A-level · Data presentation

Measures of spread, standard deviation and codingExam-style questions

Range, interquartile and interpercentile range, and — the big one — variance and standard deviation from sums Σx and Σx² (and Σfx, Σfx²). Plus coding: how a linear change of variable rescales the mean and standard deviation, and how to decode back to the original data.

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

DifficultyAll Level 2 Level 3 Level 4

1
A radio station records the length, $x$ seconds, of each of the 40 advertisements that it broadcasts one morning.

The lengths are coded using

$y = \frac{x - 30}{5}$

and the coded data are summarised by

$\sum y = 26 \qquad \sum y^2 = 212$
(a) Find the mean of $x$

(2 marks)
(b) Find the standard deviation of $x$

(3 marks)

The station manager converts each length into minutes.
(c) Find the mean and the standard deviation of the lengths in minutes.

(2 marks)
Level 2 · 7 marksTry it
2
A data set has smallest value $5$ , largest value $30$ , lower quartile $12$ and upper quartile $20$ . Find the range and the interquartile range.
Level 2 · 2 marksTry it
3
The masses, $x$ grams, of $24$ eggs from a farm are summarised as follows.

$\sum (x - 60) = -36.0 \qquad \sum (x - 60)^2 = 471.6$

Find the variance of these masses and hence find the value of $\sum x^2$ .

(4 marks)
Level 2 · 4 marksTry it
4
A company has two branches, Branch $A$ and Branch $B$ . The hourly rate of pay, £ $x$ , of each employee at the two branches is summarised in the table below.
Number of employees, $n$ Mean, $\bar{x}$ $\sum x^2$
Branch $A$ 16 13.50 2940
Branch $B$ 24 $b$ 4932
For all 40 employees at the two branches together, the mean hourly rate of pay is £13.95
(a) Show that $b = 14.25$

(2 marks)
(b) Calculate the standard deviation of the hourly rates of pay for the two branches combined.

(2 marks)

Every employee at Branch $B$ is then given a pay rise of £0.40 per hour.
(c) State, without carrying out any further calculations, how this pay rise affects each of the following. Give a reason in each case.

(i) The standard deviation of the hourly rates of pay at Branch $B$

(ii) The mean hourly rate of pay across both branches

(iii) The standard deviation of the hourly rates of pay across both branches

(3 marks)
Level 3 · 7 marksTry it
5
SECTION B: STATISTICS

A researcher records the time, $x$ seconds, taken by each of $40$ students at a school to complete a puzzle. The times are coded using $y = \dfrac{x - 60}{5}$ , and the coded values are summarised by

$\sum y = 72 \qquad\qquad \sum y^2 = 394$
(a) Find the mean and the standard deviation of the $40$ times. (3 marks)
(b) The slowest student took $118$ seconds. An outlier is defined as a value more than $2$ standard deviations from the mean. Determine whether this time is an outlier. (1 mark)
(c) The $40$ students were the first $40$ to arrive at the school library one morning. Name this method of sampling, and explain why the sample may not be representative of the students at the school. (2 marks)
Level 3 · 6 marksTry it

	Number of employees, $n$	Mean, $\bar{x}$	$\sum x^2$
Branch $A$	16	13.50	2940
Branch $B$	24	$b$	4932

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