Statistics Made Simple

196 Pages·1968·19.998 MB·English

by  H.T. Hayslett Jr.

Statistics
Made
Simple
H. T. Hayslett, Jr.
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STATISTICS
MADE
SIMPLE
HAYSLETT,
H. T. JR., M.S.
ASSISTANT PROFESSOR OF MATHEMATICS, COLBY COLLEGE
MADE SIMPLE BOOKS
DOUBLEDAY & COMPANY, INC.
GARDEN CITY, NEW YORK
ABOUT THIS BOOK
Thisbookcontainsaselection oftopicsfrom avast amountofmaterialinthefield ofstatistics.Theuseof
statistical techniques in experiments in areas ofscience asdiverse as agronomy, anthropology, and astro-
physics is becoming more and more important, to say nothing of the use of statistics in economic and
government forecasting and planning.
Nomathematicaltrainingexcepthigh-school algebraispresupposedinthisbook, although whenyou
read some sections it will be clear that themore mathematical maturity you possess the better. Perhaps
summation notation, inequalities, and the equations ofstraightlines are unfamiliar to you. Ifso, you will
find thatthesetopics aretreated wherethey areneeded sothat thebookis self-contained.
Numerous examples are included. Most of the data used in the examples are artificial. No attempt
has been made to have examples from all subject areas, nor to provide realistic examples in every case.
The author has assumed that anyone usingthisbook will be motivated tolearn some ofthe concepts and
techniquesofstatisticsbecausehemusthave someknowledgeofthesubjectforhisworkorhisstudy.The
reader can, therefore, undoubtedly supply his own specific applications after reading the explanation and
examples given. In the later chapters, where the problems become more complex, step-by-stepdirections
formakingvariousstatisticaltestsaregivenandthenthesedirectionsareillustratedbymeansofexamples.
The material in Chapters 2 through 4 is basic, and is needed throughout the book. The material in
Chapter 5 is needed specifically for Chapter 6. More generally, though, a knowledge of probability is
helpfulthroughout, althoughitisnotessential tothe abilityto performvarioustests.
IwouldliketothankProfessorWilfredJ.Combellackforhisencouragementandadvice,andforgiving
so freely of his time, energy, knowledge, and wisdom. Professor Combellack read most of the chapters,
and nearly all ofhisnumeroussuggestionsfor improvement have beenincorporated intothetext, thereby
makingit clearer and eliminating many errors.Thereader ofthisbookisin hisdebt almost as much asI.
The errors that might remain are my sole responsibility.
Finally, I wish to thank my wife, Loyce, for her help in innumerable ways. Not onlydid she read,
criticize, type, and check the entire manuscript, but she took on many of my responsibilities at home
during the months that I was engaged in writing this book while teaching full-time.This book could not
havebeen completed withouther help, and I am grateful for it.
H. T. Hayslett, Jr.
LibraryofCongressCatalogCardNumber67-10414
Copyright © 1968 by Doubleday & Company, Inc.
AllRightsReserved
PrintedintheUnited StatesofAmerica
. ..
TABLE OF CONTENTS
About This Book „ ,..., , 2 ProbabilitiesofSimpleEvents „ 36
ProbabilitiesofTwoEvents 38
CHAPTER ProbabilitiesforCombinationsofThree
1
orMoreEvents _ 43
What Is Statistics? Permutations 47
ThePresentImportanceofStatistics 6 FundamentalPrinciple 47
TwoKindsofStatistics .....„. 6 Combinations 50
MoreProbability mmm 52
CHAPTER 2 TheBinomialDistribution ..„«. 53
TheTheoreticalMeanoftheBinomial
Pictorial Description of Data Distribution _ 60
Introduction „ _ 8 TheTheoreticalVarianceoftheBinomial
SelectingaRandomSample _ 8 Distribution 63
ClassificationofData _ 9 Exercises.M ... „^._ 65
FrequencyDistributionsandCumulative
FrequencyDistributions .. .. 11 CHAPTER 6
GraphicalRepresentationofData
Histogram „ 12 The Normal Distribution
FrequencyPolygon 14 Introduction _....„ 66
Ogice 14 TheNormalDistribution 67
Exercises........... ... ..^.. 15 UseofStandardNormalTables 68
MoreNormalProbabilities „_ 73
CHAPTER 3 TheNormalApproximationtothe
Binomial „ 75
Measures of Location Theorem 77
Introduction „..„ ........ 16 Exercises.^ M_ 81
TheMidrange 16
TThheeMMeoddiean „.„ 1177 CHAPTER 7
TheArithmeticMean - 17 Some Tests of Statistical Hypotheses
TheMedianofClassifiedData ,. 18 Introduction — 82
SummationNotation » 20 TheNatureofaStatisticalHypothesis
The MeanofClassifiedData 22 TwoTypesofError 82
Exercises _•• ............. 23 TestofHo:r—toversusaSpecified
Alternative «._. 84
TestsabouttheMeanofaNormal
CHAPTER 4
Distribution 89
Measures of Variation Exercises .................._.... 91
Introduction — 24
TheRange 25
CHAPTER 8
TheMeanAbsoluteDeviation - 26
TheVarianceandtheStandardDeviation.... 27 More Tests of Hypotheses —
TheVarianceandStandardDeviationof Introduction 92
ClassifiedData - 31 TestsofHo:u.—no, NormalPopulation,
Exercises..^...........-~ — — 34 a2 Unknown _ 92
Testsaboutthe Meanofa Non-Normal
CHAPTER 5 TesPtospualbaotuitontheDifferenceofTwo _. 95
Elementary Probability and the Proportions 97
Binomial Distribution TestsabouttheDifferencesofTwoMeans... 101
Introduction................ ._ 35 Exercises - 106
. ..
TABLE OF CONTENTS
CHAPTER 9 The Rank-Correlation Coefficient 146
TheSignTest(OnePopulation) 149
Correlation and Regression The WilcoxonSigned-RankTest 152
TheSampleCorrelationCoefficient 107 The Rank-SumTest(TwoPopulations). 154
Computationofr 109 Exercises 157
TestingHypothesesaboutthePopulation
CorrelationCoefficient 110
Linear Regression 113 CHAPTER 12
FindingtheRegression(Least-Square)Line 114
The Analysis of Variance
TestingHypothesesaboutnina
RegressionProblem 117 Introduction 158
TesRteignrgesHsyipoonthPersoebsleambout/3 ina 119 OOnnee--WWaayyAAnnaallyyssiissooffVVaarriiaannccee— 159
Exercises 120 AnotherApproach 164
One-WayAnalysisofVariance, Different
SampleSizes 168
CHAPTER 10 Two-WayAnalysisofVariance 173
,
Exercises 177
,
Confidence Limtts
Introduction 121
A NoteonInequalities 122 APPENDIX
ConfidenceIntervals forn 123
Some Notes about Desk Calculators 178
Confidence Intervalfortt — 127 List of Selected Symbols 179
ConfidenceIntervalforyt\ —yti. 128 Tables
ConfidenceInterval forw\ W2. 133
Area oftheStandardNormalDistribution 180
ConfidenceIntervalforp 134
/-Distribution 182
Exercises 136
X2-Distribution 182
F-Distribution 183
CHAPTER 11 Fisher-z Values 187
SpearmanRank-CorrelationCoefficient... 187
Non-parametric Statistics WilcoxonSigned-Rank Values 187
Introduction 136 Rank-SumCriticalValues 188
TheChi-SquaredDistribution. 137 Answers to Exercises 189
ContingencyTables 143 Index 191
CHAPTER
1
WHAT
IS STATISTICS?
Inordertostudythesubjectofstatisticsintelligentlyweshouldfirstunderstandwhatthetermmeanstoday,
and know something ofits origin.
As with most other words, the word "statistics" hasdifferent meanings to different persons. When
most people hear the word theythink oftablesoffiguresgiving births,deaths, marriages,divorces, auto-
mobile accidents, and so on, such as might be found in the WorldAlmanac, for instance. This isindeed a
vitalandcorrect use oftheterm. Infact,theword "statistics" wasfirstapplied totheseaffairsofthestate,
to data that government finds necessary for effective planning, ruling, and tax-collecting. Collectors and
analyzers of this information were once called "statists," which shows much more clearly than the term
"statistician" the original preoccupation with the facts ofthe state.
Today, of course, the term "statis—tics" is applied, in this first sense, to nearly any kind of factual
informationgivenintermsofnumbers theso-called"factsandfigures." Radioandtelevisionannouncers
tell usthat they will "havethestatisticsofthegamein afew minutes," and newspapersfrequentlypublish
articles about beautycontests giving the "statistics" ofthe contestants.
The term "statistics," however, hasother meanings, and people who have notstudied the subject are
relatively unfamiliar with these other meanings. Statistics is a body of knowledge in the area of applied
mathematics, with its own symbolism, terminology, content, theorems, and techniques. When people
study the subject, statistics, they usually attempt to master some ofthese techniques.
Theterm "statistics" hasasecond meaningforthose who have been initiated into the mysteriesofthe
subject "statistics." In this second sense, "statistics" are quantities that have been calculated from sample
data; asinglequantitythathasbeensocalculatediscalled a"statistic." Forexample, the sample mean isa
statistic, as are the sample median and sample mode(alldiscussed in Chapter 3). The sample variance isa
statistic, and so is the sample range (both discussed in Chapter 4). The sample correlation coefficient
(discussed in Chapter 9) is a statistic, and so on.
Wecan summarize these meanings ofthe word "statistics":
1. The public meaning offacts and figures, graphs and charts. The word is plural when used in this
sense.
2. Thesubjectitself, withaterminology, methodology, andbodyofknowledgeofitsown.Thewordis
singular when used in this sense.
3. Quantitiescalculated fromsampledata.The word isplural when used in thissense.
In this book we will not use the word "statistics" at all in the first sense above. When we want to
refer to "facts and figures" we will use theterm "observations," or the term "data." We will occasionally
refer to a quantity that has been calculated from sampledata as a "statistic." In these cases, we will be
using the singular ofthe word "statistics," in the third sense above. Nearly always, when we use the word
"statistics" we willmean the subject itself, the body ofknowledge.
The methodology of statistics is sufficiently misunderstood to give rise to a number of humorous
comments about statistics and statisticians. For example: "A statistician is a person whodraws a mathe-
matically precise line from an unwarranted assumption to a foregoneconclusion."Thisstrikes out at two
abuses of statistical techniques, although the abuse is not by professional statisticians. In order to apply
most statistical techniques, certain assumptions must be made, the number and scope ofthe assumn.ons
varying from situation to situation. Perhaps some persons do make assumpt.ons that they
not justified, and disguise their doubt. And perhaps, also, some persons do have a conclusio! . ready
decided upon, and then choose their sample or "doctor" theirdata in order to "prove" the.r c
Each ofthese abuses, when knowinglydone, isdishonest.
6 STATISTICS MADESIMPLE
One indictment of the techniques and methodology of statistics says that "statistical analysis has
often meant the manipulation ofambiguousdata by means ofdubious methods to solve a problem that
hasnot beendenned."
Probablythe remarkthat is known best is the one attributed by MarkTwain to Disraeli: 'There are
three kinds oflies: lies, damned lies, and statistics." And yet another well-known remark,—critical of the
manner in which statistics are used, is, "He uses statistics as adrunk uses a street lamp for support,
rather than illumination."
THE PRESENT IMPORTANCE OF STATISTICS
The application ofstatistical techniques is so widespread, and the influence ofstatistics on our lives
and habits so great, that the importance ofstatistics can hardly be overemphasized.
Our present agricultural abundance can be partially ascribed to the application of statistics to the
design and analysis ofagricultural experiments. This is an area in which statistical techniques were used
relativelyearly. Some questions that the methods of statistics help answer are: Which typeofcorngives
the best yield? Which feed mixture should chickens be fed so that they will gain the most weight? What
kind ofmixtureofgrassseedsgivesthemosttonsofhayper acre? All ofthesequestions, and hundredsof
others, have adirect effect on all ofusthrough thelocal supermarket.
The methodology of statistics is also used constantly in medical and pharmaceutical research. The
effectivenessofnewdrugsisdeterminedbyexperiments,firstonanimals,andthenonhumans.Newdevelop-
mentsin medical research and newdrugs affect most ofus.
Statistics is used bythe government as well. Economicdata are studied and affect the policies ofthe
government in the areas of taxation, funds spent for public works (such as roads, dams, etc.), public
assistance funds, and so on. Statistics on unemployment affect efforts to lower the unemployment rate.
Statisticalmethods areused toevaluatethe performance ofevery sort ofmilitaryequipment, from bullets
used in pistols to huge missiles. Probability theory and statistics (especially a rather new area known as
statisticaldecisiontheory)areused asanaidinmakingextremelyimportantdecisionsatthehighestlevels.
Inprivateindustry theuses ofstatistics are nearly asimportant and theireffectsnearly aswidespread
as they are in government use. Statistical techniques are used to control the quality of products being
produced and to evaluate new products before they are marketed. Statistics are used in marketing, in
decisions to expand business, in the analysis of the effectiveness of advertising, and so oh. Insurance
companies make use ofstatistics inestablishing their rates at a realisticlevel. —
The list could go on and on. Statistics is used in geology, biology, psychology, sociology in any
areain whichdecisions must be made on the basis ofincomplete information. Statisticsis used in educa-
tionaltesting, insafetyengineering. Meteorology, thescienceofweatherprediction, isusingstatisticsnow.
Even seemingly unlikely areas use statistics. Who would think that statistics could help a literary
scholar or a historical sleuth determine the authorship ofdisputed documents? Perhaps the best-known
instance of this is the use of statistical techniques to settle the long controversy over who .wrote those
essaysin The FederalistPapers whose authorship had beendisputed.
Onthelighterside, statisticalstudieshavebeenmadeoftheeffectofthefullmoonontroutfishing; of
which oftwo kinds ofwater glasses are better for use in restaurants; and of the optimum strategies for
games ofskill and chance such as bridge, solitaire, blackjack, and baseball.
There can be littledoubt, then, ofthe effect ofstatistics and statistical techniques on each ofus.The
resultsofstatisticalstudiesareseen,butperhapsnotrealized,inourpaychecks, ournational security, our
insurance premiums, our satisfaction with products ofmany kinds, and our health.
TWO KINDS OF STATISTICS
Inadditiontoabriefconsiderationofthebasicelementsofprobability,therearetwokindsofstatistics
treated in this book. In Chapters 2, 3, and 4 we are concerned primarily with thedescription ofdata. In
Chapter2 wetreatthe pictorialdescriptionofdata; inChapters3 and 4wetreatthe numericaldescription
ofdata. The natural name for this kind ofstatistics isdescriptive statistics. The classification ofdata; the
drawingofhistogramsthatcorrespondtothefrequencydistributionsthatresultafterthedataareclassified;
WHATISSTATISTICS? 7
the representation ofdata by other sorts of graphs, such as line graphs, bar graphs, pictograms; the
computation—ofsamplemeans,medians,ormodes;thecomputationofvariances,meanabsolutedeviations,
and ranges all these activities deal with descriptive statistics. The statistical work done back in the
nineteenthcentury andtheearlypart ofthiscentury waslargelydescriptive statistics.
The secondimportantkind ofstatisticsisknown asinferentialstatistics. Statisticshasbeendescribed
asthescienceofmakingdecisionsinthefaceofuncertainty;thatis,makingthebestdecisiononthebasisof
incomplete information. In order to make a decision about a population, a sample (usuallyjust a few
members) ofthat population is selected from it.The selection is usuallyby a random process. Although
there arevariouskindsofsampling, thekind that wewill be assumingthroughoutthisbookisknown as
randomsampling.Asthetermsuggests,thisisakindofsamplinginwhichthemembersofthesampleare
selected by some sort of process that is not under the control of the experimenter. There are various
mathematicaldefinitionsofrandom sampling,butwewillconsiderit asasampleforwhicheachmember
ofthe population has anequalchance ofbeing selected, and for whichthe selection ofany one member
doesnotaffecttheselectionofanyothermember.
Onthebasisoftherandomsample,weinferthingsaboutthepopulation.Thisinferringaboutpopula-
tions on the basis ofsamplesisknown asstatisticalinference. In other words, statistical inference is the
use ofsamples to reach conclusions about the populations from which those samples have beendrawn.
Let us mention several examples of statistical inference. Suppose that a manufacturer of tricycles
buys bolts in large quantities. The manufacturer has the right to refuse to accept the shipment ifmore
than 3 per cent oftheboltsaredefective.Itisnotfeasible, ofcourse, tocheck all oftheboltsbeforethey
areused.Thiswouldtaketoolong.Neitherisitpossibletosimplylayasidethedefectiveboltsastheyare
encounteredduringthe assemblyofthetricycles.Theboltscannotbereturned aftertheyhavebeen used,
evenif20percentaredefective; and, ofcoursethetricyclemanufacturerdoesnotwanttouseashipment
ofboltsthat containsalargepercentageofdefectives, becauseitisexpensivetoattempttouse adefective
bolt,realizethatitisdefective,andthendothejobagainwithasatisfactoryboltSoforseveralreasons,the
manufacturerneedstohaveaquick,inexpensivemethodbywhichhecandeterminewhethertheshipment
containstoomanydefectives.Soheobtainsarandomsamplefromtheshipmentofbolts, and onthebasis
ofthepercentage ofdefectives in the sample, hemakesa decision about thepercentage ofdefectives in the
population (the shipment).Thisisanexampleofstatisticalinference.
Consider another example of statistical inference. A medical research worker wants to determine
whether a new drug is superior to the old one. One hundred patients in a large hospital are divided at
randomintotwogroups.Onegroupisgiventheolddrugandtheothergroupisgiventhenewdrug.Various
medicaldata are obtainedforeachpatientonthedaytheadministration ofthedrugbegan, andthe same
things measured tendayslater. By analyzingthedataforeach group, and bycomparingthedata, acon-
clusion canbereached about therelativeeffectivenessofthetwodrugs.
AsimilarexampleisdiscussedatgreaterlengthatthebeginningofChapter7,inwhichthetestingof
statistical hypothesesisfirstdiscussed.
Theusual procedurefor testing a statisticalhypothesisisthe following: Ahypothesis, known asthe
null hypothesis, is proposed about a population; a random sample isobtained from thepopulation, and
a numerical quantity, known as a statistic, is calculated from the sample data. The null hypothesis is
accepted or rejected, depending upon the value ofthe statistic. (An alternative hypothesis is formulated
atthe sametime asthenull hypothesis, andrejection ofthe nullhypothesismeans automatic acceptance
ofthe alternative hypothesis.)Thus, the testing ofa statistical hypothesis is an illustration of statistical
inference,becauseadecisionismadeaboutapopulationbymeansofasample.Chapters7,8,9,11,and12
areconcerned(somepartially,someentirely)withtestsofstatisticalhypotheses,andthereforewithstatisti-
calinference.Chapter10,inwhichanimportanttopicknownasconfidenceintervalsisdiscussed,alsodeals
with statisticalinference. •__*.
Insummary, thesubjectmatterinthisbookfallsrathernaturallyintothreecategories.Chapters2, 3,
and 4 treat the description ofdata, both graphically and numerically, and are classified as descriptive
statistics. Someverysimpletopicsfromp-robabilitytheory are discussed i-n Chapters5 and 6: elementary
probability, two importantdistributions the binomial and the normal and how thetwo are related,
andtheuseofthenormaltable.The final six chapters treat selected topics, mainly about testing hypothe-
ses,fromthatpartofthesubjectmatterofstatisticsknownasinferentialstatistics.Thefirsttwocategories,
comprising the first six chapters, arepreliminaryto thelast one.
CHAPTER
2
PICTORIAL DESCRIPTION OF DATA
INTRODUCTION
This chapter is concerned with the presentation ofsampledata. Before treating the classification ofdata
and the sketching ofhistograms, we will brieflydiscuss the idea ofarandom sample and how one can be
obtained.
Ifone is sampling from a population composed ofan infinite number ofelements, a sample selected
in such a manner that the selection ofany member ofthe populationdoes not affect the selection ofany
other member, and each member has the same chance ofbeing included in the sample, iscalled a random
sample. If one is sampling from a finite population with replacement (each member is returned to the
populationafterbeingselected, andmightbeselectedmorethanonce), arandomsampleisdefinedexactly
as above.
Ifoneis samplingfrom afinitepopulation withoutreplacement(theelements arenotreturned to the
population after they have been observed), then we say that a sample is a random sample if all other
samplesofthesamesizehaveanequalchanceofbeingselected. Nosampleisanymorelikelytobeselected
than any other.
Theword "random" indicatesthatthe sampleisselected insuch awaythatitisimpossibleto predict
which members ofthe population will be included, and that it is simply a matter ofchance that any par-
ticular member is selected. In order to applythe statistical techniquesexplained in thisbook in analyzing
sample data, it is necessary that the sample be a random one (with very few exceptions). The statistical
techniques are justified by statistical theory, which in turn rests upon probability theory, and we must
haverandom samples before the probability theory is applicable.
SELECTING A RANDOM SAMPLE
Itissometimesnot an easy matter to obtain arandom sample. Ifthe population is small, one ofthe
simplestwaysofobtainingarandomsampleistolistthemembers(onsmallpiecesofpaper, for instance)
and draw the sample "out of a hat." Perhaps you remember seeing the famous picture of the Secretary
ofWarintheearly 1940sdrawingfrom alargecontainerthenamesofthefirstmentobeinductedinto the
Armyunder thenewlypassed conscription act.Thisis an excellentexample ofdrawing a sample "from a
hat."
Whenever anintegercanbe assignedtoeachmemberofthepopulation,arandomnumbertable can be
used to obtain arandom sample.Thistable is a listing ofdigits that have been obtained by some random
process. One way of assigning an integer to each member of the population is simply to number the
members 1, 2, 3, and so on. (Sometimes the members cannot be conveniently numbered, in which case
there are other methods ofobtaining a random sample by means ofrandom numbers.) Each member of
thepopulationhasacorrespondingnumberintherandomnumbertable(orperhaps more than one corre-
spondingnumber).Toobtain a random sample, we would begin reading numbers in the random number
table at some randomly chosen place, and for each random number read, the member ofthe population
that corresponds to that number is included in the sample. For instance, ifour population consists ofa
thousand members, wecould assign them numbers from 000 to 999. If we read the numbers 027, 831,
and 415 in the random number table, we would include in the random sample those members of the
population whose numbers are 027, 831, and 415.
ThedatashowninTable2-1 arethescoresthatonehundredstudentsobtainedontheverbalportionof
the Scholastic AptitudeTest; we shall refer to these scores as the SAT-Verbal scores. The sample was
obtained from a population offreshmen students, using atableofrandom numbersto guarantee that the
sample wasrandom.
Manystatisticstextbookscontain arandomnumbertable anddiscussitsuse. See Wallisand Roberts,
Statistics:A NewApproach,orDixonandMassey,IntroductiontoStatisticalAnalysis.