A-level · Data presentation

Measures of location — mean, median, mode and quartilesExam-style questions

The three averages (mean, median, mode) and the other measures of location (quartiles and percentiles); how to find them from a list and from a frequency table; and how to estimate the median and quartiles of grouped continuous data by linear interpolation.

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

DifficultyAllLevel 2Level 3Level 4
  1. 1
    A leisure centre records the ages, in years, of the 28 people at one of its yoga classes and the 32 people at one of its spin classes. The ages are shown in the back-to-back stem and leaf diagram below.
    TotalsYogaStemSpinTotals
    (0)16 8 9(3)
    (1)921 2 3 4 4 6 7 8 9(9)
    (2)8 431 2 3 4 6 7 8(7)
    (4)8 7 5 141 2 3 5 6 8(6)
    (7)9 8 7 6 4 3 051 3 6 8(4)
    (11)9 8 7 6 4 4 4 3 2 2 162 7(2)
    (3)6 3 174(1)
    Key: 1∣6∣21 \mid 6 \mid 2 means 61 years for a person at the yoga class and 62 years for a person at the spin class
    (a) Write down the mode of the ages of the people at the spin class.

    (1 mark)

    An incomplete table of the quartiles of the two sets of ages is given below.
    YogaSpin
    Lower quartileaa25
    Median60bb
    Upper quartilecc47
    (b) Use the stem and leaf diagram to find the values of aa, bb and cc

    (3 marks)

    For the ages, xx years, of the people at the yoga class

    ∑x=1599∑x2=95157\sum x = 1599 \qquad \sum x^2 = 95157
    (c) Find, to one decimal place,

    (i) the mean age of the people at the yoga class,

    (ii) the standard deviation of the ages of the people at the yoga class.

    (3 marks)

    The leisure centre's manager describes the skewness of a set of ages using the value of

    3(mean−median)standard deviation\frac{3(\text{mean} - \text{median})}{\text{standard deviation}}
    (d) Find the value of this measure for the ages of the people at the yoga class, and describe the skewness of these ages.

    (2 marks)
    (e) Comment on one difference between the age distributions for the two classes, quoting the values of any statistics that support your comment.

    (1 mark)
    Level 4 · 10 marksTry it
  2. 2
    A charity fun run had 2121 runners. Their times, to the nearest minute, are shown below.
    StemLeaves
    22 5 7 8 9
    31 1 3 4 5 6 6 8 9
    40 1 3 4 7
    52
    6
    7
    81
    Key: 3∣43 \mid 4 means 34 minutes
    (a) Work out the median time, and the interquartile range of the times.

    (3 marks)
    (b) Use your calculator to find the mean time and the standard deviation of the times.

    (2 marks)
    (c) One of the organisers says that the median describes a typical time better than the mean does for this run.

    Give a reason why the organiser is right.

    (1 mark)
    Level 4 · 6 marksTry it