A-level · Data presentation

Histograms and frequency polygonsExam-style questions

Why a histogram for continuous data plots frequency DENSITY, not frequency, so that the AREA of each bar is proportional to the frequency — and how that one rule lets you find a missing frequency, a missing class width, or read a frequency off a drawn bar. Plus the frequency polygon, which joins class midpoints.

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16 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    The histogram summarises the volumes of water, in litres, used on one particular day by each of the 160 households in a village.0501001502002503003504004505000.10.20.30.40.50.60.70.80.9Volume (litres)Frequency densityOn that day, 35 of the households used between 100 and 150 litres of water.
    (a) Find the number of households that used between 300 and 500 litres of water on that day.

    (1 mark)
    (b) State the units of the frequency density.

    (1 mark)
    (c) The water company wants to use the histogram to learn more about the households that used the most water. Suggest one change to the histogram that would help with this.

    (1 mark)
    (d) Estimate the number of households that used between 400 and 500 litres of water on that day.

    (1 mark)
    Level 2 · 4 marksTry it
  2. 2
    The distance, dd km, that an electric scooter travels on a single charge was recorded for each of 120 scooters.

    The histogram below represents these data.102030405060701234567Distance (km)Frequency densityThe table below shows some of the frequencies.
    Distance (dd km)Frequency
    10⩽d<2010 \leqslant d < 2015
    20⩽d<2520 \leqslant d < 2520
    25⩽d<3025 \leqslant d < 30
    30⩽d<4030 \leqslant d < 4035
    40⩽d<6040 \leqslant d < 60
    (a) Use the histogram to complete the table.

    (2 marks)
    (b) Estimate the number of these scooters that travelled less than 27 km on a single charge.

    (2 marks)
    (c) Use linear interpolation to estimate the median of dd

    (2 marks)
    (d) Find an estimate for the mean of dd

    (2 marks)
    Level 3 · 8 marksTry it
  3. 3
    The journey times, tt minutes, of 100100 commuters were recorded and summarised in the table.
    Time (tt minutes)0≤t<100 \le t < 1010≤t<1510 \le t < 1515≤t<2015 \le t < 2020≤t<3020 \le t < 3030≤t<5030 \le t < 50
    Frequency882222303028281212
    A histogram is drawn, but the bar for the class 20≤t<3020 \le t < 30 has been omitted.1020304050123456Time (minutes)Frequency density(a) Determine the height, in units of frequency density, of the missing bar. (2 marks)
    (b) Calculate an estimate of the mean journey time. (3 marks)
    (c) Estimate the number of commuters whose journey took longer than 2626 minutes. (2 marks)
    (d) State one assumption needed for your estimate in part (c). (1 mark)
    Level 3 · 8 marksTry it
  4. 4
    A researcher records the time, tt minutes, that each of 120120 customers spends in a shop. The results are shown in the histogram in Figure 1.102030405012345Time, t (minutes)Frequency densityFigure 1
    (a) Find the number of customers who spent less than 1010 minutes in the shop. (2 marks)
    (b) Estimate the median time spent in the shop, giving your answer to 11 decimal place. (3 marks)
    (c) Estimate the number of customers who spent between 1515 and 2525 minutes in the shop. (3 marks)
    Level 3 · 8 marksTry it
  5. 5
    A researcher records the time, tt minutes, that each of 120120 customers spends in a shop. The results are shown in the histogram in Figure 1.102030405012345Time, t (minutes)Frequency densityFigure 1
    (a) Estimate the median time spent in the shop, giving your answer to 11 decimal place. (3 marks)
    (b) Estimate the number of customers who spent between 1515 and 2525 minutes in the shop. (3 marks)
    (c) Explain why your answers to parts (a) and (b) are only estimates. (1 mark)
    Level 3 · 7 marksTry it

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