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
    A walk-in health centre recorded how long each patient waited before being seen on one Saturday. The waiting times, tt minutes, of these 160 patients are summarised in the table below.
    Waiting time (tt minutes)Frequency
    0⩽t<40 \leqslant t < 418
    4⩽t<104 \leqslant t < 1044
    10⩽t<1510 \leqslant t < 1540
    15⩽t<2015 \leqslant t < 2020
    20⩽t<3020 \leqslant t < 3024
    30⩽t<6030 \leqslant t < 6014
    These data are to be represented by a histogram, on which the bar for the class 20⩽t<3020 \leqslant t < 30 is 2 cm wide and 2.4 cm high.
    (a) Find the area of the bar for the class 4⩽t<104 \leqslant t < 10

    (2 marks)
    (b) Find the width and the height of the tallest bar on the histogram.

    (3 marks)
    (c) Use linear interpolation to estimate

    (i) the median,

    (ii) the interquartile range

    of these waiting times.

    (4 marks)

    The skewness of a set of data can be measured using

    Q3−2Q2+Q1Q3−Q1\frac{Q_3 - 2Q_2 + Q_1}{Q_3 - Q_1}
    (d) Calculate the value of this measure for these waiting times and describe the skewness.

    (2 marks)
    Level 4 · 11 marksTry it
  2. 2
    The lengths, tt minutes, of 150 calls received by a council helpline on one Monday are summarised in the table below.
    Length (tt minutes)Frequency
    0⩽t<20 \leqslant t < 224
    2⩽t<52 \leqslant t < 545
    5⩽t<105 \leqslant t < 1050
    10⩽t<2010 \leqslant t < 2025
    20⩽t<3020 \leqslant t < 306
    (a) Give a reason to justify the use of a histogram to represent these data.

    (1 mark)
    (b) Estimate the number of these calls that lasted between 4 minutes and 12 minutes.

    (2 marks)
    (c) Use linear interpolation to estimate the median length of these calls.

    (2 marks)

    You may use ∑ft2=12 762.75\sum ft^2 = 12\,762.75
    (d) Find an estimate for

    (i) the mean,

    (ii) the standard deviation

    of the lengths of these calls.

    (4 marks)

    The helpline manager picks two of these 150 calls at random to review.
    (e) Estimate the probability that each of the two calls picked lasted between 4 minutes and 12 minutes.

    (3 marks)
    Level 4 · 12 marksTry it
  3. 3
    The masses, mm grams, of 120120 potatoes harvested from a field are summarised in the following table.
    Mass (mm grams)100⩽m<200100 \leqslant m < 200200⩽m<250200 \leqslant m < 250250⩽m<300250 \leqslant m < 300300⩽m<400300 \leqslant m < 400400⩽m<700400 \leqslant m < 700
    Frequency13131818313137372121
    (a) Draw a histogram to represent this information.

    (4 marks)
    (b) Show that the interquartile range of these masses cannot exceed 200200 grams.

    (2 marks)
    Level 4 · 6 marksTry it