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Exam questions · Maths · Statistics

Cumulative Frequency and Box Plots

  • 6 exam questions
  • 19 marks
  • 9 quick checks
  1. 1 Complete [2 marks]

    The table shows the frequencies of the heights, \(h\) cm, of 80 plants. The classes \(0 < h \leq 10\), \(10 < h \leq 20\), \(20 < h \leq 30\), \(30 < h \leq 40\), \(40 < h \leq 50\) and \(50 < h \leq 60\) have the frequencies 5, 15, 20, 20, 15 and 5. Write down the cumulative frequencies.

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    Model answer

    The running totals are 5, 20, 40, 60, 75 and 80.

    Mark scheme

    • At least four cumulative frequencies correct — M1
    • 5, 20, 40, 60, 75, 80 — A1
  2. 2 Work out [5 marks]

    The cumulative frequency graph shows the heights of 80 plants. (a) Use the graph to find an estimate for the median height. (1 mark) (b) Use the graph to find an estimate for the interquartile range. (2 marks) (c) Use the graph to find an estimate for the number of plants taller than 50 cm. (2 marks)

    A cumulative frequency graph of the heights of 80 plants, rising from 0 at 0 cm to 80 at 60 cm.
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    Model answer

    (a) The median is at cumulative frequency 40, which is 30 cm. (b) The lower quartile is at 20, which is 20 cm, and the upper quartile is at 60, which is 40 cm. The interquartile range is \(40 - 20 = 20\) cm. (c) At 50 cm the cumulative frequency is 75, so \(80 - 75 = 5\) plants are taller.

    Mark scheme

    • (a) 30 (accept 29 to 31) — B1
    • (b) Reads the lower and upper quartiles at 20 and 60 on the vertical axis — M1
    • (b) 20 (accept 18 to 22) — A1
    • (c) Reads 75 at 50 cm — M1
    • (c) 5 — A1
  3. 3 Compare [4 marks]

    The five-number summary for the heights of some plants is minimum 8 cm, lower quartile 20 cm, median 30 cm, upper quartile 40 cm and maximum 57 cm. (a) Work out the range and the interquartile range. (2 marks) A second group of plants has a median of 35 cm and an interquartile range of 12 cm. (b) Compare the heights of the two groups of plants. (2 marks)

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    Model answer

    (a) Range \(57 - 8 = 49\) cm and interquartile range \(40 - 20 = 20\) cm. (b) The second group has a higher median, 35 cm compared with 30 cm, so the plants are typically taller. Its interquartile range is smaller, 12 cm compared with 20 cm, so the heights are more consistent.

    Mark scheme

    • (a) 49 — B1
    • (a) 20 — B1
    • (b) A comparison of the medians, in context — C1
    • (b) A comparison of the interquartile ranges, in context — C1
  4. 4 Compare [2 marks]

    On Monday the median time that a bus was late was 42 seconds and the interquartile range was 14 seconds. On Tuesday the median was 38 seconds and the interquartile range was 22 seconds. Compare the lateness of the bus on the two days.

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    Model answer

    On Monday the bus was typically later, with a median of 42 seconds compared with 38 seconds. On Tuesday the lateness was less consistent, with a bigger interquartile range of 22 seconds compared with 14 seconds.

    Mark scheme

    • A comparison of the medians, in context — C1
    • A comparison of the interquartile ranges, in context — C1
  5. 5 Work out [3 marks]

    The cumulative frequencies of the masses, \(m\) kg, of 60 parcels are 4 for \(m \leq 10\), 14 for \(m \leq 20\), 34 for \(m \leq 30\), 52 for \(m \leq 40\) and 60 for \(m \leq 50\). (a) How many parcels have a mass in the class \(20 < m \leq 30\)? (1 mark) (b) How many parcels have a mass over 40 kg? (1 mark) (c) Which class contains the median? (1 mark)

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    Model answer

    (a) \(34 - 14 = 20\). (b) \(60 - 52 = 8\). (c) The median is the 30th value, which is in the class \(20 < m \leq 30\), because the cumulative frequency goes from 14 to 34.

    Mark scheme

    • (a) 20 — B1
    • (b) 8 — B1
    • (c) \(20 < m \leq 30\) — B1
  6. 6 Work out [3 marks]

    A cumulative frequency graph is drawn for 120 values. (a) At which two cumulative frequencies should you read the lower quartile and the upper quartile? (2 marks) (b) The lower quartile is 18 and the upper quartile is 33. Work out the interquartile range. (1 mark)

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    Model answer

    (a) The lower quartile is at \(\dfrac{120}{4} = 30\) and the upper quartile is at \(\dfrac{3 \times 120}{4} = 90\). (b) \(33 - 18 = 15\).

    Mark scheme

    • (a) 30 — B1
    • (a) 90 — B1
    • (b) 15 — B1

Quick check

  1. 1

    For \(n\) values, where is the median on a cumulative frequency graph?

    1. AAt cumulative frequency \(\dfrac{n}{2}\)
    2. BAt cumulative frequency \(\dfrac{n}{4}\)
    3. CAt the highest point
    4. DAt cumulative frequency \(n\)
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    A: At cumulative frequency \(\dfrac{n}{2}\)

    The median is the middle value.

  2. 2

    For 100 values, at what cumulative frequency is the lower quartile?

    1. A\(50\)
    2. B\(75\)
    3. C\(10\)
    4. D\(25\)
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    D: \(25\)

    \(\dfrac{100}{4} = 25\).

  3. 3

    The upper quartile is 70 and the lower quartile is 40. What is the interquartile range?

    1. A\(110\)
    2. B\(55\)
    3. C\(30\)
    4. D\(70\)
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    C: \(30\)

    \(70 - 40 = 30\).

  4. 4

    Frequencies 10, 15, 25, 20, 30 are added up as you go. What are the cumulative frequencies?

    1. A\(10, 15, 25, 20, 30\)
    2. B\(10, 25, 50, 70, 100\)
    3. C\(10, 25, 25, 45, 75\)
    4. D\(100, 90, 75, 50, 30\)
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    B: \(10, 25, 50, 70, 100\)

    Each is the running total.

  5. 5

    Where should the points on a cumulative frequency graph be plotted?

    1. AAt the upper boundary of each class
    2. BAt the lower boundary of each class
    3. CAt the middle of each class
    4. DAt zero
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    A: At the upper boundary of each class

    The cumulative frequency is up to the end of the class.

  6. 6

    A cumulative frequency graph of 100 students has a cumulative frequency of 90 at a mark of 80. How many scored more than 80?

    1. A\(90\)
    2. B\(80\)
    3. C\(20\)
    4. D\(10\)
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    D: \(10\)

    \(100 - 90 = 10\).

  7. 7

    Class A has an interquartile range of 30 and Class B has 15. Which is more consistent?

    1. AClass A, because its IQR is larger
    2. BThey are equally consistent
    3. CClass B, because its IQR is smaller
    4. DIt cannot be said
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    C: Class B, because its IQR is smaller

    A smaller interquartile range means more consistent.

  8. 8

    For 80 values, at what cumulative frequency is the upper quartile?

    1. A\(20\)
    2. B\(60\)
    3. C\(40\)
    4. D\(80\)
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    B: \(60\)

    \(\dfrac{3}{4} \times 80 = 60\).

  9. 9

    A box plot for Class B has a median of 55, and Class A has a median of 50. What can you say?

    1. AClass B has the higher typical mark
    2. BClass A has the higher typical mark
    3. CBoth have the same typical mark
    4. DClass B has the larger range
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    A: Class B has the higher typical mark

    A higher median means a higher typical value.