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AS Statistics · Edexcel 8MA0 · free A-level maths worksheet

Representing data

Histograms with unequal widths, box plots, cumulative frequency, outliers, comparing distributions.

12 original questions, 69 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

The lengths, l cm, of 60 leaves are summarised in the table.

Length, l (cm)Frequency
2 ≤ l < 46
4 ≤ l < 59
5 ≤ l < 615
6 ≤ l < 818
8 ≤ l < 1212
  1. Calculate the frequency density for each class.
  2. Draw a histogram to represent the data on the grid.
01234567891011120246810121416Length, l (cm)Frequency density
3 marks
Show answer
a) 3, 9, 15, 9, 3 b) bars of heights 3, 9, 15, 9, 3 over 2 to 4, 4 to 5, 5 to 6, 6 to 8 and 8 to 12, with no gaps
2

The box plot summarises the reaction times, in hundredths of a second, of a group of drivers in a simulator.

  1. Write down the median reaction time.
  2. Find the interquartile range.

An outlier is a value greater than Q3 + 1.5 × (Q3 − Q1).

  1. Show that the longest reaction time is an outlier.
  2. State one reason why a value in a data set might be an outlier.
1015202530354045Reaction time (hundredths of a second)
4 marks
Show answer
a) 22 b) 9 c) 27 + 1.5 × 9 = 40.5 and 41 > 40.5, so 41 is an outlier d) e.g. a recording error, or a driver who was distracted (a genuine but unusual value)
3

The table shows the number of hours, h, that 70 students spent watching television in one week.

Hours, hFrequency
0 ≤ h < 510
5 ≤ h < 1018
10 ≤ h < 1522
15 ≤ h < 2514
25 ≤ h < 406

A histogram is drawn to represent these data. The bar for 10 ≤ h < 15 is 2 cm wide and 8.8 cm tall.

  1. Find the width and the height of the bar for 15 ≤ h < 25
  2. Estimate the number of students who watched between 12 and 20 hours of television.
5 marks
Show answer
a) width 4 cm, height 2.8 cm b) about 20 students
4

The histogram shows the times, t minutes, taken by the runners in a 10 km race.

  1. Find the number of runners in the race.
  2. Estimate the number of runners who took less than 48 minutes.
  3. Use linear interpolation to estimate the median time.
  4. Estimate the mean time.
303540455055606570758001234567Time, t (minutes)Frequency density
7 marks
Show answer
a) 100 b) 54 c) 47.3 minutes (3 s.f.) d) 48.6 minutes (3 s.f.)
5

The masses, m kg, of 120 parcels sent from a post office are summarised below.

Mass, m (kg)Frequency
0 < m ≤ 214
2 < m ≤ 432
4 < m ≤ 640
6 < m ≤ 822
8 < m ≤ 1212
  1. On the grid, draw a cumulative frequency graph for these data, joining the points with straight lines.
  2. Use your graph to estimate the median and the interquartile range.
  3. Parcels heavier than 7 kg cost extra to send. Use your graph to estimate the number of parcels that cost extra.
0123456789101112020406080100120Mass, m (kg)Cumulative frequency
7 marks
Show answer
a) points (0, 0), (2, 14), (4, 46), (6, 86), (8, 108), (12, 120) joined with straight lines b) median about 4.7 kg, IQR about 3.4 kg c) about 23 parcels
6

Priya records the number of emails she receives each working day for 30 days. Her results are summarised below.

SmallestLower quartileMedianUpper quartileLargest
411151938

The second largest value is 26. An outlier is a value more than 1.5 × (Q3 − Q1) above Q3 or below Q1.

  1. Show that 38 is the only outlier.
  2. On the grid, draw a box plot to represent these data.
0510152025303540Number of emails
4 marks
Show answer
a) IQR = 8, limits 11 - 12 = -1 and 19 + 12 = 31; only 38 is above 31 (26 is not) and nothing is below -1 b) box from 11 to 19 with median 15, whiskers to 4 and 26, and a cross at 38
7

A café compares the delivery times, in minutes, of two delivery firms, P and Q. The box plots summarise the times for 40 deliveries by each firm.

  1. Find the interquartile range for each firm.
  2. Compare the delivery times of the two firms.
  3. The café wants a firm whose deliveries are reliably on time. State, giving a reason, which firm it should choose.
1015202530354045505560Delivery time (minutes)Firm PFirm Q
5 marks
Show answer
a) P: 7 minutes, Q: 19 minutes b) Q has a slightly lower median (26 against 28), so is a little quicker on average; Q has a much larger IQR (19 against 7) and range, so its times vary much more c) Firm P, because its times are far less spread out, so its delivery times are more predictable
8

The times, x seconds, taken by 25 students to solve a puzzle are summarised by

Σx = 1150    Σx2 = 55 250

  1. Find the mean and the standard deviation of the times.

An outlier is a value more than 2 standard deviations from the mean. The fastest time was 24 seconds and the slowest time was 63 seconds.

  1. Show that 24 is an outlier and 63 is not.
  2. The outlier is removed. Find the mean and the standard deviation of the remaining 24 times.
6 marks
Show answer
a) mean 46 s, standard deviation 9.70 s (3 s.f.) b) limits 26.6 and 65.4 (3 s.f.); 24 < 26.6 but 63 < 65.4 c) mean 46.9 s (3 s.f.), standard deviation 8.77 s (3 s.f.)
9

The speeds, s mph, of cars passing a speed camera were recorded. The table and the histogram are both incomplete.

Speed, s (mph)Frequency
20 ≤ s < 2515
25 ≤ s < 30
30 ≤ s < 3544
35 ≤ s < 4530
45 ≤ s < 609
  1. Use the histogram to complete the table.
  2. Complete the histogram.
  3. Estimate the proportion of these cars travelling at more than 40 mph.
202530354045505560012345678910Speed, s (mph)Frequency density
6 marks
Show answer
a) 36 b) a bar from 35 to 45 of height 3 c) 24/134 = 0.179 (3 s.f.)
10

The cumulative frequency graph shows the times, in seconds, taken by 80 Year 12 boys to run 400 m.

  1. Use the graph to estimate the median and the interquartile range of the times.

For the Year 12 girls the median time was 81 seconds and the interquartile range was 12 seconds.

  1. Compare the times of the boys and the girls.
  2. The coach chooses the fastest 25% of the boys for a relay squad. Estimate the slowest time that would get a boy into the squad.
606570758085909510001020304050607080Time (seconds)Cumulative frequency
6 marks
Show answer
a) median about 74 s, IQR about 9.5 s b) the boys were faster on average (median 74 s against 81 s) and their times were less spread out (IQR 9.5 s against 12 s) c) about 70 s
11

The box plot summarises the times, x minutes, taken by 31 people to assemble a flat-pack bookcase.

For these data Σx = 589 and Σx2 = 12 469

  1. Find the interquartile range.
  2. Calculate the mean and the standard deviation of the times.

Ellie defines an outlier as a value more than 2 standard deviations above the mean.

  1. State, giving a reason, whether there are any outliers in these data using Ellie’s definition.

Two more people each take 17 minutes to assemble the bookcase.

  1. State, giving a reason, the effect of including their times on (i) the median, (ii) the mean.
5101520253035404550Time, x (minutes)
8 marks
Show answer
a) 8 minutes b) mean 19 minutes, standard deviation 6.42 minutes (3 s.f.) c) yes: 19 + 2 × 6.42 = 31.8 and the longest time, 45, is above this d) (i) no change, both new times equal the median 17; (ii) the mean decreases, as both new times are below 19
12

The histogram shows the ages, a years, of the 200 members of a gym.

  1. Estimate the number of members aged between 25 and 45
  2. Use linear interpolation to estimate the median age.
  3. Two members are chosen at random. Estimate the probability that both are under 20
  4. An estimate of the mean age is 35.0 years. Explain, with reference to the histogram, why the mean is greater than the median.
1020304050607080012345678Age, a (years)Frequency density
8 marks
Show answer
a) 97 b) 31.5 years (3 s.f.) c) 0.0139 (3 s.f.) d) the classes above 40 contain a tail of older members, whose ages pull the mean up more than the median

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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