UIT 4 SOE OTHER SAPLIG SCHEES Some Other Samplg Schemes Structure 4. Itroducto Objectves 4. Itroducto to Systematc Samplg 4.3 ethods of Systematc Samplg Lear Systematc Samplg Crcular Systematc Samplg Advatages ad Dsadvatages of Systematc Samplg 4.4 Propertes of Systematc Samplg Comparso betwee Smple Radom, Stratfed ad Systematc Samplg 4.5 Itroducto to Cluster Samplg Sgfcace of Systematc Samplg Formato of Cluster 4. Propertes of Cluster Samplg 4.7 Itroducto to Two-stage Samplg 4.8 Propertes of Two-stage Samplg 4.9 Summary 4.0 Solutos / Aswers 4. ITRODUCTIO Whe oe has to mae a ferece about a large lot ad t s ot practcally possble to eame each dvdual ut.the a few uts of the lot are eamed ad o the bass of the formato of those uts, oe maes decsos about whole lot. I prevous ut, we have dscussed about stratfed radom samplg. As we stated Ut 3, a stratfed radom sample s selected radomly from the groups of populato uts amed strata, where the strata are formed of the homogeeous uts. But, f the formato of the strata s ot doe proper maer, the the results would be based. So, to overcome ths drawbac, t s suggested to adopt aother samplg method ow as Systematc samplg. The tematc radom samplg s oe of amog the med samplg schemes, whch s partly probablstc ad partly o-probablstc. A bref troducto of tematc samplg s gve Secto 4., whereas the methods of selecto of tematc radom samples are eplaed wth eamples Secto 4.3. Propertes of the tematc radom samplg terms of mea ad varace are derved Secto 4.4. I Secto 4.5 the otatos ad the methods of cluster samplg are eplaed whereas the propertes of the cluster samplg are descrbed Secto 4.. Smlarly, the bref troducto about the two stage samplg s gve Secto 4.7 ad the propertes of the two-stage samplg are dscussed Secto 4.8.
Statstcal Techques Objectves After studyg ths ut, you would be able to defe the tematc radom samplg; draw a tematc radom sample ad estmate the populato mea; obta the varace of the estmate of the populato mea; defe the cluster samplg ad obta the estmate of populato mea; ad defe ad epla the two-stage samplg scheme. 4. ITRODUCTIO TO SYSETATIC SAPLIG I prevous uts, we have dscussed those samplg techques where uts were selected radomly. I ths ut, we shall dscuss a samplg techque whch has a ce feature of selectg the whole sample wth just oe radom start. Systematc radom samplg s commoly employed f the complete ad up-to-date lst of populato uts s avalable. I tematc radom samplg oly the frst ut s selected wth the help of radom method ad the rest beg automatcally selected accordg to some predetermed patter. The tematc radom samplg s a d of med samplg, whch s partly probablstc ad partly o-probablstc. Ths s radom sce the frst ut of the sample s selected at radom ad o-radom or purposve sce the rest of uts the sample are selected by predesged patter. 4.3 ETHODS OF SYSTEATIC SAPLIG Geerally, day to day lfe, we have to obta the formato from cards or regster whch are full of formato arraged seral order. For eample, the boos lbrary, a telephoe drectory, etc. I such case tematc samplg ofte wors better tha smple radom or stratfed samplg. There are two ways of selecto of a tematc radom sample ad these are (a) Lear tematc samplg ad (b) Crcular tematc samplg. 4.3. Lear Systematc Samplg I order to draw a tematc radom sample of sze from a populato of uts, Let us suppose that samplg uts are serally umbered from to some order. Let =, where s the sample sze ad s a teger. Therefore,. I other words, order to draw a sample of sze from we dvde the total umber of uts to equal parts. Suppose each part cossts of uts. From the serally arraged to uts, draw a ut wth radom method. Let the selected ut s th ut where. The selected ut would be the frst sample ut. The select every th ut after the th ut order to select rest of (-) uts. Thus, the tematc sample of sze wll cossts of th, th, th,..., ut. The radom sample ut.e. the th ut s called the radom start. For eample, suppose there are 00 uts a populato serally umbered to 00 uts. We wll dvde the whole th
populato to 0 equal parts of 0 uts each f we have to draw a sample of sze0: 00 0 0 The we draw a ut radomly from to 0 uts ad let the selected umber s 7.e. = 7. The we select rest of 9 uts a tematc way.e. th, th, 3 th,..., 9 th ut from the serally ordered 00 uts. The the tematc sample cossts of 7 th, 7 th, 7 th, 37 th, 47 th, 57 th, 7 th, 77 th, 87 th ad 97 th uts. Some Other Samplg Schemes 4.3. Crcular Systematc Samplg Suppose a populato cossts of uts ad from ths we have to select a tematc sample of uts. Also, assume that s a multple of.e. =. The procedure s to select a radom umber, let t be such that ad the we have to select th ut from frst ut ad the every th ut.e. ( + ) th, (+) th,,( + (-) ) th postoal uts. Ths samplg techque s ow as lear tematc samplg. But geeral does ot be always a 7 multple of. For eample = 7 ad = 4, the 4. 5 has to 4 be tae as 5. ow we select a radom umber betwee ad 5 ad suppose t s 4. The the remag three uts to be selected are at postos 9, 4, 9. There s o ut the populato at seral umber 9. Hece ths stuato we ca select a sample of 3 oly stead of 4. I ths stuato, the crcular tematc samplg s used. The crcular tematc samplg s used whe sze of the populato s ot a multple of sample. I ths stuato we tae / as by roudg off / to the earest teger. Wth regard to selecto of a tematc radom sample from uts, we have to select radom umber from to. Let ths umber s. ow we select every (+j-) th ut, whe ( + j) > puttg j =,, 3, tll ut are selected. By usg the crcular tematc samplg we always get a sample of sze. For the same eample dscussed above wth = 7, = 4 ad = 5, let the radomly selected umber from to 7 s 8 ad latter the 3 th, th ad th uts are selected. Whe =, the lear ad crcular tematc samplg plas become detcal. 4.3.3 Advatages ad Dsadvatages of Systematc Samplg Advatages Systematc samplg has some advatages over other samplg schemes whch are gve as:. The tematc samplg s very smple ad s ot very epesve;. The tematc sample s uformly dstrbuted over the whole populato ad therefore, all sectos of the populato are represeted the sample; ad 3. The maageral cotrol of feld wor provdes a advatage over other samplg methods. 3
Statstcal Techques Dsadvatages Systematc samplg has some dsadvatages also alog wth the advatages whch are:. The ma dsadvatage of tematc samplg s that samples are ot geerally radom samples;. The sample sze s dfferet from that requred f s ot a multple of..; 3. I tematc radom samplg the sample mea would ot be a ubased estmate of populato mea f s ot a multple of ; 4. We caot obta a ubased estmate of the varace of the estmate of the populato mea sce t does ot provde samplg error; ad 5. It may provde hghly based estmate f the samplg frame has a perodc feature. 4.4 PROPERTIES OF SYSTEATIC SAPLIG Let j deote the j th member of the th tematc sample (where, =,, ; j=,, ). may be deoted as mea of the th sample,.e. j j,,..., ad The,.. ad S may be deoted as populato mea ad populato mea square as follows: j j,,...,.. j j ad S j.. j j j Theorem : I tematc samplg wth terval sample mea ubased estmator of populato mea.. ad ts varace s gve by S S where, S j ( ) j s a 4 s the mea square amog the uts whch le wth the same tematc sample.
Proof: We have E E = E j j = E j j Some Other Samplg Schemes = j j ow, we have =.. S j.. j = j.. j Covarace term vashes, sce Therefore, = j.. j j j.... j 0 j j S j.... j S S S S 4.4. Comparso betwee Smple Raodm, Stratfed ad Systematc Samplg If the populato cossts of lear tred ad gve by The, ;,, 3,..., 5
Statstcal Techques Therefore, We have S S.. ( ) S = S srswor 3 =.. =. = S srswor for populato of uts. I stratfed radom samplg we have st W S 4 () I our case, there are strata of sze ad we draw oe ut from each stratum so we put =, =, = ad W = / = /. st j S Sce th stratum cossts of uts, we have S = Therefore, st st. ()
For fdg out Also, we have = mea of the values of th sample =.. =... = 3... =.. =. = ( ).. j j Some Other Samplg Schemes = = = = = From equatos (), () ad (3), we get : : st 4 srswor (3) ( ) ( ) : : : : : ( )( ) : (appro) st srswor 7
Statstcal Techques Eample : I a class of Statstcs, total umber of studets s 30. Select a tematc radom sample of 0 studets.the age of 30 studets s gve below: Age: 5 5 4 3 0 3 5 3 4 4 4 0 3 0 0 5 Soluto: We have gve a populato of sze = 30 values of the age of 30 studets. ow, frst of all we arrage all the values together wth ther seral umbers. Therefore, Sr o: 3 4 5 7 8 9 0 Age: 5 5 4 3 Sr o: 3 4 5 7 8 9 0 Age: 0 3 5 3 4 4 4 Sr o. 3 4 5 7 8 9 30 Age: 0 3 0 0 5 After that we obta value as 30 0 3 From frst values.e. to 3, we select a value radomly. Let we select the age 5 whch s d place data. Therefore, our st ut whch selected the sample s = 5. ow rest of the 9 uts we wll select tematcally whch are at the posto (+), (+), (+3),, (+(-)) the gve data. So accordg to the gve data rest of the 9 uts our case would be the age gve 5 th, 8 th, th, 4 th, 7 th, 0 th, 3 rd, th, 9 th posto. Therefore, all the 0 uts whch has bee selected the sample are {5,, 3, 0, 3, 4, 4, 3, 0, }. 8 E) The formato regardg producto of wheat ( Thousad g) 5 dstrcts are collected, for a partcular seaso. Select a tematc radom sample of 7 uts from the data gve below: 3, 0,30, 37, 7, 3, 3, 3,, 58, 53, 83, 0, 5,3, 7,,, 7,, 0, 8,, 3, 7. E) A data of 50 values of heghts ( cm) accordg to the roll o. of the studets s gve as follows: 4, 5, 5, 7, 78, 80, 7,, 48, 53,, 73, 3, 4, 75, 8,, 80, 73, 85, 9, 7, 8, 73, 45, 53, 54,, 4, 70, 7, 0,, 58, 5, 3, 5, 70, 8, 58, 49, 55, 0, 50, 49, 7, 7, 9, 59, 0. Select a tematc radom sample of sze 0.
4.5 ITRODUCTIO TO CLUSTER SAPLIG Some Other Samplg Schemes The populato has bee cosdered as a group of a fte umber of dstct ad detfed uts defed as samplg uts. The smallest detty cotet a populato s ow as elemet or elemetary ut of the populato. A group of such elemetary uts s ow as cluster. Clusters are geerally made up of whch all the elemets ted to have smlar characterstcs. Whe these clusters are treated as samplg uts ad few of them are selected ether by equal or uequal probabltes the ths procedure s ow as cluster samplg. All the elemets selected clusters are to be observed, measured ad tervewed. The umber of elemets the cluster should be small ad the umber of clusters the populato should be large. For eample, f we are terested obtag the formato or data for mothly average come a coloy, the the whole coloy may be dvded to umbers of bloc ow as clusters ad a smple radom sample of blocs s to be draw. The dvduals lvg the selected clusters would be determed for tervewg to collect the formato. 4.5. Sgfcace of Cluster Samplg Followg are the varous reasos, whch cause problems the selecto of a sample of elemetary uts ad cluster samplg eables us to overcome those problems:. Whe the samplg frame s uavalable, so the detfyg ad tervewg of samplg uts s costly terms of moey, tme cosumg ad eed much labor. For eample a lst of households metro cty, lst of farmhouse owers a state, etc.;. The locato of the detfed samplg uts may be stuated far apart from oe aother ad cosume a lot of tme ad moey to survey them; 3. It may ot be possble to fd well detfable ad easly locatable elemetary uts. Thus, to overcome the above problems, cluster samplg yelds satsfactory results samplg of elemetary uts. The elemetary uts are formed groups o the bass of locato, class or area cluster samplg. 4.5. Formato of Clusters Certa precautos should be tae ecessarly whle dealg wth the clusters samplg, whch are gve as follows:. The clusters should be made le that each elemetary uts should belog to oe ad oly oe cluster;. All uts of smlar characterstcs should belog to the same cluster; 3. Each ad every ut of the populato should be cluded ay of the clusters costtutg the populato. I other words, there should ether be overlappg clusters or omsso of uts; 4. All clusters should be heterogeeous themselves; ad 5. Clusters should be as small as possble. 9
Statstcal Techques 4. PROPERTIES OF CLUSTER SAPLIG otatos = umber of clusters the populato = umber of elemets the clusters = umber of clusters the sample j be the value of characterstc uder study for the j th elemet (j=,,, ) the th cluster ( =,,, ) of a populato. j j j s the mea of the j th cluster the populato s the mea per elemet the populato Smlarly, j be the value of characterstc uder study for the j th elemet (j =,,, ) the th cluster ( =,,, ) the sample Let j j s the mea of the th cluster sample 70 ad We have S the S w th ad S b cluster S j s the mea of cluster meas the sample of sze j s the mea squar betwee elemets wth the mea square wth clusters meas the populato. Therefore, S j the populato ad E j E j j s the mea squar betwee the cluster deotes the mea square betwee elemets j j j S deote the tra-cluster correlato coeffcet. Theorem : I smple radom sample wthout replacemet of clusters each cotag elemets draw form a populato of clusters, the sample mea s a ubased estmator ad ts varace s gve by f f S S where, s the tra-cluster correlato coeffcet. b b
7 Some Other Samplg Schemes Proof: We have E E = E = j j = ad ow we have = S b = f where f = /. Usg j j j j j j j j j j j j ρ S Therefore, S f large for S f ace cluster samplg depeds o the umber of clusters the sample, the sze of the cluster, the tra-cluster correlato coeffcet ad the mea square betwee the elemets the populato. The varace of cluster samplg reduces to the varace of smple radom samplg f =. Eample : To determe the yeld rate of wheat a dstrct of Pujab, groups were costructed of plots each. The data s gve the followg table:
Statstcal Techques Plot o. Group Group Group 3 Group 4 Group 5 Group 3 4 5 8 3 3 3 5 5 5 8 8 0 0 3 7 3 7 7 5 0 0 5 5 7 8 0 Select a cluster sample of 3 clusters from the gve data ad fd sample mea. Soluto: I the gve data, groups have bee formed ad we have to draw a sample of 3 groups. Therefore, there wll be 0 possble samples of sze 3 whch may be draw from the populato of sze. Let we cosder oe sample of 3 groups draw from populato of groups s {Group, Group 3 ad Group 5}. Therefore, the sample mea wll be 3 3 5 5 3 5 where, 8 3 3 3 0 ea of Group 7 8 8 0 0 78 eaof Group3 3 3 ea of Group5 7 5 0 0 5 08 5 8 Therefore, 7 3 8 Grad ea 8 7 3 8 48 8 8 = Hece, the sample mea s. E3) A Housg board s costructg the 0 Duple o each of 5 dfferet locatos. The plot areas ( square ft) as gve the followg table: 7 S. o. locato locato locato 3 locato 4 locato 5 800 800 300 00 700 300 700 000 00 000 3 00 500 00 300 700 4 800 000 00 00 300 5 300 00 500 700 00 700 300 000 800 300 7 00 800 300 00 000 8 300 00 000 500 500 9 500 700 700 00 500 0 000 000 000 000 000
Select a sample of clusters wth the help of cluster samplg scheme ad calculate the sample mea. Some Other Samplg Schemes 4.7 ITRODUCTIO TO TWO-STAGE SAPLIG A sample survey as poted out Ut of ths bloc has certa lmtatos, maly regardg the budget ad tme avalablty. Hece survey too may elemetary uts s ofte ot possble. I stratfed radom samplg, a sample s selected of optmum sze from each stratum ad the each ad every ut selected from dfferet stratum s to be observed. I Secto 4.5 of ths ut we have dscussed cluster samplg whch the populato s dvded to some umber of clusters ad clusters were cosdered as samplg uts. All the uts the selected clusters are eumerated completely. It has bee poted out there that cluster samplg s ecoomcally better tha other samplgs but the method restrcts the spread of sample over the populato whch creased the varace of the estmator. Istead of eumeratg all the samplg uts the selected clusters, oe ca select a subsample of detfed ad specfc uts from the selected clusters by the same or dfferet samplg methods. The samplg whch cossts of selected clusters ad the select the specfed umber of uts from each selected cluster s ow as two stage samplg. I ths samplg techque, clusters beg termed as prmary stage uts ad uts wth clusters as secodary stage uts. Ths method ca be geeralzed upto three or more stages ad s termed as mult stage samplg. Whe large scale surveys o dstrct, state or atoal level are to be coducted, t s oe of the most sutable samplg schemes. For eample, suppose we would le to draw the formato about the mothly come of a household a coloy, t s better to select a sample of some blocs or wards ad the households from the selected blocs or wards. I ths procedure, the wards or blocs are the frst stage uts ad the households are the secod stage uts. 4.7. Termologes = Total umber of frst stage uts. = Sample sze of frst stage uts. = umber of secod stage uts each frst stage ut. m = umber of secod stage uts the sample from each frst stage ut. j = Observato o the j th secod stage ut belogg to th frst stage ut. ea of the th frst stage ut j j 73
Statstcal Techques Sample mea of the th frst stage ut m m j j Overall sample mea o the bass of sub samplg has bee doe each secod stage ut whe = Overall populato ea 4.8 PROPERTIES OF TWO-STAGE SAPLIG Theorem 3: If the frst stage uts ad m secod stage uts from each selected frst stage ut are selected by smple radom samplg wthout replacemet. There sample mea () s a ubased estmator of populato mea ( ) ad havg the varace where, S b S m S b w m ad Sw j Proof: Sce the uts are selected two stages by cosderg a probablty samplg each stage two stage samplg. At both stages, selecto procedures are to be cosdered dervg epected value ad varace of the sample statstc based o the umber of uts selected secod stage. For gettg the epected value ad varace we have to follow: E E E / (4) E E j (5) Where E ad are epectato ad varace over the frst stage uts ad E ad are the codtoal epectato ad varace over the secod stage ut for a gve sample of frst stage uts. ow sce we draw secod stage uts from the frst stage uts by SRSWOR so E / Therefore, from equato (4), we have E E It shows that sample mea of all elemets the sample s a ubased estmate of the populato mea. To obta the varace E E 74
where, S w E S m m S S m S w b Eample 3: Select a sample of sze from gve populato of 3 uts. The data gve Eample s dvded clusters or groups each of them havg uts. Soluto: Let us select 3 groups as frst stage uts from the gve groups. Let the selected uts are Group-, Group-3 ad Group-. Therefore the frst stage sample s: Some Other Samplg Schemes S. o. Group- Group-3 Group- 8 8 3 8 5 3 4 0 7 5 3 8 3 0 0 ow, from the selected frst stage uts, we shall select the secod stage uts. These uts are selected o the bass of ther mportace. Let we select uts each from these three selected frst stage uts of sze. Let us select d & 4 th secod stage uts from Group-, st & 4 th ut from Group-3 ad 3 rd & th ut from Group-. Therefore, the secod stage sample uts selected from 3 frst stage uts of sze are {3,, 8, 0,, 0 }. Let us aswer the gve eercse. E4) Select a frst stage sample of sze the the secod stage sample of sze 0, from the data gve E3) by two-stage samplg method. 4.9 SUARY I ths ut, we have dscussed:. Systematc radom samplg;. How to draw a tematc radom sample ad estmate the populato mea; 3. The varace of the estmate of the populato mea; 4. The cluster samplg ad the estmate of the varace of the sample mea; ad 5. ethod of drawg a two-stage sample; ad. ethod of solvg the umercal eamples 75
Statstcal Techques 4.0 SOLUTIOS / ASWERS E) Frst of all we arrage the data of producto of wheat ( Thousad g) wth ther seral order: Sr.o. : 3 4 5 7 8 9 0 3 Producto: 3 0 30 37 7 3 3 3 58 53 83 0 Sr. o. : 4 5 7 8 9 0 3 4 5 Producto: 5 3 7 7 0 8 3 7 5 ow we obta a umber 3. 5 7 We have = 5 ad = 7 So we have to tae = 4 ow, from frst 4 values serally arraged data let we select 3 rd value ( =30), so ths wll be our st sample ut selected the sample by radom method. ow remag values wll be selected tematc way.e. ( +) th, ( + ) th,, ( + ) th order value the data. So ths way, we have to select the values whch are at 7 th, th, 5 th, 9 th, 3 rd ad 7 th posto. But the data, oly 5 values are avalable. So, we have to adopt the crcular tematc samplg method for the selecto of all 7 uts. By followg the crcular tematc sample after selectg the frst ut ( = 3 rd ) betwee to 5 the remag uts would be 7 th, th, 5 th, 9 th, 3 rd ad d postoed uts. Therefore, from the populato of 5 uts the tematc radom sample of sze 7 would be {30, 3, 53, 3, 7,, 0}. E) We have = 50 ad we have to draw a tematc sample of sze 0. We arrage the values of the heghts of 50 studets correspodg to ther roll umbers from to 50. Roll o: 3 4 5 7 8 9 Heght: 4 5 5 7 78 80 7 48 Roll o: 0 3 4 5 7 8 Heght: 53 73 3 4 75 8 80 Roll o: 9 0 3 4 5 7 Heght: 73 85 9 7 8 73 45 53 54 Roll o: 8 9 30 3 3 33 34 35 3 Heght: 4 70 7 0 58 5 3 Roll o: 37 38 39 40 4 4 43 44 45 Heght: 5 70 8 58 49 55 0 50 49 7 Roll o: 4 47 48 49 50 Heght: 7 7 9 59 0
50 ow we shall obta a umber 5 that s = 5. So we 0 have to select the frst ut of the sample of sze 0 by radom selecto method from serally to 5. Let we have selected the 4 th ut order. So the st ut selected the sample s 7. ow we shall select the remag 9 ut a tematc way. We have = 4 ad = 5 so the remag 9 sample ut to be selected the sample would be 9 th,4 th, 9 th, 4 th, 9 th, 34 th, 39 th, 44 th ad 49 th order the data. Therefore, the values of tematc radom sample of sze 0 selected from the populato of sze 50 would be 7, 48, 4,73, 73, 4, 58, 8, 50, 59. Some Other Samplg Schemes E3) I cluster samplg, we ow that f populato s dvded several homogeeous groups (ow as clusters) the we have to select the clusters as sample uts for selecto of sample of sze. Here, the gve data = 5 ad =. Therefore, there wll be 5 C 0umber of possble sample of sze clusters. C Let us select the Locato ad Locato 4 as the sample uts. Therefore, the sample of locatos selected from populato of 5 locatos s Locato Sze of Plots Total Locato Locato 4 800 700 500 000 00 300 800 00 700 000 00 00 300 800 000 900 00 500 00 000 000 3000 Therefore, sample mea of frst sample ut 000 00 0 ad sample mea of secod sample ut 3000 0 300 Therefore, the sample mea 0 00 300 0 0000300 0 0 00 square ft E4) The data gve E3) are havg 50 umber of szes of plots, whch are dvded to 5 groups o the bass of the dfferet locatos,.e. Locato, Locato,, Locato 5. ow, we select a frst stage sample of two locatos. Let t be Locato ad Locato 5 as frst stage sample uts. 77
Statstcal Techques Therefore, we have Plot o. Locato Locato 5 800 700 700 000 3 500 700 4 000 300 5 00 00 300 300 7 800 000 8 00 500 9 700 500 0 000 400 Total,000 3,000 ow, from the selected frst stage uts we shall select a secod stage sample of 0 uts by selectg 5 uts each from both locatos. Let the d, 3 rd, th, 7 th ad 9 th are selected from the Locato ad 3 rd, 5 th, 7 th, 8 th ad 0 th ut are selected from the Locato 5. Therefore, the sample selected by the two-stage samplg method s {700, 500, 300, 800, 700, 700, 00, 000, 500, 400}. 78