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Grade 6 · Higher · free GCSE maths worksheet

Interquartile range

Measuring spread.

14 original questions, 61 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 box plot shows the journey times, in minutes, of 15 deliveries made by a courier.

  1. Write down the median journey time.
  2. Work out the interquartile range of the journey times.
  3. Work out the range of the journey times.
0510152025303540Journey time (minutes)
3 marks
Show answer
a) 19 minutes b) 12 minutes c) 26 minutes
2

Here are the numbers of emails Priya received on each of 11 days, written in order.

3  5  6  8  9  11  13  14  16  18  21

  1. Find the median.
  2. Find the lower quartile and the upper quartile.
  3. Work out the interquartile range.
3 marks
Show answer
a) 11 b) lower quartile 6, upper quartile 16 c) 10
3

Aisha records how many minutes late her train is on 12 days.

3   8   5   12   9   6   20   11   15   23   14   18

  1. Work out the median, the lower quartile and the upper quartile.
  2. Work out the interquartile range.
  3. Write down the five values you would need to draw a box plot of these data.
4 marks
Show answer
a) median 11.5, lower quartile 7, upper quartile 16.5 b) 9.5 c) 3, 7, 11.5, 16.5, 23
4

The stem and leaf diagram shows the marks of 15 students in a spelling test.

StemLeaves
12  5  8
20  3  4  7  9
31  4  6  8
40  2  5

Key: 2 | 3 means 23 marks

Work out the five values needed to draw a box plot for this data: the lowest mark, the lower quartile, the median, the upper quartile and the highest mark.

3 marks
Show answer
lowest) 12 lower quartile) 20 median) 29 upper quartile) 38 highest) 45
5

Two shops sell second-hand bikes. The tables summarise the prices, in pounds, of the bikes sold last month.

Shop A

Minimum60
Lower quartile95
Median130
Upper quartile180
Maximum250

Shop B: median £150, interquartile range £45.

  1. Work out the interquartile range of the prices at Shop A.
  2. Compare the prices of bikes at the two shops. Give two comparisons.
  3. Which shop has the more consistent prices? Give a reason.
4 marks
Show answer
a) £85 b) Shop B has the higher median so its bikes cost more on average; Shop A has the larger IQR so its prices vary more c) Shop B - smaller interquartile range means more consistent prices
6

Two bus companies, Northline and Citygo, record how many minutes late their buses are.

MinimumLower quartileMedianUpper quartileMaximum
Northline246915
Citygo157812

Work out the interquartile range for each company. Which company is more consistent? Which has the shorter typical delay? Give a reason for each answer.

3 marks
Show answer
Northline IQR 5, Citygo IQR 3. Citygo is more consistent (smaller IQR). Northline has the shorter typical delay (median 6 < 7).
7

The table shows the heights, h cm, of 80 sunflower seedlings.

Height (h cm)Frequency
0 < h ≤ 106
10 < h ≤ 2014
20 < h ≤ 3026
30 < h ≤ 4022
40 < h ≤ 5012
  1. Complete the cumulative frequency table.
  2. On the grid, draw a cumulative frequency graph for this information.
  3. Use your graph to find an estimate for the median height.
  4. Use your graph to find an estimate for the interquartile range of the heights.
0102030405001020304050607080Height (cm)Cumulative frequency
6 marks
Show answer
a) 6, 20, 46, 68, 80 b) points at (10,6),(20,20),(30,46),(40,68),(50,80) joined by a smooth curve through the origin c) approximately 28 cm (accept 26-29) d) approximately 16 cm (accept 14-19)
8

The table shows the times, in minutes, taken by 50 students to complete a puzzle.

Time t (minutes)0 < t ≤ 1010 < t ≤ 2020 < t ≤ 3030 < t ≤ 4040 < t ≤ 50
Frequency41119124
  1. Complete a cumulative frequency table for these data.
  2. Use the cumulative frequency to estimate the interquartile range. Use the 12.5th and 37.5th values and assume the values are spread evenly within each class. Give your answer to 1 decimal place.
4 marks
Show answer
a) 4, 15, 34, 46, 50 b) about 15.2 minutes
9

The cumulative frequency graph shows the test scores of the 60 students in Class A.

  1. Use the graph to find an estimate for the median score.
  2. Use the graph to find an estimate for the interquartile range of the scores.
  3. The 60 students in Class B had a median score of 45 and an interquartile range of 12. Compare the scores of the two classes.
01020304050600102030405060ScoreCumulative frequency
6 marks
Show answer
a) approximately 32 (accept 31-34) b) approximately 18 (accept 15-21) c) Class B has the higher median (45 vs about 32) so did better on average; Class B has the smaller IQR (12 vs about 18) so its scores are less spread out
10

Here are the times, in seconds, taken by the 15 swimmers in Team A to swim 50 metres.

41  38  45  36  50  39  43  47  35  44  40  52  37  46  42

  1. Find the median, the lower quartile and the upper quartile of the times for Team A.
  2. Team B's times had a median of 44 seconds and an interquartile range of 13 seconds. Compare the times of the two teams. Give two comparisons.
5 marks
Show answer
a) median 42, lower quartile 38, upper quartile 46 b) Team A has the lower median so was faster on average; Team A has the smaller IQR so was more consistent
11

The ages of the members of a youth orchestra are summarised below.

YoungestLower quartileMedianUpper quartileOldest
1113151824
  1. On the grid, draw a box plot for these ages.
  2. Work out the interquartile range.
8101214161820222426Age (years)
3 marks
Show answer
a) Box plot with whiskers at 11 and 24, box from 13 to 18, median line at 15 b) 5
12

80 people took part in a fun run. The table shows the cumulative frequency of their finishing times.

Time, t (minutes)≤ 10≤ 20≤ 30≤ 40≤ 50≤ 60
Cumulative frequency62046667680
  1. On the grid, draw a cumulative frequency graph for these data.
  2. Use your graph to estimate the interquartile range of the times.
010203040506001020304050607080Time (minutes)Cumulative frequency
5 marks
Show answer
a) Smooth curve through (0, 0), (10, 6), (20, 20), (30, 46), (40, 66), (50, 76), (60, 80) b) about 17 minutes (accept 16 to 18)
13

The times, in minutes, taken by 9 runners to finish a race are

21   23   24   25   26   27   28   30   31

  1. Work out the range and the interquartile range of these times.
  2. A tenth runner finishes in 58 minutes. Work out the new range and the new interquartile range.
  3. Which of the range and the interquartile range is affected less by the 58 minute time? Explain why.
5 marks
Show answer
a) range 10, IQR 5.5 b) range 37, IQR 6 c) The IQR changes much less, because it uses only the middle half of the data and ignores extreme values
14

80 students sat an exam marked out of 100. The table shows their scores, s marks.

Score (s marks)Frequency
0 < s ≤ 206
20 < s ≤ 4018
40 < s ≤ 6028
60 < s ≤ 8020
80 < s ≤ 1008
  1. Complete the cumulative frequency table.
  2. Write down the class interval that contains the median score.
  3. A cumulative frequency graph of the data gives these estimates: median 51, lower quartile 37, upper quartile 66. Work out an estimate for the interquartile range.
  4. The pass mark was 60. Work out the percentage of students who scored more than 60.
  5. Last year the same exam had a median of 46 and an interquartile range of 24. Compare this year's scores with last year's.
7 marks
Show answer
a) 6, 24, 52, 72, 80 b) 40 < s <= 60 c) 29 d) 35% e) Higher median this year (51 vs 46) so better on average; larger IQR this year (29 vs 24) so more spread out

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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