This chapter will show you. Averages. What you should already know

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1 Averges tles Grouped dt digrms Histogrms with rs of unequl width 6 Moving verges 7 Surveys Soil sttistis 9 Smpling This hpter will show you how to lulte nd use the mode, medin, men nd rnge from frequeny tles of disrete dt how to deide whih is the est verge for different types of dt how to reognise the modl lss nd lulte n estimte of the men from frequeny tles of grouped dt how to drw frequeny polygons nd histogrms how to lulte nd use moving verge how to design questions for questionnires nd surveys Wht you should lredy know How to work out the men, mode, medin nd rnge of smll sets of disrete dt How to extrt informtion from tles nd digrms Quik hek The mrks for students in mths test re,,,,, 6, 6, 6, 7, 7, 7, 7, 7,, Wht is the modl mrk? Wht is the medin mrk? Averges Questionnires Wht is the rnge of the mrks? d Wht is the men mrk? Whih verge tles Histogrms Soil sttistis Surveys Grouped dt polygons Moving verges HrperCollinsPulishers Limited 7 7

2 . Averges In this setion you will lern how to: use verges solve more omplex prolems using verges identify the dvntges nd disdvntges of eh type of verge nd lern whih one to use in different situtions Key words men mesure of lotion medin mode Averge is term we often use when desriing or ompring sets of dt. The verge is lso known s mesure of lotion. For exmple, we refer to the verge rinfll in Britin, the verge sore of tsmn, n verge weekly wge, the verge mrk in n exmintion. In eh of these exmples, we re representing the whole set of mny vlues y just one single, typil vlue, whih we ll the verge. The ide of n verge is extremely useful, euse it enles us to ompre one set of dt with nother set y ompring just two vlues their verges. There re severl wys of expressing n verge, ut the most ommonly used verges re the mode, the medin nd the men. An verge must e truly representtive of set of dt. So, when you hve to find n verge, it is ruil to hoose the orret type of verge for this prtiulr set of dt. If you use the wrong verge, your results will e distorted nd give misleding informtion. This tle, whih ompres the dvntges nd disdvntges of eh type of verge, will help you to mke the orret deision. Mode Medin Men Advntges Very esy to find Esy to find for Esy to find Not ffeted y extreme vlues ungrouped dt Uses ll the vlues Cn e used for Not ffeted y The totl for given non-numeril dt extreme vlues numer of vlues n e lulted from it Disdvntges Doesn t use ll the vlues Doesn t use ll the Extreme vlues n My not exist vlues distort it Often not understood Hs to e lulted Used for Non-numeril dt Dt with extreme Dt whose vlues re For finding the most likely vlue vlues spred in lned wy HrperCollinsPulishers Limited 7

3 EXAMPLE The ges of people ttending onferene re,, 6,,,,,, 7,, 7,,,, 6, 6, 67, 7, 7, 77 Find i the mode, ii the medin, iii the men of the dt. Whih verge est represents the ge of the people t the onferene. i The mode is, ii the medin is 6, iii the men is 9 = 6 The men is distorted euse of the few very old people t the onferene. The medin is lso distorted y the lrger vlues, so in this se the mode would e the most representtive verge. EXERCISE A Shopkeepers lwys wnt to keep the most populr items in stok. Whih verge do you think is often known s the shopkeeper s verge? A list ontins seven even numers. The lrgest numer is. The smllest numer is hlf the lrgest. The mode is nd the medin is 6. Two of the numers dd up to. Wht re the seven numers? The mrks of students in n English exmintion re s follows., 6,, 7, 6,,, 9, 9, 7,,, 69, 7,, 7,, 6,, 7,,, 6, 6, 7 Find the medin. Deide whih verge you would use for eh of the following. Give reson for your nswer. The verge mrk in n exmintion. The verge poket money for group of 6-yer-old students. The verge shoe size for ll the girls in Yer. d e f The verge height for ll the rtistes on tour with irus. The verge hir olour for pupils in your shool. The verge weight of ll neworn ies in hospitl s mternity wrd. A pk of mthes onsisted of oxes. The ontents of eh ox re s follows. 9 9 On the ox it sttes tht the verge ontents is mthes. Is this orret? HrperCollinsPulishers Limited 7 9

4 This tle shows the nnul slries for firm s employees. Chirmn Mnging diretor 7 Floor mnger Skilled worker Skilled worker Mhinist Computer engineer Seretry Offie junior 7 Wht is i the modl slry, ii the medin slry, nd iii the men slry? The mngement hs suggested py rise for ll of 6%. The shopfloor workers wnt py rise for ll of. Wht differene to the men slry would eh suggestion mke? Mr Brennn, ring mths teher, told eh pupil their individul test mrk nd only gve the test sttistis to the whole lss. He gve the lss the modl mrk, the medin mrk nd the men mrk. Whih verge would tell pupil whether he/she were in the top hlf or the ottom hlf of the lss? Whih verge tells the pupils nothing relly? Whih verge llows pupil relly to guge how well he/she hs done ompred with everyone else? A list of 9 numers hs men of 7.6. Wht numer must e dded to the list to give new men of? A dne group of 7 teengers hd men weight of. kg. To enter ompetition there needs to e people in the group with n verge weight of. kg or less. Wht is the mximum weight tht the eighteenth person ould e? The men ge of group of eight wlkers is. Jonne joins the group nd the men ge hnges to. How old is Jonne?. tles In this setion you will lern how to: lulte the mode nd medin from frequeny tle lulte the men from frequeny tle Key word frequeny tle When lot of informtion hs een gthered, it is often onvenient to put it together in frequeny tle. From this tle, you n then find the vlues of the mode, medin, men nd rnge of the dt. HrperCollinsPulishers Limited 7

5 EXAMPLE A survey ws done on the numer of people in eh r leving the Medowhll Shopping Centre, in Sheffield. The results re summrised in the tle. Clulte the mode, the medin, the men numer of people in r. Numer of people in eh r The modl numer of people in r is esy to spot. It is the numer with the lrgest frequeny (9). Hene, the modl numer of people in r is. The medin numer of people in r is found y working out where the middle of the set of numers is loted. First, dd up frequenies to get the totl numer of rs surveyed, whih omes to. Next, lulte the middle position ( + ) = You now need to dd the frequenies ross the tle to find whih group ontins the rd item. The rd item is the end of the group with in r. Therefore, the rd item must e in the group with in r. Hene, the medin numer of people in r is. The men numer of people in r is found y lulting the totl numer of people, nd then dividing this totl y the numer of rs surveyed. Numer in r Numer in these rs = 9 9 = 96 = = = = 7 Totls 6 Hene, the men numer of people in r is 6 =.9 ( signifint figures). Using your lultor The previous exmple n lso e done y using the sttistil mode whih is ville on some lultors. However, not ll lultors re the sme, so you will hve to either red your instrution mnul or experiment with the sttistil keys on your lultor. You my find one lelled d or M or or u where x is printed in lue. Try the following key strokes. d 9d du HrperCollinsPulishers Limited 7

6 EXERCISE B Find i the mode, ii the medin nd iii the men from eh frequeny tle elow. A survey of the shoe sizes of ll the Y oys in shool gve these results. Shoe size Numer of pupils This is reord of the numer of ies orn eh week over one yer in smll mternity unit. Numer of ies A survey of the numer of hildren in eh fmily of shool s intke gve these results. Numer of hildren 97 6 d Assuming eh hild t the shool is shown in the dt, how mny hildren re t the shool? Clulte the men numer of hildren in fmily. How mny fmilies hve this men numer of hildren? How mny fmilies would onsider themselves verge from this survey? A dentist kept reords of how mny teeth he extrted from his ptients. In 97 he extrted 9 teeth from 7 ptients. In 9 he extrted teeth from 96 ptients. In 99 he extrted 7 teeth from ptients. Clulte the verge numer of teeth tken from eh ptient in eh yer. Explin why you think the verge numer of teeth extrted flls eh yer. The tehers in shool were sked to indite the verge numer of hours they spent eh dy mrking. The tle summrises their replies. Numer of hours spent mrking Numer of tehers 6 6 How mny tehers re t the shool? Wht is the modl numer of hours spent mrking? Wht is the men numer of hours spent mrking? HrperCollinsPulishers Limited 7

7 Two friends often plyed golf together. They reorded their sores for eh hole over five gmes to determine who ws more onsistent nd who ws the etter plyer. The results re summrised in the tle. No. of shots to hole ll Roger Brin d e f Wht is the modl sore for eh plyer? Wht is the rnge of sores for eh plyer? Wht is the medin sore for eh plyer? Wht is the men sore for eh plyer? Whih plyer is the more onsistent nd explin why? Who would you sy is the etter plyer nd stte why? The numer of legue gols sored y footll tem over seson is given in the tle. Numer of gols sored Numer of mthes 6 7 d e f g How mny gmes were plyed tht seson? Wht is the rnge of gols sored? Wht is the modl numer of gols sored? Wht is the medin numer of gols sored? Wht is the men numer of gols sored? Whih verge do you think the tem s supporters would sy is the verge numer of gols sored y the tem tht seson? If the tem lso sored gols in ten up mthes tht seson, wht ws the men numer of gols the tem sored throughout the whole seson? The tle shows the numer of pssengers in eh of txis leving London Airport one dy. No. of pssengers in txi No. of txis x y 6 Find the vlue of x + y. If the men numer of pssengers per txi is.66, show tht x + y =. d Find the vlues of x nd y y solving pproprite equtions. Stte the medin of the numer of pssengers per txi. HrperCollinsPulishers Limited 7

8 . Grouped dt In this setion you will lern how to: identify the modl group lulte nd estimte the men from grouped tle Key words ontinuous dt disrete dt estimted men groups modl group Sometimes the informtion we re given is grouped in some wy, s in the tle in Exmple, whih shows the rnge of weekly poket money given to Y students in prtiulr lss. EXAMPLE From the dt in the tle TO PAGE write down the modl group lulte n estimte of the men weekly poket money. Poket money, p, ( ) p p p p p No. of students 9 The modl group is still esy to pik out, sine it is simply the one with the lrgest frequeny. Here the modl group is to. The men n only e estimted, sine you do not hve ll the informtion. To estimte the men, you simply ssume tht eh person in eh group hs the midwy mount, then you n proeed to uild up the tle s efore. Note how you find the midwy vlue. The two end vlues re dded together nd then divided y two. Poket money, p, ( ) ( f ) Midwy (m) f m p.. p. 7. p.. p 9.. p. 67. Totls 6 The estimted men will e 6 =. (rounded off). HrperCollinsPulishers Limited 7

9 Note the nottion used for the groups. < p mens ny mount ove p up to nd inluding. < p mens ny mount ove up to nd inluding. If you hd written..,.., et. for the groups, then the midwy vlues would hve een.,., et. Although tehnilly orret, this mkes the lultion of the men hrder nd does not hve signifint effet on the finl nswer, whih is n estimte nywy. This issue only rises euse money is disrete dt, whih is dt tht onsists of seprte numers, suh s gols sored, mrks in test, numer of hildren nd shoe sizes. Normlly grouped tles use ontinuous dt whih is dt whih n hve n infinite numer of different vlues, suh s height, weight, time, re nd pity. It is lwys rounded-off informtion. Whtever the type of dt, rememer to find the midwy vlue y dding the two end vlues of the group nd dividing y. EXERCISE C For eh tle of vlues, find the following. i the modl group ii n estimte for the men x x x x 6 x 7 x 9 y y y 9 6 y y y y 6 9 z Weeks z 6 z 7 6 z z When you opy the tles, drw them vertilly s in Exmple. Json rought peles k from the eh nd weighed them ll to the nerest grm. His results re summrised in this tle. Weight, w (grms) w 6 6 w w w w w Find the following. the modl weight of the peles n estimte of the totl weight of ll the peles n estimte of the men weight of the peles HrperCollinsPulishers Limited 7

10 One hundred light uls were tested y their mnufturer to see whether the verge life spn of the mnufturer s uls ws over hours. The tle summrises the results. Life spn, h (hours) h 7 7 h h h h 7 d Wht is the modl length of time ul lsts? Wht perentge of uls lst longer thn hours? Estimte the men life spn of the light uls. Do you think the test shows tht the verge life spn is over hours? Fully explin your nswer. The tle shows the distnes run y n thlete who is trining for mrthon. Distne, d, miles d d d d d It is reommended tht n thlete s dily verge milege should e t lest one third of the distne of the re eing trined for. A mrthon is 6. miles. Is this thlete doing enough trining? The thlete reords the times of some runs nd lultes tht her verge pe for ll runs is 6 minutes to mile. Explin why she is wrong to expet finishing time of 6. 6 minutes 7 minutes for the mrthon. The thlete lims tht the differene etween her shortest nd longest run is miles. Could this e orret? Explin your nswer. The owners of outique did survey to find the verge ge of people using the outique. The tle summrises the results. Age (yers) Wht do you think is the verge ge of the people using the outique? Three supermrkets eh limed to hve the lowest verge prie inrese over the yer. The tle summrises their verge prie inreses. Prie inrese (p) Sounduy Springfields Seto Using their verge prie inreses, mke omprison of the supermrkets nd write report on whih supermrket, in your opinion, hs the lowest prie inreses over the yer. Don t forget to justify your nswers. 6 HrperCollinsPulishers Limited 7

11 . digrms In this setion you will lern how to: drw frequeny polygons for disrete nd ontinuous dt drw histogrms for ontinuous dt with equl intervls Key words ontinuous dt disrete dt frequeny polygon histogrm polygons To help people understnd it, sttistil informtion is often presented in pitoril or digrmmti form. For exmple, you should hve seen pie hrts, r hrts nd stem-nd-lef digrms. Another method of showing dt is y frequeny polygons. polygons n e used to represent oth ungrouped dt nd grouped dt, s shown in Exmple nd Exmple respetively. They re useful to show the shpes of distriutions, nd n e used to ompre distriutions. EXAMPLE No. of hildren 6 6 This is the frequeny polygon for the ungrouped dt in the tle. Numer of hildren You simply plot the oordintes from eh ordered pir in the tle. You omplete the polygon y joining up the plotted points with stright lines. HrperCollinsPulishers Limited 7 7

12 EXAMPLE Weight, w (kilogrms) w w w w w w 7 9 This is the frequeny polygon for the grouped dt in the tle. Sore You use the midwy vlue of eh group, just s in estimting the men. You plot the ordered pirs of midwy vlues with frequeny, nmely, (., ), (7., ), (., ), (7., ), (., 7), (7., 9) You do not know wht hppens ove nd elow the groups in the tle, so do not drw lines efore (., ) or fter (7., 9). The digrm shows the shpe of the distriution. Br hrts nd histogrms You should lredy e fmilir with the r hrt in whih the vertil xis represents frequeny, nd the horizontl xis represents the type of dt. (Sometimes it is more onvenient to hve the xes the other wy.) J F M A M J Month A histogrm looks similr to r hrt, ut there re four fundmentl differenes. There re no gps etween the rs. HrperCollinsPulishers Limited 7

13 The horizontl xis hs ontinuous sle sine it represents ontinuous dt, suh s time, weight or length. The re of eh r represents the lss or group frequeny of the r. The vertil xis is lelled density, where density = of lss intervl Width of lss intervl When the dt is not ontinuous, simple r hrt is used. For exmple, you would use r hrt to represent the runs sored in test mth or the gols sored y hokey tem. Look t the histogrm elow, whih hs een drwn from this tle of times tken y people to wlk to work. Time, t (min) density t t t. t density 6 6 Time (min) Notie tht eh histogrm r strts t the lest possile time nd finishes t the gretest possile time for its group. Using your lultor Histogrms n lso e drwn on grphis lultors or y using omputer softwre pkges. If you hve ess to either of these, try to use them. HrperCollinsPulishers Limited 7 9

14 EXERCISE D The tle shows how mny students were sent from one prtiulr lss throughout the yer. Students sent Drw frequeny polygon to illustrte the dt. Clulte the men numer of senes eh lesson. The tle shows the numer of gols sored y hokey tem in one seson. Gols 9 7 Drw the frequeny polygon for this dt. Clulte the men numer of gols sored per gme in the seson. The frequeny polygon shows the mount of money spent in orner shop y the first ustomers on one dy Amount spent, i Use the frequeny polygon to omplete the tle for the mounts spent y the first ustomers Amount spent, m, m m m m m ii Work out the men mount of money spent y these ustomers. Mid morning the shopkeeper reords the mount spent y nother ustomers. The tle elow shows the dt. Amount spent, m, m m m 6 6 m m i ii On opy of the grph ove, drw the frequeny polygon to show this dt? Clulte the men mount spent y the mid-morning ustomers.. Comment on the differenes etween the frequeny polygons nd the verge mounts spent y the different sets of ustomers. HrperCollinsPulishers Limited 7

15 The tle shows the rnge of heights of the girls in Y t London shool. Height, h (m) h h 7 h h 6 6 h 7 Drw frequeny polygon for this dt. Drw histogrm for this dt Estimte the men height of the girls. A dotor ws onerned t the length of time her ptients hd to wit to see her when they me to the morning surgery. The survey she did gve her these results. Time, m (minutes) Mondy Tuesdy Wednesdy m 9 7 m 6 m 7 6 m 9 m 7 m 6 Drw frequeny polygon for eh dy on the sme pir of xes. Wht is the verge mount of time spent witing eh dy? Why might the verge time for eh dy e different?. Histogrms with rs of unequl width In this setion you will lern how to: drw nd red histogrms where the rs re of unequl width find the medin, qurtiles nd interqurtile rnge from histogrm Key words lss intervl interqurtile rnge lower qurtile medin upper qurtile Sometimes the dt in frequeny distriution re grouped into lsses whose intervls re different. In this se, the resulting histogrm hs rs of unequl width. The key ft tht you should lwys rememer is tht the re of r in histogrm represents the lss frequeny of the r. So, in the se of n unequl-width histogrm, the height to drw eh r is found y dividing its lss frequeny y its lss intervl width (r width), whih is the differene etween the lower nd upper ounds for eh intervl. Conversely, given histogrm, ny of its lss frequenies n e found y multiplying the height of the orresponding r y its width. It is for this reson tht the sle on the vertil xes of histogrms is nerly lwys lelled density, where density = of lss intervl Width of lss intervl HrperCollinsPulishers Limited 7

16 EXAMPLE 6 The heights of group of girls were mesured. The results were lssified s shown in the tle. Height, h (m) h 6 h h h 9 9 h It is onvenient to write the tle vertilly nd dd two olumns, lss width nd frequeny density. The lss width is found y sutrting the lower lss oundry from the upper lss oundry. The frequeny density is found y dividing the frequeny y the lss width. Height, h (m) Clss width density h 6 h h 7 7 h h 6 The histogrm n now e drwn. The horizontl sle should e mrked off s norml from vlue elow the lowest vlue in the tle to vlue ove the lrgest vlue in the tle. In this se, mrk the sle from m to 6 m. The vertil sle is lwys frequeny density nd is mrked up to t lest the lrgest frequeny density in the tle. In this se, is sensile vlue. Eh r is drwn etween the lower lss intervl nd the upper lss intervl horizontlly, nd up to the frequeny density vertilly. density Height (m) HrperCollinsPulishers Limited 7

17 EXAMPLE 7 This histogrm shows the distriution of heights of dffodils in greenhouse. Complete frequeny tle for the heights of the dffodils, nd show the umultive frequeny. Find the medin height. Find the interqurtile rnge of the heights. d Estimte the men of the distriution. 6 density Height, h, m 6 7 The frequeny tle will hve groups of h 6, 6 h 7, et. These re red from the height xis. The frequenies will e found y multiplying the width of eh r y the frequeny density. Rememer tht the vlue on the vertil xis is not the frequeny. Height, h (m) h 6 6 h 7 7 h h h 7 6 Cumultive frequeny 6 There re vlues so the medin will e the th vlue. Counting up the frequenies from the eginning we get the third row of the tle ove. The medin ours in the h group. There re 6 vlues efore this group nd in it. To get to the th vlue we need to go more vlues into this group. out of is one-third. One third of the wy through this group is the vlue 9 m. Hene the medin is 9 m. The interqurtile rnge is the differene etween the upper qurtile nd the lower qurtile, the qurter nd three-qurter vlues respetively. In this se, the lower qurtile is the th vlue (found y dividing, the totl numer of vlues, y ) nd the upper qurtile is the th vlue. So, in the sme wy tht you found the medin, you n find the lower (th vlue) nd upper (th vlue) qurtiles. The th vlue is t 7 m nd the th vlue is t m. The interqurtile rnge is m 7 m = m. d To estimte the men, use the tle to get the midwy vlues of the groups nd multiply these y the frequenies. The sum of these divided y will give the estimted men. So, the men is ( ) = = 9.6 m ( signifint figures) HrperCollinsPulishers Limited 7

18 EXERCISE E Drw histogrms for these grouped frequeny distriutions. Temperture, t ( C) t t t t 7 7 t t 6 Wge, w ( ) 6 w 6 w w w Age, (nerest yer) d Pressure, p (mm) 7 p 7 7 p p p 77 e Time, t (min) t 7 t t 6 6 t 6 The following informtion ws gthered out the weekly poket money given to yer olds. Poket money, p ( ) Girls Boys p 6 p p p p 6 Represent the informtion out the oys on histogrm. Represent oth sets of dt with frequeny polygon, using the sme pir of xes. Wht is the men mount of poket money given to eh sex? Comment on your nswer. The sles of the Str newspper over 6 yers re reorded in this tle. Yers Copies Illustrte this informtion on histogrm. Tke the lss oundries s 9, 9, 97, 9, 99, 99,. HrperCollinsPulishers Limited 7

19 The London trins were lwys lte, so one month survey ws undertken to find how mny trins were lte, nd y how mny minutes (to the nerest minute). The results re illustrted y this histogrm. density 6 Minutes lte How mny trins were in the survey? How mny trins were delyed for longer thn minutes? For eh of the frequeny distriutions illustrted in the histogrms i ii write down the grouped frequeny tle, stte the modl group, iii estimte the medin, iv find the lower nd upper qurtiles nd the interqurtile rnge, v estimte the men of the distriution. 7 6 density Age (yers) HrperCollinsPulishers Limited 7

20 density Temperture ( C) density Weight (kg) All the ptients in hospitl were sked how long it ws sine they lst sw dotor. The results re shown in the tle. Hours, h h h h 6 6 h h 6 6 h Find the medin time sine ptient lst sw dotor. Estimte the men time sine ptient lst sw dotor. Find the interqurtile rnge of the times. One summer, Alert monitored the weight of the tomtoes grown on eh of his plnts. His results re summrised in this tle. Weight, w (kg) 6 w w w 6 6 w 6 w d Drw histogrm for this distriution. Estimte the medin weight of tomtoes the plnts produed. Estimte the men weight of tomtoes the plnts produed. How mny plnts produed more thn kg? 6 HrperCollinsPulishers Limited 7

21 A survey ws rried out to find the speeds of rs pssing prtiulr point on the M. The histogrm illustrtes the results of the survey. density Speed (mph) Copy nd omplete this tle. Speed, v (mph) v v v 6 6 v 7 7 v v Find the numer of rs inluded in the survey. Work out n estimte of the medin speed of the rs on this prt of the M. d Work out n estimte of the men speed of the rs on this prt of the M..6 Moving verges In this setion you will lern how to: lulte moving verge nd use it to predit future trends Key words moving verge sesonl trend trend line A moving verge gives ler indition of the trend of set of dt. It smoothes out, for exmple, sesonl trends suh s monthly vritions or dily differenes. HrperCollinsPulishers Limited 7 7

22 EXAMPLE A vn rentl firm hs reord of how mny vns were hired in eh month of yer. This dt is shown in the tle. Using four-point moving verge, predit the numer of vns the firm will rent out during the following Jnury. Months Jn Fe Mr Apr My Jun Jul Aug Sep Ot Nov De Vns First, plot the rw dt. The resulting line grph shows norml vrition of usiness for the hire firm, ut does not revel the generl trend of usiness. Is the firm s usiness improving, delining or remining the sme? You n show the generl trend y first lulting the men for eh four-month spn, month on month. This is the four-point moving verge. Men for Jn, Fe, Mr nd April ( ) =. Men for Fe, Mr, Apr nd My ( ) =.7 Numer of vns hired J F M A M J J A S O N D Men for Mr, Apr, My nd Jun ( ) =. And so on, giving.,.,.7, 6.,,.7 s the remining verges. Then plot, on the first grph, eh men vlue t the midpoint of the orresponding four-month spn. This produes muh smoother grph, whih, in this se, shows slight upwrd trend. In other words, usiness is improving. Drw line of est fit (the trend line) through the dt nd red off the predited vlue of the next four-point moving verge. This is out. Let the vlue for the next Jnury e x, then ( x) = 9 + x = x = So we n predit tht the firm will rent out vns the following Jnury. Numer of vns hired J F M A M J J A S O N D In Exmple, we used n intervl of four months to onstrut moving verge ut there is nothing speil out this intervl. It ould well hve een five or six months, exept tht you would then hve needed dt for more months to give suffiient men vlues to show trend. The numer of months, weeks or even yers used for moving verges depends on the likely vritions of the dt. You would not expet to use less thn three or more thn items of dt t time. HrperCollinsPulishers Limited 7

23 EXERCISE F The tle shows the dily sles of milk t lol orner shop for month. Sun Mon Tue Wed Thu Fri St Mke tle showing the moving verge using seven-dy spn, nd drw grph to show the trend of milk sles over the month. The tle shows the mounts olleted for hrity y the students t Pope Pius Shool in the ten weeks leding up to Christms. Week Amount ( ) Plot line grph of the mounts olleted nd four-week moving verge. Comment on the trend shown. The tle shows the qurterly eletriity ill over four-yer period. First qurter Seond qurter Third qurter Fourth qurter Plot the line grph of the eletriity ills shown in the tle, nd on the sme xes plot four-qurter moving verge. Comment on the prie of eletriity over the four yers. Use the trend line of the moving verges to predit the ill for the first qurter of 6. The tle shows the telephone ills for fmily over four yers. First qurter 7 9 Seond qurter Third qurter Fourth qurter 97 7 Plot line grph showing the mounts pid eh month. Plot four-qurter moving verge. Comment on the trend shown nd give possile reson for it. d Use the trend line of the moving verges to predit the ill for the first qurter of 6. HrperCollinsPulishers Limited 7 9

24 A ftory mking omputer omponents hs the following sles figures (in hundreds) for eletri fns. Jn Fe Mr Apr My Jun Jul Aug Sep Ot Nov De Plot line grph of the sles, nd three-month moving verge. Comment on the trend in the sles. Use the trend line of the moving verges to predit the numer of eletri fn sles in Jnury 6. The tle shows the totl sles of video reorders nd DVD plyers from 999 to from n eletril store in the USA. 999 Video (thousnds) DVD (thousnds) d Plot line grph showing the sles for eh produt over these yers. On the sme digrm, plot the three-yer moving verge of eh produt. Comment on the trends seen in the sles of video reorders nd DVDs. Use the trend line of the moving verges to predit the numer of video reorders nd DVD plyers sold in 6..7 Surveys In this setion you will lern how to: ondut surveys sk good questions in order to ollet relile nd vlid dt Key words dt olletion sheet hypothesis leding question survey A survey is n orgnised wy of sking lot of people few, well-onstruted questions, or of mking lot of oservtions in n experiment, in order to reh onlusion out something. Surveys re used to test out people s opinions or to test hypothesis. 6 HrperCollinsPulishers Limited 7

25 Simple dt olletion sheet If you just need to ollet some dt to nlyse, you will hve to design simple dt olletion sheet. This setion will show you how to design ler, esy-to-fill-in dt olletion sheet. For exmple, if you wnt to find out Y students preferenes for the end-of-term trip from four options you ould sk: Where do you wnt to go for the Y trip t the end of term Blkpool, Alton Towers, The Gret Western Show or London? You would put this question, on the sme dy, to lot of Y students, nd enter their nswers stright onto dt olletion sheet, s elow. Ple Tlly Blkpool Alton Towers 6 The Gret Western Show London Notie how plenty of spe is ville for the tlly mrks, nd how the tllies re gted in groups of five to mke ounting esier when the survey is omplete. This is good, simple dt olletion sheet euse: only one question (Where do you wnt to go?) hs to e sked ll the four possile venues re listed the nswer from eh interviewee n e esily nd quikly tllied, then on to the next interviewee. Notie, too, tht sine the question listed speifi ples, they must pper on the dt olletion sheet. You would lose mny mrks in n exmintion if you just sked the open question: Where do you wnt to go? Dt sometimes needs to e olleted to otin responses for two different tegories. The dt olletion sheet is then in the form of two-wy tle. HrperCollinsPulishers Limited 7 6

26 EXAMPLE 9 The hed of shool rries out survey to find out how muh time students in different yer groups spend on their homework during prtiulr week. He sks smple of 6 students nd fills in two-wy tle with hedings s follows. Yer 7 hours hours hours More thn hours This is not good tle s the hedings overlp. A student who does hours work week ould tik either of two olumns. Response setions should not overlp, so tht there is only one possile ple to put tik. A etter tle would e: up to More thn More thn More thn hours nd up to hours nd up to hours hours Yer 7 Yer Yer 9 Yer Yer This gives lerer piture of the mount of homework done in eh yer group. Using your omputer One the dt hs een olleted for your survey, it n e put into omputer dtse. This llows the dt to e stored nd mended or updted t lter dte if neessry. From the dtse, suitle sttistil digrms n esily e drwn using softwre, nd verges lulted. The results n then e pulished in, for exmple, the shool mgzine. EXERCISE G People like the supermrket to open on Sundys. To see whether this sttement is true, design dt olletion sheet whih will llow you to pture dt while stnding outside supermrket. Does it mtter on whih dy you ollet dt outside the supermrket? The shool tuk shop wnts to know whih types of hoolte it should get in to sell plin, milk, fruit nd nut, wholenut or white hoolte. Design dt olletion sheet whih you ould use to sk pupils in your shool whih of these hoolte types re their fvourite. Invent the first entries on the hrt. Inlude spe for tllies. 6 HrperCollinsPulishers Limited 7

27 When you throw two die together, wht numer re you most likely to get? d Design dt olletion sheet on whih you n reord the dt from n experiment in whih two die re thrown together nd note the sum of the two numers shown on the die. Crry out this experiment for t lest throws. Whih sums re most likely to our? Illustrte your results on frequeny polygon. Who uses the uses the most in the mornings? Is it pensioners, mums, shoolhildren, the unemployed or some other group? Design dt olletion sheet to e used in survey of us pssengers. Design two-wy tles to show how students in different yer groups trvel to shool in the morning the type of progrmme whih different ge groups prefer to wth on TV the fvourite sport of oys nd girls Mke sure ll possile responses re overed. d how muh time students in different yer groups spend on the omputer in the evening. Invent out entries for eh one. Questionnires This setion will show you how to put together ler, esy-to-use questionnire. When you re putting together questionnire, you must think very refully out the sorts of question you re going to sk. Here re five rules tht you should lwys follow. Never sk leding question designed to get prtiulr response. Never sk personl, irrelevnt question. Keep eh question s simple s possile. Inlude questions tht will get response from whomever is sked. Mke sure the responses do not overlp nd keep the numer of hoies to resonle numer (six t the most). HrperCollinsPulishers Limited 7 6

28 The following questions re dly onstruted nd should never pper in ny questionnire. Wht is your ge? This is personl. Mny people will not wnt to nswer. It is lwys etter to give rnge of ges. Under 6 Over Slughtering nimls for food is ruel to the poor defeneless nimls. Don t you gree? This is leding question, designed to get yes response. It is etter to sk n impersonl question. Are you vegetrin? Yes No Do you go to disos when rod? This n e nswered only y people who hve een rod. It is etter to sk strter question, with follow-up question. Hve you een rod for holidy? Yes No If yes, did you go to diso whilst you were wy? Yes No When you first get up in morning nd deide to hve some sort of rekfst tht might e mde y someody else, do you feel oliged to et it ll or not? This is too-omplited question. It is etter to sk series of shorter questions. Wht time do you get up for shool? Before 7 Between 7 nd After Do you hve rekfst every dy? Yes No If No, on how mny shool dys do you hve rekfst? A questionnire is usully put together to test hypothesis or sttement. For exmple, questionnire might e onstruted to test this sttement. People uy heper milk from the supermrket s they don t mind not getting it on their doorstep. They d rther go out to uy it. A questionnire designed to test whether this sttement is true or not should inlude these questions: Do you hve milk delivered to your doorstep? Do you uy heper milk from the supermrket? Would you uy your milk only from the supermrket? One these questions hve een nswered, the responses n e looked t to see whether the mjority of people hold views tht gree with the sttement. 6 HrperCollinsPulishers Limited 7

29 EXERCISE H These re questions from questionnire on helthy eting. Fst food is d for you. Don t you gree? Strongly gree Agree Don t know Give two ritiisms of the question. Do you et fst food? Yes No If yes, how mny times on verge do you et fst food week? One or less or times or times More thn times Give two resons why these re good questions. This is question from survey on poket money. How muh poket money do you get eh week? or more Give reson why this is not good question. Rewrite the question to mke it good question. Design questionnire to test this sttement. People under 6 do not know wht is ment y ll the jrgon used in the usiness news on TV, ut the over-twenties do. Design questionnire to test this sttement. The under-twenties feel quite t ese with omputers, while the over-forties would rther not other with them. The s re ll le to use omputers effetively. Design questionnire to test this hypothesis. The older you get, the less sleep you need. A hed teher wnts to find out if her pupils think they hve too muh, too little or just the right mount of homework. She lso wnts to know the prents views out homework. Design questionnire tht ould e used to find the dt tht the hed teher needs to look t. HrperCollinsPulishers Limited 7 6

30 . Soil sttistis In this setion you will lern out: lern out soil sttistis e introdued to some of the more ommon soil sttistis in dily use Key words mrgin of error ntionl ensus polls Retil Prie Index soil sttistis time series Mny situtions our in dily life where sttistil tehniques re used to produe dt. The results of surveys pper in newsppers every dy. There re mny on-line polls nd phone-ins to vote in relity TV shows, for exmple. Results for these polls re usully given s perentge with mrgin of error, whih is mesure of how urte the informtion is. Here re some ommon soil sttistis in dily use. Generl Index of Retil Pries This is lso know s the Retil Prie Index (RPI). It mesures how muh the dily ost of living inreses (or dereses). One yer is hosen s the se yer nd given n index numer, usully. The osts of susequent yers re ompred to this nd given numer proportionl to the se yer, sy, et. Note the numers do not represent tul vlues ut just ompre urrent pries to the se yer. Time series Like the RPI, time series mesures hnges in quntity over time. Unlike the RPI the tul vlues of the quntity re used. This might mesure how the exhnge rte etween the pound nd the dollr hnges over time. Ntionl Census A ntionl ensus is survey of ll people nd households in ountry. Dt out ge, gender, religion, employment sttus, et. is olleted to enle governments to pln where to llote resoures in the future. In Britin, ntionl ensus is tken every yers. The lst ensus ws in. 66 HrperCollinsPulishers Limited 7

31 EXERCISE I In, the ost of litre of petrol ws 6p. Using s se yer, the prie index of petrol for the next yers is shown in this tle. Yer Index Prie 7p 9 Work out the prie of petrol in eh susequent yer. Give your nswers to deiml ple. The grph shows the exhnge rte for the dollr ginst the pound for eh month in. Exhnge rte of the dollr ginst the pound, Wht ws the exhnge rte in Jnury? Between whih two months did the exhnge rte fll the most? Explin why you ould not use the grph to predit the exhnge rte in Jnury 6. Rte.9..7 J F M A M J J A S O N D Month The following is tken from the UK government sttistis wesite. In mid- the UK ws home to 9. million people, of whih. million lived in Englnd. The verge ge ws.6 yers, n inrese on 97 when it ws. yers. In mid- pproximtely one in five people in the UK were ged under 6 nd one in six people were ged 6 or over. Use this extrt to nswer these questions. How mny of the popultion of the UK do not live in Englnd? By how muh hs the verge ge inresed sine 97? Approximtely how mny of the popultion re under 6? d Approximtely how mny of the popultion re over 6? The Generl Index of Retil Pries strted in Jnury 97 when it ws given se numer of. In Jnury 6 the index numer ws 9.. If the stndrd weekly shopping sket ost. in Jnury 97, how muh would it e in Jnury 6? HrperCollinsPulishers Limited 7 67

32 This time series shows r prodution in Britin from Novemer to Novemer. Cr prodution in Britin, Novemer to Novemer Why ws there shrp drop in prodution in July? 9 The verge prodution over the first three months shown ws 7 thousnd rs. i ii Work out n pproximte numer for the verge prodution over the lst three months shown. The se month for the index is Jnury when the index ws. Wht ws the pproximte prodution in Jnury? Index 7 6 N D J F M A M J J A S O N.9 Smpling In this setion you will lern how to: understnd different methods of smpling ollet unised relile dt Key words popultion rndom smple strtified unised Sttistiins often hve to rry out surveys to ollet informtion nd test hypotheses out the popultion of wide vriety of things. (In sttistis, popultion does not only men group of people, it lso mens group of ojets or events.) It is seldom possile to survey whole popultion, minly euse suh survey would ost too muh nd tke long time. Also there re popultions for whih it would e physilly impossile to survey every memer. For exmple, if you wnted to find the verge length of eels in the North Se, it would e impossile to find nd mesure every eel. So sttistiin hooses smll prt of the popultion to survey nd ssumes tht the results for this smple re representtive of the whole popultion. Therefore, to ensure the ury of survey, two questions hve to e onsidered. Will the smple e representtive of the whole popultion nd therey eliminte is? How lrge should the smple e to give results whih re vlid for the whole popultion? You will use mny of these ides in your Hndling Dt oursework. Smpling methods There re two min types of smple: rndom nd strtified. 6 HrperCollinsPulishers Limited 7

33 In rndom smple, every memer of the popultion hs n equl hne of eing hosen. For exmple, it my e the first people met in survey, or nmes piked from ht, or nmes tken t rndom from the eletorl register or telephone diretory. In strtified smple, the popultion is first divided into tegories nd the numer of memers in eh tegory determined. The smple is then mde up of memers from these tegories in the sme proportions s they re in the popultion. The required smple in eh tegory is hosen y rndom smpling. EXAMPLE A shool s pupil numers re given in the tle. The hed teher wnts to tke strtified smple of pupils for survey. Clulte the numer of oys nd girls in eh yer tht should e interviewed. Explin how the pupils ould then e hosen to give rndom smple. Shool yer Boys Girls Totl Totl numer in shool To get the orret numer in eh tegory, sy, oys in yer 7, the lultion is done s follows. = 9.6 ( deiml ple) After ll lultions re done, you should get the vlues in this tle. Shool Yer Boys Girls Oviously you nnot hve deiml point of pupil, so round off ll vlues nd mke sure tht the totl is. This gives the finl tle. Shool yer Boys Girls Totl Within eh tegory, hoose pupils to survey t rndom. For exmple, ll the yer 7 girls ould hve their nmes put into ht nd nmes drwn out or they ould e listed lphetilly nd rndom numer genertor used to pik out nmes from 6. HrperCollinsPulishers Limited 7 69

34 Smple size Before the smpling of popultion n egin, it is neessry to determine how muh dt needs to e olleted to ensure tht the smple is representtive of the popultion. This is lled the smple size. Two ftors determine smple size: the desired preision with whih the smple represents the popultion the mount of money ville to meet the ost of olleting the smple dt. The greter the preision desired, the lrger the smple size needs to e. But the lrger the smple size, the higher the ost will e. Therefore, the enefit of hieving high ury in smple will lwys hve to e set ginst the ost of hieving it. There re sttistil proedures for determining the most suitle smple size, ut these re eyond the sope of the GCSE syllus. The next exmple ddresses some of the prolems ssoited with otining n unised smple. EXAMPLE You re going to ondut survey mong n udiene of people t rok onert. How would you hoose the smple? You would not wnt to question ll of them, so you might settle for smple size of %, whih is 6 people. Assuming tht there will e s mny men t the onert s women, you would need the smple to ontin the sme proportion of eh, nmely, men nd women. Assuming tht out % of the udiene will e ged under, you would lso need the smple to ontin people ged under (% of 6) nd people ged nd over (6 or % of 6). You would lso need to selet people from different prts of the uditorium in equl proportions so s to get lned view. Sy this reks down into three equl groups of people, tken respetively from the front, the k nd the middle of the uditorium. So, you would further need the smple to onsist of people t the front, t the k nd in the middle. If you now ssume tht one reserher n survey onert-goers, you would rrive t this smpling strtegy 6 = reserhers to ondut the survey = reserhers in eh prt of the uditorium Eh reserher would need to question four men ged under, 6 men ged nd over, four women ged under nd 6 women ged nd over. 7 HrperCollinsPulishers Limited 7

35 EXERCISE J Comment on the reliility of the following wys of finding smple. d e Find out out smoking y sking people in non-smoking prt of resturnt. Find out how mny homes hve DVD plyers y sking people outside DVD hire shop. Find the most populr mke of r y ounting rs in ity r prk. Find yer representtive on shool s ounil y piking nme out of ht. Deide whether the pottoes hve ooked properly y testing one with fork. Comment on the wy the following smples hve een tken. For those tht re not stisftory, suggest etter wy to find more relile smple. d e Joseph hd disussion with his dd out poket money. To get some informtion, he sked of his friends how muh poket money they eh reeived. Dougls wnted to find out wht proportion of his shool went rod for holidys, so he sked the first people he me ross in the shool yrd. A teher wnted to know whih lesson his pupils enjoyed most. So he sked them ll. It hs een suggested tht more femles go to hurh thn mles. So Ruth did survey in her hurh tht Sundy nd ounted the numer of femles there. A group of lol people sked for rossing on usy rod. The ounil onduted survey y sking rndomly seleted people in the neighourhood. For shool projet you hve een sked to do presenttion of the soil tivities of the pupils in your shool. You deide to interview smple of pupils. Explin how you will hoose the pupils you wish to interview if you wnt your results to e relile, unised, representtive, d rndom. A fst-food pizz hin ttempted to estimte the numer of people who et pizzs in ertin town. One evening they telephoned people living in the town nd sked: Hve you eten pizz in the lst month? Eleven people sid Yes. The pizz hin stted tht % of the town s popultion et pizzs. Give three ritiisms of this method of estimtion. Adm is writing questionnire for survey out the Medowhll shopping entre in Sheffield. He is told tht fewer lol people visit Medowhll thn people from further wy. He is lso told tht the lol people spend less money per visit. Write two questions whih would help him to test these ides. Eh question should inlude t lest three options for response. People re sked to hoose one of these options. For nother survey, Adm investigtes how muh is spent t the hoolte mhines y students t his shool. The numer of students in eh yer group is shown in the tle. Explin, with lultions, how Adm should otin strtified rndom smple of students for his survey. Yer group Numers of students HrperCollinsPulishers Limited 7 7

36 Clire mde survey of pupils in her shool. She wnted to find out their opinions on the eting filities in the shool. The size of eh yer group in the shool is shown in the tle. Yer group Boys Girls Totl th form 7 6 Clire took smple of 9 pupils. Explin why she should not hve smpled equl numers of oys nd girls in the sixth form. Clulte the numer of pupils she should hve smpled in the sixth form. Using the Internet Through the Internet you hve ess to vst mount of dt on mny topis, whih you n use to rry out sttistil investigtions. This dt will enle you to drw sttistil digrms, nswer vriety of questions nd test ll mnner of hypotheses. Here re some exmples of hypotheses you n test. Footll tems re most likely to win when they re plying t home. Boys do etter thn girls t GCSE mthemtis. The numer gets drwn more often thn the numer 9 in the Ntionl Lottery. The litery rte in ountry is linked to tht ountry s verge inome. People in the north of Englnd hve lrger fmilies thn people who live in the south. The following wesites re useful soure of dt for some of the ove HrperCollinsPulishers Limited 7

37 The numer of mthes in mthoxes is shown in the tle. The tle shows the times tken for trin journey for dys. No. of mthes (m) Time tken, t (min) Clulte the men numer of mthes in the oxes. people were sked how long they hd to wit for trm. The tle shows the results. Whih lss intervl ontins the medin time? Time tken, t (min) 6 < t 7 < t < t 9 < t < t < t 6 < t < t 6 6 < t Clulte n estimte of the men journey time Some students t Highfliers Shool took mthemtis exmintion. The unfinished tle nd histogrm show some informtion out their mrks. Mrk (x%) < x < x 6 6 < x 7 7 < x 6 < x 9 Drw frequeny digrm to represent the dt. 9 < x In Junior Shool there re students who tke the Mths nd Siene KS tests. Their Ntionl Curriulum levels in these sujets re shown in the two-wy tle. 6 Level in Mths Level in Siene 6 6 density Wht is the modl level for Mths? Wht is the medin level for Mths? Show lerly how you otined your nswer. Wht is the men level for Siene? Show lerly how you otined your nswer. 6 Mrk (x%) d The teher lims tht the students re etter t Mths thn t Siene. How n you tell from the tle tht this is true? Use the informtion in the tle to opy nd omplete the histogrm. Use the informtion in the histogrm to find the missing frequeny from the tle. Edexel, Question 6, Pper A Higher, Mrh HrperCollinsPulishers Limited 7 7

38 Jk nd Jill re doing survey on fst food. This is one of Jk s questions. Give two resons why this is not good question. This is one of Jill s questions. Give two resons why this is good question. A shop sells DVD plyers. The tle shows the numer of DVD plyers sold in every three-month period from Jnury to June. Burgers re d for you nd mke you ft. Yes No How mny times, on verge, do you visit fst food outlet in week? Never or times or times More thn times Yer Months Numer of DVD plyers sold Jn Mr Apr Jun 6 Jul Sep 6 Ot De Jn Mr 6 Apr Jun 7 Clulte the set of four-point moving verges for this dt. Wht do your moving verges in prt tell you out the trend in the sle of DVD plyers? Edexel, Question, Pper B Higher, Mrh Explin wht is ment y i rndom smple, ii strtified smple. The tle shows some informtion out the memers of golf lu. Age rnge Mle Femle Totl Under 9 9 to to 7 9 Over 9 9 Totl numer of memers The lu seretry rries out survey of the memers. He hooses smple, strtified oth y ge rnge nd y gender, of 9 of the memers. Work out n estimte of the numer of mle memers, in the ge rnge to, he would hve to smple. Edexel, Question, Pper Higher, June A vet does weekly hek on the wter nimls t zoo. There re wlruses, 6 penguins nd sels. The vet is required to see % of the nimls nd to see eh type. Wht is this kind of smpling proedure lled? How mny of eh type of niml should the vet see? The msses of mrrows re mesured. Drw histogrm to show this informtion Use, your histogrm, or otherwise, to estimte the medin mss of the mrrows. The histogrm shows the numer of students t different Junior shools in Wles. density Find the medin numer of students t the shools. Find the interqurtile rnge of the numer of students t the shools. This histogrm shows the riket sores of Yorkshire Legue plyers. density Mss, m, (grms) i ii 7 9 Numer of students Sore Wht is the medin sore? Wht is the interqurtile rnge? < m 6 6 < m < m < m < m 7 HrperCollinsPulishers Limited 7

39 This histogrm is inomplete. It shows some of the riket sores for Lnshire Legue plyers. The medin sore is the sme s for the Yorkshire plyers. The upper qurtile for the Lnshire plyers is. i Wht is the lower qurtile for the Lnshire plyers? ii Complete possile histogrm. density Sore WORKED EXAM QUESTION The distnes trvelled y rs using litres of petrol is shown in the histogrm nd tle. Complete the histogrm nd the tle.. Estimte the numer of rs tht trvel etween km nd km using litres of petrol. density... Distne (km) Distne (km) Solution Set up the tle with olumns for lss width nd frequeny density nd fill in the given informtion, reding frequeny densities from the grph (e reful with sles). Now fill in the rest of the informtion Distne Clss width frequeny using f.d. = nd (km) density lss width 9. frequeny = f.d. lss width.. These vlues re shown in red. Complete the grph density Distne (km) Drw lines t nd. The numer of rs is represented y the re etween these lines. In the 6 r the re is of the totl. In the 6 r the re is of the totl. Numer of rs = + =. rs HrperCollinsPulishers Limited 7 7

40 Mr Dvies is diry frmer. Every month he reords the thousnds of litres of milk produed y his ows. For his usiness pln he ompres the mount of milk he produes in with. Monthly milk prodution in thousnds of litres Month Jn 6 Fe 6 Mr 6 Apr 6 67 My 6 7 Jun 7 Jul 7 Aug 7 6 Sep 6 7 Ot 6 7 Nov 6 7 De 6 Mr Dvies lultes three-month moving verges for nd. He plots line grphs showing the moving verges for these two yers. Help him to omplete the moving verges tle, nd the line grphs. Comment on the trends shown. -month moving verge for milk prodution in thousnds of litres Jn Fe Mr Apr My Jun Jul Aug Sep Ot Nov De Milk prodution in thousnds of litres month moving verges for nd Key J F M A M J J A S O N D Months 76 HrperCollinsPulishers Limited 7

41 Sttistis For his usiness pln Mr Dvies ompres the mount of milk he produes in, with the grphs showing the hours of sunshine nd mount of rin tht yer. Monthly rinfll in millimetres in J F M A M J J A S O N D Months of the yer Monthly sunshine in hours in J F M A M J J A S O N D Months of the yer Drw two stter grphs; one showing his monthly milk prodution nd monthly rinfll, the other showing his monthly milk prodution nd the monthly sunshine. Comment on the orreltion shown y these grphs. HrperCollinsPulishers Limited 7 77