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H18 Percentiles and Interpercentile Range

EDEXCEL 1ST0 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

For each statement about percentiles and the interpercentile range, write down whether it is True or False.

(i)It is possible to find the median of a data set by locating its 50th percentile.(1)

(ii)If a value lies at the 40th percentile of a data set, then 60% of the data lies below it.(1)

(iii)The 10-90 interpercentile range is also known as the interdecile range.(1)

(iv)Extreme outliers affect the interpercentile range in exactly the same way as they affect the range.(1)

(Total for Question 1 is 4 marks)

The distances, in km, cycled by 15 members of a cycling club during a training session are listed below, in order from shortest to longest.

4.2, 4.5, 4.8, 5.0, 5.3, 5.6, 5.8, 6.0, 6.2, 6.5, 6.7, 7.0, 7.3, 7.6, 8.0

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

(b)Find the 10th percentile of the training distances.(2)

(c)Find the 90th percentile of the training distances.(2)

(d)Hence find the 10-90 interpercentile range (the interdecile range) of the training distances.(2)

(Total for Question 2 is 7 marks)

A speed camera recorded the speeds, in mph, of 200 cars on a stretch of road one day. The results are shown in the grouped frequency table.

Speed (s mph)	Frequency	Cumulative frequency
20 < s ≤ 30	24	24
30 < s ≤ 40	55	79
40 < s ≤ 50	71	150
50 < s ≤ 60	34	184
60 < s ≤ 70	16	200

(a)In a similar study last year, the 75th percentile of speeds was found to be 47 mph. Explain what this figure means.(2)

(b)Work out the position of the 40th percentile of this year's speeds within the cumulative frequency total.(2)

(c)Write down the class interval that contains the 40th percentile.(1)

(Total for Question 3 is 5 marks)

The mass, m grams, of 150 apples picked from an orchard is shown in the grouped frequency table.

Mass (m grams)	Frequency	Cumulative frequency
100 < m ≤ 120	10	10
120 < m ≤ 140	25	35
140 < m ≤ 160	55	90
160 < m ≤ 180	40	130
180 < m ≤ 200	20	150

(a)Use linear interpolation to estimate the 10th percentile of the apple masses.(3)

(b)Use linear interpolation to estimate the 90th percentile of the apple masses.(3)

(c)Hence calculate the 10-90 interpercentile range (the interdecile range) of the apple masses.(2)

(Total for Question 4 is 8 marks)

The cumulative frequency graph shows the time, t minutes, spent by 240 visitors at a museum exhibition.

(a)Use the graph to estimate the lower quartile (Q1, the 25th percentile) of the visit times.(2)

(b)Use the graph to estimate the upper quartile (Q3, the 75th percentile) of the visit times.(2)

(c)Hence write down an estimate for the interquartile range of the visit times.(1)

(d)Use the graph to estimate the 10th and 90th percentiles of the visit times, and hence the 10-90 interpercentile range (the interdecile range).(3)

(e)The interquartile range uses only the middle 50% of the data, while the 10-90 interpercentile range uses the middle 80%. Explain why the interdecile range is larger than the interquartile range for this data.(1)

(Total for Question 5 is 9 marks)

The annual salaries of the 11 employees at a small company are listed below, in order from lowest to highest.

£18,000, £19,000, £20,000, £21,000, £22,000, £23,000, £24,000, £25,000, £26,000, £28,000, £95,000

The highest salary, £95,000, is paid to the company's founder and chief executive.

(a)Find the range of the salaries.(1)

(b)Find the 10th and 90th percentiles of the salaries, and hence find the 10-90 interpercentile range (the interdecile range).(3)

(c)Explain why the interdecile range gives a more reliable description of the spread of most employees' salaries than the range.(1)

(Total for Question 6 is 5 marks)

Elmswood and Oakfield are two nearby towns. The commute time, t minutes, to work was recorded for 150 workers from each town. The grouped frequency table for Elmswood is shown below.

Commute time (t minutes)	Frequency	Cumulative frequency
0 < t ≤ 15	5	5
15 < t ≤ 30	25	30
30 < t ≤ 45	60	90
45 < t ≤ 60	40	130
60 < t ≤ 75	20	150

(a)Use linear interpolation to estimate the 10th percentile of Elmswood's commute times.(2)

(b)Use linear interpolation to estimate the 90th percentile of Elmswood's commute times.(3)

(Total for Question 7 is 5 marks)

The grouped frequency table for Oakfield's commute times is shown below.

Commute time (t minutes)	Frequency	Cumulative frequency
0 < t ≤ 15	18	18
15 < t ≤ 30	27	45
30 < t ≤ 45	45	90
45 < t ≤ 60	35	125
60 < t ≤ 75	25	150

(a)Use linear interpolation to estimate the 10th and 90th percentiles of Oakfield's commute times.(3)

(b)Using your answers to Question 7 and part (a), calculate the 10-90 interpercentile range (the interdecile range) for both towns.(1)

(c)Hence compare the spread of commute times in the two towns, in context.(2)

(Total for Question 8 is 6 marks)

A factory machine produces metal bolts. The diameter, d mm, of 180 bolts is shown in the grouped frequency table.

Diameter (d mm)	Frequency	Cumulative frequency
8.0 < d ≤ 8.5	14	14
8.5 < d ≤ 9.0	40	54
9.0 < d ≤ 9.5	66	120
9.5 < d ≤ 10.0	42	162
10.0 < d ≤ 10.5	18	180

(a)Use linear interpolation to estimate the 15th percentile of the bolt diameters.(3)

(b)Use linear interpolation to estimate the 85th percentile of the bolt diameters.(3)

(c)State what percentage of the bolts have a diameter between your answers to (a) and (b), and name this interpercentile range.(1)

(Total for Question 9 is 7 marks)

100 Under-40 runners and 100 Over-40 runners entered a charity 10 km road race. The cumulative frequency graph shows their finishing times, t minutes.

(a)Using the graph, estimate the 10th and 90th percentiles of the Under-40 runners' finishing times, and hence their 10-90 interpercentile range.(3)

(b)Using the graph, estimate the 10th and 90th percentiles of the Over-40 runners' finishing times, and hence their 10-90 interpercentile range.(3)

(c)Compare the finishing times of the two groups, using your answers to (a) and (b).(3)

(Total for Question 10 is 9 marks)

A researcher is investigating the spread of annual income among self-employed workers in a region. Incomes in this population are known to be positively skewed, with a small number of very high earners.

(a)PLAN: Suggest a suitable statistical measure of spread for summarising this data, other than the range, and give a reason for your choice.(2)

(b)COLLECT AND PROCESS: The researcher takes a random sample of 250 self-employed workers and processes the incomes into a grouped frequency table with a cumulative frequency column. Explain why linear interpolation is needed to estimate a percentile from this table, rather than reading off an individual data value.(2)

(c)INTERPRET: The researcher estimates the 10th percentile of incomes to be £14,500 and the 90th percentile to be £68,000. Calculate the 10-90 interpercentile range, and interpret this value in context.(2)

(Total for Question 11 is 6 marks)

A factory machine fills plastic bottles with a soft drink. The volume, v ml, of the bottles is grouped into classes, but one frequency has been replaced with the letter a.

Volume (v ml)	Frequency	Cumulative frequency
480 < v ≤ 490	30	30
490 < v ≤ 500	50	80
500 < v ≤ 510	a	80 + a
510 < v ≤ 520	70	150 + a
520 < v ≤ 530	20	170 + a

(a)Show that the total number of bottles, n, can be written as n = 170 + a.(1)

(b)Given that the 90th percentile of the bottle volumes is 518 ml, and that this value lies within the class 510 < v ≤ 520, form and solve an equation to find the value of a, and hence find n.(5)

(Total for Question 12 is 6 marks)

The scores out of 100 for 12 contestants in a general knowledge quiz are listed below, in order from lowest to highest.

8, 52, 55, 58, 60, 62, 64, 66, 68, 70, 73, 76

Contestant L, who scored 8, had a technical fault with their buzzer for most of the quiz.

(a)Calculate the range of the quiz scores.(1)

(b)Calculate the interquartile range of the quiz scores.(2)

(c)Calculate the 10-90 interpercentile range (the interdecile range) of the quiz scores.(2)

(d)Contestant L's low score of 8 was due to a technical fault, not a lack of general knowledge. Explain which of the three measures - range, IQR or interdecile range - best describes how spread out the genuine scores were, and why.(2)

(Total for Question 13 is 7 marks)

The cumulative frequency graph shows the monthly electricity usage, in kWh, for 200 households in an Urban postcode area and 200 households in a Rural postcode area.

(a)Using the graph, write down the 10th and 90th percentiles of the Urban households' electricity usage, and calculate the 10-90 interpercentile range.(3)

(b)Using the graph, estimate the 10th and 90th percentiles of the Rural households' electricity usage, using linear interpolation, and calculate the 10-90 interpercentile range.(3)

(c)Compare the electricity usage of the two areas, using your answers to (a) and (b).(2)

(d)The interdecile range you found is an EXACT value for the Urban households but only an ESTIMATE for the Rural households, using this graph. Explain why.(2)

(Total for Question 14 is 10 marks)

Mark scheme · H18 Percentiles and Interpercentile Range

Question 1

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

Question 2

(a) B1 position = (k / 100) x n, oe
(a) Answer: position = (k / 100) x n
(b) M1 position = 0.1 x 15 = 1.5 (not a whole number, so round up to the 2nd value)
(b) A1 4.5 cao
(b) Answer: 4.5 km
(c) M1 position = 0.9 x 15 = 13.5 (not a whole number, so round up to the 14th value)
(c) A1 7.6 cao
(c) Answer: 7.6 km
(d) M1 7.6 - 4.5, oe (ft from (b) and (c))
(d) A1 3.1 cao
(d) Answer: 3.1 km

Question 3

(a) B1 75% of the cars in that study were travelling at 47 mph or below (oe)
(a) B1 equivalently, only 25% of cars were travelling faster than 47 mph (oe)
(a) Answer: 75% of cars were travelling at 47 mph or below (so only 25% were travelling faster than 47 mph).
(b) M1 0.4 x 200
(b) A1 80 cao
(b) Answer: 80
(c) B1 40 < s ≤ 50 cao
(c) Answer: 40 < s ≤ 50

Question 4

(a) M1 position = 0.1 x 150 = 15, identifying the class 120 < m ≤ 140
(a) M1 120 + ((15 - 10) / 25) x 20, oe
(a) A1 124 cao
(a) Answer: 124 g
(b) M1 position = 0.9 x 150 = 135, identifying the class 180 < m ≤ 200
(b) M1 180 + ((135 - 130) / 20) x 20, oe
(b) A1 185 cao
(b) Answer: 185 g
(c) M1 185 - 124, oe (ft from (a) and (b))
(c) A1 61 cao
(c) Answer: 61 g

Question 5

(a) M1 reads across from a cumulative frequency of 60 to the curve, then down to the time axis
(a) A1 25 (accept 24 - 26) minutes
(a) Answer: 25 minutes
(b) M1 reads across from a cumulative frequency of 180 to the curve, then down to the time axis
(b) A1 57 (accept 55 - 59) minutes
(b) Answer: 57 minutes
(c) B1 32 minutes, ft from (a) and (b) (accept 29 - 35)
(c) Answer: 32 minutes
(d) M1 P10 = 12 (accept 11 - 13) minutes
(d) M1 P90 = 75 (accept 73 - 77) minutes
(d) A1 interdecile range = 63 minutes, ft (accept 60 - 66)
(d) Answer: P10 = 12 min, P90 = 75 min, interdecile range = 63 min
(e) B1 the interdecile range spans a wider proportion (80%) of the data than the IQR (50%), so its lower and upper bounds must be at least as far apart as the IQR's (oe)
(e) Answer: Because it covers a wider proportion (80%) of the data than the IQR (50%), the interdecile range must span from a value at least as low up to a value at least as high, making it at least as large.

Question 6

(a) B1 77000 cao
(a) Answer: £77,000
(b) M1 positions 0.1 x 11 = 1.1 (round up to the 2nd value) and 0.9 x 11 = 9.9 (round up to the 10th value)
(b) A1 P10 = 19000 and P90 = 28000
(b) A1 interdecile range = 9000 cao
(b) Answer: P10 = £19,000, P90 = £28,000, interdecile range = £9,000
(c) B1 the range is almost entirely determined by the one outlier salary (£95,000), whereas the interdecile range ignores the most extreme 10% of salaries at each end, so it reflects the spread of the typical employees' salaries instead (oe)
(c) Answer: The range (£77,000) is dominated by the founder's unusually high salary. The interdecile range (£9,000) ignores the most extreme 10% at each end, so it better reflects how spread out most employees' salaries actually are.

Question 7

(a) M1 position = 0.1 x 150 = 15, identifying the class 15 < t ≤ 30; 15 + ((15-5)/25) x 15
(a) A1 21 cao
(a) Answer: 21 minutes
(b) M1 position = 0.9 x 150 = 135, identifying the class 60 < t ≤ 75
(b) M1 60 + ((135-130)/20) x 15, oe
(b) A1 63.75 cao
(b) Answer: 63.75 minutes

Question 8

(a) M1 P10: position = 15 falls within the class 0 < t ≤ 15; 0 + (15/18) x 15
(a) M1 P90: position = 135 falls within the class 60 < t ≤ 75; 60 + ((135-125)/25) x 15
(a) A1 P10 = 12.5 and P90 = 66, both correct
(a) Answer: P10 = 12.5 minutes, P90 = 66 minutes
(b) B1 Elmswood 42.75 minutes (ft) and Oakfield 53.5 minutes (ft), both correct
(b) Answer: Elmswood: 42.75 minutes; Oakfield: 53.5 minutes
(c) B1 Oakfield has a larger interdecile range (53.5 min) than Elmswood (42.75 min)
(c) B1 so commute times are more variable/spread out among Oakfield's workers than Elmswood's, even after the most extreme 10% at each end are excluded (oe)
(c) Answer: Oakfield's commute times are more spread out than Elmswood's, since Oakfield's interdecile range (53.5 min) is larger than Elmswood's (42.75 min).

Question 9

(a) M1 position = 0.15 x 180 = 27, identifying the class 8.5 < d ≤ 9.0
(a) M1 8.5 + ((27-14)/40) x 0.5, oe
(a) A1 8.66 (awrt) cao
(a) Answer: 8.66 mm (awrt)
(b) M1 position = 0.85 x 180 = 153, identifying the class 9.5 < d ≤ 10.0
(b) M1 9.5 + ((153-120)/42) x 0.5, oe
(b) A1 9.89 (awrt) cao
(b) Answer: 9.89 mm (awrt)
(c) B1 70% of the bolts; this is the 15-85 interpercentile range (oe)
(c) Answer: 70% of the bolts (the 15-85 interpercentile range)

Question 10

(a) M1 P10 = 45 (accept 44 - 46) minutes
(a) M1 P90 = 70 minutes (read directly from the curve)
(a) A1 interdecile range = 25 minutes, ft
(a) Answer: P10 = 45 min, P90 = 70 min, interdecile range = 25 min
(b) M1 P10 = 50 minutes (read directly from the curve)
(b) M1 P90 = 77.5 (accept 76 - 79) minutes
(b) A1 interdecile range = 27.5 minutes, ft (accept 26 - 29)
(b) Answer: P10 = 50 min, P90 = 77.5 min, interdecile range = 27.5 min
(c) B1 the Over-40 group has both a higher P10 (50 vs 45) and a higher P90 (77.5 vs 70), so Over-40 runners generally took longer to finish than Under-40 runners (oe)
(c) B1 the Over-40 group has a slightly larger interdecile range (27.5 min) than the Under-40 group (25 min)
(c) B1 so finishing times were slightly more variable/spread out among the Over-40 group than the Under-40 group (oe)
(c) Answer: Over-40 runners typically took longer to finish (higher P10 and P90) and had slightly more variable times (larger interdecile range: 27.5 min vs 25 min) than Under-40 runners.

Question 11

(a) B1 the (10-90) interpercentile range, or the interquartile range (oe)
(a) B1 because the distribution is skewed with some very high earners (potential outliers), and this measure ignores the extreme tails, so it is not distorted by them (oe)
(a) Answer: The 10-90 interpercentile range (or IQR), because it ignores the most extreme incomes at each end, so it is not distorted by the small number of very high earners.
(b) B1 the incomes have been grouped into class intervals, so the exact individual values within each class are no longer known (oe)
(b) B1 linear interpolation estimates where a percentile falls within its class by assuming the data are evenly spread across that class (oe)
(b) Answer: Because the data are grouped, individual incomes within each class are unknown; linear interpolation estimates the percentile's position by assuming the incomes are evenly spread across the class.
(c) B1 interdecile range = 68000 - 14500 = 53500
(c) B1 the middle 80% of self-employed workers in the sample earn between £14,500 and £68,000 (a spread of £53,500), ignoring the lowest and highest 10% of earners (oe)
(c) Answer: £53,500 - the middle 80% of self-employed workers earn between £14,500 and £68,000.

Question 12

(a) B1 30 + 50 + a + 70 + 20 = 170 + a, as required
(a) Answer: n = 170 + a
(b) M1 position = 0.9n = 0.9(170 + a) = 153 + 0.9a
(b) M1 518 = 510 + ((153 + 0.9a - (80 + a)) / 70) x 10, oe interpolation equation formed
(b) M1 equation simplified: 8 x 70 = (73 - 0.1a) x 10, i.e. 560 = 730 - a
(b) A1 a = 170 cao
(b) A1 n = 340 cao
(b) Answer: a = 170, n = 340

Question 13

(a) B1 68 cao
(a) Answer: 68
(b) M1 Q1 = (55+58)/2 = 56.5 and Q3 = (68+70)/2 = 69
(b) A1 IQR = 12.5 cao
(b) Answer: 12.5
(c) M1 P10 = 52 (position 1.2 rounds up to the 2nd value) and P90 = 73 (position 10.8 rounds up to the 11th value)
(c) A1 interdecile range = 21 cao
(c) Answer: 21
(d) B1 the interdecile range (21) gives the best description (accept IQR with valid justification)
(d) B1 because it excludes the affected outlier score of 8, while still using more of the data than the IQR does, unlike the range, which is entirely determined by the faulty score (oe)
(d) Answer: The interdecile range (21) is the most reliable, since it excludes the faulty outlier score (8) but still reflects more of the spread of genuine scores than the IQR, unlike the range, which is dictated entirely by the fault.

Question 14

(a) B1 P10 = 100 kWh
(a) B1 P90 = 300 kWh
(a) B1 interdecile range = 200 kWh
(a) Answer: P10 = 100 kWh, P90 = 300 kWh, interdecile range = 200 kWh
(b) M1 P10: 100 + ((20-10)/40) x 100
(b) M1 P90: 300 + ((180-150)/40) x 100
(b) A1 P10 = 125 kWh, P90 = 375 kWh, interdecile range = 250 kWh, all correct
(b) Answer: P10 = 125 kWh, P90 = 375 kWh, interdecile range = 250 kWh
(c) B1 Rural households generally use more electricity than Urban households (higher P10 and P90: 125 vs 100, 375 vs 300)
(c) B1 Rural households' usage is also more variable/spread out than Urban's (larger interdecile range: 250 kWh vs 200 kWh)
(c) Answer: Rural households typically use more electricity (higher P10 and P90) and have more variable usage (larger interdecile range: 250 kWh vs 200 kWh) than Urban households.
(d) B1 for the Urban curve, the 10th and 90th percentiles happen to fall exactly on plotted (known) points, so no interpolation was needed (oe)
(d) B1 for the Rural curve, the 10th and 90th percentiles fall between plotted points, so linear interpolation (which assumes the data are evenly spread between them) had to be used, which only gives an estimate (oe)
(d) Answer: For Urban, P10 and P90 land exactly on known plotted points, so the values are exact. For Rural, they fall between plotted points, so linear interpolation is needed - this assumes an even spread of data and only gives an estimate.

Percentiles and Interpercentile Range - Worksheets, Questions and Revision