aonccpt!; ofpopuidtion and sanpl

8/14/2019 Aonccpt!; OfpopuIdtion and Sanpl.

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aonccpt!; ofpopuIdtion and sanpl.Rdndom, sr-ati.fied, sy:ttematic dnd quota sdn'plcsAdvantages und disadt,antages ofthe rarious sanpling nefhodsSANIPLE AND POPULATION

In many practical statislical illvestigations, \re are iilerested in gathering information about alarge group ofindividuals or obtects (knowr') as .lpqpg!41i9!). It should be notcd that in anyinvestigalion, the population nccds to bc clcarly dcfincd. Howevcr, in many cases, it is notpossible to obtain infbrmation about all members of a population for the following reasons:

The collectio or tesling ofinfotmution on every individual or object ofthe populationtqt destroy the popalation.The population ntay be inJinite or very large.The collection ol ittlbtuation on all ntembers in tlle population muy be too costly ortoo ti te consu ting.

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In such cases, wc gather inlbnnation about tl'le population based on data denved from a!4!IIp!9 of the population.A sample is a linile suhset ol the population'fo gather rseful infomation, the sample musl be r"epresentdtire of lhe population. A sampleis clescribed as rzpresdrrairiyd ifjt is believed that it :t'airly and complehcnsivcly represents thepopulation from u'hich it is drawn. A sample is llosey' ifit over-represcnts or under-repr-esentsa parlicular subgroup. The words 'unbiased' and 'rqrresentative' are usually interchaDgeabic.SAMPLING M-ETTIODSSaDpling is coilecting datd Aom a representative sample ofa population. [n the following, vr'eintroduce 4 tlpes ofsampling, nalnely,. Random sampling. Stratified sampling. Systcrnatic sampling. Quota sampliDgSome general comments before the details of each method :The f,rst 3 t]?es are refered to as probabilistic sampling in which every member of thepopulation has an equal chance of beins selected. The sample obtained is called a !q!!!q!4sample. In each ofthese sampling methods, a !4!!p!Ig-!A!E-(sec section below) is needed.Quota sampling is a t)?e of nonprobabilistic sampling and the samplc obtained is non-tundom. A sampling frame is ol eeded in quota sampling.We are more likely to represent the population well through probability sampling. ln general,researcheN prefer probability sampling methods and consider them to be more accurate andrigorcus.

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Eranrple l.lwhich ofthc following is a random sample'l(i) John picks out 2 idcntical balls frol! a bag of2l, without replacenent.(ji) Mark picks out 2 identical balls from a bag of 2 l. with replacetncnt.(iii) Matthew picks oLlt 2 identical balls lron 2 bags, onc from a bag of l0 al1d onc fionl abag of l l.(iv) Lukc picks out 3 idcntical balls liom 3 bags of7, onc from each bag.SolutionRecall:In a sarnpling method, if every member of the popllation h^s an cq!4!. !L4!e! !LLb!i!!zre1ectel, thell the salnplc obtained is a random sample.(D The probability of selecting thc first ball is 1/21. Sincc the tirst ball is not replaccdbcfore the second ball is drawn, the probability of selecling lhe second ball is l,/20.Thus, rve see that cach me'mber ofthe population does not llave an cqual chance ofbeing sclccted and we conclude thal tire sample is non-randoln.(ii) Since the lirst ball is rcplaccd belbre the seconcl ball is drarvn, lhe probabilily ofselecting tlle sccond ball is l/21. Ihus, we see llut elch nember ofthe populatioi hasilll ealual chancc ofbeing selected and we conclude thal the sample is rand()]n.(iji) Thc probability of selecting a bdll liom the bag of l0 is li l0 and the probability ofselcctirg a ball liom the bag of 11 is 1/11. Thus, wc scc that cach mcmbcr of thepopulation does not have an equal chance of being selecled and we conclude that thesample is lon-random.

The probability of selecting each ball from a bag of 7 js l/7. Thus, we sec that eachrnernber of the population has an equal chance of being selected and we conclude thatthe saDpie is rand(nn.

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Examplc 1.2 (Do it yourself)(Adapted from IJC Prelin Exam 2007 P2 Q5)A kindergarten has niDc Ycar One classes and six Year 'l wo classes with dillerent class sizes.Thc Principal of the kindergaden wishes to take a sample of 30 childrcn to mcct up with aministry oflicial for a shofi conversation- She chooses 2 children at random fi-om each of thei5 classes. Statc, with a reason, whether this gives a random sample of30 children from thekindergartcn.Solution


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The Sampling Franrc1'he sampling fiane is a qgn\llglLIt! (to bc ideal), in somc lbrm. of all the mcrntrers in thcpopulation. fro:n which a samplc can be selected.For examplc, in a surv_ey oi employces of a company commissioned by ils managenleDt, it isrcasonable to expcct to be able to obtain a comprchensive, up{o-date and accuratc list of 4Llcmployees, with suffi cicnt classiflcatory infonnation-Other cxanrples ilclude population rcgistcrs or electoral registers of countds. registem ofschools and rccords ofserial numbers ii manuiactured goods.llowever, in nlany situations, it may be dilficult to maintain a complctc and up-to-date iist olthc populatiorl. Even if such a list cxists, due to the general trend towards grcatcr fomalprotcction lor privacy ofthc individual. infomtation may be withheld.I sanpling franc i.t nrcded itl any satnpling lctholl that is random.Due to diflicultics in obtaining a valid sampling liamc, thcrc is a need lor us to studynonprobabjlistic sampling rnethods such as quota samplinS.(1) Simple Random Sampling

In simple random sampling. cach individual in lhe sample is chosen liom the populaiionpurcly by chance. Each mcmbcr ofthe populatiol1 has an equal chance of being selected.To obtain a aandom sample ofsizc z frorn a populiltio[ ofsize N, wel) obtain a sampllng frame 01:the population,2) number the members ofthe population, say from I to N,3) tnake a random selection of n ofthese l1umben, and4) identi{y thc members ofthe populrtion leprcsented by the numhcrs sclcctcd.Note that randon seiection of the numbe$ can be done mechanically through wdtingach nrunber that represents a member of thc population on a slip of papcr, mix themreally well in a container and drawing the requircd numbcr of slips out. Alternatively, wecan usc computers to genemte the random numbcrs selected.Advantagcs(i) Aralysis ofdata is rclatively casy.(ii)The data collected is not biased.Disadvantages(D It is dil'ficult or impossible to idcntily cvcry mcmbcr of the population, to beissued a number(difficulty in obtaiDiog the sampliDg frame).(it Even if we can identily every member ofthe population, we may not be able toget access to some members who have been chosen fbr the sample.

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(2) Stratified SamplingIn cerlain invcstigations, vaiablcs such as age and gender in the population havc aninllucnce on the results. For exarnple, iD a study on thc tastc of music, age of theindividuals studied will inlluence thc responses signilicantly. In such r case, i1 wi]l beuseful to ensure that the sample represents the proportions of nutually exclusivesubgroups (by age), in the population.Stratifi ed Sampling involvesI ) dividing thc polulation into mutually exclusive subgroups/s(rala, and2) randomb selectins(using lha samplitlg fiz,ze) a sample liom each stratum, withsample size proportional to the relative size ofthe stratum-

Suppose that in a company thc proportions of staff in diffcrcnt age-groups are aslbllows:Aged under 40 Agcd bctwccn 40 and 60 Aged over 6038% 10% 2204

To obtain a sanple oi'200 stafl, we draw random samples Aom the age-groups withsample size in the same proportion as the size ofeach age-group.Ag-group Undcr 40 Betwecn 40 and 60 Over 60

Sample sjze 38% of200(:76) ,10% of200(:80) 22Yo ol 20()(:44)Advantages(D It is more likely lo give a good representative sample ofthc population.(i0 The data obtaiDcd from each stratum can be analyzed separately, and thisusually gives more accurate estimates of the population parametrs, comparedto simple random sampling.Disadvantages(i) lt may be hard to obtain the sampling frame needed(ii)Morc difficult to conduct as compared with (simple) random sampling. 'fhere may not

be sufficient information about the population to delineate the subgroups..(iii) 'Ihe strata may not be clearly delined.(it, lt is relatircly more time consuming.

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(3) SystemnlicSrmplingSystematic sarnpling is done tluough the following steps:l) Obtain a samplirg l'rame ofthe population.2) Nurnbcr the metnbers ofthe poplrlation, sry from I to ly'-3) l o draw a sample of size n Aon a population of size N, let /r:rVh.4) Randomly select an integer bctwcen I to k .5) Then takc cve.y lr' integer aller that.6) Identily the members ofthe populalion reprcsented by thc numbers selected.For example, to d.aw a sample of 250 tickels lrom a population of 5300 lickets, we nay takc /r5300:20 (--). In this casc, wy srlect every 20rh ticket commencing with, say the 8'h ticket.250(Thc nurnber 8 is randomly delennined.) In other words, we select the 8'h, 28th, 48't', (r8th, ......

Advantagcs(D It is casicr to conducl as compared with other types ofsarnpling(ii) It is morc cvenly spread over the populalion. The use of wide spacing of k individualslJuards against the list consisting ofclusters of similar irdividuals.

Disadvantages(i) It is not always possible to obtain a valid sampling frame of the population and torlrnnber each in{liv;dral(ii) 'lhe samplc obtained can be biased when the rrrembers ol the populatioD havc aperiodic or cyclic pattcm of occurrlrncc jn tho sampling framc. As an cxarnple, if asampling tame consists of a list ol rnanied couples in the order of husband. wifc,husbanci, wife, husband, wifc-...ctc, thcn if evcry tcnth pcrson is selected afterchoosing lhe lirst ofthe husbands, then the sample will likely to consist only ofrnalesl(4) Quota SamplingQuota sampling is similar to stratified sampling. ln quota sampling, the populatioD is alsodivided into mutually exclusive subgroups, but the sample is non-mndom.

For exanpie, in a market sufley, an interviewer is given an assignment to interview 20 pcoplein the street based upon quotas that specify or define the sample to be drawn, e.g.Sex Ag" Sociai classMale Fernale 20-29 4 High 29 tl 3o-44 6 Middle 4

45-64 '/ Low t,l65+ 3

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Thc choice ofthe sample is subjectivc. Thc jnten'ierver chooses potential respotdents to mcctthe rcquired quola. A possible oulcome oflhe profilc ofthe respondents is as lbllows:

Social classHich Middl e Lou,

Age/Scx M Ij M F M F20-29 I30-44 345-61 2 3651 2

Advanlases(D Cost is lower bccausr- thc sample size can be smaller to meet the objectivc.(ii) Thc intbmration can be collected quickly.(iii) Thc sampling does not rcquire a sampling iia1ne.Disadvantages(i) It is not a good reproscntativc of the population as compared with othe. t)?es ofsampliDg.(ii) It is non-random(iii) bias as the interviewer may simply sclcct thosc who arc casicst to interviewNB: Quota sampling is vcry useful and convcnicnt though it is non-mndom in natue.Infonnation gathered from this t)?e ofsample should be treated with caution.f,xample 1.3A samplc of5 is to be sclccted from a class of 15 boys and 10 girls. Describe how you wouldchoose the committeo using(i) random sarnpling(ii) stratilied sampling(iii)quota san'rpling(iv) systematic samplingSolutiol(t) To form a sampling Aame, assign a number to each student in the class. Write the

numbers I to 25 on stips of pape6, put them in a containet mix them up well anddrarv 5 numbers out. Select the 5 students assigned to these 5 selected numbers.(iD Since the propofiions ofboys and girls in the class are 15/25 and 10/25 respectively,we ned to randomly seloct (15/25)x5=3 boys and (10/25)!5 : 2girls. We assign thenumbers I to 15 to thl- boys and randomly selcct (as in part (i)) 3 numbers. We select

the 3 boys that are assigned these 3 numbers. Similarly, we assign the numbers I to l0to thc girls and random select 2 numbers bcfore selecting dre coresponding girls.

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Wc decidc on the proportions of boys ancl girls in the sample of 5. 'l'hey need not bcl5/25and 10/25. Suppose we decided to sclcct 3 girls and 2 boys, we caD selesl any 3girls. i'rorn l0 in thc class, aDd any 2 boys, from l5in thc class, that arc tnostcooperative or convenient to us. They are not randomly selectcd within their subgroupsaunliLc slrclr ic(l silmplinB).Number the studeDts in the class Aom I to 25 (formiDg the sampling frame). Sincesample size is 5 and population size is 25, f: 2515= 5. Select any number from 1to I(=5), say we select 3. Thcn thc studcnts to be selecled are those numbered 3, 3+k,3 f2k,. -..-i.e. those numbered 3,5,13, 18, and 23.

Example 1.,{An eftployment agcncy wtuits to estimate the number of unemployed people in a HDB newtown, and so decide to obtain infbnnation lrom a samplo of its adult who is of working age.Cive one reason rvhy it would not be appropriate to obtain thc san]ple b1'stoppingadults at the town ceutre's supcrmarkcl during one working day.SolutionIt misses out pcople who do not shop at that supeflna.ket(biased sample).Examplc 1,5In l citr. the fr,lbwin infbnnation is aveilal)le fiom the re stry officcNumber of residents

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A sarnple of40 residents is to bl- chosen for an inteNiew. Discuss the tdllowing samplingmethods:(i) Stratified SampJing(ii) Quota SamplingStatc the advantages aDd disadvdntages ofeach methocl.Solution(i) ln stratified samplr'ng, we \eed to randomly select (74,1/930)x40 Chinese, (93/930)x40Malays, (65/930)x40 Indians and (28i930)xa0 Others. Thus, we randomlv select 32Chinese liorn 744 Chinese,4 Malays from 93 Malays,3 Indians ftom 65 Indians and Iperson from 28 pcople ofothff races.

Advantaees: (1) The sanple represents the population bettcr than samples obtainedthrough othcr sampling methods.(2) The data from each race group can be analyzed separately to give moredetails and thc rcsults can be mote representative ofthe population.Disadvantages:(l) It is more difficult to conduct. 'Ihe sampling frame may not be available and thus itmay be impossible to have random selection within each race group.(2) It is more time consuming. c/- 1

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(ii) In quota sampling, the proportions ofthe different races in the sample need not be thesame as those in the population. Suppose we decide to select 10 people from eachsubgroup. The l0 people in each subgroup need not be randomly selected. we canselect those that are most available.Advantages:(l) Data can be collected faster .(2) Sampling fiame is not required.Disadvantaqesi(l)The sample obtained does not represent the population as wll as samples obtainedthrough other methods.(2)The sample is not mrdomly selected.

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aonccpt!; ofpopuidtion and sanpl

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