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

A decorator compares two brands of paint, Brand P and Brand Q. In each test, a sample of the paint is applied to a board and the time taken for it to become dry to the touch is recorded, to the nearest minute. The decorator carries out 33 tests on Brand P and 29 tests on Brand Q.

The drying times are shown in the back-to-back stem and leaf diagram below, in which

k

Totals	Brand P	Stem	Brand Q	Totals
(0)		1	8	(1)
(2)	7 3	2	1 3 4 6 8 9	(6)
(5)	9 8 6 4 1	3	0 2 3 4 5 5 6 $k$ 8 9	(10)
(11)	9 8 7 7 6 5 4 3 2 1 0	4	0 1 3 4 6 7 9	(7)
(8)	8 7 6 5 3 2 1 0	5	2 5	(2)
(4)	5 3 1 0	6	1 4 8	(3)
(2)	2 0	7		(0)
(1)	8	8		(0)

5 \mid 4 \mid 1

means 45 minutes for Brand P and 41 minutes for Brand Q

For these Brand Q tests, the median drying time is 37 minutes.
(a) Find the value of

k

(1 mark)
(b) Find

Q_1

Q_3

for the drying times of Brand P.

(2 marks)

The decorator classes a drying time as an outlier if it is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

Q_1 - 1.5 \times (Q_3 - Q_1)

(c) Show that exactly one of the drying times for Brand P is an outlier.

(2 marks)

The drying times for Brand Q are summarised in the box plot below.(d) Using the same scale, draw a box plot for the drying times of Brand P. State the values that you plot.

(4 marks)
(e) Give one difference between the drying times of the two brands of paint. Support your answer with the values of suitable statistics.

(1 mark)

The decorator later carries out four more tests on Brand Q. The drying times, to the nearest minute, are 22 minutes,

a

minutes, 42 minutes and

(2a + 4)

22 < a < 42 < 2a + 4

When these four times are included, the box plot for Brand Q does not change.
(f) Find the possible values of

a

Show your working clearly.

(3 marks)

Level 5 · 13 marksTry it

2
At the end of a ten-week keyboard skills course, the typing speed, in words per minute, of each of the 35 students on the course is measured.

The results are shown in the stem and leaf diagram below.
Stem Leaf Totals
0 9 (1)
1 (0)
2 1 4 6 7 8 9 (6)
3 0 1 1 2 3 3 4 5 5 6 6 7 8 8 9 (15)
4 0 1 1 2 3 4 5 6 7 8 (10)
5 0 2 (2)
6 3 (1)
Key: $3 \mid 4$ means 34 words per minute
(a) For these results, find

(i) the median,

(ii) the lower quartile and the upper quartile.

(3 marks)

The course tutor decides to treat any value below $Q_1 - 1.5 \times (Q_3 - Q_1)$ or above $Q_3 + 1.5 \times (Q_3 - Q_1)$ as an outlier.
(b) Show that there are exactly two outliers in these results.

(3 marks)

The same 35 students were tested in the same way at the start of the course. The box plot below summarises their typing speeds at that time.(c) Using the same scale, draw a box plot for the results at the end of the course, stating the values that you plot.

(4 marks)
(d) Compare the results at the start of the course with the results at the end of the course, interpreting your comparisons in context.

(3 marks)
Level 5 · 13 marksTry it