Cumulative frequency diagrams — Paper 4 · IGCSE Mathematics 0580

4 200 180 160 140 120 Cumulative frequency 100 ( h) 80 60 40 20 0 0 10 20 30 40 50 60 70 80 Number of hours worked (h) 200 people record the number of hours they work in a week. The cumulative frequency graph shows this information. (a) Use the graph to find (i) the median, [1] (ii) the upper quartile, [1] (iii) the inter-quartile range, [1] (iv) the number of people who work more than 60 hours in a week. [2] (b) Omar uses the graph to make the following frequency table. Hours 0IhY10 10IhY20 20IhY30 30IhY40 40IhY50 50IhY60 60IhY70 70IhY80 worked (h) Frequency 12 34 36 30 38 30 p q (i) Use the graph to find the values of p and q. [2] (ii) Calculate an estimate of the mean number of hours worked in a week. [4] (c) Shalini uses the graph to make a different frequency table. Hours worked (h) 0IhY30 30IhY40 40IhY50 50IhY80 Frequency 82 30 38 50 When she draws a histogram, the height of the column for the interval 30IhY40 is 9 cm. Calculate the height of each of the other three columns. [4]

15 marks

Mark scheme: 4 (a) (i) 36 (36.0–36.4) B1 (ii) 50 (50.0–50.4) B1 (iii) 29 (28.6–29.4) B1 (iv) 20 B2 If B0, SC1 for 19 or 21 or 180 seen (b) (i) p = 16, q = 4 B1,B1 If B0, SC1 if p and q add up to 20 (ii)  7220    = 36.1 cso www4 B4 Answer 36 scores 4 marks after some  200  correct working shown with no incorrect working seen M1 for using mid-values at least four correct from 5, 15, 25, 35, 45, 55, 65, 75 M1 (dep on correct mid values or mid- values ±0.5) for (at least four ∑fx correct products) M1 (dependent on 2nd M1) for dividing sum by 200 or 180 + their p + their q (c) 8.2 (8.19–8.20), 11.4, 5 (5.00–5.01) B4 B3 for 2 correct or B2 for 1 correct After B0, SC2 for fd’s 2.7(3…) o.e., 3.8 o.e, 1.6(6...) o.e. or SC1 for 2 of fd’s correct (15)

This question in 0580/41 May/June 2008

9 Sam asked 80 people how many minutes their journey to work took on one day. For Examiner′s The cumulative frequency diagram shows the times taken (m minutes). Use 80 70 60 50 Cumulative 40 frequency 30 20 10 m 0 10 20 30 40 50 Time (minutes) (a) Find (i) the median, Answer(a)(i) … min [1] (ii) the lower quartile, Answer(a)(ii) … min [1] (iii) the inter-quartile range. Answer(a)(iii) … min [1] (b) One of the 80 people is chosen at random. For Examiner′s Use Find the probability that their journey to work took more than 35 minutes. Give your answer as a fraction. Answer(b) … [2] (c) Use the cumulative frequency diagram to complete this frequency table. Time (m minutes) 0 < m Y 10 10 < m Y 15 15 < m Y 30 30 < m Y 40 40 < m Y 50 Frequency 30 12 18 [2] (d) Using mid-interval values, calculate an estimate of the mean journey time for the 80 people. Answer(d) … min [3] (e) Use the table in part (c) to complete the histogram to show the times taken by the 80 people. One column has already been completed for you. 4 3 Frequency 2 density 1 m 0 10 20 30 40 50 Time (minutes) [5] _____________________________________________________________________________________

15 marks

Mark scheme: 9 (a) (i) 14 1 (ii) 8 1 (iii) 30 – their (ii) 1FT 11 69 (b) 2 SC1 for 80 80 (c) 16, 4 2 B1 for each correct value (d) 18.0625 rot to 3sf or better or 3 M1 for Σmf for m as mid values of 5, 12.5, 22.5, 18.1 www 35 and 45 (= 1445) and M1 dep for Σmf ÷ 80, dep on M1 earned (e) Correct widths with no gaps 1 2nd block w = 5, fd = 2.4 1 3rd block w = 15 fd = 1.2 1 4th block w = 10 and fd = 1.6 1FT Strict FT from their (c) 5th block w = 10 and fd = 0.4 1FT Strict FT from their (c) After 0 scored for blocks, SC1 for 4 correct fds soi by correct heights

This question in 0580/43 May/June 2013

9 80 70 60 50 Cumulative frequency 40 30 20 10 t 0 10 20 30 40 50 Time (minutes) The times (t minutes) taken by 80 people to complete a charity swim were recorded. The results are shown in the cumulative frequency diagram above. (a) Find (i) the median, Answer(a)(i) … min [1] (ii) the inter-quartile range, Answer(a)(ii) … min [2] (iii) the 70th percentile. Answer(a)(iii) … min [2] (b) The times taken by the 80 people are shown in this grouped frequency table. Time (t minutes) 0 < t Ğ 20 20 < t Ğ 30 30 < t Ğ 45 45 < t Ğ 50 Frequency 12 21 33 14 (i) Calculate an estimate of the mean time. Answer(b)(i) … min [4] (ii) Draw a histogram to represent the grouped frequency table. 4 3 Frequency density 2 1 t 0 10 20 30 40 50 Time (minutes) [4] __________________________________________________________________________________________

13 marks

Mark scheme: 9 (a) (i) 37.5 to 38.5 1 (ii) 19.5 to 20.5 nfww 2 B1 for [LQ =] 23.5 to 24 or [UQ =] 43.5 to 44 (iii) 43 2 B1 for 56 seen or horizontal line drawn at cf = 56 (b) (i) 31.8[4…] nfww 4 M1 for midpoints soi (condone 1 error or omission) and M1 for use of ∑ft with t in correct interval including both boundaries (condone 1 further error or omission) and M1 (dep on 2nd M1) for ∑ft ÷ 80 (2547.5 ÷ 80) (ii) Correct histogram 4 B1 for each correct block with correct width and height If B0 then SC1 for four correct f.d.s or four correct widths

This question in 0580/41 May/June 2014

3 Leo measured the rainfall each day, in millimetres, for 120 days. The cumulative frequency table shows the results. Rainfall (r mm) r  20 r  25 r  35 r  40 r  60 r  70 Cumulative 5 13 72 90 117 120 frequency (a) On the grid below, draw a cumulative frequency diagram to show these results. 120 100 80 Cumulative frequency 60 40 20 r 0 10 20 30 40 50 60 70 Rainfall (mm) [3] (b) (i) Find the median. Answer(b)(i) … mm [1] (ii) Use your diagram to find the number of days when the rainfall was more than 50 mm. Answer(b)(ii) … [2] (c) Use the information in the cumulative frequency table to complete the frequency table below. Rainfall (r mm) 0  r  20 20  r  25 25  r  35 35  r  40 40  r  60 60  r  70 Frequency 5 59 3 [2] (d) Use your frequency table to calculate an estimate of the mean. You must show all your working. Answer(d) … mm [4] (e) In a histogram drawn to show the information in the table in part (c), the frequency density for the interval 25  r  35 is 5.9 . Calculate the frequency density for the intervals 20  r  25 , 40  r  60 and 60  r  70 . Answer(e) 20  r  25 … 40  r  60 … 60  r  70 … [4]

16 marks

Mark scheme: 3 (a) Correct diagram 3 B1 for correct vertical plots and B1 for correct horizontal plots and B1 dep on at least B1 for reasonable increasing curve or polygon through their 6 points If zero scored, SC1 for 5 out of 6 correct plots (b) (i) 32 to 34 1 (ii) 120 – reading at r = 50 2FT B1FT for reading at r = 50 seen (c) 8 18 27 2 B1 for 2 correct 1 (d) 4 M1 for mid-values soi 35.2 or 35 or 35.16 to 35.17 6 with x in the correct interval M1 FT for ∑fx nfww including boundaries 120 M1dep for ∑fx ÷ dependent on second M1 earned (e) 1.6 4FT FT from (c) their 8 ÷ 5 and their 27 ÷ 20 1.35 B3FT for any 2 correct 0.3 or B2FT for first or second answer correct or B1 for 0.3 only 0 6

This question in 0580/42 Oct/Nov 2015

7 The table shows information about the time taken by 400 people to complete a race. Time taken 45 1 m G 50 50 1 m G 60 60 1 m G 70 70 1 m G 90 90 1 m G 100 100 1 m G 120 (m minutes) Frequency 23 64 122 136 26 29 (a) Calculate an estimate of the mean time taken. … min [4] (b) (i) Complete the cumulative frequency table. Time taken m G 50 m G 60 m G 70 m G 90 m G 100 m G 120 (m minutes) Cumulative 23 400 frequency [2] (ii) On the grid, draw a cumulative frequency diagram to show this information. 400 300 Cumulative 200frequency 100 0 m 40 50 60 70 80 90 100 110 120 Time taken (minutes) [3] (iii) Use your diagram to estimate (a) the median, … min [1] (b) the inter-quartile range, … min [2] (c) the 60th percentile. … min [2]

14 marks

Mark scheme: 7 (a) 72.7 or 72.70 to 72.71 nfww 4 M1 for midpoints soi (condone 1 error or omission) (47.5, 55, 65, 80, 95, 110) M1 for use of ∑fx with x in correct interval including both boundaries (condone 1 further error or omission) (1092.5, 3520, 7930, 10880, 2470, 3190) M1 (dep on 2nd M1) for ∑fx ÷ 400 (b) (i) [23] 87 209 345 371 [400] 2 B1 for 2 or 3 correct (ii) Correct graph 3 B1FT their (b)(i) for 6 correct heights B1 for 6 points at upper ends of intervals on correct vertical line B1FT (dep on at least B1) for increasing curve or polygon through 6 points After 0 scored, SC1FT their (b)(i) for 5 correct points plotted (iii) (a) 69 to 70 1 (b) 20 to 23 2FT FT their cumulative freq curve M1 for correct UQ or LQ for their cumulative freq curve (c) 72 to 75 2 M1 for 240 soi

This question in 0580/42 Feb/March 2017

2 The time taken for each of 90 cars to complete one lap of a race track is shown in the table. Time (t seconds) 70 1 t G 71 71 1 t G 72 72 1 t G 73 73 1 t G 74 74 1 t G 75 Frequency 17 24 21 18 10 (a) Write down the modal time interval. … 1 t G … [1] (b) Calculate an estimate of the mean time. … s [4] (c) (i) Complete the cumulative frequency table. Time (t seconds) t G 71 t G 72 t G 73 t G 74 t G 75 Cumulative frequency 17 [2] (ii) On the grid, draw a cumulative frequency diagram to show this information. 90 80 70 60 50 Cumulative frequency 40 30 20 10 0 t 70 71 72 73 74 75 Time (seconds) [3] (iii) Find the median time. … s [1] (iv) Find the inter-quartile range. … s [2] (d) One lap of the race track measures 3720 metres, correct to the nearest 10 metres. A car completed the lap in 75 seconds, correct to the nearest second. Calculate the upper bound for the average speed of this car. Give your answer in kilometres per hour. … km/h [4]

17 marks

Mark scheme: 2(a) 71 < t ⩽ 72 1 2(b) 72.3 or 72.27 to 72.28 nfww 4 M1 for midpoints soi (condone 1 error or omission) M1 for use of ∑fx with x in correct interval including both boundaries M1 (dep on 2nd M1) for ∑fx ÷ 90 2(c)(i) 41, 62, 80, 90 2 B1 for 2 correct values 2(c)(ii) Correct curve 3 B1FT their (c)(i) for 5 correct heights B1 for 5 points plotted at upper ends of intervals B1FT (dep on at least B1) for increasing curve or increasing polygon through 5 points If zero scored, SC1FT for 4 correct points plotted 2(c)(iii) 72.1 to 72.4 1 2(c)(iv) 1.9 to 2.2 2 M1 for UQ = 73.2 to 73.4 or LQ = 71.2 to 71.3 2(d) 180 cao nfww 4 B3 for 50 [m/s] nfww OR 3725 ÷ 1000 M3 for 74.5 ÷ 3600 OR M2 for 3725 ÷ 74.5 or M1 for 3725 or 74.5 seen or for (3715 to 3725) ÷ (74.5 to 75.5) M1 indep for multiply by 3.6 oe

This question in 0580/41 May/June 2017

3 (a) 200 students estimate the capacity, x millilitres, of a cup. The results are shown in the frequency table. Capacity (x ml) 0 1 x G 100 100 1 x G 150 150 1 x G 200 200 1 x G 250 250 1 x G 400 Frequency 20 55 66 35 24 (i) Calculate an estimate of the mean. … ml [4] (ii) Complete the histogram. 1.5 1 Frequency density 0.5 0 x 0 100 200 300 400 Capacity (ml) [4] (b) The 200 students also estimate the mass, m grams, of a small rock. The results are shown in the cumulative frequency table. Mass (m grams) m G 50 m G 100 m G 150 m G 200 m G 250 Cumulative frequency 28 64 104 168 200 (i) On the grid, draw a cumulative frequency diagram. 200 150 Cumulative frequency 100 50 0 m 0 50 100 150 200 250 Mass (g) [3] (ii) Find (a) the 65th percentile, … g [1] (b) the number of students who estimated more than 75 g. … [2]

14 marks

Mark scheme: 3(a)(i) 175.5 nfww 4 M1 for at least four of 50, 125, 175, 225, 325 soi M1 for Σ fx with x inside or on boundary of each interval their Σ fx M1 (dep on second M1) for 200 3(a)(ii) Fully correct histogram 4 B1 for each correct bar If zero scored, B1 for 0.2, 1.32, 0.7, 0.16 seen 3(b)(i) Fully correct cumulative 3 B1 for correct horizontal plots frequency diagram B1 for correct vertical plots B1FT dep on at least B1 earned for points joined with smooth increasing curve or polygon If zero scored, SC1 for 4 correct plotted points 3(b)(ii)(a) 170 to 175 1 3(b)(ii)(b) 152 to 158 2 M1 for 42 to 48 written

This question in 0580/42 May/June 2017

6 (a) There are 100 students in group A. The teacher records the distance, d metres, each student runs in one minute. The results are shown in the cumulative frequency diagram. 100 90 80 70 60 Cumulative 50 frequency 40 30 20 10 0 d 100 200 300 400 Distance (metres) Find (i) the median, … m [1] (ii) the upper quartile, … m [1] (iii) the inter-quartile range, … m [1] (iv) the number of students who run more than 350 m. … [2] (b) There are 100 students in group B. The teacher records the distance, d metres, each of these students runs in one minute. The results are shown in the frequency table. Distance 100 1 d G 200 200 1 d G 250 250 1 d G 280 280 1 d G 32 0 320 1 d G 400(d metres) Number of 20 22 30 16 12students (i) Calculate an estimate of the mean distance for group B. … m [4] (ii) Complete the histogram to show the information in the frequency table. 1 0.8 0.6 Frequency density 0.4 0.2 0 d 100 200 300 400 Distance (metres) [4] (c) For the 100 students in group B, the median is 258 m. Complete the statement. On average, the students in group A run … than the students in group B. [1]

14 marks

Mark scheme: 6(a)(i) 280 1 6(a)(ii) 320 1 6(a)(iii) 90 1 6(a)(iv) 10 2 M1 for 90 written 6(b)(i) 250.2 nfww cao 4 M1 for at least 4 correct mid-values M1 for Σfx M1 dep on second M1 for Σfx ÷ 100 6(b)(ii) Correct completion of 4 B1 for each correct block histogram If zero scored, then SC1 for correct frequency densities seen 6(c) [22 m] further oe 1

This question in 0580/42 Oct/Nov 2017

2 The time taken for each of 120 students to complete a cooking challenge is shown in the table. Time (t minutes) 20 1 t G 25 25 1 t G 30 30 1 t G 35 35 1 t G 40 40 1 t G 45 Frequency 44 32 28 12 4 (a) (i) Write down the modal time interval. … 1 t G … [1] (ii) Write down the interval containing the median time. … 1 t G … [1] (iii) Calculate an estimate of the mean time. … min [4] (iv) A student is chosen at random. Find the probability that this student takes more than 40 minutes. … [1] (b) (i) Complete the cumulative frequency table. Time (t minutes) t G 20 t G 25 t G 30 t G 35 t G 40 t G 45 Cumulative 0 44 frequency [2] (ii) On the grid, draw a cumulative frequency diagram to show this information. 120 110 100 90 80 70 Cumulative frequency 60 50 40 30 20 10 0 t 20 25 30 35 40 45 Time (minutes) [3] (iii) Find the median time. … min [1] (iv) Find the interquartile range. … min [2] (v) Find the number of students who took more than 37 minutes to complete the cooking challenge. … [2]

17 marks

Mark scheme: 2(a)(i) 20 [< t ⩽] 25 1 2(a)(ii) 25 [< t ⩽] 30 1 2(a)(iii) 28.3 or 28.33.. 4 M1 for 22.5, 27.5, 32.5, 37.5, 42.5 soi M1 for ∑fx where x is in the correct interval including boundaries M1dep for ∑fx ÷ 120 or ∑fx ÷ (44 + 32 + 28 + 12 + 4) 2(a)(iv) 4 1 oe isw 120 2(b)(i) 76, 104, 116, 120 2 B1 for one error FT other values or for 3 correct 2(b)(ii) Correct curve 3 B1 for correct horizontal placement for 6 plots B1FT for correct vertical placement for 6 plots B1FT dep on at least B1 for reasonable increasing curve or polygon through their 6 points If 0 scored SC1FT for 5 out of 6 points correctly plotted 2(b)(iii) 27 to 27.5 1 2(b)(iv) 8.5 to 9.5 2 B1 for [UQ=] 32 to 32.5 or [LQ=] 23 to 23.5 2(b)(v) 8, 9, 10, 11 or 12 2 B1 for 108 to 112 seen or B1FT their graph reading at 37 mins seen

This question in 0580/42 May/June 2018

4 A school nurse records the height, h cm, of each of 180 children. The table shows the information. Height 60 1 h G 70 70 1 h G 90 90 1 h G 100 100 1 h G 110 110 1 h G 115 115 1 h G 125 (h cm) Frequency 8 26 35 67 28 16 (a) Calculate an estimate of the mean. Give your answer correct to 1 decimal place. … cm [4] (b) In a histogram showing the information, the height of the bar for the interval 60 1 h G 70 is 0.4 cm. Calculate the height of the bar for each of the following intervals. 115 1 h G 125 … cm 110 1 h G 115 … cm 70 1 h G 90 … cm [3] (c) Complete the cumulative frequency table below. Height h G 70 h G 90 h G 100 h G 110 h G 115 h G 125 (h cm) Cumulative 180 frequency [2] (d) On the grid opposite, draw a cumulative frequency diagram. 180 160 140 120 100 Cumulative frequency 80 60 40 20 0 h 60 70 80 90 100 110 120 130 Height (cm) [3] (e) Use your cumulative frequency diagram to find an estimate of (i) the interquartile range, … cm [2] (ii) the 70th percentile, … cm [2] (iii) the number of children with height greater than 106 cm. … [2]

18 marks

Mark scheme: 4(a) 100.2 nfww 4 M1 for midpoints soi 65, 80, 95, 105, 112.5, 120 M1 for use of ∑fx with x in correct interval including both boundaries M1dep for ∑fx ÷ 180 dep on previous M1 4(b) 0.8 3 B1 for each 2.8 If zero scored, SC1 for 1.6, 5.6 and 1.3 0.65 seen 4(c) 8 34 69 136 164 2 B1 for one error FT other values or for 3 or 4 correct 4(d) Correct diagram 3 B1FT for correct vertical placement for 6 plots B1 for correct horizontal placement for 6 plots B1FT dep on at least B1 for reasonable increasing curve or polygon through their 6 points If zero scored, SC1FT for 5 out of 6 correct plots 4(e)(i) 15 to 17 2 B1 for [LQ =] 93 to 94 or [UQ =] 109 to 110 4(e)(ii) 107 to 109 2 B1 for 126 seen 4(e)(iii) 66 to 72 2 FT their graph for 2 marks B1 for answer 106 to 114 or B1FT their graph reading at 106 cm seen

This question in 0580/41 Oct/Nov 2018

7 (a) 20 students each record the mass, p grams, of their pencil case. The table below shows the results. Mass 0 1 p G 50 50 1 p G 100 100 1 p G 125 125 1 p G 150 150 1 p G 200 ( p grams) Frequency 2 5 4 6 3 (i) Calculate an estimate of the mean mass. … g [4] (ii) Use the frequency table above to complete the cumulative frequency table. Mass p G 50 p G 100 p G 125 p G 150 p G 200 ( p grams) Cumulative 20 frequency [2] (iii) A student is chosen at random. Find the probability that this student has a pencil case with a mass greater than 150 g. … [1] (b) Some students each record the mass, m kg, of their school bag. Adil wants to draw a histogram to show this information. Complete the table below. Mass (m kg) 0 1 m G 4 4 1 m G 6 6 1 m G 7 7 1 m G 10 Frequency 32 42 Height of bar on 1.6 2 1.2 2.8 histogram (cm) [2] (c) The frequency table below shows information about the number of books read by some students in a reading marathon. Number of 1 2 3 4 5 6 7 8 books read Frequency 2 2 16 10 9 4 x 2 (i) The mean number of books read is 4.28 . Find the value of x. x = … [3] (ii) Write down the mode. … [1] (iii) Write down the median. … [1]

14 marks

Mark scheme: 7(a)(i) 111.25 4 M1 for midpoints soi (25, 75, 112.5, 137.5, 175) M1 for ∑fx with x in correct interval including both boundaries M1 (dep on 2nd M1) for ∑fx ÷ 20 7(a)(ii) 2 7 11 17 2 B1 for three correct 7(a)(iii) 3 1 oe 20 7(b) 20 6 2 B1 for one correct value or [SF = ] 5 or 1 oe 5 7(c)(i) 5 nfww 3 M2 for ∑fx ÷ ∑f = 4.28 oe or M1 for 179 + 7x oe or 4.28 × (45 + x ) oe seen 7(c)(ii) 3 1 7(c)(iii) 4 1

This question in 0580/42 Feb/March 2019

6 (a) The cumulative frequency diagram shows information about the times taken by 200 students to solve a problem. 200 180 160 140 120 Cumulative 100 frequency 80 60 40 20 0 0 10 20 30 40 50 60 Time (minutes) Use the cumulative frequency diagram to find an estimate for (i) the median, … min [1] (ii) the interquartile range, … min [2] (iii) the number of students who took more than 40 minutes. … [2] (b) Roberto records the value of each of the coins he has at home. The table shows the results. Value (cents) 1 2 5 10 20 50 Frequency 3 1 3 2 4 2 (i) Find the range. … cents [1] (ii) Find the mode. … cents [1] (iii) Find the median. … cents [1] (iv) Work out the total value of Roberto’s coins. … cents [2] (v) Work out the mean. … cents [1] (c) The histogram shows information about the masses of 100 boxes. 10 9 8 7 6 Frequency 5 density 4 3 2 1 0 0 5 10 15 20 25 30 Mass (kilograms) Calculate an estimate of the mean. … kg [6]

17 marks

Mark scheme: 6(a)(i) 34 1 6(a)(ii) 18 2 B1 for [l.q. = ] 25 or [u.q. = ] 43 seen 6(a)(iii) 60 2 M1 for 140 written 6(b)(i) 49 1 6(b)(ii) 20 1 6(b)(iii) 10 1 6(b)(iv) 220 2 M1 for 3 × 1 + 1 × 2 + 3 × 5 + 2 × 10 + 4 × 20 + 2 × 50 6(b)(v) 14.7 or 14.66 to 14.67 1 FT their (iv) ÷ 15 6(c) 13.25 nfww 6 B2 for frequencies 30, 40, 30 soi or B1 for 2 of these M1 for 5, 12.5, 22.5 M1 Σ fx with their frequencies (if seen) and each x in correct interval including boundaries Σfx M1 dependent for (dependent on 100 second M1) OR Alternative Method B2 for frequencies 15, 15, 40, 10, 10, 10 soi or B1 for 2 of 15, 40, 10 M1 for 2.5, 7.5, 12.5, 17.5, 22.5, 27.5 M1 Σ fx with their frequencies (if seen) and each x in correct interval including boundaries Σfx M1 dependent for (dependent on 100 second M1)

This question in 0580/41 Oct/Nov 2019

2 The cumulative frequency diagram shows information about the time taken, t seconds, for a group of girls to each solve a maths problem. 80 70 60 50 Cumulative 40 frequency 30 20 10 0 t 0 10 20 30 40 50 60 70 80 90 100 Time (seconds) (a) Use the cumulative frequency diagram to find an estimate for (i) the median, … s [1] (ii) the interquartile range, … s [2] (iii) the 20th percentile, … s [1] (iv) the number of girls who took more than 66 seconds to solve the problem. … [2] (b) (i) Use the cumulative frequency diagram to complete the frequency table. Time (t seconds) 0 1 t G 20 20 1 t G 40 40 1 t G 60 60 1 t G 80 80 1 t G 100 Frequency 6 4 [2] (ii) Calculate an estimate of the mean time. … s [4] (c) A group of boys solved the same problem. The boys had a median time of 60 seconds, a lower quartile of 46 seconds and an upper quartile of 66 seconds. (i) Write down the percentage of boys with a time of 66 seconds or less. … % [1] (ii) Howard says The boys’ times vary more than the girls’ times. Explain why Howard is incorrect. … … [2]

15 marks

Mark scheme: 2(a)(i) 54 1 2(a)(ii) 29 2 M1 for [UQ =] 65 or [LQ =] 36 2(a)(iii) 32 1 2(a)(iv) 17, 18 or 19 2 M1 for 61 to 63 written or for decimal answer in range 17 to 19 2(b)(i) 18, 26, 26 2 B1 for 1 or 2 correct 2(b)(ii) 51 nfww 4 M1 for 10 , 30 , 50 , 70 , 90 soi M1 for Σ fx M1 dep for their ∑ fx ÷ ∑ f 2(c)(i) 75 1 2(c)(ii) IQR is bigger for the girls with 2 FT their IQR from (a)(ii) [boys =] 20 seen oe M1 for IQR for boys = 20 isw or for girls IQR is bigger than boys IQR oe isw FT their IQR from (a)(iii)

This question in 0580/42 Oct/Nov 2019

5 The cumulative frequency diagram shows information about the distance, d km, travelled by each of 60 male cyclists in one weekend. 60 50 40 Cumulative 30 frequency 20 10 0 d 0 20 40 60 80 100 120 Distance (km) (a) Use the cumulative frequency diagram to find an estimate of (i) the median, … km [1] (ii) the lower quartile, … km [1] (iii) the interquartile range. … km [1] (b) For the same weekend, the interquartile range for the distances travelled by a group of female cyclists is 40 km. Make one comment comparing the distribution of the distances travelled by the males with the distribution of the distances travelled by the females. … … [1] (c) A male cyclist is chosen at random. Find the probability that he travelled more than 50 km. … [2] (d) (i) Use the cumulative frequency diagram to complete this frequency table. Distance (d km) Number of male cyclists 0 1 d G 40 18 40 1 d G 50 9 50 1 d G 60 60 1 d G 70 70 1 d G 90 90 1 d G 120 2 [2] (ii) Calculate an estimate of the mean distance travelled. … km [4]

12 marks

Mark scheme: 5(a)(i) 52 1 5(a)(ii) 36 1 5(a)(iii) 26 1 FT 62 – their (a)(ii) evaluated correctly 5(b) Valid comment 1 Strict FT their (a)(iii), e.g. distances for females are more varied 5(c) 11 2 27 oe M1 for 27 written or answer of oe 20 60 5(d)(i) [18 9] 14 12 5 [2] 2 B1 for 1 correct value 5(d)(ii) 48.75 nfww 4 M1 for midpoints soi M1 for use of ∑fx with their frequencies M1 (dep on 2nd M1) for ∑fx ÷ (60 or by their ∑f )

This question in 0580/43 Oct/Nov 2019

2 The heights, h metres, of the 120 boys in an athletics club are recorded. The table shows information about the heights of the boys. Height (h metres) 1.3 1 h G 1.4 1.4 1 h G 1.5 1.5 1 h G 1.6 1.6 1 h G 1 .7 1.7 1 h G 1.8 1.8 1 h G 1.9 Frequency 7 18 30 24 27 14 (a) (i) Write down the modal class. … 1 h G … [1] (ii) Calculate an estimate of the mean height. … m [4] (b) (i) One boy is chosen at random from the club. Find the probability that this boy has a height greater than 1.8 m. … [1] (ii) Three boys are chosen at random from the club. Calculate the probability that one of the boys has a height greater than 1.8 m and the other two boys each have a height of 1.4 m or less. … [4] (c) (i) Use the frequency table on page 4 to complete the cumulative frequency table. Height h G 1.4 h G 1.5 h G 1.6 h G 1.7 h G 1.8 h G 1.9(h metres) Cumulative 7 25frequency [2] (ii) On the grid, draw a cumulative frequency diagram to show this information. 120 110 100 90 80 Cumulative 70 frequency 60 50 40 30 20 10 0 h 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Height (m) [3] (d) Use your diagram to find an estimate for (i) the median height, … m [1] (ii) the 40th percentile. … m [2]

18 marks

Mark scheme: 2(a)(i) 1.5 < h ⩽ 1.6 1 2(a)(ii) 1.62 or 1.623… nfww 4 M1 for 1.35, 1.45, 1.55, 1.65, 1.75 1.85 soi M1 for Σfx M1 dep for their Σfx ÷ 120 2(b)(i) 14 1 oe 120 2(b)(ii) 21 4  14 7 6  oe M3 for 3  × ×  20060  120 119 118  14 7 6 or M2 for × × isw 120 119 118 14 7 6 or M1 for , , 120 119 118 343 After 0 scored, SC1 for answer or 864000 343 oe 288000 2(c)(i) 55, 79, 106, 120 2 B1 for 2 or 3 correct 2(c)(ii) Correct diagram 3 B1 for correct horizontal plots B1FT for correct vertical plots B1FT dep on at least B1 for reasonable increasing curve or polygon through their 6 points If 0 scored SC1 for 5 out of 6 points correctly plotted 2(d)(i) 1.62 to 1.63 1 2(d)(ii) 1.57 to 1.58 2 B1 for 48 soi

This question in 0580/41 May/June 2020

3 The speed, v km/h, of each of 200 cars passing a building is measured. The table shows the results. Speed (v km/h) 0 1 v G 20 20 1 v G 40 40 1 v G 45 45 1 v G 50 50 1 v G 60 60 1 v G 80 Frequency 16 34 62 58 26 4 (a) Calculate an estimate of the mean. … km/h [4] (b) (i) Use the frequency table to complete the cumulative frequency table. Speed (v km/h) v G 20 v G 40 v G 45 v G 50 v G 60 v G 80 Cumulative frequency 16 50 196 200 [1] (ii) On the grid, draw a cumulative frequency diagram. 200 180 160 140 120 Cumulative frequency 100 80 60 40 20 0 v 0 10 20 30 40 50 60 70 80 Speed (km/h) [3] (iii) Use your diagram to find an estimate of (a) the upper quartile, … km/h [1] (b) the number of cars with a speed greater than 35 km/h. … [2] (c) Two of the 200 cars are chosen at random. Find the probability that they both have a speed greater than 50 km/h. … [2] (d) A new frequency table is made by combining intervals. Speed (v km/h) 0 1 v G 40 40 1 v G 50 50 1 v G 80 Frequency 50 120 30 On the grid, draw a histogram to show the information in this table. 15 10 Frequency density 5 0 v 0 10 20 30 40 50 60 70 80 Speed (km/h) [3]

16 marks

Mark scheme: 3(a) 41.4 4 M1 for 10, 30, 42.5, 47.5, 55, 70 M1 for Σ fx where x lies in or on the boundary of each interval. Σfx M1 dep for dep on second M1 200 3(b)(i) 112, 170 1 3(b)(ii) Correct diagram 3 B1 for correct horizontal plot B1FT for correct vertical plots B1 FT dep on at least B1 earned for reasonable increasing curve or polygon through their 6 points If 0 scored SC1FT for 5 out of 6 points plotted correctly 3(b)(iii)(a) 48 1 3(b)(iii)(b) 160 2 M1 for 40 seen 3(c) 87 2 30 29 oe M1 for × oe 3980 200 199 3(d) Correct histogram 3 B1 for each column If 0 scored SC1 for correct frequency densities soi 1.25, 12, 1

This question in 0580/42 May/June 2020

4 The height, h cm, of each of 120 plants is measured. The cumulative frequency diagram shows this information. 120 100 80 Cumulative 60 frequency 40 20 0 0 10 20 30 40 50 h Height (cm) (a) Use the cumulative frequency diagram to find an estimate of (i) the median, … cm [1] (ii) the interquartile range, … cm [2] (iii) the 60th percentile, … cm [1] (iv) the number of plants with a height greater than 40 cm. … [2] (b) The information in the cumulative frequency diagram is shown in this frequency table. Height, h cm 0 1 h G 10 10 1 h G 20 20 1 h G 30 30 1 h G 50 Frequency 2 18 62 38 (i) Calculate an estimate of the mean height. … cm [4] (ii) A histogram is drawn to show the information in the frequency table. The height of the bar representing the interval 10 1 h G 20 is 7.2 cm. Calculate the height of the bar representing the interval 30 1 h G 50 . … cm [2]

12 marks

Mark scheme: 4(a)(i) 25 1 4(a)(ii) 10 nfww 2 B1 for [lq =] 22 or [uq =] 32 4(a)(iii) 27 1 4(a)(iv) 6 2 B1 for 114 written 4(b)(i) 27.9 or 27.91 to 27.92 nfww 4 M1 for mid-values M1 where x lies within or on forfx boundary of correct interval M1 dep ÷ 120 dep on second M1 fx 4(b)(ii) 7.6 2 18 38 M1 for oe or oe 10 20 or B1 for [multiplier] 4 or ¼

This question in 0580/42 Oct/Nov 2020

8 (a) The table shows information about the mass, in kilograms, of each of 50 children. Mass (k kg) 0 1 k G 10 10 1 k G 25 25 1 k G 35 35 1 k G 40 40 1 k G 50 Frequency 3 19 21 5 2 (i) Complete the cumulative frequency table. Mass (k kg) k G 10 k G 25 k G 35 k G 40 k G 50 Cumulative frequency [2] (ii) On the grid, draw a cumulative frequency diagram to show this information. 50 40 30 Cumulative frequency 20 10 0 0 10 20 30 40 50 k Mass (kg) [3] (iii) Use your diagram to find an estimate of the number of children with a mass of 32 kg or less. … [1] (b) This cumulative frequency diagram shows information about the height, in metres, of each of 100 students. 100 80 60 Cumulative frequency 40 20 0 1.4 1.5 1.6 1.7 1.8 1.9 Height (m) The height of the tallest student is 1.83 metres. The height of the shortest student is 1.45 metres. 1.4 1.5 1.6 1.7 1.8 1.9 Height (m) On this grid, draw a box-and-whisker plot for the heights of the 100 students. [4]

10 marks

Mark scheme: 8(a)(i) 3 22 43 48 50 2 B1 for 4 correct or M1 for one error in adding. 8(a)(ii) correct diagram 3 B1FT their (a)(i) for 5 correct heights B1 for 5 points at upper ends of intervals on correct vertical line B1FT dep on at least B1 for increasing curve through their 5 points After 0 scored, SC1 for 4 of their points correctly plotted 8(a)(iii) 35 to 38 1 8(b) Correct box-and-whisker diagram 4 B1 for median 1.64 drawn B1 for LQ 1.57 drawn B1 for UQ 1.71 drawn If 0 scored SC1 for 1.64, 1,57 or 1.71 seen 1.45 1.57 1.64 1.71 1.83

This question in 0580/41 May/June 2021

3 (a) Zoe’s test scores last term were 6 7 7 7 8 9 9 10 10. Find (i) the range, … [1] (ii) the mode, … [1] (iii) the median. … [1] (b) The cumulative frequency diagram shows information about the time taken by each of 200 students to solve a problem. 200 180 160 140 120 Cumulative 100 frequency 80 60 40 20 0 0 2 4 6 8 10 12 14 16 18 20 Time (minutes) Use the diagram to find an estimate of (i) the median, … min [1] (ii) the interquartile range. … min [2] (c) The test scores of 200 students are shown in the table. Score 5 6 7 8 9 10 Frequency 3 10 43 75 48 21 Calculate the mean. … [3] (d) The height, in cm, of each of 200 plants is measured. The histogram shows the results. 4 3 Frequency 2 density 1 0 50 60 70 80 90 100 110 120 130 140 150 160 Height (cm) Calculate an estimate of the mean height. You must show all your working. … cm [6]

15 marks

Mark scheme: 3(a)(i) 4 1 3(a)(ii) 7 1 3(a)(iii) 8 1 3(b)(i) 14 1 3(b)(ii) 4 2 B1 for [ l.q. =] 11 or [u.q =] 15 3(c) 8.09 3 M1 for 5 × 3 + 10 × 6 + 43 × 7 + 75 × 8 + 48 × 9 + 21 × 10 M1 dep ÷ 200 3(d) 30, 70, 40, 36, 24 seen B2 B1 for 3 or 4 correct or M1 for 1 × (80 – 50), 3.5 × (100 – 80), 4 × (110 – 100), 3.6 × (120 – 110) and 0.6 × (160 – 120) oe (their 30 × 65 + their 70 × 90 M3 M1 for midpoints soi + their 40 × 105 + their 36 × 115 M1 for Σfx , x in interval or boundary of + their 24 × 140) ÷ 200 interval M1 dep on second M1 for ÷ 200 99.75 A1

This question in 0580/43 May/June 2021

3 The cumulative frequency diagram shows information about the mass, m kg, of each of 80 boys. 80 60 Cumulative frequency 40 20 0 m 30 40 50 60 70 80 90 Mass (kg) (a) m 30 40 50 60 70 80 90 Mass (kg) On the grid, draw a box-and-whisker plot to show the information in the cumulative frequency diagram. [4] (b) Use the cumulative frequency diagram to find an estimate of (i) the 30th percentile, … kg [2] (ii) the number of boys with a mass greater than 75 kg. … [2] (c) (i) Use the cumulative frequency diagram to complete this frequency table. Mass 30 1 m G 40 40 1 m G 50 50 1 m G 60 60 1 m G 70 70 1 m G 80 80 1 m G 90(m kg) Frequency 8 12 14 10 [1] (ii) Calculate an estimate of the mean mass of the boys. … kg [4] (iii) Two boys are chosen at random from those with a mass greater than 70 kg. Find the probability that one of them has a mass greater than 80 kg and the other has a mass of 80 kg or less. … [3]

16 marks

Mark scheme: 3(a) Correct box-and-whisker plot 4 B1 for lowest value and highest value at 30 and 90 B1 for LQ and UQ at 50 and 72 B1 for median at 63 3(b)(i) 56 2 M1 for 24 soi 3(b)(ii) 16 2 B1 for 64 written 3(c)(i) 14, 22 1 3(c)(ii) 61.5 4 M1 for 35, 45, 55, 65, 75, 85 soi M1 for Σ fx M1 dep for their Σ fx ÷ (8 + 12 +their 14 + their 22 + 14 +10) or Σ fx ÷ 80 3(c)(iii) 35 3  10 14  oe M2 for [2]  ×  oe 69  24 23  10 14 or M1 for or oe seen 24 24 35 If 0 scored, SC1 for answer oe 72

This question in 0580/41 Oct/Nov 2021

2 (a) The cumulative frequency diagram shows information about the floor area, a m 2, of each of 80 houses. 80 70 60 50 Cumulative 40frequency 30 20 10 0 a 40 60 80 100 120 140 160 180 200 Floor area (m2) Use the diagram to find an estimate of (i) the median, … m2 [1] (ii) the lower quartile, … m2 [1] (iii) the interquartile range, … m2 [1] (iv) the number of houses with a floor area greater than 120 m2. … [2] (b) The information about the 80 floor areas is shown in this frequency table. Floor area 2 40 1 a G 60 60 1 a G 80 8 0 1 a G 100 100 1 a G 130 130 1 a G 160 160 1 a G 200 ( a m ) Frequency 14 17 18 15 9 7 (i) Calculate an estimate of the mean floor area. … m2 [4] (ii) Complete the histogram to show the information in the frequency table. 1.0 Frequency 0.5density 0 a 40 60 80 100 120 140 160 180 200 Floor area (m2) [4] (iii) Two of the houses are picked at random. Find the probability that one of the houses has a floor area greater than 130 m 2 and the other has a floor area 60 m 2 or less. … [3]

16 marks

Mark scheme: 2(a)(i) 90 1 2(a)(ii) 68 1 2(a)(iii) 52 1 FT 120 – their (a)(ii) 2(a)(iv) 20 2 B1 for 60 in working or as answer 2(b)(i) 97.5 4 M1 for mid-points soi (50, 70, 90, 115, 145, 180) M1 for use of Σfm with m in correct interval including both boundaries M1 for (dep on 2nd M1) for Σfm ÷ 80 2(b)(ii) Bars with heights 0.9, 0.5, 0.3, 0.175 4 B1 for each correct bar and with correct widths If 0 scored, SC1 for 3 or 4 correct frequency densities 2(b)(iii) 28 3 16 14 oe M2 for [2 ×] × oe 395 80 79 16 16 14 or M1 for or oe or oe or 80 79 80 14 oe seen 79 7 If 0 scored, SC1 for answer oe 100

This question in 0580/42 Oct/Nov 2021

5 The table shows information about the mass, m grams, of each of 120 letters. Mass (m grams) 0 1 m G 50 50 1 m G 100 100 1 m G 200 200 1 m G 500 Frequency 43 31 25 21 (a) Calculate an estimate of the mean mass. … g [4] (b) Iraj draws a histogram to show this information. He makes the height of the first bar 17.2 cm. Calculate the height of each of the remaining bars. height of bar for 50 1 m G 100 … cm height of bar for 100 1 m G 200 … cm height of bar for 200 1 m G 500 … cm [3] (c) Complete the cumulative frequency table. Mass (m grams) m G 50 m G 100 m G 200 m G 500 Cumulative frequency [2] (d) Draw a cumulative frequency diagram. 120 100 80 Cumulative 60 frequency 40 20 0 m 0 100 200 300 400 500 Mass (g) [3] (e) Use the cumulative frequency diagram to find an estimate for (i) the median, … g [1] (ii) the upper quartile, … g [1] (iii) the 40th percentile, … g [2] (iv) the number of letters with a mass m where 250 1 m G 400 . … [2]

18 marks

Mark scheme: 5(a) 5 4 M1 for midpoints soi 121 or 120.8… or 120 6 M1 for use of ∑fx with x in correct interval including both boundaries but not if x is 50, 50, 100 and 300 M1 (dep on 2nd M1) for ∑fx ÷ 120 5(b) 12.4 5 1.4 3 B1 for each If 0 scored SC1 for fd’s [0.86,] 0.62, 0.25 and 0.07 oe 5(c) 43 74 99 120 2 B1 for 2 or 3 correct 5(d) Correct diagram 3 B1 for correct horizontal placement for 4 plots B1FT for correct vertical placement for 4 plots B1FT dep on at least B1 for reasonable increasing curve or polygon through their 4 points If 0 scored SC1 FT for 3 out of 4 points correctly plotted 5(e)(i) Strict FT their median reading 1 5(e)(ii) Strict FT their UQ reading 1 5(e)(iii) Strict FT their reading at 40th 2 B1 for 48 written or mark at cf = 48 on graph percentile 5(e)(iv) Strict FT their reading at 400 2 B1 for either correct reading at 250 or 400 – their reading at 250

This question in 0580/42 Feb/March 2022

7 Information about the mass, m kg, of each of 150 children is recorded in the frequency table. Mass (m kg) 0 1 m G 10 10 1 m G 20 20 1 m G 25 25 1 m G 40 40 1 m G 50 Frequency 12 38 32 50 18 (a) Calculate an estimate of the mean mass. … kg [4] (b) Draw a histogram to show the information in the table. 8 7 6 5 Frequency density 4 3 2 1 0 m 0 10 20 30 40 50 Mass (kg) [4] (c) (i) Use the frequency table to complete this cumulative frequency table. Mass (m kg) m G 10 m G 20 m G 25 m G 40 m G 50 Cumulative frequency [2] (ii) Calculate the percentage of children with a mass greater than 10 kg. … % [2]

12 marks

Mark scheme: 7(a) 25.2 or 25.23… 4 M1 for midpoints soi M1 for use of ∑fx with x in correct interval including both boundaries M1 (dep on 2nd M1) for ∑fx ÷ 150 7(b) 5 correct blocks 4 B3 for 4 correct blocks or B2 for 3 correct blocks or B1 for 2 correct blocks or block widths 10, 10, 5, 15, 10 If 0 scored SC1 for 4 correct frequency densities from 1.2, 3.8, 6.4, 3.33[3…] and 1.8 oe soi 7(c)(i) 12, 50, 82, 132, 150 2 B1 for 3 or 4 correct 7(c)(ii) 92 2 M1 for 150 −12 oe seen If 0 scored, SC1 for answer 8[%]

This question in 0580/42 May/June 2022

5 The time, t minutes, taken by each of 80 people to travel to work is recorded. The table shows information about these times. Time 0 1 t G 5 5 1 t G 10 10 1 t G 20 20 1 t G 35 35 1 t G 60 (t minutes) Frequency 3 7 18 28 24 (a) (i) Write down the class interval containing the median time. … 1 t G … [1] (ii) Calculate an estimate of the mean time. … min [4] (b) (i) One of these 80 people is chosen at random. Find the probability that this person took longer than 10 minutes to travel to work. Give your answer as a fraction in its simplest form. … [2] (ii) Two people are chosen at random from those taking 20 minutes or less to travel to work. Calculate the probability that one of these people took 5 minutes or less and the other took more than 5 minutes. … [3] (c) (i) Use the frequency table on page 8 to complete the cumulative frequency table. Time t G 5 t G 10 t G 20 t G 35 t G 60 (t minutes) Cumulative 3 10 80 frequency [1] (ii) On the grid, draw a cumulative frequency diagram to show this information. 80 70 60 50 Cumulative frequency 40 30 20 10 0 0 10 20 30 40 50 60 t Time (minutes) [3] (iii) Find an estimate for the 80th percentile. … min [2] (iv) Find an estimate for the percentage of people who took longer than 45 minutes to travel to work. Show all your working. … % [3]

19 marks

Mark scheme: 5(a)(i) 20 < t ⩽ 35 1 5(a)(ii) 28 nfww 4 M1 for midpoints soi M1 for use of ∑fm with m in correct interval including both boundaries M1 (dep on 2nd M1) for ∑fm ÷ 80 5(b)(i) 7 2 18  28  24 cao M1 for oe 8 80 5(b)(ii) 25 3  3 25  oe M2 for [2 ]  or [2 ]   126  28 27   25 3   oe    28 27  3 25 or M1 for either or or 28 27 25 3 or 28 27 75 If 0 scored, SC1 for answer oe 392 5(c)(i) 28 and 56 1 5(c)(ii) Correct diagram 3 B1FT their (c)(i) for plots at 5 correct heights B1 for 5 plots at upper ends of intervals on correct vertical line B1FT (dep on at least B1) for increasing curve or polygon through 5 points After 0 scored, SC1FT for 4 correct points plotted 5(c)(iii) Strict FT their reading at 80th percentile 2 B1 for 64 written or a mark at cf = 64 on for an increasing curve/polygon graph or a mark on curve at (t, 64) 5(c)(iv) Correct integer reading at t = 45 M1 FT their cf graph for all three marks 80  (their readingat t  45) M1 [ 100] 80 (their readingat t  45) or  100 80 Percentage consistent with their reading A1 If no working shown then SC1 for a correct percentage that follows from a correct reading from their graph.

This question in 0580/43 May/June 2022

5 (a) 100 students each record the time, t minutes, taken to eat a pizza. The cumulative frequency diagram shows the results. 100 80 60 Cumulative frequency 40 20 0 t 0 5 10 15 Time (minutes) Find an estimate of (i) the median, … min [1] (ii) the interquartile range, … min [2] (iii) the number of students taking more than 11 minutes to eat a pizza. … [2] (b) 150 students each record how far they can throw a tennis ball. The table shows the results. Distance 0 1 d G 20 20 1 d G 30 30 1 d G 35 35 1 d G 45 45 1 d G 60 (d metres) Frequency 4 38 40 53 15 (i) Calculate an estimate of the mean. … m [4] (ii) A histogram is drawn to show this information. The height of the bar representing 30 1 d G 35 is 12 cm. Calculate the height of each of the other bars. Distance (d metres) Frequency Height of bar (cm) 0 1 d G 20 4 20 1 d G 30 38 30 1 d G 35 40 12 35 1 d G 45 53 45 1 d G 60 15 [3] (iii) Two students are chosen at random. Find the probability that they both threw the ball more than 45 m. … [2]

14 marks

Mark scheme: 5(a)(i) 9.4 1 5(a)(ii) 2.4 2 B1 for [uq =] 10.4 or [lq =] 8 but not as final answer 5(a)(iii) 18 2 B1 for 82 seen 5(b)(i) 13 4 M1 for midpoints 10, 25, 32.5, 40, 52.5 34.65 or 34 soi 20 M1 for fx where values of x are in interval or on boundary fx M1 dep on second M for 150 5(b)(ii) 0.3, 5.7, ..., 7.95, 1.5 3 B2 for any two correct or B1 for one correct or for at least three frequency densities seen 0.2, 3.8, 8, 5.3, 1 oe or M1 for [factor] 1.5 5(b)(iii) 7 2 15 14 oe M1 for  745 150 149

This question in 0580/41 Oct/Nov 2022

5 (a) There are 160 people in a village. The cumulative frequency diagram shows information about their ages. 160 140 120 100 Cumulative 80frequency 60 40 20 0 0 10 20 30 40 50 60 70 Age (years) (i) Find an estimate for (a) the median age … [1] (b) the lower quartile … [1] (c) the number of people who are 50 or more years of age … [2] (d) the 65th percentile. … [2] (ii) The youngest person in the village is 1 year old and the oldest is 70 years old. (a) Draw a box-and-whisker plot to show the distribution of ages in the village. 0 10 20 30 40 50 60 70 80 Age (years) [3] (b) Write down an estimate of the percentage of people in the village that are younger than the median age. … % [1] (b) The frequency table shows information about the age of each person in another village. Age (n years) 0 1 n G 20 20 1 n G 30 30 1 n G 50 50 1 n G 80 Frequency 52 37 24 60 On the grid, complete the histogram to show this information. The first block has been drawn for you. 4 3 Frequency 2density 1 0 n 0 10 20 30 40 50 60 70 80 Age (years) [3]

13 marks

Mark scheme: 5(a)(i)(a) 25 1 5(a)(i)(b) 17 to 18 1 5(a)(i)(c) 12 2 B1 for 148 seen 5(a)(i)(d) 30 2 B1 for 104 seen 5(a)(ii)(a) correct diagram or correct for 3 B1 for whiskers at 1 and at 70 their median and LQ B1 for with median and LQ at their (a)(i)(a) and (a)(i)(b) B1 for UQ at 34 Maximum 2 marks if diagram incorrect If 0 scored SC1 for their 5 correct ages plotted 5(a)(ii)(b) 50 1 5(b) correct histogram 3 B1 for each correct block width 10 height 3.7 width 20 height 1.2 width 30 height 2 If 0 scored SC1 for correct frequency densities 3.7, 1.2, 2 oe

This question in 0580/41 May/June 2023

4 The table shows information about the heights of 80 children. Height 1.2 1 h G 1.4 1.4 1 h G 1.5 1.5 1 h G 1.65 1.65 1 h G 1. 8 1.8 1 h G 1.9 (h metres) Frequency 2 13 24 32 9 (a) (i) Write down the interval containing the median. … 1 h G … [1] (ii) Calculate an estimate of the mean height. … m [4] (b) (i) One of these children is chosen at random. Calculate the probability that they have a height of 1.4 m or less. … [1] (ii) Two of these children are chosen at random. Calculate the probability that both children are taller than 1.5 m but only one of them is taller than 1.8 m. … [3] (c) (i) Complete the cumulative frequency table for the heights. Height h G 1.4 h G 1.5 h G 1.65 h G 1.8 h G 1.9 (h metres) Cumulative 2 frequency [2] (ii) On the grid, draw the cumulative frequency diagram. 80 70 60 50 Cumulative 40frequency 30 20 10 0 h 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Height (m) [3] (d) Use your diagram to find an estimate of (i) the interquartile range … m [2] (ii) the 60th percentile. … m [2]

18 marks

Mark scheme: 4(a)(i) 1.65 < h ≤ 1.8 1 4(a)(ii) 1.63875 4 M1 for midpoints soi M1 for use of ∑fh with h in correct interval including both boundaries M1dep on 2nd M1 for ∑fh ÷ 80 4(b)(i) 1 1 oe 40 4(b)(ii) 63 3 56 9 oe M2 for  [ 2] oe 395 80 79 56 9 9 56 or B1 for or or or oe seen 80 79 80 79 If 0 or B1 scored, instead award SC2 for 117 answer oe 632 63 or SC1 for answer oe 400 4(c)(i) 15, 39, 71, 80 2 B1 for 3 correct or M1 for 1 error in addition with other values then consistent 4(c)(ii) Correct curve 3 B1 for correct horizontal placement for 5 plots B1FT for correct vertical placement for 5 plots B1FT dep on at least B1 for reasonable increasing curve or polygon through their 5 points If 0 scored SC1 FT for 4 out of 5 points correctly plotted 4(d)(i) Strict FT their UQ – their LQ 2dep B1dep for their UQ or their LQ seen Dep on increasing curve/polygon for 2 marks or B1 4(d)(ii) Strict FT their reading at 48 2dep B1 for 48 written

This question in 0580/42 May/June 2023

5 (a) The cumulative frequency diagram shows information about the distance travelled by each of 80 motorists in a month. 80 60 Cumulative 40 frequency 20 0 0 400 800 1200 1600 2000 2400 Distance (km) (i) Use the cumulative frequency diagram to find an estimate for (a) the median … km [1] (b) the interquartile range … km [2] (ii) One of these motorists is picked at random. Find the probability that this motorist travels more than 1800 km. … [2] (b) The distance around a racing track is 5.104 km. The time taken by a car to complete one lap of the track is 1 min 18 s. Calculate the average speed of the car. Give your answer in km/h. … km/h [3] (c) The top speed, v km/h, of each of 160 cars is recorded. The histogram shows this information. 2.0 1.5 Frequency 1.0 density 0.5 0 v 100 140 180 220 260 300 Top speed (km / h) (i) Show that there are 8 cars with a top speed in the interval 120 1 v G 160 . [1] (ii) Calculate an estimate of the mean top speed. You must show all your working. … km/h [6]

15 marks

Mark scheme: 5(a)(i)(a) 1480 1 5(a)(i)(b) 440 2 M1 for [UQ =] 1600 soi or [LQ =] 1160 soi 5(a)(ii) 8 2 oe M1 for 72 or 8 written 80 5(b) 236 or 235.5 to 235.6 3 5.104 M2 for  60 oe 1.3 5.104 or M1 for their time 5(c)(i) (160 – 120)  0.2 [ = 8] 1 with no errors seen 5(c)(ii) 22, 36, 64, 30 seen B2 B1 for 2 or 3 correct frequencies or M1 for three of 1.1  (180 – 160), 1.8  (200 – 180), 1.6  (240 – 200) and 0.5  (300 – 240) oe (8 × 140 + their22 × 170 M3 + their36 × 190 + their64 × 220 M1 for midpoints soi + their30 × 270) ÷ 160 M1 for fx , x in interval or boundary of interval M1 dep on second M1 for fx ÷ 160 211.75 B1

This question in 0580/42 Oct/Nov 2024