Measures of central tendency and spreadAQA A-Level Maths: Subtopic test

1. 1
   Six students' times, in minutes, to complete a puzzle are 12, 15, 15, 18, 22 and 26. In this question the standard deviation uses divisor $n$.

   (a)

   What is the median time?

   [1 mark]

   • A15
   • B18
   • C16.5
   • D17

   (b)

   What is the standard deviation of the times, correct to 3 significant figures?

   [1 mark]

   • A4.73
   • B5.18
   • C22.3
   • D18.6

   (c)

   A seventh student takes exactly 18 minutes. State, with a reason, the effect of including this time on (i) the mean and (ii) the standard deviation.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A courier firm records 25 delivery times, $x$ minutes. The summary statistics are $\sum x=400$ and $\sum x^2=7000$. The standard deviation uses divisor $n$.

   (a)

   What is the mean delivery time?

   [1 mark]

   • A280
   • B16
   • C7000
   • D0.0625

   (b)

   What is the standard deviation, to 3 significant figures?

   [1 mark]

   • A24
   • B16.7
   • C280
   • D4.90

   (c)

   One recorded time of 40 minutes was a mistake: it should have been 20 minutes. Find the correct mean.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A factory has two machines, A and B, that fill bags of sugar labelled 1000 g. For a sample of bags from machine A the mean mass is 1004 g and the standard deviation is 3.2 g. For a sample from machine B the mean is 1001 g and the standard deviation is 1.1 g.

   (a)

   Compare the two machines and state which you would recommend, giving reasons.

   [3 marks]

   (b)

   A sample of 40 bags from a third machine, C, has $\sum x=40120$ and $\sum x^2=40240520$. Calculate the mean and the standard deviation of this sample.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The numbers of goals scored in 30 football matches are: 0 goals in 3 matches, 1 goal in 7, 2 goals in 9, 3 goals in 6, 4 goals in 3 and 5 goals in 2. The standard deviation uses divisor $n$.

   (a)

   (i) Show that the mean number of goals per match is $2.17$ to 3 significant figures.
   (ii) Calculate the standard deviation.
   (iii) State the median.

   [6 marks]

   (b)

   In a further 20 matches the summary statistics are $\sum x=44$ and $\sum x^2=118$.
   (i) Find the mean and standard deviation for these 20 matches.
   (ii) Find the mean number of goals for all 50 matches.
   (iii) Compare the two sets of matches.

   [6 marks]

   Total for question 4: 12 marks

End of questions