A-level · Data presentation

Cumulative frequency, box plots and outliersExam-style questions

Building a cumulative frequency diagram and reading the median, quartiles and percentiles off it by plotting against the UPPER class boundary; testing for outliers with the 1.5×IQR rule or the mean ± 2 standard deviations rule; drawing a box plot whose whiskers stop at the most extreme value that is NOT an outlier; and comparing two distributions with a measure of location and a measure of spread.

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

DifficultyAllLevel 3Level 4Level 5
  1. 1
    A taxi driver records the length of time, to the nearest minute, of each of the 39 journeys she made in one week. The times are shown in the stem and leaf diagram below, in which ww is a digit.
    StemLeafTotals
    05 7 8 9(4)
    10 1 2 2 3 4 4 5 6 7 8 9 9(13)
    20 1 1 2 3 4 5 6 7 8 9(11)
    30 ww ww ww 7 8(6)
    41 4 6 9(4)
    5(0)
    6(0)
    72(1)
    Key: 3∣73 \mid 7 means 37 minutes
    (a) Find the range of these times.

    (1 mark)
    (b) Find the median of these times.

    (1 mark)

    The times have an interquartile range of 21 minutes.
    (c) Find the value of ww

    (3 marks)

    For these times, tt minutes,

    ∑t=947∑t2=30337\sum t = 947 \qquad \sum t^2 = 30337
    (d) Use these sums to show that the standard deviation of the times is 13.7 minutes to 3 significant figures.

    (2 marks)

    Any time that is less than Q1−1.5×(Q3−Q1)Q_1 - 1.5 \times (Q_3 - Q_1) or more than Q3+1.5×(Q3−Q1)Q_3 + 1.5 \times (Q_3 - Q_1) is to be treated as an outlier.
    (e) Show that exactly one of these times is an outlier.

    (2 marks)
    (f) Describe, giving a reason, the skewness of these times.

    (2 marks)
    Level 4 · 11 marksTry it
  2. 2
    1020304050102030405060Time, t (minutes)Cumulative frequencyFigure 2

    Figure 2 is the cumulative frequency graph for the times, tt minutes, taken by 6060 people to complete a task.
    (a) Use the graph to estimate the median and the interquartile range. (4 marks)
    (b) An outlier is defined as a value more than 1.5×1.5 \times the interquartile range above the upper quartile or below the lower quartile. Show that this data set contains no outliers. (2 marks)
    Level 4 · 6 marksTry it
  3. 3
    The heights, in cm, of a random sample of plants of variety AA are summarised below.

    MinimumLower quartileMedianUpper quartileMaximum814192545\begin{array}{|c|c|c|c|c|}\hline \text{Minimum} & \text{Lower quartile} & \text{Median} & \text{Upper quartile} & \text{Maximum} \\ \hline 8 & 14 & 19 & 25 & 45 \\ \hline \end{array}

    An outlier is defined as a value that is more than 1.5×1.5 \times the interquartile range above the upper quartile, or more than 1.5×1.5 \times the interquartile range below the lower quartile.
    (a) Show that the maximum height of 4545 cm is an outlier. (2 marks)

    Figure 2 shows a box plot for the heights of a random sample of plants of variety BB.101520253035Height (cm)Figure 2
    (b) Compare the distributions of the heights of the two varieties. State three differences. (3 marks)
    (c) The heights of variety AA are coded using y=x−105y = \dfrac{x - 10}{5}, where xx cm is the original height.

    Find the median and the interquartile range of the coded heights. (3 marks)
    (d) State, giving a reason, whether the heights of variety AA are positively or negatively skewed. (1 mark)
    Level 4 · 9 marksTry it
  4. 4
    On one day, 1919 telephone calls were made from a small office. Their lengths, in minutes, are shown below.
    StemLeaves
    04
    1
    23 6 8
    30 1 4 5 5 7 9
    41 2 2 6 8
    50 3 8
    Key: 2∣32 \mid 3 means 23 minutes
    (a) The lengths are to be shown on a box-and-whisker plot whose whiskers reach the shortest and the longest call.

    Write down the five values needed to draw the plot: the median, the lower and upper quartiles, and the two ends of the whiskers.

    (4 marks)
    (b) The office manager wants to find the mean length of a typical call, and intends first to remove any lengths that are outliers.

    Use a suitable test for outliers to decide which calls, if any, would be left out, and comment on whether leaving them out is sensible.

    (3 marks)
    Level 4 · 7 marksTry it