Statistics A B C D E F G H I J K L M N O. Review set 5. Contents:

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1 5 Sttistics Contents: A B C D E F G H I J K L M N O Descriing dt Collecting informtion Rndom smpling Presenting nd interpreting dt Grouped discrete dt Continuous (intervl) dt Mesures of centres of distriutions Mesuring the spred of dt Compring dt Wht smple tells us out popultion Dt sed investigtion Reltive frequency Norml distriutions Properties of norml distriutions Technology nd norml distriutions Review set 5

2 242 STATISTICS (Chpter 5) (T6) The erlier chpter Dt in Context discussed how to collect, collte, nd orgnise informtion. This chpter extends from tht work. Once informtion hs een collected nd collted, the dt it yields cn e nlysed nd interpreted. Mthemtics provides numer of tools for tht purpose. A conclusion sed on mthemticl interprettion of numericl dt is clled sttistic. Mthemticl conclusions sed on numericl dt re clled sttistics. There re numer of steps involved when considering ny sttisticl prolem: ² identify nd stte the prolem ² develop method to e used in the investigtion ² decide how to collect the dt ² collect the dt ² orgnise nd nlyse the dt ² interpret the dt nd form conclusion ² reconsider the underlying ssumptions of the investigtion. Before we look t how to use mthemtics to interpret informtion, we egin with how numericl informtion cn e collected nd orgnised. A DESCRIBING DATA TYPES OF DATA Dt re individul oservtions of vrile. A vrile is quntity tht cn hve vlue recorded for it or to which we cn ssign n ttriute or qulity. There re two types of vrile tht we commonly del with: CATEGORICAL VARIABLES A ctegoricl vrile is one which descries prticulr qulity or chrcteristic. The dt cn e divided into ctegories. The informtion collected is clled ctegoricl dt. Exmples of ctegoricl vriles re: ² Getting to school: the ctegories could e trin, us, cr nd wlking. ² Colour of eyes: the ctegories could e lue, rown, hzel, green, grey. ² Gender: mle nd femle. QUANTITATIVE (NUMERICAL) VARIABLES A quntittive vrile is one which hs numericl vlue nd is often clled numericl vrile. The informtion collected is clled numericl dt. There re two types of quntittive vriles; discrete nd continuous. A quntittive discrete vrile tkes exct numer vlues nd is often result of counting.

3 Exmples of discrete quntittive vriles re: ² The numer of people in cr: the vrile could tke the vlues 1, 2, 3,... ² The score out of 50 on test: the vrile could tke the vlues 0, 1, 2, 3,..., 50. A quntittive continuous vrile tkes numericl vlues within certin continuous rnge. It is usully result of mesuring. Exmples of quntittive continuous vriles re: ² The weight of neworn pups: the vrile could tke ny vlue on the numer line ut is likely to e in the rnge 0:2 kg to 1:2 kg. STATISTICS (Chpter 5) (T6) 243 ² The heights of Yer 10 students: the vrile would e mesured in centimetres. A student whose height is recorded s 163 cm could hve exct height etween 162:5 cm nd 163:5 cm. Exmple 1 Self Tutor Clssify these vriles s ctegoricl, quntittive discrete or quntittive continuous: the numer of heds when 4 coins re tossed the fvourite vriety of fruit eten y the students in clss c the heights of group of 16 yer old students. c The vlues of the vriles re otined y counting the numer of heds. The result cn only e one of the vlues 0, 1, 2, 3 or 4. It is quntittive discrete vrile. The vrile is the fvourite vriety of fruit eten. It is ctegoricl vrile. This is numericl dt otined y mesuring. The results cn tke ny vlue etween certin limits determined y the degree of ccurcy of the mesuring device. It is quntittive continuous vrile. EXERCISE 5A 1 For ech of the following possile investigtions, clssify the vrile s ctegoricl, quntittive discrete or quntittive continuous: c d e the numer of gols scored ech week y netll tem the numer of children in n Austrlin fmily the numer of red rolls ought ech week y fmily the pets owned y students in yer 10 clss the numer of leves on the stems of ottle rush species

4 244 STATISTICS (Chpter 5) (T6) f g h i j k l m n o p q the numer of hours of sunshine in dy the numer of people who die from cncer ech yer in Austrli the mount of rinfll in ech month of the yer the countries of origin of immigrnts the most populr colours of crs the time spent doing homework the mrks scored in clss test the items sold t the school cnteen the resons people use txis the sports plyed y students in high schools the stopping distnces of crs doing 60 km/h the pulse rtes of group of thletes t rest. 2 For the ctegoricl vriles in question 1, write down two or three possile ctegories. Discuss your nswers. For ech of the quntittive vriles (discrete nd continuous) identified in question 1, discuss s clss the rnge of possile vlues you would expect. B COLLECTING INFORMATION Informtion is collected y sking questions. Designing questions (nd questionnires) requires considerle skill ecuse of the type of informtion tht n nswer cn provide. Wherever possile, questions should e cler nd simple. Cre must e tken in the wy in which question is phrsed. Leding questions cn mnipulte the opinions of the person nswering, producing misleding results. For exmple, if the Government is considering rellocting funds from eduction to helth, poll sking Should the Government cut funding to schools? is likely to produce negtive response to the relloction, wheres poll sking Should the Government increse funding to hospitls? is likely to produce positive response. A poll sking Should the Government move funds from eduction to helth? is more likely to produce lnced response. CHECKING QUESTION RESPONSES Sometimes it is good ide to sk the sme question ut from slightly different perspective so tht the responses cn e checked. This process cn help determine if person is consistent. For exmple, suppose tht student work in sttistics course is ssessed y three different ssessment tsks. How students vlue prticulr tsk my e reflected in the wy they think it should e rted. A first question might e: On scle of 0 to 100, rte the vlue you plce on ech of these tsks.

5 STATISTICS (Chpter 5) (T6) 245 Written tests: Investigtions: Projects: Answers to this question cn tell resercher wht vlue student plced on ech tsk. To check this, resercher my wnt to put similr question lter in the questionnire. Continuing with the exmple ove, if students vlue prticulr tsk highly then it seems resonle to think tht they my expect tht their work in tht tsk should lso e weighted highly in their overll ssessment. A check question could e: Wht percentge of your overll ssessment should e given for your work in Written tests? This could e repeted for the other types of tsk. Depending on the responses, the tsk of the resercher is to then refine his or her understnding of the resons for ny differences in wht the person indicted. The qulity of the questions used determines the qulity of the sttistics tht come from them. This is prticulrly the cse for ny usiness which relies on ccurte informtion in order to meet the needs of its clients. Creful survey preprtion gives the results crediility. For tht reson resercher will often tril questionnire to see if it meets its ims. This process in turn improves the vlue of the questionnire. Summry: As you lredy know dt comes in two types, discrete nd continuous. Idelly dt is collected y interview using questions given in writing to person. In this wy the resercher cn void swying respondent with their voice nd mnner. Sometimes it is good ide to repet question to check the reliility of response. Finlly, from mthemtics point of view, questions must: ² provide dt ² e frmed so tht only one type of dt response is collected from ech question ² not led the person to predetermined nswer ² e designed to e unmiguous in their interprettion.

6 246 STATISTICS (Chpter 5) (T6) RANDOM SAMPLING Informtion is collected from popultion. A popultion is usully understood to e the people who live in prticulr country. In sttistics, however, the word popultion refers to ll the memers of prticulr group eing considered. Tht my men ll the customers or clients of prticulr usiness, or it could e s specific s the numer of ll erings produced y series of mchines. Studying popultion will provide informtion out ll its memers. A popultion is the entire set out which we wnt to drw conclusion. Dt cn e collected from every memer of popultion in wht is clled census. Every 5 yers the Austrlin government crries out census in which it seeks sic informtion from the whole popultion. However, it is often too expensive or imprcticl to otin informtion from every memer of the popultion. Often informtion out the whole popultion cn e successfully gthered from smple, or prt, of the popultion. A smple is selection from the popultion. For exmple, efore n election smple of voters is sked how they will vote. With this informtion prediction is mde on how the popultion of eligile voters will vote. In collecting smples, gret cre nd expense is usully tken to mke the selection s free from prejudice s possile, nd lrge enough to e representtive of the whole popultion. A ised smple is one in which the dt hs een unduly influenced y the collection process nd is not representtive of the whole popultion. To void is in smpling, mny different smpling procedures hve een developed. A rndom smple is smple in which ll memers of the popultion hve n equl chnce of eing selected. We discuss four commonly used rndom smpling techniques. These re: ² simple rndom smpling ² systemtic smpling ² cluster smpling ² strtified smpling. SIMPLE RANDOM SAMPLING Mking sure tht smple is representtive of whole popultion cn e difficult prolem. The im is to select the memers of smple in rndom wy, tht is, ech memer of popultion is eqully likely to e chosen for the smple. This rndom selection process ims to void is.

7 STATISTICS (Chpter 5) (T6) 247 A simple rndom smple of size n is smple chosen in such wy tht every set of n memers of the popultion hs the sme chnce of eing chosen. To select five students from your clss to form committee, the clss techer cn drw five nmes out of ht contining ll the nmes of students in your clss. SYSTEMATIC SAMPLING Suppose we wish to find the views on extended shopping hours of shoppers t huge supermrket. As people come nd go, simple rndom process is not prcticl. In such sitution systemtic smpling my e used. In this process, the first memer of the smple is chosen t rndom nd every other memer is chosen ccording to set pttern, for exmple, every fourth person fter tht. To otin k% systemtic smple the first memer is chosen t rndom, nd from then on every 100 k th memer from the popultion. If we need to smple 5% of n estimted 1600 shoppers t the supermrket, i.e., 80 in ll, then s =20, we pproch every 20th shopper. The method is to rndomly select numer etween 1 nd 20. If this numer were 13 sy, we would then choose for our smple the 13th, 33rd, 53rd, 73rd,... person entering the supermrket. This group forms our systemtic smple. CLUSTER SAMPLING Suppose we need to nlyse smple of 300 iscuits. The iscuits re in pckets of 15 nd form lrge tch of 1000 pckets. It is costly, wsteful nd time consuming to tke ll the iscuits from their pckets, mix them up nd then tke the smple of 300. Insted, we would rndomly choose 20 pckets nd use their contents s our smple. This is clled cluster smpling where cluster is one pcket of iscuits. To otin cluster smple the popultion must e in smller groups clled clusters nd rndom smple of the clusters is tken. All memers of ech cluster re used. STRATIFIED RANDOM SAMPLING Often popultion is mde up of diverse groups of vrying size. A strtified smple ims to reflect the sme proportions nd prticulr diversity of the roder popultion. Suppose the student leders of very lrge high school wish to survey the students to sk their opinion on lirry use fter school hours. Asking only yer 12 students their opinion is uncceptle s the requirements of the other yer groups would not e ddressed. Consequently, sugroups from ech of the yer levels need to e smpled. These sugroups re clled strt. If school of 1135 students hs 238 yer 8 s, 253 yer 9 s, 227 yer 10 s, 235 yer 11 s nd 182 yer 12 s nd we wnt smple of 15% of the students, we must rndomly choose:

8 248 STATISTICS (Chpter 5) (T6) 15% of 238 = 36 yer 8 s 15% of 253 = 38 yer 9 s 15% of 227 = 34 yer 10 s 15% of 235 = 35 yer 11 s 15% of 182 = 27 yer 12 s To otin strtified rndom smple, the popultion is first split into pproprite groups clled strt nd rndom smple is selected from ech in proportion to the numer in ech strt. It is not lwys possile to select rndom smple. Dieticins my wish to test the effect fish oil hs on lood pltelets. To test this they need people who re prepred to go on specil diets for severl weeks efore ny chnges cn e oserved. The usul procedure to select smple is to dvertise for volunteers. People who volunteer for such tests re usully not typicl of the popultion. In this cse they re likely to e people who re diet conscious, nd hve proly herd of the supposed dvntge of eting fish. The dieticin hs no choice ut to use those tht volunteer. EXERCISE 5B A convenient smple is smple tht is esy to crete. 1 In ech of the following stte the popultion, nd the smple. A pollster sks 500 people if they pprove of Mr John Howrd s prime minister of Austrli. Fisheries officers ctch 200 whiting fish to mesure their size. c A memer of consumer group uys sket of red, utter nd milk, met, rekfst cerel, fruit nd vegetles from supermrket. d A dieticin sks 12 mle volunteers over the ge of 70 to come in every morning for 2 weeks to et muffin hevily enriched with fire. e A promoter offers every shopper in supermrket slice of mettwurst. 2 For ech of the following descrie smple technique tht could e used. Five winning tickets re to e selected in clu rffle. A sergent in the rmy needs six men to crry out dirty, tiresome tsk. c The deprtment of tourism in Victori wnts visitors opinions of its fcilities tht hve een set up ner the Twelve Apostles long the Gret Ocen Rod. d Cinem owners wnt to know wht their ptrons think of the ltest lockuster they hve just seen. e A reserch tem wnts to test new diet to lower glucose in the lood of dietics. To get sttisticlly significnt results they need 30 women etween the ges of 65 nd 75 who suffer from type II dietes. f When legion disgrced itself in the Romn rmy it ws decimted; tht is, 10% of the soldiers in the legion were selected nd killed. g A council wnts to know the opinions of residents out uilding swimming pool in their neighourhood.

9 STATISTICS (Chpter 5) (T6) In ech of the following, stte: i the intended popultion ii the smple iii ny possile is the smple might hve. A recretion centre in suurn re wnts to enlrge its fcilities. Nery residents oject strongly. To support its cse the recretion centre sks ll persons using the centre to sign petition. Tom hs to complete his sttistics project y Mondy morning. He is keen on sport nd hs chosen s prt of his project oxygen det in exercise. As mesure of oxygen det he hs decided to mesure the time it tkes for the hert rte to return to norml fter 25 m sprint. Unfortuntely he hs not collected ny dt nd he persudes six of his footll friends to come long on Sturdy fternoon to provide him with some numers. c A telephone survey conducted on ehlf on motor cr compny contcts 400 households etween the hours of 2 nd 5 o clock in the fternoon to sk wht rnd of cr they drive. d A council sends out questionnires to ll residents sking out proposl to uild new lirry complex. Prt of the proposl is tht residents in the wrds tht will enefit most from the lirry hve to py higher rtes for the next two yers. 4 A sles promoter decides to visit 10 houses in street nd offer specil discounts on new window tretment. The street hs 100 houses numered from 1 to 100. The sles promoter selects rndom numer etween 1 nd 10 inclusive nd clls on the house with tht street numer. After this the promoter clls on every tenth house. Wht smpling technique is used y the sles promoter? Explin why every house in the street hs n equl chnce of eing visited. c How is this different from simple rndom smple? 5 Tissue pper is mde from wood pulp mixed with glue. The mixture is rolled over huge hot roller tht dries the mixture into pper. The pper is then rolled into rolls metre or so in dimeter nd few metres in width. When the roll comes off the mchine qulity controller tkes smple from the end of the roll to test it. Explin why the smples tken y the qulity controller could e ised. Explin why the qulity controller only smples the pper t the end of the roll. INVESTIGATION 1 STATISTICS FROM THE INTERNET In this investigtion you will e exploring the we sites of numer of orgnistions to find out the topics nd the types of dt tht they collect nd nlyse. Note tht the we ddresses given here were opertive t the time of writing ut there is chnce tht they will hve chnged in the mentime. If the ddress does not work, try using serch engine to find the site of the orgnistion. Wht to do: Visit the site of world orgnistion such s the United Ntions ( or the World Helth Orgnistion ( nd see the ville types of dt nd sttistics. The Austrlin Bureu of Sttistics ( lso hs lrge collection of dt.

10 250 STATISTICS (Chpter 5) (T6) C When tking smple it is hoped tht the informtion gthered is representtive of the entire popultion. We must tke certin steps to ensure tht this is so. If the smple we choose is too smll, the dt otined is likely to e less relile thn tht otined from lrger smples. For ccurte informtion when smpling, it is essentil tht: ² the individuls involved in the survey re rndomly chosen from the popultion ² the numer of individuls in the smple is lrge enough. Note: RANDOM SAMPLING How to choose rndom smple using rndom numers is covered in Yer Mths Applictions. For exmple: Mesuring group of three fifteen-yer-olds would not give very relile estimte of the height of fifteen-yer-olds ll over the world. We therefore need to choose rndom smple tht is lrge enough to represent the popultion. Note tht conclusions sed on smple will never e s ccurte s conclusions mde from the whole popultion, ut if we choose our smple crefully, they will e good representtion. Cre should e tken not to mke smple too lrge s this is costly, time consuming nd often unnecessry. A lnce needs to e struck so tht the smple is lrge enough for there to e confidence in the results ut not so lrge tht it is too costly nd time consuming to collect nd nlyse the dt. THE SIZE OF A SAMPLE ² For n extremely lrge popultion where the popultion size is unknown: To e very confident tht smple ccurtely reflects the popultion within r%, we tke smple of size n where n r 2 ² For popultion size known to e N: To e very confident tht smple ccurtely reflects the popultion within r%, we tke smple of size n where 9600N n Nr 2 12 Exmple 2 Self Tutor To exmine the effect of new dietes drug, smple of users needs to e tken. How lrge smple must e tken to e very confident tht the smple ccurtely reflects the popultion of users within 2% if the popultion size is unknown? As the popultion size is unknown, n = 9600 r 2 = = 2400 So, to e very confident, within 2%, smple of out 2400 needs to e tken.

11 STATISTICS (Chpter 5) (T6) 251 Exmple 3 Self Tutor A reporter for the Western Suurs ws seeking nswers to the following question: Do you wnt more money spent on rods? How could he investigte this sttisticlly if there re voters on the electorl roll nd he wishes to e very confident of ccurcy within 1: 5%? It would e imprcticl to survey every voter on the electorl roll, so rndom smple could e used. The smple size should e clculted using: n = = 9600N Nr (87 694) 1: So, smple of out 4070 should e tken. Click on the icon to otin smple size clcultor. DEMO You my wish to progrm your grphics clcultor to otin these results. EXERCISE 5C 1 A clothing mnufcturer produces 450 shirts per week. Ech week 25 shirts re rndomly selected y the qulity control stff nd checked. 4 were found to e defective in one week. How mny shirts per week form the popultion? Estimte the totl numer of shirts ech week which re defective. c Estimte the percentge of shirts produced ech week which re stisfctory. 2 A fctory produces 5000 microprocessors per week. A rndom smple of 400 reveled tht 2 were fulty. Wht size is the popultion? Wht size is the smple? c Estimte the totl numer of microprocessors produced in week tht re not fulty householders were selected t rndom from the electorl roll nd sked whether they would vote for the Austrlin Lor Prty. The survey reveled tht 620 nswered yes. If there re 12:6 million people in Austrli over the ge of 18, estimte how mny of them would nswer no. Wht percentge of Austrlins over 18 would nswer yes in your estimtion? 4 Discuss how you would rndomly select: first nd second prize in cricket clu rffle three numers from 0 to 37 on roulette wheel.

12 252 STATISTICS (Chpter 5) (T6) 5 In conducting survey to find out the percentge of people who elieve the AFL grnd finl should lwys e plyed t the MCG (Melourne), it would e good ide to sk section of the crowd t this yer s clsh etween Melourne nd Sint Kild. Discuss A newspper conducts survey of Austrlins to determine whether they elieve Austrli should e more restrictive in its immigrtion policy. How mny people must e surveyed to e very confident tht the survey will e ccurte within 15% : if the popultion size is unknown? A government survey is to e held on the question of wter use from our river systems. How mny people should e surveyed to e very confident of ccurcy within 08% :? To determine whether memers of locl clu would e willing to py higher fees in order to fund the instlltion of new equipment, smple of the memers is surveyed. Given tht there re 748 memers t the clu, how lrge smple must e tken to e very confident tht the smple ccurtely mesures the views of ll the memers within 2%? To find the proportion of cpsicums in crop which re suitle for sle, smple of them will e tested. How mny cpsicums must e tested to e very confident tht the smple ccurtely reflects the qulity of ll cpsicums in the crop (within 3% ), if there re cpsicums in the crop? D PRESENTING AND INTERPRETING DATA ORGANISING CATEGORICAL DATA A tlly nd frequency tle cn e used to orgnise ctegoricl dt. For exmple, survey ws conducted on 200 rndomly chosen victims of sporting injuries, to find which sport they plyed. The vrile sport plyed is ctegoricl vrile ecuse the informtion collected cn only e one of the five ctegories listed. The dt hs een counted nd orgnised in the given frequency tle: Sport plyed Frequency Aussie rules 57 Netll 43 Rugy 41 Cricket 21 Other 38 Totl 200

13 STATISTICS (Chpter 5) (T6) 253 DISPLAYING CATEGORICAL DATA Acceptle grphs to disply the sporting injuries ctegoricl dt re: Verticl column grph Horizontl r grph frequency Aussie rules Netll Rugy Cricket Other Aussie rules Netll Rugy Cricket Other Pie chrt Segmented r grph Other Aussie rules Cricket Aussie rules Netll Rugy Cricket Other Rugy Netll THE MODE The mode of set of ctegoricl dt is the ctegory which occurs most frequently. This tle shows the colours of crs recorded during usy time on min rod. Red occurs more often thn ny other colour. So, the mode is Red. Colour Frequency White 50 Red 72 Yellow 38 Blck 32 Blue 21 Green 18 Silver 47 GRAPHING SOFTWARE ² ² Click on the icon to otin n esy to use grphing pckge for ctegoricl dt. Click on the icon for instructions for using computer spredsheet to grph ctegoricl dt. STATISTICS PACKAGE SPREADSHEET

14 254 STATISTICS (Chpter 5) (T6) EXERCISE 5D.1 1 Wht is the mode of the sporting injuries dt? 2 Wht ctegories could e / re used for: egg sizes wine ottle sizes c eye colour? 3 Ech girl t high school hs nmed the sport tht she would most like to ply. The tle shows this dt. Wht percentge would like to ply tennis? Wht is the mode of the dt? c Use technology to otin for the dt: i verticl column grph ii pie chrt. Sport Frequency Tennis 68 Netll 87 Bsketll 48 Volleyll 43 Bdminton 59 Sqush 17 4 There re six ctegories of memership, A to F, t golf clu. A B C D E F Scle: 1 mm 1 person Wht is the mode? In pie chrt, wht sector ngle would e used for ctegory: i A ii D? c Drw pie chrt of the dt. ORGANISING DISCRETE NUMERICAL DATA OPENING PROBLEM A frmer wishes to investigte whether 0 new food formul increses egg production from his lying hens. To test this he feeds 60 hens with the current formul nd 60 with the new one. The hens were rndomly selected from the 1486 hens on his property. Over period he collects nd counts the eggs lid y the individul hens. All other fctors such s exercise, wter, etc re kept the sme for oth groups. The results of the experiment were: Current formul New formul

15 STATISTICS (Chpter 5) (T6) 255 For you to consider: ² Cn you stte clerly the prolem tht the frmer wnts to solve? ² How hs the frmer tried to mke fir comprison? ² How could the frmer mke sure tht his selection is t rndom? ² Wht is the est wy of orgnising this dt? ² Wht re suitle methods of disply? ² Are there ny normlly high or low results nd how should they e treted? ² How cn we est indicte the most numer of eggs lid? ² How cn we est indicte the spred of possile numer of eggs lid? ² Wht is the est wy to show numer of eggs lid nd the spred? ² Cn stisfctory conclusion e mde? In the ove prolem, the discrete quntittive vrile is: The numer of eggs lid. To orgnise the dt tlly/frequency tle could e used. We count the dt systemticlly nd use j to indicte ech dt vlue. Rememer tht jjjj represents 5. The reltive frequency of n event is the frequency of tht event expressed s frction (or deciml equivlent) of the totl frequency. Below is the tle for the new formul dt: Numer of eggs lid Tlly Frequency Reltive frequency 3 j =0:017 4 jj =0:033 5 jjjj =0:067 6 jjjj jjjj jjjj jjjj 21 j =0:350 7 jjjj jjjj jjjj jjjj 22 jj =0:367 8 jjjj 7 jj 7 60 =0:117 9 jj =0: j =0:017 Totl 60 1 A column grph of the frequencies or the reltive frequencies cn e used to disply results. Column grph of frequencies of new formul dt Column grph of reltive frequencies of new formul dt 25 frequency 0.4 reltive frequency numer of eggs/hen numer of eggs/hen

16 256 STATISTICS (Chpter 5) (T6) Cn you explin why the two grphs re similr? DESCRIBING THE DISTRIBUTION OF THE DATA SET It is useful to e le to recognise nd clssify common shpes of distriutions. These shpes often ecome clerer if curve is drwn through the columns of column grph or histogrm. Common shpes re: ² Symmetric distriutions One hlf of the grph is roughly the mirror imge of the other hlf. Heights of 18 yer old women tend to e symmetric. ² Negtively skewed distriutions The left hnd, or negtive, side is stretched out. This is sometimes descried s hving long, negtive til. The time people rrive for concert, with some people rriving very erly, ut the ulk close to the strting time, hs this shpe. ² Positively skewed distriutions The right hnd, or positive, side is stretched out. This is sometimes descried s hving long, positive til. The life expectncy of nimls nd light gloes hve this shpe. negtive side stretched positive side stretched ² Bimodl distriutions The distriution hs two distinct peks. The heights of mixed clss of students where girls re likely to e smller thn oys hs this shpe.

17 STATISTICS (Chpter 5) (T6) 257 OUTLIERS Outliers re dt vlues tht re either much lrger or much smller thn the generl ody of dt. Outliers pper seprted from the ody of dt on frequency grph. For exmple, in the egg lying dt, the frmer found one hen lid 14 eggs, which is clerly well ove the rest of the dt. So, 14 is sid to e n outlier. On the column grph outliers pper well seprted from the reminder of the grph. Outliers which re genuine dt vlues should e included in ny nlysis. However, if they re result of experimentl or humn error, they should e deleted nd the dt re-nlysed Column grph of frequencies of new formul dt frequency outlier numer of eggs/hen MISLEADING PRESENTATION Sttisticl dt cn lso e presented in such wy tht misleding impression is given. ² A common wy of doing this is y mnipulting the scles on the xes of line grph. For exmple, consider the grph shown. profit ($1000 s) The verticl scle does not strt t zero. So the 17 increse in profits looks lrger thn it relly is. 16 The rek of scle on the verticl xis should 15 hve een indicted y profit ($1000 s) Jn Fe Mr Apr month The grph should look like tht shown longside. This grph shows the true picture of the profit increses nd proly should e lelled A modest ut stedy increse in profits. ² These two chrts show the results of survey of shoppers preferences for different rnds of sop. Both chrts egin their verticl scles t zero, ut chrt 1 does not use uniform scle long the verticl xis. The scle is compressed t the lower end nd enlrged t the upper end. 14 Profits skyrocket! Jn Fe Mr Apr month Chrt 1 Shopper preferences A B C D E Chrt 2 Shopper preferences A B C D E

18 258 STATISTICS (Chpter 5) (T6) This hs the effect of exggerting the difference etween the rs on the chrt. The r for rnd B, the most preferred rnd, hs lso een drkened so tht it stnds out more thn the other rs. Chrt 2 hs used uniform scle nd hs treted ll the rs in the sme wy. Chrt 2 gives more ccurte picture of the survey results. ² The rs on r chrt (or column grph) re given lrger ppernce y dding re or the ppernce of volume. The height of the r represents frequency. For exmple, consider the grph compring sles of three different types of soft drink. sles ($m s) By giving the rs the ppernce of volume the sles of Kick drinks look to e out eight times the sles of Fizz drinks. sles ($m s) On r chrt, frequency (sles in this cse) is proportionl to the height of the r only. The grph should look like this: It cn e seen from the r chrt tht the sles of Kick re just type of drink over twice the sles of Fizz. Fizz Kick Cool There re mny different wys in which dt cn e presented so s to give misleding impression of the figures. The people who use these grphs, chrts, etc., need to e creful nd to look closely t wht they re eing shown efore they llow the picture to tell thousnd words. EXERCISE 5D.2 Fizz Kick Cool type of drink 1 Stte whether these quntittive (or numericl) vriles re discrete or continuous: the time tken to run 1500 metre rce the minimum temperture reched on July dy c the numer of tooth picks in continer d the weight of hnd luggge tken on ord n ircrft e the time tken for ttery to run down f the numer of ricks needed to uild grge g the numer of pssengers on trin h the time spent on the internet per dy dults were chosen t rndom nd sked How mny children do you hve? The results were: Wht is the vrile in this investigtion? Is the vrile discrete or continuous? Why? c Construct column grph to disply the dt. Use heding for the grph, nd scle nd lel the xes. d How would you descrie the distriution of the dt? (Is it symmetricl, positively skewed or negtively skewed? Are there ny outliers?) e Wht percentge of the dults hd no children? f Wht percentge of the dults hd three or more children?

19 STATISTICS (Chpter 5) (T6) For n investigtion into the Numer of phone clls mde y teengers numer of phone clls mde 20 y teengers, smple of 80 sixteen-yer-olds ws sked 15 the question How mny phone clls did you mke yesterdy? 10 The following col- 5 umn grph ws constructed 0 from the dt: Wht is the vrile in this investigtion? numer of clls Explin why the vrile is discrete numericl. c Wht percentge of the sixteen-yer-olds did not mke ny phone clls? d Wht percentge of the sixteen-yer-olds mde 3 or more phone clls? e Copy nd complete: The most frequent numer of phone clls mde ws... f How would you descrie the dt vlue 12? g Descrie the distriution of the dt. frequency 4 The numer of mtches in ox is stted s 50 ut the ctul numer of mtches hs een found to vry. To investigte this, the numer of mtches in ox hs een counted for smple of 60 oxes: Wht is the vrile in this investigtion? Is the vrile continuous or discrete numericl? c Construct frequency tle for this dt. d Disply the dt using r chrt. e Descrie the distriution of the dt. f Wht percentge of the oxes contined exctly 50 mtches? E GROUPED DISCRETE DATA A locl high school is concerned out the numer of vehicles pssing y etween 8:45 m nd 9:00 m. Over 30 consecutive week dys they recorded dt. The results were: 48, 34, 33, 32, 28, 39, 26, 37, 40, 27, 23, 56, 33, 50, 38, 62, 41, 49, 42, 19, 51, 48, 34, 42, 45, 34, 28, 34, 54, 42 We cn orgnise the dt into frequency tle. In situtions like this we group the dt into clss intervls. It seems sensile to use clss intervls of length 10 in this cse. The tlly/frequency tle is: Numer of crs Tlly Frequency 10 to 19 j 1 20 to 29 jjjj 5 30 to 39 jjjj jjjj to 49 jjjj jjjj 9 50 to 59 jjjj 4 60 to 69 j 1 Totl 30

20 260 STATISTICS (Chpter 5) (T6) COLUMN GRAPHS A verticl column grph cn e used to disply grouped discrete dt. For exmple, consider the locl high school dt. The frequency tle is: Numer of crs Frequency 10 to to to to to to 69 1 Note tht once dt hs een grouped in this mnner there could e loss of useful informtion for future nlysis. EXERCISE 5E 1 The dt set given is the test scores (out of 100) for Science test for 42 students The column grph for this dt is: Construct tlly nd frequency tle for this dt using clss intervls 0-9, 10-19, 20-29,..., Wht percentge of the students scored 50 or more for the test? c Wht percentge of students scored less thn 60 for the test? d Copy nd complete the following: More students hd test score in the intervl... thn in ny other intervl. e Drw column grph of the dt. STEM-AND-LEAF PLOTS frequency numer of crs A stem-nd-lef plot (often clled stemplot) is wy of writing down the dt in groups. It is used for smll dt sets. A stemplot shows ctul dt vlues. It lso shows comprison of frequencies. For numers with two digits, the first digit forms prt of the stem nd the second digit forms lef. For exmple, ² for the dt vlue 27, 2 is recorded on the stem, 7 is lef vlue. ² for the dt vlue 116, 11 is recorded on the stem nd 6 is the lef. The stem-nd-lef plot is: Stem Lef Note: 2 j 3 mens 23 The ordered stem-nd-lef plot is Stem Lef

21 The ordered stemplot rrnges ll dt from smllest to lrgest. Notice tht: ² ll the ctul dt is shown ² the minimum (smllest) dt vlue is 19 ² the mximum (lrgest) dt vlue is 62 ² the thirties intervl (30 to 39) hs the highest frequency ² no dt is lost ² the stemplot shows the spred of dt for ech clss ² the stemplot provides horizontl histogrm ² the stemplot utomticlly cretes frequency tle of the dt. Note: Unless otherwise stted, stem-nd-lef plot, or stemplot, mens ordered stem-ndlef plot. 2 Following is n ordered stem-nd-lef plot of the numer of gols kicked y individuls in n Aussie rules footll tem during seson. Find: Stem Lef f How would you descrie the distriution of the dt? Hint: Turn your stemplot on its side. c d e the minimum numer kicked the mximum numer kicked the numer of plyers who kicked greter thn 25 gols the numer plyers who kicked t lest 40 gols the percentge of plyers who kicked less thn 15 gols: 3 The test score, out of 50 mrks, is recorded for group of 45 Geogrphy students Construct n unordered stem-nd-lef plot for this dt using 0, 1, 2, 3, 4 nd 5 s the stems. Redrw the stem-nd-lef plot so tht it is ordered. c Wht dvntge does stem-nd-lef plot hve over frequency tle? d Wht is the i highest ii lowest mrk scored for the test? e If n A ws wrded to students who scored 42 or more for the test, wht percentge of students scored n A? f Wht percentge of students scored less thn hlf mrks for the test? 4 The stemplot elow shows the results of test for group of students. The test ws mrked out of 35. Stem Lef f j 6=16 STATISTICS (Chpter 5) (T6) 261 How mny students scored 32? Wht percentge of students scored 30 or higher? c Descrie the distriution of the results.

22 262 STATISTICS (Chpter 5) (T6) The dt ove ws re-orgnised into smller clsses. Dt from 10 to 14 is recorded with stem 1, nd dt from 15 to 19 is recorded with stem 1, etc. Stem Lef f d e f Descrie the distriution of the results. Compre your response with prt c, ove. Wht is the rnge of scores for the test? Is the dt evenly spred? F CONTINUOUS (INTERVAL) DATA Recll tht: Continuous dt is numericl dt which hs vlues within continuous rnge. For exmple, if we consider the weights of students in netll trining squd we might find tht ll weights lie etween 40 kg nd 90 kg. Suppose 2 students lie in the 40 kg up to ut not including 50 kg, 5 students lie in the 50 kg up to ut not including 60 kg, 11 students lie in the 60 kg up to ut not including 70 kg, 7 students lie in the 70 kg up to ut not including 80 kg, 1 student lies in the 80 kg up to ut not including 90 kg. The frequency tle is shown elow: Weight intervl Frequency 40 - < < < < < 90 1 We could use histogrm to represent the dt grphiclly Weights of the students in the netll squd frequency weight (kg) HISTOGRAMS A histogrm is verticl column grph used to represent continuous grouped dt. There re no gps etween the columns in histogrm s the dt is continuous. The r widths must e equl nd ech r height must reflect the frequency.

23 STATISTICS (Chpter 5) (T6) 263 Exmple 4 Self Tutor The time, in minutes (ignoring ny seconds) for shoppers to exit shopping centre on given dy is s follows: Orgnise this dt on frequency tle. Use time intervls of 0 -, 10 -, 20 -, etc. Drw histogrm to represent the dt. Time int. Tlly Freq. 0 - < 10 jjj < 20 jjjj < 30 jjjj jjjj < 40 jjjj jj < 50 jjjj < 60 j < 70 jj frequency time (min) Note: ² The continuous dt hs een grouped into clsses. ² The clss with the highest frequency is clled the modl clss. ² The size of the clss is clled the clss intervl. In the ove exmple it is 10. ² As the continuous dt hs een plced in groups it is sometimes referred to s intervl dt. INVESTIGATION 2 CHOOSING CLASS INTERVALS When dividing dt vlues into intervls, the choice of how mny intervls to use, nd hence the width of ech clss, is importnt. DEMO Wht to do: 1 Click on the icon to experiment with vrious dt sets. You cn chnge the numer of clsses. How does the numer of clsses lter the wy we cn red the dt? 2 Write rief ccount of your findings. As rule of thum we generlly use pproximtely p n clsses for dt set of n individuls. For very lrge sets of dt we use more clsses rther thn less. EXERCISE 5F 1 The weights (kg) of plyers in oy s hockey squd were found to e: Using clsses 40 - < 50, 50 - < 60, 60 - < 70, 70 - < 80, 80 - < 90, tulte the dt using columns of weight, tlly, frequency. How mny students re in the 60 - < 70 clss?

24 264 STATISTICS (Chpter 5) (T6) c d How mny students weighed less thn 70 kg? Find the percentge of students who weighed 60 kg or more. 2 A group of young thletes ws invited to prticipte in hmmer throwing competition. The following results were otined: Distnce (metres) 10 - < < < < < 60 No. of thletes How mny thletes threw less thn 20 metres? Wht percentge of the thletes were le to throw t lest 40 metres? 3 G A plnt inspector tkes rndom smple of two week old seedlings from nursery nd mesures their height to the nerest mm. The results re shown in the tle longside. c How mny of the seedlings re 150 mm or more? Wht percentge of the seedlings re in the < 150 mm clss? The totl numer of seedlings in the nursery is Estimte the numer of seedlings which mesure: i less thn 150 mm ii etween 149 nd 175 mm. Height (mm) 50 - < 75 Freq < < < < < MEASURES OF CENTRES OF DISTRIBUTIONS Interested to know how your performnce in mthemtics is going? Are you out verge or ove verge in your clss? How does tht compre with the other students studying the sme suject in the stte? To nswer questions such s these you need to e le to locte the centre of dt set. The word verge is commonly used word tht cn hve different menings. Sttisticins do not use the word verge without stting which verge they men. Two commonly used mesures for the centre or middle of distriution re the men nd the medin. The men of set of scores is their rithmetic verge otined y dding ll the scores nd dividing y the totl numer of scores. The men is denoted x. The medin of set of scores is the middle score fter they hve een plced in order of size from smllest to lrgest. In every dy lnguge, verge usully mens the men, ut when the Austrlin Bureu of Census nd Sttistics reports the verge weekly income it refers to the medin income. Note: For smple contining n scores, in order, the medin is the n+1 2 th score. n +1 If n =11, =6, nd so the medin is the 6th score. 2 n +1 If n =12, =6:5, nd so the medin is the verge of the 6th nd 7th scores. 2

25 STATISTICS (Chpter 5) (T6) 265 Exmple 5 Self Tutor The numer of typing errors on the pges of Nigel s ssignment were: 4, 1, 3, 7, 2, 6, 5, 3, 8, 6 nd 1. Find his: men numer of errors medin numer of errors. men = = :18 In order of size: medin =6th score =4 f n +1 2 = =6g Exmple 6 Self Tutor In llet clss, the ges of the students re: 17, 13, 15, 12, 15, 14, 16, 13, 14, 18. Find the men ge the medin ge of the clss memers men = 10 = =14:7 The ordered dt set is: 12, 13, 13, 14, 14, 15, 15, 16, 17, 18 {z } middle scores There re two middle scores, 14 nd 15. So the medin is 14:5. ftheir vergeg INTERPRETING THE MEDIAN Regrdless of its ctul vlue, the medin is the score in the middle of the dt. One hlf (50%) of the dt is elow it nd one hlf ove it. Consider this tle of incomes of compny directors: Income ($ 0 000) 100 -< < < < < < 400 No. of directors

26 266 STATISTICS (Chpter 5) (T6) Notice tht this dt is not evenly spred throughout the ctegories. The vlues t the top end seem to distort the dt. The medin of this dt is out $ medin rnge STATISTICS USING A COMPUTER Click on the icon to enter the sttistics pckge on the CD. Enter dt set 1: Enter dt set 2: Exmine the side-y-side column grphs. Click on the Box-nd-Whisker spot to view the side-y-side oxplots. Click on the Sttistics spot to otin the descriptive sttistics. Click on Print to otin print-out of ll of these on one sheet of pper. Notice tht the pckge hndles the following types of dt: ² ungrouped discrete ² grouped continuous Consider the dt 2, 3, 5, 4, 3, 6, 5, 7, 3, 8, 1, 7, 5, 5, 9: For TI-83 Dt is entered in the In ² ungrouped continuous ² lredy grouped discrete STATISTICS USING A GRAPHICS CALCULATOR STAT EDIT menu. Press STAT 1 to select 1:Edit L1, delete ll existing dt. Enter the new dt. Press 2 ENTER then 3 ENTER etc, until ll dt is entered. STATISTICS PACKAGE ² grouped discrete ² lredy grouped continuous To otin the descriptive sttistics Press STAT to select the STAT CALC menu. Press 1 to select 1:1 Vr Stts Pressing 2nd 1(L 1) ENTER gives the men x =4:87 (to 3 sf) Scrolling down y pressing repetedly gives the medin =5

27 STATISTICS (Chpter 5) (T6) 267 For Csio From the Min Menu, select STAT. In List 1, delete ll existing dt nd enter the new dt. Press 2 EXE then 3 EXE etc until ll dt is entered To otin the descriptive sttistics Press F6 ( ) if the GRPH icon is not in the ottom left corner of the screen. Press F2 (CALC) F1 (1VAR) which gives the men x = 4:87 (to 3 sf) Scrolling down y pressing repetedly gives the medin =5 EXERCISE 5G In the following exercise you should use technology. You should use oth forms of technology ville. The rel world uses computer pckges. 1 Below re the points scored y two sketll tems over 14 mtch series: Tem A: 91, 76, 104, 88, 73, 55, 121, 98, 102, 91, 114, 82, 83, 91 Tem B: 87, 104, 112, 82, 64, 48, 99, 119, 112, 77, 89, 108, 72, 87 Which tem hd the higher men score? 2 A survey of 40 students reveled the following numer of silings per student: 2, 0, 0, 3, 2, 0, 0, 1, 3, 3, 4, 0, 0, 5, 3, 3, 0, 1, 4, 5, 0, 1, 1, 5, 1, 0, 0, 1, 2, 2, 1, 3, 2, 1, 4, 2, 0, 0, 1, 2 Wht is the men numer of silings per student? Wht is the medin numer of silings per student? 3 The selling prices of the lst 10 houses sold in certin district were s follows: $ , $ , $ , $ , $ , $ , $ , $ , $ , $ Clculte the men nd medin selling price nd comment on the results. Which mesure would you use if you were: i vendor wnting to sell your house ii looking to uy house in the district? 4 Towrds the end of seson, sketller hd plyed 12 mtches nd hd n verge of 18:5 points per gme. In the finl two mtches of the seson the sketller scored 23 points nd 18 points. Find the sketller s new verge.

28 268 STATISTICS (Chpter 5) (T6) GROUPED DISCRETE DATA Exmple 7 The distriution otined y counting the contents of 25 mtch oxes is shown: Find the: men numer of mtches per ox medin numer of mtches per ox. Numer of mtches Frequency Self Tutor For TI-83 Press STAT 1 to select 1:Edit. Key the vrile vlues into L1 nd the frequency vlues into STAT CALC menu. L2. Press STAT 1 to select 1:1 Vr Stts from the Enter L1, L2y pressing 2nd 1(L) 1, 2nd 2(L) 2 ENTER The men is Scroll down, nd the medin is 49. Note: If you do not include L2you will get screen of sttistics for L1only. For Csio From the Min Menu, select STAT. Key the vrile vlues into List 1 nd the frequency vlues into List 2. Press F6 ( ) if the GRPH icon is not in the ottom left corner of the screen. Press F2 (CALC) F6 (SET) (LIST) 2 to chnge the frequency vrile to List 2. Press EXIT F1 (1VAR) The men is Scroll down, nd the medin is 49.

29 STATISTICS (Chpter 5) (T6) 269 Use technology to nswer these questions. 5 A hrdwre store mintins tht pckets contin 60 screws. To test this, qulity control inspector tested 100 pckets nd found the following distriution: Find the men nd medin numer of screws per pcket. Comment on these results in reltion to the store s clim. c Which of these two mesures is more relile? Comment on your nswer pckets of Choc Fruits were opened nd their contents counted. The following tle gives the distriution of the numer of Choc Fruits per pcket smpled. Find the men nd medin of the distriution. Numer of screws Frequency Totl 100 Numer in pcket Frequency The tle longside compres the mss t irth of some guine pigs with their mss when they were two weeks old. Wht ws the men irth mss? Wht ws the men mss fter two weeks? c Wht ws the men increse over the two weeks? Guine Pig Mss (g) t irth Mss (g) t 2 weeks A B C D E F G H GROUPED CLASS INTERVAL DATA When dt hs een grouped into clss intervls, it is not possile to find the mesure of the centre directly from frequency tles. In these situtions estimtes cn e mde using the midpoint of the clss to represent ll scores within tht intervl. The midpoint of clss intervl is the men of its endpoints. For exmple, the midpoint for continuous dt of clss 40 - < 50 is The midpoint of discrete dt of clss is =14: =45.

30 270 STATISTICS (Chpter 5) (T6) The modl clss is the clss with the highest frequency. Exmple 8 The histogrm displys the distnce in metres tht 28 golf lls were hit y one golfer. c d Construct the frequency tle for this dt nd dd ny other columns necessry to clculte the men nd medin. Find the modl clss for this dt. Find the men for this dt. Find the medin for this dt. Clss Intervl Midpt. (x) Freq. (f) < : < : < : < : < : < : < : < :5 1 Totl 28 For TI-83 We enter the midpoints into L nd the frequencies into L2: We then proceed using the instructions s in Exmple 7 to get frequency distnce (m) There were 7 hits of distnce etween 260 m nd 265 m, which is more thn in ny other clss. The modl clss is therefore the clss etween 260 nd 265 metres. For Csio Self Tutor We enter the midpoints into List 1 nd the frequencies into List 2: We then proceed using the instructions s in Exmple 7 to get c The men is 260 m. c The men is 260 m. d The medin is 262:5 m. d The medin is 262:5 m. Note: The medin is given here s one of the midpoints entered. Why?

31 8 Find the pproximte men for ech of the following distriutions: Score (x) Frequency (f) Score (x) Frequency (f) students sit mthemtics test nd the results re s follows: Score Frequency STATISTICS (Chpter 5) (T6) 271 Find the pproximte vlue of the men score. 10 The tle shows the weight of neworn ies t hospitl over one week period. Find the pproximte men weight of the neworn ies. Weight (kg) Frequency 1:0 - < 1:5 1 1:5 - < 2:0 2 2:0 - < 2:5 6 2:5 - < 3:0 17 3:0 - < 3:5 11 3:5 - < 4:0 8 4:0 - < 4:5 0 4:5 - < 5: The tle shows the petrol sles in one dy y numer of city service sttions. How mny service sttions were involved in the survey? Estimte the numer of litres of petrol sold for the dy y the service sttions. c Find the pproximte men sles of petrol for the dy. Litres (L) Frequency < < < < < < INVESTIGATION 3 EFFECTS OF OUTLIERS In set of dt n outlier, or extreme vlue, is vlue which is much greter thn, or much less thn, the other vlues. Your tsk: Exmine the effect of n outlier on the two mesures of centrl tendency. Wht to do: 1 Consider the following set of dt: 1, 2, 3, 3, 3, 4, 4, 5, 6, 7. Clculte: the men the medin. 2 Now introduce n extreme vlue, sy 100, to the dt. Clculte: the men the medin.