OpenRevise

Exam questions · Maths

Statistics

  • 30 exam questions
  • 94 marks
  • 45 quick checks

Averages and Range

Just this lesson
  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\(7.5\)
    Show answerHide answer

    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\(22\)
    Show answerHide answer

    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\(8\)
    Show answerHide answer

    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
    Show answerHide answer

    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\(36\)
    Show answerHide answer

    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\(4\)
    Show answerHide answer

    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\(30\)
    Show answerHide answer

    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\(10\)
    Show answerHide answer

    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\(31\)
    Show answerHide answer

    B: \(9\)

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

Pie Charts, Bar Charts and Stem-and-Leaf Diagrams

Just this lesson
  1. 1 Work out [3 marks]

    Yasmin asks 30 people which fruit they like best. 12 choose apple, 9 choose banana, 6 choose pear and 3 choose grape. She draws a pie chart. Work out the angle for each sector.

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    Model answer

    Each person is \(\dfrac{360}{30} = 12^\circ\). Apple: \(12 \times 12 = 144^\circ\). Banana: \(9 \times 12 = 108^\circ\). Pear: \(6 \times 12 = 72^\circ\). Grape: \(3 \times 12 = 36^\circ\).

    Mark scheme

    • \(360 \div 30 = 12\) — M1
    • At least two angles correct — A1
    • \(144^\circ, 108^\circ, 72^\circ, 36^\circ\) — A1
  2. 2 Work out [4 marks]

    The pie chart shows how 36 students travel to school. The angle for cycle is not shown. (a) Work out how many students walk. (2 marks) (b) Work out the angle for cycle. (1 mark) (c) Work out how many students cycle. (1 mark)

    A pie chart of how 36 students travel to school with angles of 140 degrees for bus, 100 degrees for walk, 60 degrees for car, and cycle unlabelled.
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    Model answer

    (a) Each student is \(\dfrac{360}{36} = 10^\circ\), so the number who walk is \(\dfrac{100}{10} = 10\). (b) \(360 - 140 - 100 - 60 = 60^\circ\). (c) \(\dfrac{60}{10} = 6\).

    Mark scheme

    • (a) \(\dfrac{100}{360} \times 36\) or \(100 \div 10\) — M1
    • (a) 10 — A1
    • (b) \(60^\circ\) — B1
    • (c) 6 — B1 (follow through from (b))
  3. 3 Work out [4 marks]

    The stem-and-leaf diagram shows the times, in minutes, that 14 runners took to finish a race. (a) Work out the median time. (2 marks) (b) Work out the range of the times. (2 marks)

    A stem-and-leaf diagram of the times of 14 runners with a key where 1 bar 2 means 12 minutes.
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    Model answer

    (a) There are 14 values, so the median is halfway between the 7th and 8th values, which are 16 and 18, giving 17 minutes. (b) \(36 - 5 = 31\) minutes.

    Mark scheme

    • (a) 7th and 8th values 16 and 18 identified — M1
    • (a) 17 — A1
    • (b) \(36 - 5\) — M1
    • (b) 31 — A1
  4. 4 Explain [2 marks]

    A newspaper shows a bar chart of the sales of a phone. The vertical axis starts at 98 instead of 0, and the headline says “Sales soar”. Explain why the chart may be misleading.

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    Model answer

    The vertical axis does not start at zero, so a very small difference in sales looks like a large change.

    Mark scheme

    • States that the scale does not start at zero — B1
    • States that a small difference looks large or exaggerated — B1
  5. 5 Work out [3 marks]

    A stem-and-leaf diagram shows the ages, in years, of some members of a club. The stem 1 has leaves 2, 5 and 8. The stem 2 has leaves 1, 4, 4 and 9. The stem 3 has leaves 0 and 3. The key says that 2 bar 1 means 21 years. For these ages, find (a) the number of members, (b) the median age, (c) the range.

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    Model answer

    (a) \(3 + 4 + 2 = 9\) members. (b) In order the ages are 12, 15, 18, 21, 24, 24, 29, 30, 33, so the 5th value is 24. (c) \(33 - 12 = 21\).

    Mark scheme

    • (a) 9 — B1
    • (b) 24 — B1
    • (c) 21 — B1
  6. 6 Work out [3 marks]

    Pie chart A shows the travel to school of 40 students. The angle for cycle is \(90^\circ\). Pie chart B shows the travel to school of 120 students. The angle for cycle is \(60^\circ\). Kim says, “The cycle sector is bigger on chart A, so more students cycle in school A.” Is Kim correct? You must show your working.

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    Model answer

    In school A, \(\dfrac{90}{360} \times 40 = 10\) students cycle. In school B, \(\dfrac{60}{360} \times 120 = 20\) students cycle. Kim is not correct. A bigger fraction cycle in school A, but more students cycle in school B, because there are more students.

    Mark scheme

    • \(\dfrac{90}{360} \times 40 = 10\) — M1
    • \(\dfrac{60}{360} \times 120 = 20\) — M1
    • Kim is not correct, with a reason using the totals — C1

Quick check

  1. 1

    60 students choose a sport. 24 choose football. What is the angle for football on a pie chart?

    1. A\(24^\circ\)
    2. B\(120^\circ\)
    3. C\(144^\circ\)
    4. D\(60^\circ\)
    Show answerHide answer

    C: \(144^\circ\)

    Each student is \(\dfrac{360}{60} = 6^\circ\), so \(24 \times 6 = 144^\circ\).

  2. 2

    A pie chart shows 40 people. How many degrees represent each person?

    1. A\(40^\circ\)
    2. B\(9^\circ\)
    3. C\(4^\circ\)
    4. D\(90^\circ\)
    Show answerHide answer

    B: \(9^\circ\)

    \(\dfrac{360}{40} = 9\).

  3. 3

    A pie chart shows 36 people. A sector is \(100^\circ\). How many people does it represent?

    1. A\(10\)
    2. B\(100\)
    3. C\(36\)
    4. D\(3.6\)
    Show answerHide answer

    A: \(10\)

    \(\dfrac{100}{360} \times 36 = 10\).

  4. 4

    What fraction of a pie chart is a \(90^\circ\) sector?

    1. A\(\dfrac{1}{3}\)
    2. B\(\dfrac{1}{2}\)
    3. C\(\dfrac{9}{10}\)
    4. D\(\dfrac{1}{4}\)
    Show answerHide answer

    D: \(\dfrac{1}{4}\)

    \(\dfrac{90}{360} = \dfrac{1}{4}\).

  5. 5

    What must every stem-and-leaf diagram have?

    1. AA title only
    2. BGaps between the rows
    3. CA key
    4. DLeaves in a random order
    Show answerHide answer

    C: A key

    Without a key the values cannot be read.

  6. 6

    A stem-and-leaf diagram has 14 ordered values. The 7th is 26 and the 8th is 29. What is the median?

    1. A\(26\)
    2. B\(27.5\)
    3. C\(29\)
    4. D\(28\)
    Show answerHide answer

    B: \(27.5\)

    \(\dfrac{26 + 29}{2} = 27.5\).

  7. 7

    The values in a stem-and-leaf diagram run from 12 to 45. What is the range?

    1. A\(33\)
    2. B\(57\)
    3. C\(28.5\)
    4. D\(45\)
    Show answerHide answer

    A: \(33\)

    \(45 - 12 = 33\).

  8. 8

    Which of these makes a bar chart misleading?

    1. AThe bars have labels
    2. BThe chart has a title
    3. CThe bars are the same width
    4. DThe vertical scale does not start at zero
    Show answerHide answer

    D: The vertical scale does not start at zero

    A scale that does not start at zero exaggerates differences.

  9. 9

    On a pie chart for 60 people, a sector is \(54^\circ\). What percentage is that?

    1. A\(54\%\)
    2. B\(9\%\)
    3. C\(15\%\)
    4. D\(30\%\)
    Show answerHide answer

    C: \(15\%\)

    \(\dfrac{54}{360} = 0.15\).

Scatter Graphs

Just this lesson
  1. 1 Write down [2 marks]

    Write down the type of correlation between (a) the number of hours of sunshine and the sales of sun cream, (b) the age of a car and its value.

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    Model answer

    (a) Positive correlation. (b) Negative correlation.

    Mark scheme

    • (a) Positive — B1
    • (b) Negative — B1
  2. 2 Estimate [5 marks]

    The scatter graph shows the hours of sunshine in a week and the number of plants sold by a garden centre in that week. A line of best fit is drawn. (a) Describe the correlation. (1 mark) (b) In one week there were 20 hours of sunshine. Use the line of best fit to estimate the number of plants sold. (2 marks) (c) Gavin uses the line to estimate the number of plants sold in a week with 60 hours of sunshine. Comment on the reliability of his estimate. (2 marks)

    A scatter graph of hours of sunshine and plants sold with positive correlation and a line of best fit through 10 at 0 hours and 50 at 20 hours.
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    Model answer

    (a) Positive correlation: as the hours of sunshine increase, the number of plants sold increases. (b) Going up from 20 hours to the line and across gives 50 plants. (c) The estimate is not reliable. 60 hours is outside the range of the data, so this is extrapolation and the pattern may not continue.

    Mark scheme

    • (a) Positive, in context — B1
    • (b) A line drawn up from 20 to the line of best fit and across — M1
    • (b) 50 (accept 48 to 52) — A1
    • (c) States that the estimate is not reliable — B1
    • (c) Explains that 60 is outside the range of the data — B1
  3. 3 Work out [3 marks]

    A line of best fit for the revision time, \(x\) hours, and the test score, \(y\), has the equation \(y = 3x + 5\). (a) Use the equation to estimate the score for 8 hours of revision. (1 mark) (b) What does the number 3 in the equation represent? (1 mark) (c) What does the number 5 in the equation represent? (1 mark)

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    Model answer

    (a) \(3 \times 8 + 5 = 29\). (b) The score goes up by about 3 marks for each extra hour of revision. (c) The estimated score with no revision.

    Mark scheme

    • (a) 29 — B1
    • (b) The score increases by 3 for each extra hour — B1
    • (c) The estimated score with 0 hours of revision — B1
  4. 4 Explain [2 marks]

    There is a positive correlation between ice cream sales and the number of people with sunburn. Hay says, “Eating ice cream causes sunburn.” Comment on what Hay says.

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    Model answer

    Hay is not necessarily correct. Correlation does not mean causation. Hot, sunny weather causes both ice cream sales and sunburn to increase.

    Mark scheme

    • States that correlation does not prove causation — B1
    • Gives a third factor such as hot weather — B1
  5. 5 Write down [2 marks]

    These are points on a scatter graph: \((1, 3)\), \((2, 4)\), \((3, 5)\), \((4, 6)\) and \((5, 15)\). (a) Write down the outlier. (1 mark) (b) Say how this should affect the line of best fit. (1 mark)

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    Model answer

    (a) \((5, 15)\), as it is far from the pattern of the other points. (b) The outlier should be ignored when the line of best fit is drawn.

    Mark scheme

    • (a) \((5, 15)\) — B1
    • (b) It should be ignored — B1
  6. 6 Work out [4 marks]

    A line of best fit goes through the points \((10, 30)\) and \((30, 70)\). (a) Work out the equation of the line. (3 marks) (b) Use your equation to estimate \(y\) when \(x = 25\). (1 mark)

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    Model answer

    (a) The gradient is \(\dfrac{70 - 30}{30 - 10} = 2\). Then \(30 = 2 \times 10 + c\), so \(c = 10\) and \(y = 2x + 10\). (b) \(2 \times 25 + 10 = 60\).

    Mark scheme

    • (a) Gradient \(\dfrac{40}{20}\) — M1
    • (a) Substitutes a point to find \(c\) — M1
    • (a) \(y = 2x + 10\) — A1
    • (b) 60 — B1 (follow through from (a))

Quick check

  1. 1

    What type of correlation has points sloping downwards from left to right?

    1. APositive
    2. BNone
    3. CPerfect
    4. DNegative
    Show answerHide answer

    D: Negative

    One variable falls as the other rises.

  2. 2

    Which statement describes positive correlation?

    1. AAs one variable increases, the other decreases
    2. BThe points are scattered at random
    3. CAs one variable increases, the other increases
    4. DThe points form a curve
    Show answerHide answer

    C: As one variable increases, the other increases

    Positive correlation slopes upwards.

  3. 3

    What is an outlier?

    1. AThe mean of the points
    2. BA point that does not fit the pattern of the others
    3. CA point on the line of best fit
    4. DThe first point plotted
    Show answerHide answer

    B: A point that does not fit the pattern of the others

    Outliers are far from the trend.

  4. 4

    A line of best fit has equation \(y = 4x + 8\). Estimate \(y\) when \(x = 12\).

    1. A\(56\)
    2. B\(20\)
    3. C\(60\)
    4. D\(48\)
    Show answerHide answer

    A: \(56\)

    \(4 \times 12 + 8 = 56\).

  5. 5

    Data runs from 2 to 16 hours of revision. What do we call an estimate for 25 hours?

    1. AInterpolation
    2. BCorrelation
    3. CCausation
    4. DExtrapolation
    Show answerHide answer

    D: Extrapolation

    Outside the range of the data is extrapolation.

  6. 6

    Which estimate is likely to be more reliable?

    1. AOne far outside the range of the data
    2. BOne from a graph with no correlation
    3. COne inside the range of the data
    4. DOne from only two points
    Show answerHide answer

    C: One inside the range of the data

    Interpolation is more reliable than extrapolation.

  7. 7

    Ice cream sales and sunburn are positively correlated. What is the best explanation?

    1. AIce cream causes sunburn
    2. BHot weather increases both
    3. CSunburn causes ice cream sales
    4. DThey are completely unrelated
    Show answerHide answer

    B: Hot weather increases both

    A third factor can cause both.

  8. 8

    The line of best fit is \(y = 4x + 8\), where \(x\) is hours of revision and \(y\) is the test score. What does the gradient mean?

    1. AEach extra hour of revision adds about 4 marks
    2. BThe score with no revision is 4
    3. CThe maximum score is 8
    4. DEach mark takes 4 hours
    Show answerHide answer

    A: Each extra hour of revision adds about 4 marks

    The gradient is the change in score for each extra hour.

  9. 9

    In \(y = 4x + 8\), what does 8 represent?

    1. AThe number of hours needed
    2. BThe gradient
    3. CThe maximum possible score
    4. DThe estimated score with no revision
    Show answerHide answer

    D: The estimated score with no revision

    It is the \(y\)-intercept, the value when \(x = 0\).

Cumulative Frequency and Box Plots

Just this lesson
  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.

    Show answerHide answer

    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.
    Show answerHide answer

    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)

    Show answerHide answer

    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)

    Show answerHide answer

    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 \(n\)
    Show answerHide answer

    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\(25\)
    Show answerHide answer

    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\(70\)
    Show answerHide answer

    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\(100, 90, 75, 50, 30\)
    Show answerHide answer

    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
    Show answerHide answer

    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\(10\)
    Show answerHide answer

    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
    Show answerHide answer

    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\(80\)
    Show answerHide answer

    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
    Show answerHide answer

    A: Class B has the higher typical mark

    A higher median means a higher typical value.

Histograms and Sampling

Just this lesson
  1. 1 Work out [3 marks]

    The table shows the times of some people. For the class \(0 < x \leq 4\) the frequency is 8, for \(4 < x \leq 10\) it is 30, and for \(10 < x \leq 20\) it is 40. Work out the frequency density for each class.

    Show answerHide answer

    Model answer

    The widths are 4, 6 and 10. The frequency densities are \(\dfrac{8}{4} = 2\), \(\dfrac{30}{6} = 5\) and \(\dfrac{40}{10} = 4\).

    Mark scheme

    • Class widths 4, 6 and 10 used — M1
    • At least two frequency densities correct — A1
    • 2, 5 and 4 — A1
  2. 2 Work out [4 marks]

    The histogram shows the times, in minutes, that some people took to complete a puzzle. (a) Work out the number of people who took more than 20 minutes and up to 40 minutes. (2 marks) (b) Work out the number of people who took more than 40 minutes. (2 marks)

    A histogram of times with four bars of frequency densities 2, 3.5, 2.5 and 1 over the classes 0 to 10, 10 to 20, 20 to 40 and 40 to 60.
    Show answerHide answer

    Model answer

    (a) The frequency is the area of the bar: \(2.5 \times 20 = 50\). (b) The bar from 40 to 60 has a frequency density of 1 and a width of 20, so \(1 \times 20 = 20\).

    Mark scheme

    • (a) \(2.5 \times 20\) — M1
    • (a) 50 — A1
    • (b) \(1 \times 20\) — M1
    • (b) 20 — A1
  3. 3 Work out [2 marks]

    On a histogram the bar for the class \(10 < x \leq 30\) has a frequency density of 3. Work out an estimate for the number of values in the class \(10 < x \leq 25\).

    Show answerHide answer

    Model answer

    The width of \(10 < x \leq 25\) is 15, so the estimate is \(3 \times 15 = 45\).

    Mark scheme

    • \(3 \times 15\) — M1
    • 45 — A1
  4. 4 Work out [3 marks]

    A school has 200 pupils in Year 9, 240 pupils in Year 10 and 160 pupils in Year 11. Hira takes a stratified sample of 60 pupils. Work out the number of pupils from each year group in her sample.

    Show answerHide answer

    Model answer

    The school has \(200 + 240 + 160 = 600\) pupils. Year 9: \(\dfrac{200}{600} \times 60 = 20\). Year 10: \(\dfrac{240}{600} \times 60 = 24\). Year 11: \(\dfrac{160}{600} \times 60 = 16\).

    Mark scheme

    • \(\dfrac{200}{600} \times 60\) or the total 600 seen — M1
    • At least two numbers correct — A1
    • 20, 24 and 16 — A1
  5. 5 Work out [3 marks]

    A scientist catches 50 fish in a lake, marks them and puts them back. Later she catches 80 fish, and 10 of them are marked. (a) Work out an estimate for the number of fish in the lake. (2 marks) (b) State one assumption you have made. (1 mark)

    Show answerHide answer

    Model answer

    (a) \(\dfrac{10}{80} = \dfrac{50}{N}\), so \(N = \dfrac{50 \times 80}{10} = 400\). (b) The marked fish have mixed evenly with the others, and the number of fish in the lake has not changed.

    Mark scheme

    • (a) \(\dfrac{50 \times 80}{10}\) — M1
    • (a) 400 — A1
    • (b) The marked fish have mixed evenly, or the population has not changed — B1
  6. 6 Explain [2 marks]

    Ravi wants to find the favourite sport of the students in his school. He asks the first 20 students who leave the sports hall. (a) Give a reason why his sample may be biased. (1 mark) (b) Describe a better way to choose a sample. (1 mark)

    Show answerHide answer

    Model answer

    (a) The students leaving the sports hall are more likely to like sport. (b) Number all the students and use random numbers to choose the sample.

    Mark scheme

    • (a) A reason relating to the students being more likely to like sport — B1
    • (b) A random sample from all the students — B1

Quick check

  1. 1

    What is the formula for frequency density?

    1. AFrequency times class width
    2. BFrequency divided by class width
    3. CClass width divided by frequency
    4. DTotal frequency divided by the number of classes
    Show answerHide answer

    B: Frequency divided by class width

    Frequency density \(= \dfrac{\text{frequency}}{\text{class width}}\).

  2. 2

    What does the area of a bar in a histogram show?

    1. AThe frequency
    2. BThe frequency density
    3. CThe class width
    4. DThe mean
    Show answerHide answer

    A: The frequency

    Area \(=\) frequency density \(\times\) class width \(=\) frequency.

  3. 3

    A histogram bar has class width 20 and frequency density 3. What is the frequency?

    1. A\(23\)
    2. B\(6.7\)
    3. C\(17\)
    4. D\(60\)
    Show answerHide answer

    D: \(60\)

    \(3 \times 20 = 60\).

  4. 4

    A class of width 10 has frequency 30. What is its frequency density?

    1. A\(300\)
    2. B\(0.3\)
    3. C\(3\)
    4. D\(40\)
    Show answerHide answer

    C: \(3\)

    \(\dfrac{30}{10} = 3\).

  5. 5

    A survey about exercise only asks people leaving a gym. What is wrong with the sample?

    1. AIt is too random
    2. BIt is biased towards people who exercise
    3. CIt is too small to be a sample
    4. DIt is a stratified sample
    Show answerHide answer

    B: It is biased towards people who exercise

    People at a gym are not typical of the whole population.

  6. 6

    A school has 600 pupils, 240 in Year 10. A stratified sample of 50 is taken. How many Year 10 pupils are in the sample?

    1. A\(20\)
    2. B\(12\)
    3. C\(24\)
    4. D\(240\)
    Show answerHide answer

    A: \(20\)

    \(\dfrac{240}{600} \times 50 = 20\).

  7. 7

    50 fish are marked and released. A second sample of 40 fish contains 8 marked fish. What is the estimate of the population?

    1. A\(10\)
    2. B\(400\)
    3. C\(62.5\)
    4. D\(250\)
    Show answerHide answer

    D: \(250\)

    \(\dfrac{50 \times 40}{8} = 250\).

  8. 8

    A histogram bar from 30 to 40 has frequency 40. Estimate the number of values from 30 to 35.

    1. A\(40\)
    2. B\(5\)
    3. C\(20\)
    4. D\(35\)
    Show answerHide answer

    C: \(20\)

    35 is halfway through the class, so about half the frequency: 20.

  9. 9

    Why does a larger sample usually give a better estimate?

    1. AIt removes all bias
    2. BIt is more likely to represent the whole population
    3. CIt always gives exact answers
    4. DIt makes the population smaller
    Show answerHide answer

    B: It is more likely to represent the whole population

    Bigger samples are less affected by chance.