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

DifficultyAll Level 3 Level 4 Level 5

1
SECTION B: STATISTICS

Figure 1 shows a cumulative frequency diagram for the times, $t$ minutes, taken by $80$ students to complete a puzzle.Figure 1
(a) Use Figure 1 to estimate the median time. (2 marks)
(b) Use Figure 1 to estimate the interquartile range of the times. (3 marks)
(c) Estimate the number of students who took more than $25$ minutes. (2 marks)
Level 3 · 7 marksTry it
2
Figure 2 shows a box plot for the daily rainfall, in mm, recorded at weather station $A$ on $30$ days.Figure 2

For weather station $B$ over the same $30$ days, the daily rainfall had minimum $0$ mm, lower quartile $1.2$ mm, median $3.5$ mm, upper quartile $7.8$ mm and maximum $22$ mm.

An outlier is defined as a value more than $1.5 \times$ the interquartile range above the upper quartile.
(a) Show that the maximum value of $22$ mm at station $B$ is an outlier. (2 marks)
(b) Compare the rainfall distributions at the two stations. State three differences. (3 marks)
(c) The rainfall values at station $A$ are coded using $y = 2x + 1$ , where $x$ mm is the recorded rainfall.

Write down the median and the interquartile range of the coded values. (3 marks)
Level 3 · 8 marksTry it
3
A cumulative frequency graph for $80$ values is used to estimate the median. The median is the value at a cumulative frequency of how many, and which class does it lie in if the cumulative frequency reaches $35$ at $x = 20$ and $52$ at $x = 30$ ?
Level 3 · 2 marksTry it
4
A cumulative frequency distribution has $n = 60$ . The lower quartile is estimated at the $15$ th value and the upper quartile at the $45$ th. Given $Q_1 = 18$ and $Q_3 = 34$ , find the interquartile range.
Level 3 · 2 marksTry it
5
A box plot has $Q_1 = 25$ and $Q_3 = 45$ . An outlier is a value more than $1.5\times\text{IQR}$ beyond a quartile. The smallest value in the data is $-8$ . Determine whether it is an outlier.
Level 3 · 2 marksTry it

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