extreme values i the dataset, whike the median and the mode are not Other measures of contal tendency are the weighted mean, the geometric mean and the harmonic mean (60 Probs 2.7 10 29),

= ĐESCRIPTIVE SFATRSTICS chen > [EXAMPLE 3 ‘The mean grade forthe population on the 10 quizzes given in Example 1, sing the forma for

‘This calealtion coal be simplified by ending (se Prob 26

observations)

= 39 number of observations oF tems

3 sum of feguenis upto but not ectaling the median cass

reguency of the ian cas ith of clas inter

As noted in Prab 24 the mean, mada, and mode for grouped data ae estimates sed when oa the grouped ta tse oveotle or tcofee eainations wih olarpe waproped date ợc

2.3 MEASURES OF DISPERSION

Dispersion refers to the variability or spread inthe data The most important measures of disper sion are (I) the average deviation, (2) the variance, and (3) the standard deviation, We will mea-

‘sure these for populations and samples, as well as for grouped and ungrouped data,

1 Average dation The average deviation (AD), aso called the mean absolute deviation (MAD),

is given by

p= ZH =H fr poputtins (2%

v=)

where the two vertical bars indicate the absolute value o the values omitting the sgn, withthe other symbols having the same meaning as in Sse.22- For grouped data

Ap-E/F= Gr populations an) ant Ap-EIE-T „.ee~ am

where f refers to the frequency ofeach class and X tothe elas midpoints,

Variance ‘The population variance o (the Greck letter sigma squared) and the sample variances for uigrouped data are given by

4 The coeficin of variation (V) measures relative dispersion:

Y= tor poputations (2.129)

EXAMPLE 8 ‘The average deviation, vrtanc, standard deviation and eofient of variation for he ungroupe

‘ata gn in Example I sam be ound with thea of Table 28 (n= 720 Cramp 3

“ ĐESCRIPTIVE SFATRSTICS chen >

Thhk 35 Calolsta onthe Data a Example 1

[EXAMPLE 6 ‘The average deviation, variance, standard deviation, and eoficient of variation for the frequency

‘Siribaion of weighs (grouped data) sve in Tale 23 can be found mith the aid of Table 2 (1 ~ 3905 se Eames:

ap -EAE=H1 96 osig00 _$

" `

"" — ố

3.2 03843 0z Vago SE woos, or 196%

“Table 26 Calesations othe Data Tale 24

CHAP 3) DESCRIPTIVE STATISTICS 1s

24 SHAPE OF FREQUENCY DISTRIBUTIONS

‘The shape ofa distribution refers to (I) its symmetry or lack of it (skewness) and (2) its peak ness kurt)

1 Stewness A distribution hac aưo skewness if itis symmetrical about its mean For a

‘symmetrical (unimodal) distribution, the mean, median, and mode are equal "A distribution {s potively skewed ifthe ight tails longer Then, mean > median > mode A dsteibution is negatively skewed if the lf tails longer Then, mode > median > mean (se Fig 23)

at A: Syme rac Pc) tend Pent: Netty

Fig 2 Sewness cam he measuted by the Pearson cocci! of skewness:

sx EL ~ =0 for populations (2140)

For symmetric distributions, Sk = 0

2 Kuwtoss A peaked curve is call eptokutic, as opposed toa fat one (plarpkuri) relative to

‘one that is mesohuric (ce Fig 2-4) Kustosis ean be measured by the fourth moment (the numerator of Eq 2

6 ĐESCRIPTIVE SFATRSTICS chen >

"` ` ` (eas)

3 Joon moment The comoverent of two separate distributions can be measured by covariance:

EX M=T) EQN _ Ty for populations

S(Y-WJ -Ÿ)_ BQT) yp

cor.)

{A positive covariance indicates that Y and Y move together in elation to their means A negative covariance indicates that they move in opposite directions

[EXAMPLE 7 We can nd the Peron coficint of skewnes for the grade gven in Example I by ing = 7, med = 6.5 (oe Example 3), and o = 148 oe Example 3

24 Table 27 gies the grades on a gue for class of 40 students, (a) Arrange these gradss (raw

‘data et) into an aray from the lowest grade tothe highest grade () Construct a table showing

‘lass intervals and class midpoints and the absolute, relative, and cumulative frequencies for each rade (0) Present the data in the form ofa histogram, relativefrequency histogram, frequency polygon, and ogive

(o) See Table 28,

“Table 25, Data Aray of Grades

KEO RE 0 0 0 9 0 10

CHAP 3] DESCRIPTIVE STATISTICS ” () See Table 29 Note that since we ae dealing here with dsrete data (Le, data expres in who somber) we usd the actual grads as the lass midpoint

ĐESCRIPTIVE SFATRSTICS chen > {A sample of 28 workers in a plant rssive the hourly wages given in Table 210 (2) Arrange

‘hose raw data into an array from the lowest to the highest wage (6) Group the data into classes (¢) Present the data inthe form ofa histogram, relative-frequency histogram frequency polygon, and ogive

‘ame 210 Hoary Wages etary

‘Table 212, Frauency Ditton of Wages

(0) SeeFie 2.6 Another way of ting he opt ito plot the cumalativeFreqeenis 5798 ano ono toed the per ht ofeach cla) The vale 95,398, 3795, wp 03595, 5495, are

‘fen refer to the ls andar oF exact mits, Noe tat the clas pis are obtained

‘ding ogee the ower and upper ts boundaries ‘pnt en by (3595-4 3408/3 — 7300/3 ~ 268 (ne Table 313) nd ding by 2 For example, the cod clas

CHAP 3] DESCRIPTIVE STATISTICS ”

[MEASURES OF CENTRAL TENDENCY

2.3 Find the mean, median, and mode (a) forthe grades on the qui for the class of 40 students given in Table 27 (the ungrouped data) and (0) for the grouped data ofthese grades given in Table 29,

(4) Since we are deslng with ll rads, we wart the populacon mea:

‘qua 06 the median i 6 "The modes 7 (the vale that oars ot Frequently in the data et) (8) Weean tad the population mea for the rouped ata i Table 29 withthe aid of Table 213:

E30 2

‘Tiss the same mean we found forthe ungroupe dat Not thatthe wm ofthe frequencies, Sf

‘ual the mumber of oburvations i the popelaion, N, and EY Sof The mean forthe rouped data of Ta 313 en By

=55406 617

ĐESCRIPTIVE SFATRSTICS chen >

“4; =2- fequeny of the modal clas, 8, mins he Feqoency ofthe previons cas, 6

«4, =4 = fequeny of the modal class, 8, mio the Frequency of the following clas, 4

dt of the cas tra

[Note that while the mean called from the grouped dat in this case Mental wo the move

‘lta for the umgrooped dat, the median andthe made are ony (good appronimatone

“Table 2.13 Caen ofthe Poptaton Meas forthe Groped Dat a

y_EY 96497990894 480% s56 `

Median = 5395 value ofthe (n+ 1}/2= (25+ 1) = 13th item in the data aray in Tale 211)

“Mode ~ 53.98 and S408, sine ter are threo ach of hee wages Thos the distribution iy Boal

‘Thos T calelated ‘acute for the ungrouped att Ie the real word meen hate only the groupe data, o i we from the grouped data sony a very good approximation forthe tre vale of T havea very large boy of ungroupod data wl sve on calculations to erie the mea by fin _rouping the dat

‘compared wih the tue median of $3.98 found fom the ungrouped ata se part

` ata

‘3 compared withthe trae mods of $39 and S408 found fom the ungrouped data (ee part oh

‘Sometimes the mode i simply ven as the mit ofthe modal cas,

34085 or stax

‘alle 214 Cacation ofthe Sample Mean for the Groped Dat in

(0) Theadvantapes ofthe mean are_(1) is famine and understood by virally everyone, 2) al he

‘bjeratons ithe dita are taker ito account, and (1) uso ie performing many other Stmoicalprosedes and ets The dnadvantoges ofthe mean are (I) alerted hy extreme

“alae, Wis tne consuming 10 compute fora large boy of ungrouped data and (3) W cannot

be cakulated wher the lant clas of groupe datas ope-ended (sade the Lowe it ofthe Uist cans "and over)

() Theadvantagss of the median are (D itis pot alfcted by extreme values, (2) itseasly understood (Ge, bal the data are sal than the median and half ae peter, and (3) ican Be acute che

‘shun the ls ls open-nded and when the ata are guatativ rather than guanine The sadvantages of the mean ate (1) does ot we mech ofthe information aval, and (2)

‘oie that sbvervstion be arrange! into an array, ich ie tie onroming For are Rody of soproeped dat,

(0) The advantages of the mode are the same as those forthe median, The disadvantages of the rode are (1) as for the median, the made doer not wie mich of the information avaible, {2nd 2) sometimes no vale ofthe data repeatod more than once 50 that there sno mode, while 3e time the may Be mary modes Te poner the mea she most Fenty wed meas of

‘Stra tendency an the mode the eet a

27

ĐESCRIPTIVE SFATRSTICS chen > Find the mean forthe grouped data in Table 2.12 by coding (ie, by asigning the valve of y=

to the Ath or Sth clases and = 1, jp = ~2, et to each lower cass and ys

sạch lang chss and thơn púng the Foemula

Fa xy + Elbe = sas + Š (s9 10) — 3885 Lit = s3.85-+ (8019) + S010 = S395 8

“for the grouped data formod by coding is identical to that found i Prob, 2.48 without coding

‘Coding ciminates the problem of having to deal with possibly large and inconvenient class ripoints; thus it may smp the eaculations

A irm pays a wage of $4 per hour to its 25 unskilled workers, $6 to it 15 semishillod workers, and

‘stots 10shlled workers What ithe weighed average oF weahted mean, wage pad By ths?

Te fing the weighed mean, or weighted average, ofa population, po sample, T, the weights, w

‘nave the sme fonction asthe exons) nding the meas forthe proud data Thur

A nation faces rate of inflation of 2% in one year, Sin the second year and 12.5% in the thử

‘year Find the geometric mean ofthe inflation rate (the geometric mean ju, oF X,of a set of Positive numbers isthe mth root of their product and is used mainly to average rates of change and index numbers):

‘The goometric mean i wed primary inthe mathematics of nance and franca managemce

‘A commuter drives 10mi on the ịghway at 60h and I0 mủ on lcal sticets at ISmih Find the harmonic mean The harmonic mean yy is used primarily to average ratios:

ascompured with y= Y/N = (00 + 18)/2 = 78/2 [Note that if the commuter had averaged S7.Smi woukd have taken her (20037 Sma = 32min to dive the 30ml Instead she es

‘nn on the ihe (10m at 6) and Din oe nal sets (101 a Sth) Fora tual of SO min,

‘and hs the core answer we et by ting y 24B Thạt Oe 24eih) ~ €Oeain = SO (8) Fer the ungrouped data in Table 27, find the ist scond, and third quartiles andthe third

‘deciles and sntith percentiles (b) Do the same forthe grouped data in Table 2.12 (Quarter divide the data into 4 parts, deciles into 10 pars, and percents into 100 pats.)

2) Q, (Set quart) = 4 (the average of the 10th ad 110 yas in Table 25)

0: (cond quartile) = 6 = the vale of he 208th item = the median

7.11 7.5 =the vale of he 30 tor

1, (third deve) = $= the vale ofthe 12h ta

‘Po (iteth percentile) =7 = the ve of the 245 item

™ DescRIPTive statistics chen >

2.11 (a) Find the rang forthe ungrouped ‘data in Table 210 and forthe groupod data in Table 2.12 (2) What are the advantages and data in Table 27 (0) Eind the range for the ungrouped disadvantages of the range?

(0) The range for ungrouped ata sequal to the vale ofthe largest observation minus the vale of he smallest observation inthe dataset The range Yor the ungsouped datas Table 274 fom 20 10,088 manh

() The range forthe ungrouped data in Table 210i fom $3.48 1 9436 or S071, Fae grouped data, he range extends fom the lowe Ht ofthe salle clas to e vper ướt he agest ls, For the

‘roped data ig Tale 212, the ana extends from S180 €9 $129

(0) The advantages ofthe range ae tht it ea) to fl a ndrtand ts disadvantages ae that it comers oy the lowest and hight vals of a rbuton, peat inacnoat by exreme ales,

‘ad it cannot be found for opted dstibutions Because of thee dadvantags, the range i ltd useless except in quay corel

For the ngroopa dat in Table 2.7, IR = 7.54 3S pints ating the valves of Qy and found

in Prob 2.10 Note thatthe nenuarbe range no ated by extreme vals Baas ies

‘aly themida femur of duperson For the guar devon, of he data tis tus beter than the rane, Buta ak Wiel Used asthe oer

en

“Therefore, QD = (75 —4)/2= 3.5/2 = 1.5 point Quartile devistion ears the average range of

‘eerourth ofthe data

() R=, Q, ~ S08 $3.83 — 50.25 foising the wales of Qs amd Q, found in Prob 210467

a

[Not thatthe average deviation takes every observation nto account measure the average ofthe sixelue detuton oi cach abuervatioa fra the mean, It takes the able vale Gate by the

‘90 vertical tas) Beate (1 ~ i) = (ee Example 3

(0) We an find the average deviation forthe same groupal dita with head of Tale 216

the ame as we found forthe grouped data,

‘able 2.16 Caan fr the Average Deaton foe the Groped Dat ia Table 29

Find the average deviation forthe grouped data in Table 212

We ca ind the average deviation fo the group data of houty wages ia Table 212 with the aid of Table 217 (T= #395, se Prob 240)

Noe that the average deviton found or the grouped dita san etna of the “ive” average deviation that cou be found forthe ungroupe data 1 sual ifr sghly from the trae average deviation bate we Ue the etme ofthe mean forthe grouped data our esculationsfompare the vas of {ound ia Prob 2) 308

‘able 217 Caledatiows fr the Aveate Deviation fo the Grouped Data ia Tae 212

+ ĐESCRIPTIVE SFATRSTICS chen >

“ĐI Find the variance and the standard deviation for (a) the ungrouped data in Table 2.7 and () the grouped data in Table 29._ (c) What isthe advantage of the standard deviation ovee the variance?

“ PEA aa nat Gee ab 2)

oF ELLER a A 8 points squared

and = Vai = Va% = 2.19poims

the same as we found forthe unroupod data,

‘Table 218° Caleations fr the Variance and Standard Deviation forthe Data TaM 29

“The standard deviation is by far the most widely wed measire of (absolute) dispersion

246 Find the variance and the standard deviation forthe grouped data in Table 2.10,

‘We cam fi he varia and the andar dition fi the prone ata of hry ange with hei

‘of Table 219 [= 8398 se Prob 240)

2/11 TỶ „9 xonye san quel

ant 1» PLE oman

[Note tht in the forma for # and sn I ater than ws win the denominator The reson fr this

is that i we ake many samples foe + population the average ofthe ample variances dos ot tend to

‘sta popsstion variance, yale we sa ‘Sidon thsi Chap 5) Fatherore, ands for the grouped data are stnaes forthe rue? atl ¢ nin the dnomintor of the formula For more wil Be

‘that could be found forthe ungrouped ata hecake we the the extimate of from the proupot data i

we can gt y ng gang xí ison he oma for nổ A wh nthe mame an 8

Ege Eat gee ba epi

> ĐESCRIPTIVE SFATRSTICS chen > fo) ỶTMS49439136441641404164494.93281 41004 16436436

F164 3640 FISHES A945 255364481 HONG 16 S14 16426449 464594 36-640 81 4 100625,

the ame as in part and Prob, 2.15,

Toble 220 Calelmloe or the Variance and Standard Deviation for th Groped Data

‘Table 29

2.19 Find the variance andthe standard deviation for the grouped data in Table 212 using the simple

‘computational formula given in Prob 2.170)

“Tae LAI Caeaation for the Variance and Standard Deslaton fr the Grouped Data te

4304| dat SÁU |iWAGS| - 36lsg

Sa-% EAP = sss

0) The coeficent of variation measures the relative dispersion inthe data and is expesed as » pure umber without ay unis This to be conzasie with standard deviation and othr eases of

‘hoisedipersio, which ae exprsed i the urs ofthe problem Thus the colo! of Yaron

‘an be uid to compare the elaine daperion of 9 or mufe itbations expr in ellen wis

“Tabie 27 greater than hatin Table 2.12 Th coelicent of arson alo can be ue to compare the reat diperson of the sare peo ata ver diferent me periods (when of Vand ors change)

‘SHAPE OF FREQUENCY DISTRIBUTIONS

221 Find the Pearson cooicient of skewness forthe (grouped) data in (a) Table 29 and_ (6) Table 20D

(0) With = 6 ed = 617 fice Prob 24h), ando % 2.19 (ee Prob 2.1),

3 — mg) „ 16—617) 0.7) 939 (a pure numbed

2 Tig Eig 02 (orate umber [Note hat mean is reatr than mean and thatthe dtrtbution hy gan} dhevel(€ FZ (6) With ¥ = $395, mad ~ $3.97 [se Prob, 2.4, ad 5280.18 (ce Prob, 2.16)

(20 = med) 4395-397) _ 002)

(om Fie 260)

3 ĐESCRIPTIVE SFATRSTICS chen > 2.22 Using the formula for skewness based on the third moment, find the coeicent of skewness for the dats in (a) Table 2.9-and_(b) Table 212

(@) Wecan fd the ooeflcet of hownes fr the data in Table 29 wing the Formula base onthe hind moment with he ad of Table 22

see Baw,

—m “ịp "Hang

“This inicates that this distribution is negatively skewed, but the degree of skewness is measured

<iererty han in Prob 221,

Grade | Midpoior x | Frequency [Mean | =e =a] form

[Note that regards ofthe measure of skewness we, the dstibtions ofthe data in Tables 29 and

212 are neptvel showed, with the ater more nepatvely skewed than the forme

‘Table 223° Catesatio for Shen forthe Data in Table 212

apes,s |MapomeY,S| rmgwmy/ | ẲVS | s |X-V| - /(V-W

2.23 Find the coefisiont of kurtosis forthe data in (a) Table 29 and (b) Table 2.12

(6) W6 san Ind the eeđfidert of bsloi for the data i Table 29 with the ai of Table 224

unig EL 5 208 2085p me

‘Thos the dstibtion of grades is very peaked (polar see Fig 2-6)

Table 224 Calelations for Keron for the Data iTable 29

Waee,S |Mdmierx,sỈ Esgamy/ | Ts |S [V-TÍ| - /WV-TỢC

xuam | 3# T fas | ow | ome nnse

2 ĐESCRIPTIVE SFATRSTICS chen > 2.24 Find the covariance between how

‘he data in Table 226 wage Vand education ¥, measured in years of schooling in

‘Table 226 Employee Hourly Wages and Years of Schtag

From the cautions ip Table 227, co, VỊ = (10385/10)

tatow thei means, covariance creased” When ¥ and Y move in oppnitedrthos relative to thet ream emlcgee ) cneafteeeicdeemaed Seo th ceeen(V F) > Y and Y sue ether

‘ate tothe mean

Enpme] Maww | Yeon of

Nuober | Wages |shosegv] (VY) | @-T| | DOT)

228 Compute the covariance from Table 226 using the alternate formal Computations “ase phen is Table 228 oy(Y¥) = (17285/10) ~

16295 = 10388

1 779)(158) = 1288 — Table 22 Caleaton for Covariance sh Ahonse Foal

226 Table 229 gives the froqueny for sasotne ricer at stations ina town Preset the ata inthe form of histogram a relatnetreqnney histogram eyocte ro) ga and an one

“alle 229 Freeney Disebation of