[discia] GCSE Maths › Statistics › Cumulative Frequency

Cumulative Frequency

discia.org 10 questions · 33 marks
Name Date
  1. Either paper 2 marks

    A post office recorded the mass of each parcel handed in one morning. The cumulative frequency table gives information about 54 parcels.

    Mass (\(m\) kg)Cumulative frequency
    \(m \le 5\)\(5\)
    \(m \le 10\)\(12\)
    \(m \le 15\)\(25\)
    \(m \le 20\)\(33\)
    \(m \le 25\)\(44\)
    \(m \le 30\)\(54\)

    Calculate the number of parcels with a mass of more than 15 kg.

    Show answer
    \(29\)
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  2. Either paper 3 marks

    A teacher recorded how long each student took to finish a puzzle. The table shows the results.

    time (minutes)Frequency
    \(0 < t \le 10\)\(3\)
    \(10 < t \le 20\)\(6\)
    \(20 < t \le 30\)\(12\)
    \(30 < t \le 40\)\(6\)
    \(40 < t \le 50\)\(3\)

    The cumulative frequency graph for the same information is drawn below. The dashed lines show one reading taken from it.

    1020304050510152025300time (minutes)cumulative frequencymedian
    (a)

    How many of the students had a time of \(20\) minutes or less?

    (b)

    Use the graph to find an estimate for the median time.

    Show answer
    (a) \(9\ \text{students}\)
    (b) \(25\) — accept \(23\) to \(27\ \text{minutes}\)
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  3. Either paper 3 marks

    A vet recorded the mass of each dog brought in for a check-up one week. The frequency table shows the results.

    mass (kg)FrequencyCumulative frequency
    \(0 < m \le 10\)\(6\)
    \(10 < m \le 20\)\(10\)
    \(20 < m \le 30\)\(12\)
    \(30 < m \le 40\)\(11\)
    (a)

    Complete the cumulative frequency column of the table.

    (b)

    Owen used the completed table to draw a cumulative frequency graph. For the class \(0 < m \le 10\), he plotted the point at \(m = 5\), the middle of the class, instead of at the end of it.

    Why is this wrong?

    Show answer
    (a) 6, 16, 28, 39
    (b) A cumulative frequency point is always plotted at a class's upper boundary, not its midpoint — the running total of 6 is not reached until every one of the dogs in that class has been counted.
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  4. Either paper 2 marks

    A nurse recorded how long each patient waited at a health centre. The cumulative frequency table shows the results.

    waiting time (minutes)Cumulative frequency
    \(0 < w \le 10\)\(2\)
    \(0 < w \le 20\)\(12\)
    \(0 < w \le 30\)\(18\)
    \(0 < w \le 40\)\(26\)
    \(0 < w \le 50\)\(28\)

    On the grid, draw a cumulative frequency graph for this information.

    1020304050102030400waiting time (minutes)cumulative frequency
    Show answer
    Points at \((10, 2)\), \((20, 12)\), \((30, 18)\), \((40, 26)\) and \((50, 28)\), joined with a smooth curve
    1020304050102030400waiting time (minutes)cumulative frequency
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  5. Either paper 6 marks

    A PE teacher measured the height of each student in Year 11. The table shows the results.

    Height (cm)Frequency
    \(140 < h \le 150\)\(12\)
    \(150 < h \le 160\)\(16\)
    \(160 < h \le 170\)\(16\)
    \(170 < h \le 180\)\(12\)
    \(180 < h \le 190\)\(8\)
    (a)

    Work out the cumulative frequency for each of the five classes, in order, starting with the class \(140 < h \le 150\).

    (b)

    On the grid, draw a cumulative frequency graph for this information.

    14015016017018019001020304050607080height (cm)cumulative frequency
    (c)

    Use your graph to find an estimate for the median height.

    Show answer
    (a) 12, 28, 44, 56, 64
    (b) Points at \((140, 0)\) and at the upper boundary of each class, with heights 12, 28, 44, 56, 64 at 150, 160, 170, 180, 190, joined with a curve or straight lines
    (c) \(162.5\) — accept \(161\) to \(164\ \text{cm}\)
    (b)14015016017018019001020304050607080height (cm)cumulative frequency
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  6. Either paper 4 marks

    The cumulative frequency graph shows information about the heights of 80 plants.

    2040608010012001020304050607080height (cm)cumulative frequency
    (a)

    Use the graph to find an estimate for the median height.

    (b)

    Use the graph to find an estimate for the interquartile range.

    Show answer
    (a) \(70\) — accept \(65\) to \(75\ \text{cm}\)
    (b) \(38\) — accept \(28\) to \(48\ \text{cm}\)
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  7. Either paper 4 marks

    The cumulative frequency graph shows how long each of the students took to finish a puzzle. There were 40 students altogether.

    10203040505101520253035400time (minutes)cumulative frequency
    (a)

    Use the graph to estimate how many of the students had a time of less than \(22\) minutes.

    (b)

    Use the graph to estimate how many of the students had a time of more than \(22\) minutes.

    Show answer
    (a) \(19\) — accept \(18\) to \(20\ \text{students}\)
    (b) \(21\) — accept \(20\) to \(22\ \text{students}\)
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  8. Either paper 3 marks

    A charity recorded the amount each person gave in one collection. The classes are not all the same width.

    amount (pounds)Frequency
    \(0 < a \le 10\)\(3\)
    \(10 < a \le 20\)\(4\)
    \(20 < a \le 40\)\(26\)
    \(40 < a \le 60\)\(4\)
    \(60 < a \le 100\)\(3\)

    The cumulative frequency graph is drawn below. Each point is plotted at the top end of its class, so the points are not evenly spaced.

    1020304050607080901005101520253035400amount (pounds)cumulative frequency
    (a)

    State the class interval that contains the median.

    (b)

    Use the graph to find an estimate for the median amount.

    Show answer
    (a) \(20 < a \le 40\)
    (b) \(30\) — accept \(28\) to \(32\ \text{pounds}\)
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  9. Either paper 3 marks

    An allotment society recorded the mass of each pumpkin entered into its autumn competition. Two of the frequencies are missing from the table, but the cumulative frequency column is complete.

    mass (kg)FrequencyCumulative frequency
    \(10 < m \le 20\)\(8\)\(8\)
    \(20 < m \le 30\)\(17\)
    \(30 < m \le 40\)\(4\)\(21\)
    \(40 < m \le 50\)\(35\)
    \(50 < m \le 60\)\(4\)\(39\)
    (a)

    Use the cumulative frequency column to work out the two missing frequencies.

    (b)

    Which class contains the most pumpkins?

    Show answer
    (a) 9 and 14
    (b) \(40 < m \le 50\)
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  10. Either paper 3 marks

    The cumulative frequency graph shows the masses of \(220\) onions.

    406080100120140022446688110132154176198220mass (g)cumulative frequency

    Onions with a mass of \(80\) g or less cost \(35\)p each.

    Onions with a mass of more than \(80\) g cost \(46\)p each.

    Find an estimate for the total cost of the \(220\) onions. Give your answer in pounds.

    Show answer
    \(\text{£}91.52\)
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discia.org Cumulative Frequency · sheet 1
Answers Cumulative Frequency · sheet 1
  1. \(29\)
  2. (a) \(9\ \text{students}\)
    (b) \(25\) — accept \(23\) to \(27\ \text{minutes}\)
  3. (a) 6, 16, 28, 39
    (b) A cumulative frequency point is always plotted at a class's upper boundary, not its midpoint — the running total of 6 is not reached until every one of the dogs in that class has been counted.
  4. Points at \((10, 2)\), \((20, 12)\), \((30, 18)\), \((40, 26)\) and \((50, 28)\), joined with a smooth curve
    1020304050102030400waiting time (minutes)cumulative frequency
  5. (a) 12, 28, 44, 56, 64
    (b) Points at \((140, 0)\) and at the upper boundary of each class, with heights 12, 28, 44, 56, 64 at 150, 160, 170, 180, 190, joined with a curve or straight lines
    (c) \(162.5\) — accept \(161\) to \(164\ \text{cm}\)
    (b)14015016017018019001020304050607080height (cm)cumulative frequency
  6. (a) \(70\) — accept \(65\) to \(75\ \text{cm}\)
    (b) \(38\) — accept \(28\) to \(48\ \text{cm}\)
  7. (a) \(19\) — accept \(18\) to \(20\ \text{students}\)
    (b) \(21\) — accept \(20\) to \(22\ \text{students}\)
  8. (a) \(20 < a \le 40\)
    (b) \(30\) — accept \(28\) to \(32\ \text{pounds}\)
  9. (a) 9 and 14
    (b) \(40 < m \le 50\)
  10. \(\text{£}91.52\)
discia.org Cumulative Frequency · sheet 1