AQA A Level Maths: Statistics

Topic Questions

2.1 Statistical Measures

1a
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4 marks

Students’ marks, given as a percentage, on their recent statistics test were:

38         41         19         33         22         0            27         19         10         99

Find the mode, range, mean and median of the students’ marks.

1b
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1 mark

Give a reason why the median is an appropriate measure of location for these data.

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2a
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3 marks

Two sets of data are given below:

Set 1

1

2

3

4

5

6

7

8

9

Set 2

1

5

5

5

5

5

5

5

9

For set 1,

(i)
Calculate the mean, x with bar on top, of the data.
(ii)
Calculate the variance, sigma squared, of the data using the formula:
sigma squared equals fraction numerator sum open parentheses x minus space x with bar on top close parentheses squared over denominator n end fraction
2b
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3 marks

For set 2,

(i)
Calculate the mean, x with bar on top, of the data.
(ii)
Calculate the variance, sigma squared, of the data using the formula:
sigma squared equals fraction numerator sum x squared over denominator n end fraction minus x with bar on top squared

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3a
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3 marks

Seven friends decide to see how long they can hold their breath underwater. Their times, in seconds, are shown below.

                59          72          69          105           77          81          92

Write down the

(i)
median, Q2,

(ii)
lower quartile, Q1,

(iii)
and the upper quartile, Q3,


of the data.

3b
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2 marks

One more friend comes along and decides to join in. He holds his breath for 85 seconds. Comment on how this 8th value will affect your answers in part (a).

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4a
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3 marks

Lucy is working with some grouped, continuous data. For a set of 100 items of data, she has calculated thatspace sum x f equals 357 and sum left parenthesis x minus x with bar on top right parenthesis squared f equals 42, where f is the frequency for each group.

(i)
Give a reason why Lucy has decided to group her data.
(ii)
Briefly explain what is meant by sum x f and sum left parenthesis x minus x with bar on top right parenthesis squared f.
4b
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3 marks

To calculate the standard deviation, Lucy could choose to use either of the following two formulae.

standard deviation =  square root of fraction numerator sum open parentheses x minus x with bar on top close parentheses squared f over denominator sum f end fraction end root     or     square root of fraction numerator sum x squared f over denominator sum f end fraction end root minus x with bar on top squared where x with bar on top equals fraction numerator sum x f over denominator sum f end fraction

(i)
Calculate the mean of Lucy’s data.
(ii)
Using the appropriate formula from above, calculate the standard deviation of Lucy’s data.

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5
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4 marks

As part of her veterinary course, Harriet measured the weight, x grams, of 50 new-born kittens and summarised their data as sum x space equals space 6342 and sum x squared space equals 879013.

Calculate the mean and standard deviation of the weights of the kittens.

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6a
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3 marks

Katie is collecting information on Jupiter’s moons for a research project. She collects data on the diameters of 78 of Jupiter’s known moons and organises the information into the table below.

Diameter  (km)

Number of moons bold italic f

 0 space less than space d space less or equal than space 1

6

 1 space less than space d space less or equal than space 2

20

 2 space less than space d space less or equal than space 5

23

 5 space less than space d space less or equal than space 50

17

 50 space less than space d space less or equal than space 1000

8

 1000 space less than space d space less or equal than space 6000

4

(i)
Write down the modal class interval.
(ii)
Write down the class interval that contains the median.
(iii)
Katie discovers another moon, Valetudo, which has a diameter of 1 km. Write down the class interval which should include the diameter of Valetudo.
6b
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1 mark

Katie calculates the mean diameter of Jupiter’s moons to be 6500 km. Explain how you know Katie is incorrect.

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7a
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2 marks

The number of goals scored by the 24 teams that played in the first 44 games of the UEFA Euro cup 2020 can be summarised in the table below.

Goals scored 0 - 1 2 - 3 4 - 5 6 - 7 8 - 9 10 - 11
Frequency f 3 5 5 6 4 1

Estimate the mean number of goals scored by each team.

7b
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2 marks

Find the standard deviation of the number of goals scored by each team.

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8a
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2 marks

The Mythical Creatures Research Centre measures the heights, h, of nine unicorns to the nearest centimetre. The heights are shown below:

            276       219      198       154        213       243       192       161      218

Use your calculator to find the mean and standard deviation of the nine heights.

8b
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3 marks

Before calculating the mean, the researchers choose to form a new variable, y, using the formula y=h-200.

(i)
Write down the nine values of y.

(ii)
Use your calculator to find the mean and standard deviation of the nine values of y.
8c
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2 marks

By comparing your answers to (a) and (b)(ii), describe how subtracting a value from each piece of data affects the mean and standard deviation.

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9a
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1 mark

A random sample of data from the large data set relating to the CO2 emissions of Volkswagen cars in 2016 is given below.

113 space space space space space space space space 118 space space space space space space space 107 space space space space space space space space 110 space space space space space space space space space 119 space space space space space space space space space space space 120 space space space space space space space space 107 space space space space space space space space 106 space space space space space space 119 space space space space space space space space 113

Using your knowledge of the large data set, state the units that are used to measure C02 emissions.

9b
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4 marks
(i)
Find the value of the median of the data.
(ii)
Find the value of the lower quartile of the data.
(iii)
Calculate the interquartile range of the data.

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1
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4 marks

a, bc and d are 4 integers written in order of size, starting with the smallest. 

The sum of a, b and c is 70
The mean of a, bc and d is 25
The range of the 4 integers is 14.

Work out the median of a, bc and d

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2a
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2 marks

The speeds (s), to the nearest mile per hour, of 80 vehicles passing a speed camera were recorded and are grouped in the table below. 

Speed, s
(mph)

20 ≤ s <25 25 ≤ s <30 30 ≤ s <35 s ≥ 35
Number of vehicles 23 48 7 2

(i)

Write down the modal class for this data.

(ii)
Write down the class group that contains the median.
2b
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4 marks
(i)
Assuming that ≥35 means ‘at least 35 mph but less than 40 mph’, calculate an estimate for the mean speed of the 80 vehicles.

(ii)
It is now discovered that ≥35 means ‘at least 35 mph but less than 60 mph’. Without further calculation, state with a reason whether this would cause an increase, a decrease or no change to the value of the estimated mean.

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3a
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4 marks

A veterinary nurse records the weight of puppies (in kg) at birth and again at their eight week check-up.  The table below summarises the weight gain of 50 small breed puppies over their first eight weeks.

Weight gain w (kg) Number of puppies f
0.0 ≤ w < 0.5 1
0.5 ≤ w < 1.0 8
1.0 ≤ w < 1.5 19
1.5 ≤ w < 2.0 18
2.0 ≤ w < 2.5 4

Use linear interpolation to estimate the median and interquartile range of the weight gain of the 50 puppies.

3b
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1 mark

Give a reason why it is not possible to determine the exact median for this data.

3c
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2 marks

The head veterinarian directs the nurse to monitor all puppies whose weight gain lies within the bottom 20% of the data set.

Explain why, without further information, the veterinary nurse would have to monitor over 50% of the puppies to be sure of satisfying this instruction.

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4
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4 marks

Whilst in lockdown, 100 people were asked to record the length of time, rounded to the nearest minute, that they spent exercising on a particular day. 

The results are summarised in the table below:

Time mins Frequency f
0 ≤ t ≤10 1
10 < t ≤20 12
20 < t ≤30 25
30 < t ≤40 a
40 < t ≤50 b
50 < t ≤60 14

Given that the estimate of the mean time spent exercising based on this table is 35.4 minutes, find the values of a and b.

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5a
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3 marks

The ages, x years, of 200 people attending a vaccination clinic in one day are summarised by the following:  straight capital sigma x= 7211  and  straight capital sigma x squared= 275 360.

Calculate the mean and standard deviation of the ages of the people attending the clinic that day.

5b
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4 marks

One person chooses not to get the vaccine, so their data is discounted. The new mean is exactly 36.  Calculate the age of the person who left and the standard deviation of the remaining people.

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6a
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2 marks

In January of 2021, the UK government announced a nationwide lockdown to control the spread of the coronavirus.  The table below shows the means and standard deviations of the average amounts of time spent indoors per day by some people in London, UK and in Wellington, New Zealand, in January of 2021.

  Number of people Mean (hr) Standard deviation
London 25 20.9 1.51
Wellington 15 15.1 2.87

Suggest a reason, in the context of the question, for why

(i)
the mean in London is higher than the mean in Wellington

(ii)

the standard deviation in London is lower than the standard deviation in Wellington.
6b
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1 mark

Based on the data in the table, do you think the government in New Zealand had imposed the same restrictions as those in the UK?  Give a reason for your answer.

6c
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2 marks

Calculate the overall mean for the average amounts of time spent indoors by all 40 people.

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7a
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6 marks
Engine size
(cm3)
1998 1997 1995 1995 1499 1499 0 2993 2993 1995
CO2
Emissions
(g/km)
49 149 124 114 49 126 0 142 131 139

The large data set contains data on cars from 2002 and 2016. A selection of data relating to the engine size and CO2 emissions often BMW cars from 2016 is given above.

(i)
Using your knowledge of the large data set, give two possible reasons for the CO2 emissions value of 0 g/km. Given that the corresponding engine size is also recorded as 0 cm3, state which of your reasons you think is most likely.
(ii)
Considering your answer from part (i), use the rest of the values to find the mean and standard deviation of the CO2 emissions of the BMW cars from 2016.
7b
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1 mark

Given that in 2002 the mean CO2 emissions of BMW cars was 215.5 g/km, use your knowledge of the large data set to give a possible reason for the change in the mean CO2 emissions from 2002 to 2016.

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1
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4 marks

A pharmacy sells face masks in a variety of sizes.  Their sales over a week are recorded in the table below:

  Kids Adults
Size Small Large S M L XL
Frequency f 29 4 8 24 15 4

(i)

Write down the mode for this data.

(ii)
Explain why, in this case, the mode from part (i) would not be particularly helpful to the shop owner when reordering masks.

(iii)
Given that the shop is open every day of the week, calculate the mean number of masks sold per day.

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2
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4 marks

Fran sits three Maths papers and six Science papers during her final A Level exams.  She achieves a mean score of 62.7% across the three Maths exam papers, and needs an overall mean score of 78.5% across all nine papers to get into her chosen University.  After getting the results of four out of her six Science papers, her mean score in Science is 84.2%.

Given that each of the nine papers is weighted equally when working out the mean scores, calculate the mean score she must achieve on her final two science papers in order to gain a place at University.

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3a
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4 marks

Coffee4Life manufactures reusable coffee cups out of coffee plant waste.  Coffee cups are tested to see how many times they can be used before they begin to disintegrate.  A sample of 15 cups are tested, giving the following results for numbers of uses:

                     31    36    41    43    47

                     49    51    56    58    62

                     62    63    68    69    72

(i)
Write down the modal number of times a cup can be used.

(ii)
Find the values of the lower quartile, median and upper quartile.
3b
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2 marks

The advertising department at Coffee4Life designs an advert which says;

“If used once a day,  3 over 4 of our cups last longer than 9 weeks.”

Explain the mistake that the advertising department has made, and state how the advert could be reworded to make it correct.

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4
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3 marks

The lengths (l cm) of a sample of nine otters, measured to the nearest centimetre by a wildlife research team, are:

                        76     77      91      65       63      83      92      61      88

Calculate the mean and standard deviation of the nine recorded lengths.

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5a
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2 marks

During initial small aircraft pilot training candidates must sit an aptitude test. Grades for the latest 28 candidates are shown in the table below (0 is the lowest grade, 8 is the highest):

Grade Frequency bold italic f
0 2
1 3
2 2
3 3
4 4
5 2
6 5
7 4
8 3


Candidates in the bottom 25% are disqualified.

Calculate the grade candidates must achieve to avoid disqualification.

5b
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3 marks

Those who score in the top 25% move on to the next stage of training while the rest (other than those who have been disqualified) must re-sit the test.

One of the candidates, Amelia, achieves grade 6. Determine whether Amelia will need to re-sit the test or will be moved on to the next stage of training.

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6a
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1 mark

A random sample of 50 students were asked how long they spent revising for their Maths exam in the 24 hours before the exam.  The results are shown in the table below:

Time t (minutes) Number of students f
0 ≤ t < 60 5
60 ≤ t < 120 6
120 ≤ t < 180 17
180 ≤ t < 240 14
240 ≤ t < 300 8

For this data, use linear interpolation to estimate the median.

6b
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3 marks

Using x to represent the mid-point of each class, straight capital sigma x f= 8340 and straight capital sigma x squared f= 1 636 200.

Estimate the mean and the standard deviation of the amount of time students spent revising.

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7
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5 marks
CO2
Emissions
(g/km)
187 165 163 159 178 145 165 192 145 264
CO
Emissions
(g/km)
0.375 0.518 0.518 0.09 0.656 0.347 0.518 0.427 0.109 0.341

   
The large data set contains data on cars from 2002 and 2016. A selection of data relating to the CO2 and CO emissions of ten Ford cars from 2002 is given above.

Using the large data set it can be calculated that in 2016 the mean CO2 emissions of Ford cars was 121.9 g/km and the mean CO emissions of Ford cars was 0.350 g/km. Ford engineers are constantly working to reduce exhaust emissions and they claim that between 2002 and 2016 they reduced emissions of Ford cars by over 30%.

For the data on the 2002 cars given above, calculate the mean emissions for both CO2 and CO and state whether or not this data supports Ford's claim that they reduced exhaust emissions by over 30% between 2002 and 2016.

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1
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4 marks

Whilst in lockdown, a group of people were asked to record the length of time, t hours, they spent browsing the internet on a particular day. 

The results are summarised in the table below.

Time, t (hours) Frequency, f
t ≤ 2 3
2 < t ≤ 4 5
4 < t ≤ 6 a
6 < t ≤ 8 10
8 < t ≤10 2

From this data an A Level Statistics student calculated that the estimated mean time spent browsing the internet is 5 hours and 15 minutes. Calculate the value of a and find the estimated variance of the length of time spent browsing the internet.

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2
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5 marks

Two friends, Anna and Connor, are playing a gaming app on their phones.  As they play, they can choose from three different booster options.  They are unaware that each of the three options are charging them automatically from their mobile accounts.  The number of in-app purchases they each make are shown in the table below.

  Super-charge Re-energise Level-up
Anna 4 0 2
Connor 3 6 1

(i)

The mean and standard deviation of the cost of Anna’s in-app purchases are
£0.50 and £0 respectively.  Write down the cost of a single in-app purchase to ‘Level-up’.

(ii)
Given that the mean cost of Connor’s in-app purchases is £0.38, find the standard deviation of the costs of Connor’s purchases.

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3a
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3 marks

Wildlife researchers are studying the swimming speeds, x kmph, of two species of penguin, the emperor penguin and the gentoo penguin. The mean swimming speed of 40 gentoo penguins was found to be 31.4 kmph and the standard deviation was found to be 3.8 kmph.

Allowing x subscript G to represent the swimming speeds of the gentoo penguins, show that straight capital sigma x subscript G=1256  and calculate the value of  straight capital sigma x subscript G squared.

3b
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4 marks

The swimming speeds of 20 emperor penguins (x subscript E) were also recorded and the mean swimming speed of all 60 penguins surveyed was found to be 24.1 kmph. Given that  straight capital sigma x squared=41891,  calculate the mean and standard deviation of the 20 emperor penguins.

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4a
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3 marks

Some entomologists were studying the amount of time two different species of butterflies spent cocooned.  The table shows the means and standard deviations of the time spent cocooned, measured in days, by 15 Monarch butterflies and 25 Common Blue butterflies.

Species Mean Standard deviation
Monarch   1.51
Common Blue 13.4 1.24

Given that the overall mean time for all 40 butterflies was 11.93 days, calculate the mean number of days the Monarch butterflies spent cocooned and complete the table.

4b
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4 marks

Calculate the overall standard deviation of the time spent cocooned by all 40 butterflies.

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5a
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1 mark

Hattie’s homeroom teacher decides to summarise the number of minutes, t, she has been late to school during the last year in preparation for a parents’ meeting.  The results are shown in the table below.

Time t (mins) Frequency f
-10 ≤ t < -5 3
-5 ≤ t < 0 19
0 ≤ t < 5 32
 5 ≤ t < 10 a
10 ≤ t < 20 53
20 ≤ t < 60 24

Write down, in the context of the question, what the time interval -10 ≤ t < -5  represents.

5b
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3 marks
(i)
Using x to represent the mid-point of each class, write an expression in terms of a for  straight capital sigma f x, giving your answer in simplified form. 

 

(ii)
Given that  straight capital sigma f x= 2132.5  and  straight capital sigma f x squared= 53568.75,  calculate the estimated mean and standard deviation of the amount of time Hattie was late for school last year.
5c
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2 marks

Hattie happens to notice that on three of the days she was recorded as being 40 minutes late for school, she had actually arrived 40 minutes early.  Calculate the corrected estimate for the mean amount of time she was late for school last year.

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6a
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5 marks

The table below gives the mass, rounded to the nearest kg, of all available data on convertible cars from 2002 using the large data set. The data has been taken straight from the large data set and has been grouped as follows.

Mass (kg) Frequency bold italic f
1250 - 1299 6
1300 - 1399 3
1500 1549 a
1550 - 1599 10
1600 - 1700 2
1400 - 1499 3


Given that the estimated mean of the actual mass of the vehicles is calculated to be 1383.172 kg, rounded to 3 decimal places, find the value of a.

6b
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2 marks

Calculate an estimate for the standard deviation of the mass of the convertible cars from 2002.

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7a
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2 marks

Zisien measures the speeds, x miles per hour, of a number of cars passing her house one day.  She knows that the speed limit is 30 miles per hour.

She finds that  straight capital sigma open parentheses straight x minus 30 close parentheses=13.4  and  straight capital sigma open parentheses straight x minus 30 close parentheses squared= 1470.

Is it more likely that more of the cars in Zisien’s sample were going over or under the speed limit that day? Give a reason for your answer.

7b
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3 marks

Given that the mean speed of the cars in Zisien’s sample, x with bar on top, is 30.67 miles per hour, calculate the standard deviation of x.

7c
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2 marks

Zisien's sister, Ying, worked out the median of the speeds of the cars and found it was 27.43 miles per hour.

Use this information to comment on your answer to (a).

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8a
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5 marks

Botanists at a research centre are carrying out research on a new type of fertiliser.  They collect data on the heights of one group of geraniums growing without the fertiliser (the control group) and of another group growing with the fertiliser (the experimental group).  They take care to keep all other growing conditions the same for both groups.

The table below shows the heights of the control group of geraniums 15 weeks after planting. 

Height (cm) < 10 < 15 < 20 < 25 < 30 < 35 < 40
Cumulative Frequency 2 7 12 19 34 39 40

 

(i)
Write down the modal class for the heights of geraniums in the control group.

(ii)
Find the smallest and largest possible values for the interquartile range of the heights of the control group.

(iii)
Use linear interpolation to calculate the median height.
8b
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2 marks

The data for the group of geraniums growing in the experimental group were summarised as follows:

 Q1=23.4 cm                 Q2=27.1 cm                 Q3=28.5 cm

The shortest plant in the experimental group was 15.2 cm and the tallest was 33.5 cm. 

Compare the distribution of the heights of the plants in the two groups.

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