9 Data Aalysis 9. Mea, Media, Mode ad Rage I Uit 8, you were lookig at ways of collectig ad represetig data. I this uit, you will go oe step further ad fid out how to calculate statistical quatities which summarise the importat characteristics of the data. The mea, media ad mode are three differet ways of describig the average. To fid the mea, add up all the umbers ad divide by the umber of umbers. To fid the media, place all the umbers i order ad select the middle umber. The mode is the umber which appears most ofte. The rage gives a idea of how the data are spread out ad is the differece betwee the smallest ad largest values. Worked Example Fid the mea the media the mode (d) the rage of this set of data. 5, 6,, 4, 7, 8, 3, 5, 6, 6 Solutio The mea is 5+ 6+ + 4+ 7+ 8+ 3+ 5+ 6+ 6 0 = 5 0 = 5.. To fid the media, place all the umbers i order., 3, 4, 5, 5, 6, 6, 6, 7, 8 As there are two middle umbers i this example, 5 ad 6, media = 5+ 6 = = 55.. (d) From the list above it is easy to see that 6 appears more tha ay other umber, so mode = 6. The rage is the differece betwee the smallest ad largest umbers, i this case ad 8. So the rage is 8 6 =. 45
9. MEP Pupil Text 9 Worked Example Five people play golf ad at oe hole their scores are 3, 4, 4, 5, 7. For these scores, fid the mea the media the mode (d) the rage. Solutio The mea is 3+ 4+ 4+ 5+ 7 5 = 3 5 = 46.. (d) The umbers are already i order ad the middle umber is 4. So The score 4 occurs most ofte, so, media = 4. mode = 4. The rage is the differece betwee the smallest ad largest umbers, i this case 3 ad 7, so rage = 7 3 = 4. Exercises. Fid the mea media, mode ad rage of each set of umbers below. 3, 4, 7, 3, 5,, 6, 0 8, 0,, 4, 7, 6, 5, 7, 9, 7, 8, 6, 7, 7, 4,, 5, 6, 7, 4, (d) 08, 99,,, 08 (e) 64, 66, 65, 6, 67, 6, 57 (f), 30,, 6, 4, 8, 6, 7. Twety childre were asked their shoe sizes. The results are give below. 8, 6, 7, 6, 5, 4, 7, 6, 8, 0 7, 5, 5 8, 9, 7, 5, 6, 8 6 For this data, fid the mea the media the mode (d) the rage. 46
3. Eight people work i a shop. They are paid hourly rates of, 5, 5, 4, 3, 4, 3, 3. Fid (i) the mea (ii) the media (iii) the mode. Which average would you use if you wated to claim that the staff were: (i) well paid (ii) badly paid? What is the rage? 4. Two people work i a factory makig parts for cars. The table shows how may complete parts they make i oe week. Worker Mo Tue Wed Thu Fri Fred 0 0 Harry 30 5 36 8 Fid the mea ad rage for Fred ad Harry. Who is most cosistet? Who makes the most parts i a week? 5. A gardeer buys 0 packets of seeds from two differet compaies. Each pack cotais 0 seeds ad he records the umber of plats which grow from each pack. Compay A 0 5 0 0 0 6 0 0 0 8 Compay B 7 8 5 6 8 8 7 5 7 8 (d) Fid the mea, media ad mode for each compay's seeds. Which compay does the mode suggest is best? Which compay does the mea suggest is best? Fid the rage for each compay's seeds. 6. Adria takes four tests ad scores the followig marks. 65, 7, 58, 77 What are his media ad mea scores? If he scores 70 i his ext test, does his mea score icrease or decrease? Fid his ew mea score. Which has icreased most, his mea score or his media score? 7. Richard keeps a record of the umber of fish he catches over a umber of fishig trips. His records are:, 0,, 0, 0, 0,, 0,, 0, 0,, 8, 0,, 0,. Why does he object to talkig about the mode ad media of the umber of fish caught? 47
9. MEP Pupil Text 9 What are the mea ad rage of the data? Richard's fried, Najir, also goes fishig. The mode of the umber of fish he has caught is also 0 ad his rage is 5. What is the largest umber of fish that Najir has caught? 8. A garage ower records the umber of cars which visit his garage o 0 days. The umbers are: 04, 30, 79, 34, 57, 30, 3, 6, 308, 7. Fid the mea umber of cars per day. The ower hopes that the mea will icrease if he icludes the umber of cars o the ext day. If 5 cars use the garage o the ext day, will the mea icrease or decrease? 9. The childre i a class state how may childre there are i their family. The umbers they state are give below.,,, 3,,,, 4,,,, 3,,,,,,, 7, 3,,,,,,,, 3 Fid the mea, media ad mode for this data. Which is the most sesible average to use i this case? 0. The mea umber of people visitig Jae each day over a five-day period is 8. If 0 people visit Jae the ext day, what happes to the mea?. The table shows the maximum ad miimum temperatures recorded i six cities oe day last year. City Maximum Miimum Los Ageles C C Bosto C 3 C Moscow 8 C 9 C Atlata 7 C 8 C Archagel 3 C 5 C Cairo 8 C 3 C Work out the rage of temperature for Atlata. Which city i the table had the lowest temperature? Work out the differece betwee the maximum temperature ad the miimum temperature for Moscow. (LON). The weights, i grams, of seve potatoes are 60, 5, 05, 40, 3, 05, 4. What is the media weight? (SEG) 48
3. Here are the umber of goals scored by a school football team i their matches this term. 3,, 0,,, 0, 3, 4, 3, Work out the mea umber of goals. Work out the rage of the umber of goals scored. (LON) 4. The weights, i kilograms, of the 8 members of Hereward House tug of war team at a school sports are: 75, 73, 77, 76, 84, 76, 77, 78. Calculate the mea weight of the team. The 8 members of Nelso House tug of war team have a mea weight of 64 kilograms. Which team do you thik will wi a tug of war betwee Hereward House ad Nelso House? Give a reaso for your aswer. (MEG) 5. Pupils i Year 8 are arraged i eleve classes. The class sizes are 3, 4, 4, 6, 7, 8, 30, 4, 9, 4, 7. What is the modal class size? Calculate the mea class size. The rage of the class sizes for Year 9 is 3. What does this tell you about the class sizes i Year 9 compared with those i Year 8? (SEG) 6. A school has to select oe pupil to take part i a Geeral Kowledge Quiz. Kim ad Pat took part i six trial quizzes. The followig lists show their scores. Kim 8 4 7 4 6 Pat 33 9 6 3 34 8 Kim had a mea score of 5 with a rage of 7. Calculate Pat's mea score ad rage. Which pupil would you choose to represet the school? Explai the reaso for your choice, referrig to the mea scores ad rages. (MEG) Iformatio The study of statistics was begu by a Eglish mathematicia, Joh Graut (60 674). He collected ad studied the death records i various cities i Britai ad, despite the fact that people die radomly, he was fasciated by the patters he foud. 49
9. MEP Pupil Text 9 7. Eight judges each give a mark out of 6 i a ice-skatig competitio. Oksaa is give the followig marks. 5.3, 5.7, 5.9, 5.4, 4.5, 5.7, 5.8, 5.7 The mea of these marks is 5.5, ad the rage is.4. The rules say that the highest mark ad the lowest mark are to be deleted. 5.3, 5.7, 5.9, 5.4, 4.5, 5.7, 5.8, 5.7 (i) Fid the mea of the six remaiig marks. (ii) Fid the rage of the six remaiig marks. Do you thik it is better to cout all eight marks, or to cout oly the six remaiig marks? Use the meas ad the rages to explai your aswer. The eight marks obtaied by Toya i the same competitio have a mea of 5. ad a rage of 0.6. Explai why oe of her marks could be as high as 5.9. (MEG) 9. Fidig the Mea from Tables ad Tally Charts Ofte data are collected ito tables or tally charts. This sectio cosiders how to fid the mea i such cases. Worked Example A football team keep records of the umber of goals it scores per match durig a seaso. No. of Goals Frequecy 0 8 0 3 3 4 5 5 Fid the mea umber of goals per match. Solutio The table above ca be used, with a third colum added. The mea ca ow be calculated. Mea = 73 40 =. 85. No. of Goals Frequecy No. of Goals Frequecy 0 8 0 8 = 0 0 0 = 0 = 4 3 3 3 3 = 9 4 5 4 5 = 0 5 5 = 0 TOTALS 40 73 (Total matches) (Total goals) 50
Worked Example The bar chart shows how may cars were sold by a salesma over a period of time. Frequecy 6 5 4 3 0 3 4 5 Cars sold per day Fid the mea umber of cars sold per day. Solutio The data ca be trasferred to a table ad a third colum icluded as show. Cars sold daily Frequecy Cars sold Frequecy 0 0 = 0 4 4 = 4 3 3 = 6 3 6 3 6 = 8 4 3 4 3 = 5 5 = 0 TOTALS 0 50 (Total days) (Total umber of cars sold) Worked Example 3 Mea = 50 0 = 5. A police statio kept records of the umber of road traffic accidets i their area each day for 00 days. The figures below give the umber of accidets per day. 4 3 5 5 5 4 3 0 3 3 0 5 3 3 6 6 3 5 4 4 3 3 4 7 3 3 0 5 4 3 3 4 3 4 5 3 5 4 4 6 5 4 5 5 3 0 3 3 4 5 3 3 4 4 3 5 5 6 6 4 6 5 8 5 3 3 5 4 Fid the mea umber of accidets per day. 5
9. MEP Pupil Text 9 Solutio The first step is to draw out ad complete a tally chart. The fial colum show below ca the be added ad completed. Number of Accidets Tally Frequecy No. of Accidets Frequecy 0 4 0 4 = 0 0 0 = 0 = 44 3 3 3 3 = 69 4 6 4 6 = 64 5 7 5 7 = 85 6 6 6 6 = 36 7 7 = 7 8 8 = 8 TOTALS 00 33 Exercises Mea umber of accidets per day = 33 00 = 33... A survey of 00 households asked how may cars there were i each household The results are give below. No. of Cars Frequecy 0 5 70 3 3 4 Calculate the mea umber of cars per household.. The survey of questio also asked how may TV sets there were i each household. The results are give below. No. of TV Sets Frequecy 0 30 5 3 8 4 5 5 3 Calculate the mea umber of TV sets per household. 5
3. A maager keeps a record of the umber of calls she makes each day o her mobile phoe. Number of calls per day 0 3 4 5 6 7 8 Frequecy 3 4 7 8 0 4 3 Calculate the mea umber of calls per day. 4. A cricket team keeps a record of the umber of rus scored i each over. No. of Rus Frequecy 0 3 3 6 4 5 5 4 6 7 8 Calculate the mea umber of rus per over. 5. A class coduct a experimet i biology. They place a umber of m by m square grids o the playig field ad cout the umber of worms which appear whe they pour water o the groud. The results obtaied are give below. 6 3 3 3 0 0 3 4 0 0 4 3 3 3 7 Calculate the mea umber of worms. How may times was the umber of worms see greater tha the mea? 6. As part of a survey, a statio recorded the umber of trais which were late each day. The results are listed below. 0 4 0 0 3 0 0 0 0 5 3 0 0 0 0 3 0 0 0 Costruct a table ad calculate the mea umber of trais which were late each day. 53
9. MEP Pupil Text 9 7. Haah drew this bar chart to show the umber of repeated cards she got whe she opeed packets of football stickers. q y Frequecy 0 8 6 4 0 3 4 5 6 Number of repeats Calculate the mea umber of repeats per packet. 8. I a seaso a football team scored a total of 55 goals. The table below gives a summary of the umber of goals per match. Goals per Match Frequecy 0 4 6 3 8 4 5 I how may matches did they score goals? Calculate the mea umber of goals per match. 9. A traffic warde is tryig to work out the mea umber of parkig tickets he has issued per day. He produced the table below, but has accidetally rubbed out some of the umbers. Tickets per day Frequecy No. of Tickets Frequecy 0 0 3 7 4 0 5 6 TOTALS 6 7 Fill i the missig umbers ad calculate the mea. 54
0. Here are the weights, i kg, of 30 studets. 45, 5, 56, 65, 34, 45, 67, 65, 34, 45, 65, 87, 45, 34, 56, 54, 45, 67, 84, 45, 67, 45, 56, 76, 57, 84, 35, 64, 58, 60 Copy ad complete the frequecy table below usig a class iterval of 0 ad startig at 30. Weight Rage (w) Tally Frequecy 30 w < 40 Which class iterval has the highest frequecy? (LON). The umber of childre per family i a recet survey of families is show. 3 4 3 3 4 3 What is the rage i the umber of childre per family? Calculate the mea umber of childre per family. Show your workig. A similar survey was take i 960. I 960 the rage i the umber of childre per family was 7 ad the mea was.7. Describe two chages that have occurred i the umber of childre per family sice 960. (SEG) 9.3 Calculatios with the Mea This sectio cosiders calculatios cocered with the mea, which is usually take to be the most importat measure of the average of a set of data. Worked Example The mea of a sample of 6 umbers is 3.. A extra value of 3.9 is icluded i the sample. What is the ew mea? 55
9.3 MEP Pupil Text 9 Solutio Worked Example Total of origial umbers = 6 3. = 9. New total = 9. + 3. 9 = 3. New mea = 3. 7 = 33. The mea umber of a set of 5 umbers is.7. What extra umber must be added to brig the mea up to 3.? Solutio Total of the origial umbers = 5. 7 = 63. 5 Total of the ew umbers = 6 3. = 78. 6 So the extra umber is 5.. Differece = 78. 6 63. 5 = 5. Exercises. The mea height of a class of 8 studets is 6 cm. A ew girl of height 49 cm jois the class. What is the mea height of the class ow?. After 5 matches the mea umber of goals scored by a football team per match is.8. If they score 3 goals i their 6th match, what is the mea after the 6th match? 3. The mea umber of childre ill at a school is 3.8 per day, for the first 0 school days of a term. O the st day 8 childre are ill. What is the mea after days? 4. The mea weight of 5 childre i a class is 58 kg. The mea weight of a secod class of 9 childre is 6 kg. Fid the mea weight of all the childre. 5. A salesma sells a mea of 4.6 coservatories per day for 5 days. How may must he sell o the sixth day to icrease his mea to 5 sales per day? 6. Adria's mea score for four tests is 64%. He wats to icrease his mea to 68% after the fifth test. What does he eed to score i the fifth test? 7. The mea salary of the 8 people who work for a small compay is 5 000. Whe a extra worker is take o this mea drops to 4 000. How much does the ew worker ear? 56
8. The mea of 6 umbers is.3. Whe a extra umber is added, the mea chages to.9. What is the extra umber? 9. Whe 5 is added to a set of 3 umbers the mea icreases to 4.6. What was the mea of the origial 3 umbers? 0. Three umbers have a mea of 64. Whe a fourth umber is icluded the mea is doubled. What is the fourth umber? 9.4 Mea, Media ad Mode for Grouped Data The mea ad media ca be estimated from tables of grouped data. The class iterval which cotais the most values is kow as the modal class. Worked Example The table below gives data o the heights, i cm, of 5 childre. Class Iterval 40 h < 50 50 h < 60 60 h < 70 70 h < 80 Frequecy 6 6 8 Estimate the mea height. Estimate the media height. Fid the modal class. Solutio To estimate the mea, the mid-poit of each iterval should be used. Class Iterval Mid-poit Frequecy Mid-poit Frequecy 40 h < 50 45 6 45 6 = 870 50 h < 60 55 6 55 6 = 480 60 h < 70 65 65 = 3465 70 h < 80 75 8 75 8 = 400 Totals 5 85 Mea = 85 5 = 6 (to the earest cm) The media is the 6th value. I this case it lies i the 60 h < 70 class iterval. The 4th value i the iterval is eeded. It is estimated as 4 60 + 0 = 6 (to the earest cm) The modal class is 60 h < 70 as it cotais the most values. 57
9.4 MEP Pupil Text 9 Also ote that whe we speak of someoe by age, say 8, the the perso could be ay age from 8 years 0 days up to 8 years 364 days (365 i a leap year!). You will see how this is tackled i the followig example. Worked Example The age of childre i a primary school were recorded i the table below. Age 5 6 7 8 9 0 Frequecy 9 40 38 Estimate the mea. Estimate the media. Fid the modal age. Solutio To estimate the mea, we must use the mid-poit of each iterval; so, for example for '5 6', which really meas 5 age < 7, the mid-poit is take as 6. Class Iterval Mid-poit Frequecy Mid-poit Frequecy 5 6 6 9 6 9 = 74 7 8 8 40 8 40 = 30 9 0 0 38 0 38 = 380 Totals 07 874 Mea = 874 07 = 8. (to decimal place) The media is give by the 54th value, which we have to estimate. There are 9 values i the first iterval, so we eed to estimate the 5th value i the secod iterval. As there are 40 values i the secod iterval, the media is estimated as beig 5 40 of the way alog the secod iterval. This has width 9 7 = years, so the media is estimated by 5 40 = 5. from the start of the iterval. Therefore the media is estimated as 7 +. 5 = 8. 5 years. The modal age is the 7 8 age group. 58
Worked Example uses what are called cotiuous data, sice height ca be of ay value. (Other examples of cotiuous data are weight, temperature, area, volume ad time.) The ext example uses discrete data, that is, data which ca take oly a particular value, such as the itegers,, 3, 4,... i this case. The calculatios for mea ad mode are ot affected but estimatio of the media requires replacig the discrete grouped data with a approximate cotiuous iterval. Worked Example 3 The umber of days that childre were missig from school due to sickess i oe year was recorded. Number of days off sick 5 6 0 5 6 0 5 Frequecy 0 4 3 Estimate the mea Estimate the media. Fid the modal class. Solutio The estimate is made by assumig that all the values i a class iterval are equal to the midpoit of the class iterval. Class Iterval Mid-poit Frequecy Mid-poit Frequecy 5 3 3 = 36 6 0 8 8 = 88 5 3 0 3 0 = 30 6 0 8 4 8 4 = 7 5 3 3 3 3 = 69 Totals 40 395 Mea = 395 40 = 9. 95 days. As there are 40 pupils, we eed to cosider the mea of the 0th ad st values. These both lie i the 6 0 class iterval, which is really the 5.5 0.5 class iterval, so this iterval cotais the media. As there are values i the first class iterval, the media is foud by cosiderig the 8th ad 9th values of the secod iterval. As there are values i the secod iterval, the media is estimated as beig of the way alog the secod iterval. 85. 59
9.4 MEP Pupil Text 9 But the legth of the secod iterval is 0. 5 5. 5 = 5, so the media is estimated by 85. 5 = 386. from the start of this iterval. Therefore the media is estimated as 55. + 386. = 936.. The modal class is 5, as this class cotais the most etries. Exercises. A door to door salesma keeps a record of the umber of homes he visits each day. Homes visited 0 9 0 9 0 9 30 39 40 49 Frequecy 3 8 4 60 Estimate the mea umber of homes visited. Estimate the media. What is the modal class?. The weights of a umber of studets were recorded i kg. Mea (kg) 30 w < 35 35 w < 40 40 w < 45 45 w < 50 50 w < 55 Frequecy 0 5 7 4 Estimate the mea weight. Estimate the media. What is the modal class? 3. A stopwatch was used to fid the time that it took a group of childre to ru 00 m. Time (secods) 0 t < 5 5 t < 0 0 t < 5 5 t < 30 Frequecy 6 6 8 Is the media i the modal class? Estimate the mea. Estimate the media. (d) Is the media greater or less tha the mea? 4. The distaces that childre i a year group travelled to school is recorded. Distace (km) 0 d < 0. 5 05. d < 0. 0. d < 5. 5. d < 0. Frequecy 30 9 8 Does the modal class cotai the media? Estimate the media ad the mea. Which is the largest, the media or the mea? 60
5. The ages of the childre at a youth camp are summarised i the table below. Age (years) 6 8 9 4 5 7 Frequecy 8 9 5 Estimate the mea age of the childre. 6. The legths of a umber of leaves collected for a project are recorded. Legth (cm) 5 6 0 5 6 5 Frequecy 8 0 4 Estimate the mea the media legth of a leaf. 7. The table shows how may ights people sped at a campsite. Number of ights 5 6 0 5 6 0 5 Frequecy 0 6 3 5 Estimate the mea. Estimate the media. What is the modal class? 8. A teacher otes the umber of correct aswers give by a class o a multiple-choice test. Correct aswers 0 0 30 3 40 4 50 Frequecy 8 5 3 (i) Estimate the mea. (ii) Estimate the media. (iii) What is the modal class? Aother class took the same test. Their results are give below. Correct aswers 0 0 30 3 40 4 50 Frequecy 3 4 0 (i) Estimate the mea. (ii) Estimate the media. (iii) What is the modal class? How do the results for the two classes compare? Iformatio A quartile is oe of 3 values (lower quartile, media ad upper quartile) which divides data ito 4 equal groups. A percetile is oe of 99 values which divides data ito 00 equal groups. The lower quartile correspods to the 5th percetile. The media correspods to the 50th percetile. The upper quartile correspods to the 75th percetile. 6
9.4 MEP Pupil Text 9 9. 9 childre are asked how much pocket moey they were give last week. Their replies are show i this frequecy table. Pocket moey Frequecy f 0. 00 0. 00 9 0. 300 6 30. 400 Which is the modal class? Calculate a estimate of the mea amout of pocket moey received per child. (NEAB) 0. The graph shows the umber of hours a sample of people spet viewig televisio oe week durig the summer. 40 Number of people 30 0 0 0 0 0 30 40 50 60 70 Viewig time (hours) Copy ad complete the frequecy table for this sample. Viewig time Number of (h hours) people 0 h < 0 3 0 h < 0 7 0 h < 30 33 30 h < 40 40 h < 50 50 h < 60 Aother survey is carried out durig the witer. State oe differece you would expect to see i the data. Use the mid-poits of the class itervals to calculate the mea viewig time for these people. You may fid it helpful to use the table below. 6
Viewig time Mid-poit (h hours) Mid-poit Frequecy Frequecy 0 h < 0 5 3 65 0 h < 0 5 7 405 0 h < 30 5 33 85 30 h < 40 35 40 h < 50 45 50 h < 60 55 (SEG). I a experimet, 50 people were asked to estimate the legth of a rod to the earest cetimetre. The results were recorded. Legth (cm) 0 3 4 5 6 7 8 9 Frequecy 0 4 6 7 9 0 7 5 0 Fid the value of the media. Calculate the mea legth. I a secod experimet aother 50 people were asked to estimate the legth of the same rod. The most commo estimate was 3 cm. The rage of the estimates was 3 cm. Make two comparisos betwee the results of the two experimets. (SEG). The followig list shows the maximum daily temperature, i F, throughout the moth of April. 56. 49.4 63.7 56.7 55.3 53.5 5.4 57.6 59.8 5. 45.8 55. 4.6 6.0 6.9 60. 57. 48.9 63. 68.4 55.5 65. 47.3 59. 53.6 5.3 46.9 5.3 56.7 64.3 Copy ad complete the grouped frequecy table below. Temperature, T Frequecy 40 < T 50 50 < T 54 54 < T 58 58 < T 6 6 < T 70 Use the table of values i part to calculate a estimate of the mea of this distributio. You must show your workig clearly. Draw a histogram to represet your distributio i part. (MEG) 63
9.5 Cumulative Frequecy Cumulative frequecies are useful if more detailed iformatio is required about a set of data. I particular, they ca be used to fid the media ad iter-quartile rage. The iter-quartile rage cotais the middle 50% of the sample ad describes how spread out the data are. This is illustrated i Example. Worked Example For the data give i the table, draw up a cumulative frequecy table ad the draw a cumulative frequecy graph. Solutio The table below shows how to calculate the cumulative frequecies. Height (cm) Frequecy 90 < h 00 5 00 < h 0 0 < h 0 30 0 < h 30 3 30 < h 40 8 40 < h 50 6 Height (cm) Frequecy Cumulative Frequecy 90 < h 00 5 5 00 < h 0 5 + = 7 0 < h 0 30 7 + 30 = 57 0 < h 30 3 57 + 3 = 88 30 < h 40 8 88 + 8 = 06 40 < h 50 6 06 + 6 = A graph ca the be plotted usig poits as show below. 0 00 (50,) (40,06) q y Cumulative Frequecy 80 60 (0,57) (30,88) 40 0 (0,7) 0 (90,0) (00,5) 90 00 0 0 30 40 50 Height (cm) 64
Note A more accurate graph is foud by drawig a smooth curve through the poits, rather tha usig straight lie segmets. 0 00 (50,) (40,06) Cumulative Frequecy 80 60 (0,57) (30,88) 40 0 (0,7) Worked Example (90,0) (00,5) 0 90 00 0 0 30 40 50 Height (cm) The cumulative frequecy graph below gives the results of 0 studets o a test. 0 00 Cumulative Frequecy 80 60 40 0 0 0 0 40 60 80 00 Test Score 65
9.5 MEP Pupil Text 9 Use the graph to fid: the media score, the iter-quartile rage, the mark which was attaied by oly 0% of the studets, (d) the umber of studets who scored more tha 75 o the test. Solutio Sice of 0 is 60, the media ca be foud by startig at 60 o the vertical scale, movig horizotally to the graph lie ad the movig vertically dow to meet the horizotal scale. I this case the media is 53. Cumulative Frequecy 0 00 80 60 Start at 60 40 0 0 0 Score Media = 53 0 40 60 80 00 To fid out the iter-quartile rage, we must cosider the middle 50% of the studets. To fid the lower quartile, start at of 0, which is 30. 4 This gives Lower Quartile = 43. To fid the upper quartile, start at 3 of 0, which is 90. 4 This gives Upper Quartile = 67. The iter-quartile rage is the Cumulative Frequecy 0 00 80 60 40 0 0 0 90 30 Upper quartile = 67 0 40 60 80 00 Lower quartile = 43 Test Score Iter - quartile Rage = Upper Quartile Lower Quartile = 67 43 = 4. 66
Cumulative Frequecy 0 00 80 60 40 08 Here the mark which was attaied by the top 0% is required. 0%of 0 = so start at 08 o the cumulative frequecy scale. This gives a mark of 79. 0 0 0 0 40 60 80 00 Test Score 79 0 03 (d) To fid the umber of studets who scored more tha 75, start at 75 o the horizotal axis. 00 80 This gives a cumulative frequecy of 03. Cumulative Frequecy 60 So the umber of studets with a score greater tha 75 is 0 03 = 7. As i Worked Example, a more accurate estimate for the media ad iter-quartile rage is obtaied if you draw a smooth curve through the data poits. Exercises. Make a cumulative frequecy table for each set of data give below. The draw a cumulative frequecy graph ad use it to fid the media ad iter-quartile rage. 40 0 0 0 0 40 60 80 00 Test Score 75 Joh weighed each apple i a large box. His results are give i this table. Weight of apple (g) 60 < w 80 80 < w 00 00 < w 0 0 < w 40 40 < w 60 Frequecy 4 8 33 7 8 Pasi asked the studets i his class how far they travelled to school each day. His results are give below. Distace (km) 0 < d < d < d 3 3 < d 4 4 < d 5 5 < d 6 Frequecy 5 5 6 5 3 67
9.5 MEP Pupil Text 9 A P.E. teacher recorded the distaces childre could reach i the log jump evet. His records are summarised i the table below. Legth of jump (m) < d < d 3 3 < d 4 4 < d 5 5 < d 6 Frequecy 5 5 6 5. A farmer grows a type of wheat i two differet fields. He takes a sample of 50 heads of cor from each field at radom ad weighs the grais he obtais. Mass of grai (g) 0< m 5 5< m 0 0 < m 5 5< m 0 0 < m 5 5< m 30 Frequecy Field A 3 8 0 4 3 Frequecy Field B 0 34 4 0 Draw cumulative frequecy graphs for each field. Fid the media ad iter-quartile rage for each field. Commet o your results. 3. A cosumer group tests two types of batteries usig a persoal stereo. Lifetime (hours) < l 3 3 < l 4 4 < l 5 5 < l 6 6 < l 7 7 < l 8 Frequecy Type A 3 0 8 4 Frequecy Type B 0 38 6 0 Use cumulative frequecy graphs to fid the media ad iter-quartile rage for each type of battery. Which type of battery would you recommed ad why? 4. The table below shows how the height of girls of a certai age vary. The data was gathered usig a large-scale survey. Height (cm) 50 < h 55 55 < h 60 60 < h 65 65 < h 70 70 < h 75 75 < h 80 80 < h 85 Frequecy 00 300 400 300 700 50 50 A doctor wishes to be able to classify childre as: Category Percetage of Populatio Very Tall 5% Tall 5% Normal 60% Short 5% Very short 5% Use a cumulative frequecy graph to fid the heights of childre i each category. 68
5. The maager of a double glazig compay employs 30 salesme. Each year he awards bouses to his salesme. Bous Awarded to 500 Best 0% of salesme 50 Middle 70% of salesme 50 Bottom 0% of salesme The sales made durig 995 ad 996 are show i the table below. Value of sales ( 000) 0 < V 00 00 < V 00 00 < V 300 300 < V 400 400 < V 500 Frequecy 996 0 5 0 3 Frequecy 995 8 8 0 Use cumulative frequecy graphs to fid the values of sales eeded to obtai each bous i the years 995 ad 996. 6. The histogram shows the cost of buyig a particular toy i a umber of differet shops. 7 6 5 Frequecy 4 3 0.00.0.40.60.80 3.00 Price ( ) Draw a cumulative frequecy graph ad use it to aswer the followig questios. (i) How may shops charged more tha.65? (ii) What is the media price? (iii) How may shops charged less tha.30? (iv) How may shops charged betwee.0 ad.60? (v) How may shops charged betwee.00 ad.50? Commet o which of your aswers are exact ad which are estimates. 69
9.5 MEP Pupil Text 9 7. Laura ad Joy played 40 games of golf together. The table below shows Laura's scores. Scores (x) 70 < x 80 80 < x 90 90 < x 00 00 < x 0 0 < x 0 Frequecy 4 5 7 3 O a grid similar to the oe below, draw a cumulative frequecy diagram to show Laura's scores. 40 q y 30 Cumulative Frequecy 0 0 60 70 80 90 00 0 0 Score Makig your method clear, use your graph to fid (i) Laura's media score, (ii) the iter-quartile rage of her scores. Joy's media score was 03. The iter-quartile rage of her scores was 6. (i) Who was the more cosistet player? Give a reaso for your choice. (ii) The wier of a game of golf is the oe with the lowest score. Who wo most of these 40 games? Give a reaso for your choice. (NEAB) 8. A sample of 80 electric light bulbs was take. The lifetime of each light bulb was recorded. The results are show below. Lifetime (hours) 800 900 000 00 00 300 400 Frequecy 4 3 7 0 4 0 Cumulative Frequecy 4 7 Copy ad complete the table of values for the cumulative frequecy. Draw the cumulative frequecy curve, usig a grid as show below. 70
q y Cumulative Frequecy 90 80 70 60 50 40 30 0 0 (d) (e) 0 800 900 000 00 00 300 400 500 Lifetime (hours) Use your graph to estimate the umber of light bulbs which lasted more tha 030 hours. Use your graph to estimate the iter-quartile rage of the lifetimes of the light bulbs. A secod sample of 80 light bulbs has the same media lifetime as the first sample. Its iter-quartile rage is 90 hours. What does this tell you about the differece betwee the two samples? (SEG) 9. The umbers of joureys made by a group of people usig public trasport i oe moth are summarised i the table. Number of joureys 0 0 0 30 3 40 4 50 5 60 6 70 Number of people 4 7 8 6 3 4 0 Copy ad complete the cumulative frequecy table below. Number of joureys 0 0 30 40 50 60 70 Cumulative frequecy (i) Draw the cumulative frequecy graph, usig a grid as below. 40 q y Cumulative Frequecy 30 0 0 0 0 0 30 40 50 60 70 Number of joureys 7
9.5 MEP Pupil Text 9 (ii) (iii) Use your graph to estimate the media umber of joureys. Use your graph to estimate the umber of people who made more tha 44 joureys i the moth. The umbers of joureys made usig public trasport i oe moth, by aother group of people, are show i the graph. 40 Cumulative Frequecy 30 0 0 0 0 0 30 40 50 60 70 Number of joureys Make oe compariso betwee the umbers of joureys made by these two groups. (SEG) 0. The cumulative frequecy graph below gives iformatio o the house prices i 99. The cumulative frequecy is give as a percetage of all houses i Eglad. 00 q y ( ) Cumulative Frequecy 80 60 40 0 0 00000 House prices i 99 i ( ) 00000 40000 This grouped frequecy table gives the percetage distributio of house prices (p) i Eglad i 993. 7