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Exam questions · Maths · Statistics

Averages and Range

  • 6 exam questions
  • 21 marks
  • 9 quick checks
  1. 1 Work out [3 marks]

    Here are seven numbers: 6, 2, 9, 4, 9, 7, 12. Work out (a) the mode, (b) the median, (c) the range.

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    Model answer

    (a) 9 appears twice, so the mode is 9. (b) In order: 2, 4, 6, 7, 9, 9, 12, so the median is 7. (c) \(12 - 2 = 10\).

    Mark scheme

    • (a) 9 — B1
    • (b) 7 — B1
    • (c) 10 — B1
  2. 2 Work out [6 marks]

    The table shows the number of goals scored by a team in each of 20 matches. (a) Write down the mode. (1 mark) (b) Work out the median. (2 marks) (c) Work out the mean. (3 marks)

    A frequency table showing the goals scored in 20 matches: 0 goals 3 times, 1 goal 7 times, 2 goals 5 times, 3 goals 3 times and 4 goals 2 times.
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    Model answer

    (a) The highest frequency is 7, for 1 goal, so the mode is 1. (b) The 10th value is 1 and the 11th value is 2, so the median is \(\dfrac{1 + 2}{2} = 1.5\). (c) \(0 \times 3 + 1 \times 7 + 2 \times 5 + 3 \times 3 + 4 \times 2 = 0 + 7 + 10 + 9 + 8 = 34\). The mean is \(\dfrac{34}{20} = 1.7\).

    Mark scheme

    • (a) 1 — B1
    • (b) Uses the 10th and 11th values, or cumulative frequencies 3, 10, 15 — M1
    • (b) 1.5 — A1
    • (c) Products \(f \times x\) with at least 4 correct — M1
    • (c) \(\dfrac{34}{20}\) or total 34 — M1
    • (c) 1.7 — A1
  3. 3 Compare [2 marks]

    Class A has a mean mark of 56 and a range of 20. Class B has a mean mark of 52 and a range of 12. Compare the marks of the two classes.

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    Model answer

    On average Class A scored higher, with a mean of 56 compared with 52. Class B's marks were more consistent, because its range of 12 is smaller than 20.

    Mark scheme

    • A comparison of the means, in context — C1
    • A comparison of the ranges, in context — C1
  4. 4 Work out [3 marks]

    The mean of 4 numbers is 11. A fifth number is added and the mean of the 5 numbers is 12. Work out the fifth number.

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    Model answer

    The total of the four numbers is \(4 \times 11 = 44\). The total of the five numbers is \(5 \times 12 = 60\). The fifth number is \(60 - 44 = 16\).

    Mark scheme

    • \(4 \times 11 = 44\) or \(5 \times 12 = 60\) — M1
    • \(60 - 44\) — M1
    • 16 — A1
  5. 5 Work out [4 marks]

    The table shows the masses of 30 parcels. For the classes \(0 < w \leq 10\), \(10 < w \leq 20\), \(20 < w \leq 30\) and \(30 < w \leq 40\) the frequencies are 3, 9, 12 and 6. (a) Write down the modal class. (1 mark) (b) Work out an estimate for the mean mass. (3 marks)

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    Model answer

    (a) The highest frequency is 12, so the modal class is \(20 < w \leq 30\). (b) The mid-points are 5, 15, 25 and 35. \(5 \times 3 + 15 \times 9 + 25 \times 12 + 35 \times 6 = 15 + 135 + 300 + 210 = 660\). The estimate of the mean is \(\dfrac{660}{30} = 22\).

    Mark scheme

    • (a) \(20 < w \leq 30\) — B1
    • (b) Mid-points 5, 15, 25, 35 used — M1
    • (b) \(\dfrac{660}{30}\) or total 660 — M1
    • (b) 22 — A1
  6. 6 Work out [3 marks]

    The mean mass of 5 boys is 60 kg. The mean mass of 3 girls is 52 kg. Work out the mean mass of all 8 children.

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    Model answer

    The total mass of the boys is \(5 \times 60 = 300\) kg and of the girls is \(3 \times 52 = 156\) kg. The total is 456 kg, so the mean is \(\dfrac{456}{8} = 57\) kg.

    Mark scheme

    • \(5 \times 60 = 300\) or \(3 \times 52 = 156\) — M1
    • \(\dfrac{300 + 156}{8}\) — M1
    • 57 kg — A1

Quick check

  1. 1

    What is the median of 3, 5, 6, 8, 9, 12?

    1. A\(6\)
    2. B\(7\)
    3. C\(8\)
    4. D\(7.5\)
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    B: \(7\)

    With 6 values, take the mean of the 3rd and 4th: \(\dfrac{6 + 8}{2} = 7\).

  2. 2

    What is the range of 3, 8, 11, 20?

    1. A\(17\)
    2. B\(10.5\)
    3. C\(8\)
    4. D\(22\)
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    A: \(17\)

    \(20 - 3 = 17\).

  3. 3

    What is the mean of 4, 6, 8, 10, 12?

    1. A\(6\)
    2. B\(10\)
    3. C\(40\)
    4. D\(8\)
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    D: \(8\)

    \(\dfrac{40}{5} = 8\).

  4. 4

    Which average is least affected by an outlier?

    1. AThe mean
    2. BThe range
    3. CThe median
    4. DNone of them
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    C: The median

    The median depends only on the middle value.

  5. 5

    The numbers 0, 1, 2, 3, 4 have frequencies 4, 4, 6, 4, 2. What is the mean?

    1. A\(2\)
    2. B\(1.8\)
    3. C\(3.6\)
    4. D\(36\)
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    B: \(1.8\)

    \(\dfrac{0 + 4 + 12 + 12 + 8}{20} = \dfrac{36}{20} = 1.8\).

  6. 6

    What is the mode of the numbers 0, 1, 2, 3, 4 with frequencies 4, 4, 6, 4, 2?

    1. A\(2\)
    2. B\(6\)
    3. C\(1.8\)
    4. D\(4\)
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    A: \(2\)

    The highest frequency, 6, is for the value 2.

  7. 7

    What is the mid-point of the class \(20 < w \leq 40\)?

    1. A\(20\)
    2. B\(40\)
    3. C\(10\)
    4. D\(30\)
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    D: \(30\)

    \(\dfrac{20 + 40}{2} = 30\).

  8. 8

    Classes \(0 < w \leq 20\), \(20 < w \leq 40\) and \(40 < w \leq 60\) have frequencies 5, 10 and 5. What is the estimated mean?

    1. A\(20\)
    2. B\(40\)
    3. C\(30\)
    4. D\(10\)
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    C: \(30\)

    Using mid-points: \(\dfrac{10 \times 5 + 30 \times 10 + 50 \times 5}{20} = \dfrac{600}{20} = 30\).

  9. 9

    The mean of five numbers is 8. Four of them are 5, 7, 9 and 10. What is the fifth?

    1. A\(8\)
    2. B\(9\)
    3. C\(7.75\)
    4. D\(31\)
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    B: \(9\)

    The total is \(5 \times 8 = 40\), and \(40 - 31 = 9\).