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F05 Percentiles

EDEXCEL 1ST0 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

For each statement about percentiles, write down whether it is True or False.

(i)Percentiles divide an ordered set of data into 100 equal-sized groups.(1)

(ii)The 90th percentile of a data set is smaller than its 10th percentile.(1)

(iii)Linear interpolation can be used to estimate a percentile from grouped (continuous) data.(1)

(Total for Question 1 is 3 marks)

In a health survey, the resting heart rates of a large group of adults were recorded. The 65th percentile of the resting heart rates was found to be 72 beats per minute (bpm).

(a)Circle the statement that correctly describes what this means.(1)

A) 65% of adults had a resting heart rate of exactly 72 bpm.
B) 65% of adults had a resting heart rate of 72 bpm or below.
C) 72% of adults had a resting heart rate of 65 bpm or below.
D) 65 out of every 72 adults had a slower heart rate.

(b)Explain why statement A is not correct.(1)

(Total for Question 2 is 2 marks)

The table shows some measures used to summarise a set of data. Complete the table by writing the missing quartile name or the missing percentile in each gap.

Measure	Percentile
Lower quartile (Q1)	?
?	50th percentile
Upper quartile (Q3)	?

(Total for Question 3 is 3 marks)

Baby growth charts compare a baby's weight with the weights of other babies of the same age, using percentiles. At his 6-month check-up, Noah's weight is on the 91st percentile for his age. At her 6-month check-up, Grace's weight is on the 60th percentile for her age.

(a)Explain what it means for Noah's weight to be on the 91st percentile for his age.(2)

(b)State which baby, Noah or Grace, is heavier compared with other babies of the same age. Give a reason.(2)

(Total for Question 4 is 4 marks)

The times, in seconds, taken by 15 sprinters to run 100 m at a school sports day are listed below, in order from fastest to slowest.

12.1, 12.4, 12.6, 12.9, 13.0, 13.2, 13.3, 13.5, 13.6, 13.8, 13.9, 14.1, 14.3, 14.6, 15.0

(a)Write down the formula used to find the position of the kth percentile within an ordered list of n data values.(1)

(b)Use the formula to find the 20th percentile of the sprint times.(2)

(c)Find the 70th percentile of the sprint times.(2)

(d)Using the same method, find the lower quartile (Q1, the 25th percentile) and the upper quartile (Q3, the 75th percentile) of the sprint times. Hence write down the interquartile range.(3)

(Total for Question 5 is 8 marks)

The times, t (seconds), taken by 40 members of a swimming club to swim 50 m freestyle are shown in the table.

Time (t seconds)	Frequency	Cumulative frequency
30 < t ≤ 35	6	6
35 < t ≤ 40	14	20
40 < t ≤ 45	12	32
45 < t ≤ 50	8	40

(a)Work out the position of the 35th percentile within the cumulative frequency total.(2)

(b)Write down the class interval that contains the 35th percentile.(1)

(Total for Question 6 is 3 marks)

The time, t (minutes), taken by 80 people to complete a jigsaw puzzle is shown in the grouped frequency table.

Time (t minutes)	Frequency	Cumulative frequency
0 < t ≤ 10	6	6
10 < t ≤ 20	14	20
20 < t ≤ 30	22	42
30 < t ≤ 40	20	62
40 < t ≤ 50	12	74
50 < t ≤ 60	6	80

(a)Work out the position of the 60th percentile within the cumulative frequency total.(2)

(b)Write down the class interval that contains the 60th percentile.(1)

(c)Use linear interpolation to estimate the 60th percentile of the completion times.(3)

(Total for Question 7 is 6 marks)

The weight, w (kg), of recycling collected from 60 households in one week is shown in the table.

Weight (w kg)	Frequency	Cumulative frequency
0 < w ≤ 5	8	8
5 < w ≤ 10	16	24
10 < w ≤ 15	20	44
15 < w ≤ 20	10	54
20 < w ≤ 25	6	60

(a)Write down the position of the median within the cumulative frequency total.(1)

(b)Which class interval contains the median?(1)

(c)Use linear interpolation to estimate the median weight of recycling collected per household.(3)

(Total for Question 8 is 5 marks)

The marks, out of 50, scored by 20 students in a mock statistics exam are listed below, in order.

18, 22, 24, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 45, 47, 49

(a)Find the 10th percentile of the marks.(2)

(b)Find the 90th percentile of the marks.(2)

(c)A teacher claims that 'at least 1 in 10 students scored more than 46 marks.' Use your answer to part (b) to comment on whether the teacher's claim is correct.(1)

(Total for Question 9 is 5 marks)

Two athletes, Amara and Ben, recorded their times, in seconds, for their last 8 sprints. The times are listed below, in order.

Amara: 11.8, 11.9, 12.0, 12.2, 12.3, 12.5, 12.6, 12.9
Ben: 11.5, 11.7, 12.1, 12.4, 12.5, 12.8, 13.0, 13.4

(a)Find the median (the 50th percentile) of each athlete's sprint times.(2)

(b)The interquartile range of Amara's times is 0.6 seconds and Ben's is 1.0 seconds. Using this, together with your answer to part (a), compare the two athletes' sprint times. Give TWO comparisons.(2)

(Total for Question 10 is 4 marks)

The cumulative frequency graph shows the length, in minutes, of 100 calls made to a call centre one morning.

(a)Use the graph to estimate the 40th percentile of the call lengths.(2)

(b)A call lasted 24 minutes. Use the graph to estimate what percentile of the call lengths this represents.(2)

(c)Write down an estimate for the lower quartile, Q1, of the call lengths.(1)

(d)Write down an estimate for the upper quartile, Q3, of the call lengths.(1)

(e)Hence estimate the interquartile range of the call lengths, and explain what this value tells us about the call lengths.(2)

(Total for Question 11 is 8 marks)

Priya is investigating how long Year 11 students at her school spend on their phones each evening. There are 180 Year 11 students, split unevenly across 6 tutor groups.

(a)Priya wants her sample to be representative of all Year 11 students. Describe a suitable sampling method she could use, and explain why it would give a representative sample.(2)

(b)Priya collects screen-time data (t minutes) from a sample of 50 students, shown in the table.

Screen time (t minutes)	Frequency	Cumulative frequency
0 < t ≤ 30	4	4
30 < t ≤ 60	8	12
60 < t ≤ 90	20	32
90 < t ≤ 120	10	42
120 < t ≤ 150	8	50

Work out the position of the 80th percentile within the cumulative frequency total.(2)

(c)Write down the class interval that contains the 80th percentile.(1)

(d)Use linear interpolation to estimate the 80th percentile of the screen-time data.(3)

(e)Priya read a claim online that 'at least 80% of Year 11 students spend under 180 minutes (3 hours) on their phone each evening.' Use your answer to part (d) to comment on this claim.(2)

(Total for Question 12 is 10 marks)

The delivery times, t (minutes), for 120 parcels sent by a courier company are shown in the table. Some of the cumulative frequencies are missing.

Delivery time (t minutes)	Frequency	Cumulative frequency
0 < t ≤ 10	8	8
10 < t ≤ 20	20	?
20 < t ≤ 30	34	?
30 < t ≤ 40	32	?
40 < t ≤ 50	18	?
50 < t ≤ 60	8	120

(a)Complete the cumulative frequency column of the table.(2)

(b)A parcel was delivered in 35 minutes. By first estimating the cumulative frequency up to 35 minutes using linear interpolation, find the percentile of the delivery-time distribution that this represents.(3)

(Total for Question 13 is 5 marks)

Two supermarket checkouts, Checkout A and Checkout B, were each observed for 40 customers. The waiting time, w (minutes), for each customer was recorded.

Checkout A

Waiting time (w minutes)	Frequency	Cumulative frequency
0 < w ≤ 5	10	10
5 < w ≤ 10	20	30
10 < w ≤ 15	6	36
15 < w ≤ 20	4	40

Checkout B

Waiting time (w minutes)	Frequency	Cumulative frequency
0 < w ≤ 5	4	4
5 < w ≤ 10	20	24
10 < w ≤ 15	12	36
15 < w ≤ 20	4	40

(a)Show that the median waiting time at Checkout A is 7.5 minutes.(3)

(b)Use linear interpolation to estimate the median waiting time at Checkout B.(3)

(c)Estimate the interquartile range of waiting times at Checkout A.(3)

(d)The interquartile range of waiting times at Checkout B is 6 minutes. Using this, together with your answers to parts (a) to (c), compare the waiting times at the two checkouts. Give TWO comparisons, in context.(2)

(Total for Question 14 is 11 marks)

The distance, d (km), travelled to work by 60 employees at a company is shown in the table.

Distance (d km)	Frequency	Cumulative frequency
0 < d ≤ 10	8	8
10 < d ≤ 20	20	28
20 < d ≤ 30	22	50
30 < d ≤ 40	10	60

(a)Use linear interpolation to estimate the 30th percentile of the distances travelled to work.(3)

(b)Explain why your answer to part (a) is an ESTIMATE of the 30th percentile, rather than an exact value.(2)

(Total for Question 15 is 5 marks)

Mark scheme · F05 Percentiles

Question 1

(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 True cao
(iii) Answer: True

Question 2

(a) B1 B cao
(a) Answer: B) 65% of adults had a resting heart rate of 72 bpm or below.
(b) B1 a percentile does not mean that percentage of people share one exact value; it means that percentage of the data lies at or below that value (oe)
(b) Answer: A percentile is about the proportion of data at or below a value, not about everyone sharing that exact value.

Question 3

B1 25th (percentile)
B1 Median
B1 75th (percentile)
Answer: Lower quartile = 25th percentile; Median = 50th percentile; Upper quartile = 75th percentile.

Question 4

(a) B1 91% of babies of the same age weigh the same as or less than Noah (oe)
(a) B1 so Noah is heavier than (about) 91 out of every 100 babies of his age (oe)
(a) Answer: About 91% of babies of Noah's age weigh the same as or less than he does - Noah is one of the heavier babies for his age.
(b) B1 Noah
(b) B1 because his percentile (91st) is higher than Grace's (60th), so a greater percentage of babies weigh less than Noah than weigh less than Grace (oe)
(b) Answer: Noah, because his percentile (91st) is higher than Grace's (60th).

Question 5

(a) B1 position = (k / 100) x n, oe
(a) Answer: position = (k / 100) x n
(b) M1 position = 0.2 x 15 = 3 (a whole number, so average the 3rd and 4th values)
(b) A1 12.75 cao
(b) Answer: 12.75 seconds
(c) M1 position = 0.7 x 15 = 10.5, round up to 11
(c) A1 13.9 cao
(c) Answer: 13.9 seconds
(d) B1 Q1 = 12.9 (position 3.75 rounds up to the 4th value)
(d) B1 Q3 = 14.1 (position 11.25 rounds up to the 12th value)
(d) B1 IQR = 1.2 cao
(d) Answer: Q1 = 12.9 s, Q3 = 14.1 s, IQR = 1.2 s

Question 6

(a) M1 0.35 x 40
(a) A1 14 cao
(a) Answer: 14
(b) B1 35 < t ≤ 40 cao
(b) Answer: 35 < t ≤ 40

Question 7

(a) M1 0.6 x 80
(a) A1 48 cao
(a) Answer: 48
(b) B1 30 < t ≤ 40 cao
(b) Answer: 30 < t ≤ 40
(c) M1 (48 - 42) / 20 x 10, oe
(c) M1 30 + ...
(c) A1 33 cao
(c) Answer: 33 minutes

Question 8

(a) B1 30 cao
(a) Answer: 30
(b) B1 10 < w ≤ 15 cao
(b) Answer: 10 < w ≤ 15
(c) M1 (30 - 24) / 20 x 5, oe
(c) M1 10 + ...
(c) A1 11.5 cao
(c) Answer: 11.5 kg

Question 9

(a) M1 position = 0.1 x 20 = 2 (whole, so average the 2nd and 3rd values)
(a) A1 23 cao
(a) Answer: 23
(b) M1 position = 0.9 x 20 = 18 (whole, so average the 18th and 19th values)
(b) A1 46 cao
(b) Answer: 46
(c) B1 correct (oe), since the 90th percentile is 46, exactly 2 students (10%) scored more than 46 marks
(c) Answer: The claim is correct - exactly 2 of the 20 students (10%) scored more than 46 marks.

Question 10

(a) B1 Amara median = 12.25 cao
(a) B1 Ben median = 12.45 cao
(a) Answer: Amara: 12.25 s; Ben: 12.45 s
(b) B1 Amara has a lower median time (12.25 s) than Ben (12.45 s), so Amara tends to sprint faster (oe)
(b) B1 Amara has a smaller interquartile range (0.6 s) than Ben (1.0 s), so Amara's times are more consistent / less variable (oe)
(b) Answer: Amara is typically faster (lower median) and more consistent (smaller IQR) than Ben.

Question 11

(a) M1 reads across from a cumulative frequency of 40 to the curve, then down to the time axis
(a) A1 15 (accept 14 - 16) minutes
(a) Answer: 15 minutes
(b) M1 reads up from t = 24 to the curve, then across to the cumulative frequency axis
(b) A1 63rd percentile (accept 60th - 66th)
(b) Answer: 63rd percentile
(c) B1 10 minutes cao
(c) Answer: 10 minutes
(d) B1 30 minutes cao
(d) Answer: 30 minutes
(e) B1 IQR = 30 - 10 = 20 minutes
(e) B1 the middle 50% of calls varied in length by about 20 minutes (oe interpretation in context)
(e) Answer: IQR = 20 minutes - the middle half of calls lasted between 10 and 30 minutes, a spread of 20 minutes.

Question 12

(a) M1 stratified sampling, sampling from each tutor group in proportion to its size (oe)
(a) A1 this ensures every tutor group is represented in the sample in the same proportion as in the whole year group, so the sample reflects the whole year group (oe)
(a) Answer: Stratified sampling: take a number of students from each tutor group in proportion to its size, so the sample fairly represents every group.
(b) M1 0.8 x 50
(b) A1 40 cao
(b) Answer: 40
(c) B1 90 < t ≤ 120 cao
(c) Answer: 90 < t ≤ 120
(d) M1 (40 - 32) / 10 x 30, oe
(d) M1 90 + ...
(d) A1 114 cao
(d) Answer: 114 minutes
(e) B1 114 < 180, so at least 80% of students in the sample spend under 180 minutes (oe)
(e) B1 the claim is supported by the data (oe justified conclusion)
(e) Answer: The claim is supported - the 80th percentile (114 minutes) is already well under 180 minutes, so at least 80% of students spend under 180 minutes on their phone.

Question 13

(a) M1 running totals found (at least 3 of 28, 62, 94, 112 correct)
(a) A1 all four correct: 28, 62, 94, 112
(a) Answer: 28, 62, 94, 112
(b) M1 62 + (5 / 10) x 32, oe
(b) M1 78 -:- 120 x 100
(b) A1 65% (65th percentile) cao
(b) Answer: 65th percentile

Question 14

(a) M1 position = 0.5 x 40 = 20
(a) M1 5 + ((20 - 10) / 20) x 5
(a) A1 = 7.5, as required
(a) Answer: 7.5 minutes
(b) M1 position = 0.5 x 40 = 20
(b) M1 5 + ((20 - 4) / 20) x 5
(b) A1 9 cao
(b) Answer: 9 minutes
(c) M1 Q1 = 5 (position 10, in class 0 < w ≤ 5)
(c) M1 Q3 = 10 (position 30, in class 5 < w ≤ 10)
(c) A1 IQR = 5 cao
(c) Answer: IQR = 5 minutes (Q1 = 5, Q3 = 10)
(d) B1 Checkout A has a lower median waiting time (7.5 min) than Checkout B (9 min), so customers typically wait less time at Checkout A (oe)
(d) B1 Checkout A has a smaller interquartile range (5 min) than Checkout B (6 min), so waiting times at Checkout A are more consistent / less spread out (oe)
(d) Answer: Checkout A is typically faster (lower median) and more consistent (smaller IQR) than Checkout B.

Question 15

(a) M1 position = 0.3 x 60 = 18
(a) M1 10 + ((18 - 8) / 20) x 10
(a) A1 15 cao
(a) Answer: 15 km
(b) B1 the individual distances within each class interval are not known, only the grouped frequencies (oe)
(b) B1 interpolation assumes the data are spread evenly across the class interval, which may not be true, so the true percentile could differ from 15 km (oe)
(b) Answer: Because the raw data values are not known, only how many fall in each class, interpolation assumes they are evenly spread across each class interval - which may not be exactly true, so the estimate may differ from the real 30th percentile.

Percentiles - Worksheets, Questions and Revision