A Level Mathematics Questionbanks



1. A group of 13 students sat a test in numeracy. Their marks out of a possible score of 50 were

20, 45, 25, 37, 45, 22, 36, 45, 41, 50, 38, 9, 29

Determine

a) the mode

[1]

b) the median

[2]

c) the interquartile range

[4]

d) Hence comment on the skewness of the distribution of the marks, explaining how you arrived at your answer.

[2]

2. A local parcel delivery service sets out the number of orders received each week over a 52 week period.

|Number of orders received |30 |31 |32 |33 |34 |35 |

|Number of children |4 |7 |6 |5 |3 |2 |

a) Using linear interpolation, calculate

i) the median ii) the 87th percentile

of this data

[7]

b) Calculate the mean of this data

[2]

A second class of children took the same test. Their marks were found to have a mean of 52 and a median of 44.

c) Compare the distributions of the marks of the marks of the two classes, explaining your comments.

[3]

4. The table below gives the number of hours that it takes for a car to be repaired at a garage.

|Time (nearest hour) |0-4 |5-9 |10-14 |15-19 |20-24 |25-29 |

|Frequency |66 |51 |12 |6 |3 |2 |

a) Determine, by calculation, the median and interquartile range of the time taken.

[10]

The following summary statistics for time taken for repairs were obtained at a second garage:

Q1= 1.3 hours Q2= 4 hours Q3= 16.5 hours Shortest time = 0.5 hours Longest time = 35 hours

b) On the same diagram, draw labelled box plots for the two garages.

[5]

c) Compare the distribution of times at the two garages

[4]

5. A class of 30 pupils sit a test, and their marks are recorded. For these data Σx = 1512 and Σx2 = 83838.

a) Calculate the mean and variance of the marks.

[3]

b) One pupil missed the test. He sat it late, and obtained a mark of 68. Find the new mean and variance of

the class’s marks when this pupil is included

[5]

The teacher later discovered that he had recorded the mark of one of the pupils incorrectly; he had recorded

the mark as 26, when it should have been 62.

c) Without carrying out any further calculations, state what effect this will have on the mean and variance of

the marks and explain your reasoning.

[4]

6. A group of 20 people have a mean age of 22.4 years with standard deviation 2.1 years. A group of 30 people have

a mean age of 25.7 years with a standard deviation of 1.9 years.

a) Find the mean age of all 50 people

[3]

b) For each group of people, find the sums of the squares of their ages

[4]

c) Hence obtain the standard deviation of the ages of all 50 people

[3]

7. The table below show the number of minutes late of 30 trains at a station.

|Lateness (nearest minute) |1-3 |4-6 |7-11 |12-21 |22-41 |

|Number of trains |9 |12 |5 |2 |2 |

a) Give a reason to justify the use of a histogram to represent this data

[1]

b) Draw a histogram to represent this data

[5]

8. The following data are recorded on the value of the orders (to the nearest pound) obtained by forty door-to-door

salesmen from a particular company.

|Sales Value (£) |11-50 |51-100 |101-200 |201-300 |301-500 |501-1000 |(1001 |

|Number of salesmen |3 |11 |17 |5 |2 |2 |0 |

The mean and standard deviation for these figures are to be calculated using the coding [pic],

where X is the sales value

a) Find ΣfY and show that ΣfY2 = 400.03

[6]

b) Calculate the mean and standard deviation of Y

[3]

c) Hence find the mean and standard deviation of the sales

[5]

d) Comment on the suitability of the mean and standard deviation to represent this data, explaining your reasons.

[2]

9. A student completing a project records the following data

|Value |2 |3 |4 |5 |6 |

|Frequency |7 |12 |2 |3 |6 |

a) Calculate the mean and variance of this data

[3]

In fact, the student’s table did not record the actual data; he was using “x” to stand for “5.00x”, so that the

value “2” represented “5.002” etcetera.

b) Use your answer to a) to find the mean and variance of the actual figures, giving your answers correct to

four significant figures

[5]

10. The table below contains data on the heights of 24 children, recorded to the nearest 10cm

|Height |80-100 |110 |120 |130 |140-150 |

|Number of children |6 |2 |9 |3 |4 |

This data is to be represented by a histogram.

a) Explain why a histogram is suitable for this purpose, and give one advantage of using a histogram

[2]

The bar to represent the children of height 140-150 cm is of width 1cm and height 8cm.

b) State the width and height of the bar which represents

i) children of height 110cm ii) children of height 80-100cm

[8]

c) Use the coding [pic], where H is the height of the children, to find an estimate for the mean and

standard deviation of the heights

[10]

d) Give one reason why your answer to c) is only an estimate

[1]

11. a) Find the mean and standard deviation of: n-3, n-2, n-1, n, n+1, n+2, n+3

[5]

b) Hence deduce the mean and standard deviation of

i) 5, 6, 7, 8, 9, 10, 11

[2]

ii) 11, 13, 15, 17, 19, 21, 23

[3]

iii) a – 6b, a – 4b, a – 2b, a, a+2b, a+4b, a+6b

[3]

The value n – 1.5 is added to the data set in part a)

c) Without further calculations, state the effect this would have on the mean and standard deviation,

explaining your reasoning.

[4]

12. The data below are the marks (out of 60) obtained by a class of 30 pupils

25 32 58 47 47 52 28 34 30 44

47 39 36 21 58 40 47 34 37 31

29 36 17 28 56 38 47 30 40 42

a) Construct a stem-and-leaf diagram to represent this data

[3]

b) Find the median and interquartile range of this data

[6]

c) Draw a box-plot to represent this data, and hence comment on its skewness

[4]

The diagram below shows a box-plot for the performance of another class on the same test

|25 | | | | |30 | |

|Number of children |5 |8 |13 |8 |6 |0 |

a) Explain why it would be incorrect to treat this as continuous data

[1]

b) Construct a suitable cumulative frequency diagram to represent this data

[4]

14. a) State two advantages of using

i) a histogram ii) a box plot iii) a stem and leaf diagram

to represent statistical data.

[6]

b) State the types of data for which each of the above types of diagram are suitable

[3]

15. The table below gives data on the number of people of various ages at a social club.

|Age (in completed years) |18-20 |21-25 |26-30 |31-40 |41-50 |51+ |

|Number of people |7 |10 |15 |20 |7 |1 |

a) Construct a cumulative frequency polygon to represent this data

[7]

b) Hence obtain an estimate for the median

[2]

For a normal distribution with mean μ and standard deviation σ, approximately the central 95% of the data

lie between μ - 2σ and μ + 2σ.

c) By using your diagram to find suitable percentiles, find the values between which the central 95% of the

above data lie.

[4]

d) Given that the mean and variance of the above ages are 31.6 and 70.19 respectively, use your answer to c)

to comment on the suggestion that the given data follows a normal distribution.

[6]

16. The stem-and-leaf diagram below represent the lengths in centimetres to 1 decimal place of two

species of lizards.

|Species B | |Species A |

|9 |8 |0 0 1 5 7 |

| 8 5 3 |9 |0 3 3 6 7 7 8 |

|8 8 6 4 1 |10 |1 2 2 2 2 5 6 6 9 |

|9 9 8 7 7 6 5 4 3 1 1 |11 |0 1 5 5 6 6 |

|8 7 5 5 4 2 1 0 |12 |0 2 |

|5 4 3 0 0 |13 | |

a) Explain what is meant by the 12 |2 in this diagram

[1]

b) Find the median length for each species of lizard.

[3]

c) Use the diagram to compare the distributions of lengths of the two species of lizard.

[3]

d) Suggest one other type of diagram that would be suitable for representing this data.

[1]

e) Give one advantage and one disadvantage of using your choice of diagram rather than a stem and

leaf diagram for this data.

[2]

17. At a children’s party, there were 15 four-year olds, 9 five-year olds, two mothers, aged 30 and 35, and

a grandfather aged 70.

a) Find the mean, median and mode of the ages of the people at the party.

[4]

b) Suggest which of these would be the most useful “average”, explaining your choice

[2]

c) Without carrying out any further calculations, state the effect on these figures if two five-year olds left

the party, explaining your reasons.

[4]

At a second party, the mean age of the 20 children attending was 5.05 and the mean age of the 3 adults

present was 40 years.

d) Calculate the mean age of all those attending the second party.

[3]

18. The following data were obtained on the ages of cars on which repairs were being carried out at a garage.

|Age (completed years) |0-1 |2-3 |4-5 |6-7 |8-9 |10-12 |13+ |

|Number of cars |5 |4 |5 |10 |12 |3 |1 |

a) Use linear interpolation to calculate estimates for the median and third decile of the ages of the cars.

[9]

b) Explain what is meant by the third decile

[1]

c) Explain why the figures calculated are only estimates

[1]

d) Comment on the likely accuracy of the estimates, giving reasons for your answers

[2]

19. A set of five data items have mean μ and standard deviation σ.

A set of ten data items have mean 2μ and standard deviation 3σ

a) Find the mean of all fifteen data items.

[3]

b) If μ = 2σ

i) show that the standard deviation of all fifteen data items is[pic]

[10]

ii) State which of the two data sets is likely to contain the lowest value, explaining your reasoning.

[2]

A new item of data which has value μ +[pic]σ, is added to the first set of data

c) Without performing any calculations, state the effect this would have on the mean and standard deviation

of this data set. Explain your reasoning.

[4]

20. Shown below is a histogram which represents the weights (in kilograms) of 50 adults.

a) Use the histogram to construct a frequency table for the weights.

[4]

b) Calculate the mean and standard deviation of the weights.

[4]

c) Explain why the mean and standard deviation may not be the best statistics to use, and suggest suitable

alternative measures of location and spread.

[3]

21. A data set containing 20 values have mean 12, standard deviation 3, median 10 and interquartile range 5.

State the effect on each of these statistics if

a) All the data values are increased by 5

[4]

b) All the data values are increased by 5%

[1]

c) The highest data value increases by 10

[2]

22. a) Explain how the median and quartiles may be used to describe a distribution.

[3]

The table below shows the times taken by a class of children to complete an arithmetic problem correctly

|Time (nearest 20 seconds) |20 |40 |60 |80 |100 |120 |

|Number of children |2 |5 |13 |8 |1 |1 |

b) Treating the data as discrete, find the median and quartiles of the times.

[6]

c) Treating the data as continuous, find estimates for the median and quartiles of the times

[7]

d) Explain why your answers to b) and c) were not the same, and explain which values would be preferable to use

[3]

A second class were given the same arithmetic problem, and their times were recorded. The following statistics were obtained: Q1 = 35 Q2 = 50 Q3 = 65

e) Compare the performance of the two classes

[4]

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Mark

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