Practise Averages and spread. 4 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
Here are the annual salaries of the five people who work in a small business. £20 000 £21 000 £22 000 £23 000 £150 000
(a) Write down the median salary.[1]
(b) The owner says, 'The mean salary is £47 200, so our staff are well paid.' Explain why the mean is not a good average to represent these salaries.[1]
Show the answer and mark scheme
(a)Answer: £22 000
B1 for £22 000
Worked solution: The middle value of the five ordered salaries is £22 000.
(b)Answer: The £150 000 salary is an extreme value (outlier) that makes the mean much higher than four of the five salaries; the median is more typical.
C1 for explaining that the £150 000 is an outlier / extreme value that raises the mean above almost every salary
Worked solution: Four of the five people earn between £20 000 and £23 000. The one very large salary pulls the mean up to £47 200, which is not typical of the staff.
Question 2Medium5 marks
The table gives information about the times, \(t\) minutes, that 40 people took to complete a puzzle.
[object Object]
(a) Work out an estimate for the mean time.[3]
(b) Explain why your answer to part (a) is only an estimate.[1]
(c) Mo says, 'The median time is in the class \(20 \lt t \le 30\).' Is Mo correct? Give a reason for your answer.[1]
(b)Answer: The exact times are not known; the midpoint of each class is used instead.
C1 for stating that the actual times are not known (midpoints are used)
Worked solution: The data are grouped, so each person's time is taken to be the midpoint of their class.
(c)Answer: Yes: the median is the 20th/21st time (the 20.5th), and the cumulative frequencies are 4, 15, 31, so it is in \(20 \lt t \le 30\).
C1 for 'yes' with a reason, e.g. the 20th and 21st values lie in this class since 15 people take 20 minutes or less and 31 take 30 minutes or less
Worked solution: The median is between the 20th and 21st times. 15 people took 20 minutes or less and 31 took 30 minutes or less, so both are in \(20 \lt t \le 30\).
Question 3Hard4 marks
Five positive whole numbers have a mean of 6, a median of 5, a mode of 3 and a range of 9. Find the five numbers. You must show your working.[4]
Show the answer and mark scheme
Answer: 3, 3, 5, 7, 12
P1 for a total of \(5 \times 6 = 30\)
P1 for the middle number 5 with two 3s below it (the mode is 3, so 3 appears at least twice, and both must be below the median)
P1 for the largest number \(3 + 9 = 12\)
A1 for 3, 3, 5, 7, 12
Worked solution: In order: \(a \le b \le 5 \le d \le e\). The mode is 3, so 3 appears at least twice; it can only be \(a\) and \(b\): \(a = b = 3\). Range 9, so \(e = 3 + 9 = 12\). Total \(= 30\), so \(d = 30 - 3 - 3 - 5 - 12 = 7\). The numbers are 3, 3, 5, 7, 12.