OCR GCSE Maths
Topic questions Statistics Averages & spread
65 questions · 330 marks · about 5 hours
Here is a list of numbers.
34 42 17 26 34 12 27
Write down the mode.
A student says,
The median is 26 as it is the middle number.
Explain why the student is not correct.
Also involves Forming and solving equations
Here is a list of three numbers.
8 15 10
A fourth number is added to the list. The mean of the four numbers is 13.
Work out the fourth number.
Also involves Cumulative frequency graphs
The cumulative frequency graph shows the distribution of the examination scores for 120 students.
A student says that they had an examination score of 34 and so were in the top half of the 120 students.
Is the student correct? Give a reason for your decision.
The top 10% of the 120 students are awarded a commendation.
Find the minimum mark required for a commendation.
Also involves Inequalities
The mass of some parcels are recorded. The table and the histogram each show part of this information.
| Mass (m kg) | 0 < m ⩽ 2 | 2 < m ⩽ 5 | 5 < m ⩽ 10 | 10 < m ⩽ 20 |
| Frequency | 12 | 18 |
Complete the table and the histogram.
The heaviest parcel is 19.5 kg.
Complete the inequality to represent the range, R kg, of the masses of the parcels.
Two Year 11 classes, 11G and 11K, record their times for a race. The box plots show the distribution of the times, in seconds, for each class.
Write down the percentage of students in 11G who took longer than 14.6 seconds.
Which class, 11G or 11K, was fastest on average? Include values to support your decision.
Also involves Linear equations
Here is a list of five numbers.
14 2 26 43 9
Find the range.
A sixth number is added to the list. The mean of the six numbers is 23.5.
Work out the sixth number.
A farmer has 60 pear trees. The table shows the heights, h metres, of the pear trees.
Calculate an estimate of the mean height of the 60 pear trees.
Explain why it is not possible to use the information from this table to calculate the exact value of the mean height.
Also involves Probability
A farmer grows pumpkins. The farmer records the masses, m kilograms, of 80 of their pumpkins. The table shows the results.
| Mass (m kg) | 0 < m ⩽ 5 | 5 < m ⩽ 10 | 10 < m ⩽ 15 | 15 < m ⩽ 20 | 20 < m ⩽ 25 |
| Frequency | 10 | 22 | 28 | 14 | 6 |
Complete the cumulative frequency table.
| Mass (m kg) | m ⩽ 5 | m ⩽ 10 | m ⩽ 15 | m ⩽ 20 | m ⩽ 25 |
| Cumulative frequency | 10 | 32 |
Draw the cumulative frequency graph to represent these results.
Write down the upper quartile of the mass of the 80 pumpkins.
The farmer picks a pumpkin at random.
Find an estimate for the probability that the pumpkin has a mass greater than 18 kg.
Here is a list of five numbers.
10 12 4 3 6
Write down the median.
A sixth number is added to the list. The range of the six numbers is 15.
Work out a possible value for the sixth number.
50 students are asked how many books they have read this month. The table shows the results.
Work out the mean number of books read this month by the 50 students.
The box plot shows the distribution of the waiting time of cars at a road junction.
Write down the median waiting time.
What percentage of the waiting times were less than 30 seconds?
Given that a randomly chosen car waited for more than 30 seconds, write down the probability that the car waited for more than 90 seconds.
Also involves Fractions, decimals and percentages
Jack travels to work each day by train.
He records whether
• the train is on time or late
• there are seats available or no seats available.
Jack’s results are shown on this partly completed frequency tree.
Find the relative frequency of there being no seats available on Jack’s train journey.
Jack says
If the train is late, travellers are less likely to find seats available than if the train was on time.
Does Jack’s data suggest he is correct? Show how you decide.
Also involves Forming and solving equations
Three adults, Amos, Beth and Charlie, are comparing their age in years.
Amos is 32. Beth is 48. The mean age of Amos, Beth and Charlie is 50.
Work out the age of Charlie.
The box plot shows the distribution of the times, in seconds, taken by class 11P to complete a
The times, in seconds, taken by class 11Q to complete the same problem are summarised below.
• median = 52
• lower quartile = 42
• interquartile range = 18
• range = 70
• highest score = 86
Show the distribution of 11Q’s times as a box plot on the diagram above.
Which class has the more consistent times? Give a reason for your answer.
Also involves Cumulative frequency graphs
A fitness centre records how long each customer spends in the gym. This cumulative frequency table summarises the results.
Calculate an estimate of the mean time the customers spend in the gym. You must show your working. You may use the table above to help present your work.
Kai has four differently numbered cards.
• The range of the numbers is 14.
• The median of the numbers is 9.
• All the numbers are prime numbers.
• The lowest number is 5.
Work out the numbers on the cards. Write the numbers in order of size.
........,........,........,........ 5
lowest
Also involves Forming and solving equations
A six-sided numbered spinner is thrown 50 times. The score for each throw is recorded. Some of the results are shown in the table.
An 8 was thrown f times. An unknown number on the spinner is represented by n.
| Score | Frequency |
| 1 | 12 |
| 3 | 2 |
| 5 | 9 |
| 6 | 16 |
| 8 | f |
| n | 4 |
| Total | 50 |
The mean score of the 50 throws is 5.5.
Find the value of f and the value of n.
A six-sided numbered spinner is thrown 50 times. The score for each throw is recorded. Some of the results are shown in the table.
An 8 was thrown f times. An unknown number on the spinner is represented by n.
| Score | Frequency |
| 1 | 12 |
| 3 | 2 |
| 5 | 9 |
| 6 | 16 |
| 8 | f |
| n | 4 |
| Total | 50 |
The mean score of the 50 throws is 5.5.
Find the value of f and the value of n.
A running club records the distances run by each member during December. The results are shown in this histogram.
18 members run less than 20 km.
Work out the number of members who run more than 30 km.
Finley says
To estimate the range, I subtracted the smallest possible value from the largest possible value. So, 80 – 0 = 80 km.
Explain why Finley’s method is likely to overestimate the true value of the range.
This box plot shows the distribution of the distance run by each member of the running club during July.
During December,
• the median distance run was 30 km
• the interquartile range of the distance run was 20 km.
Make two comparisons between the distances run during December and the distances run during July. Include values to support your comparisons.
Also involves The four operations
At the end of each day, a driver works out the mean distance they have driven so far that week.
At the end of 5 days, the mean distance they have driven is 185.5 miles per day. At the end of 6 days, the mean distance they have driven is 190 miles per day.
Work out how many miles the driver drove on the sixth day of that week.
Also involves Cumulative frequency graphs
The cumulative frequency graph shows the distribution of the ages of the members of a tennis
The table summarises the ages of the members of a cycling club.
| Age (a years) | 0 < a ⩽ 20 | 20 < a ⩽ 30 | 30 < a ⩽ 40 | 40 < a ⩽ 50 | 50 < a ⩽ 70 | 70 < a ⩽ 80 |
| Frequency | 8 | 14 | 8 | 12 | 17 | 5 |
On the graph above, draw the cumulative frequency graph of the ages of the members of the
cycling club.
Find out which club has younger members on average. Give evidence to support your decision.
60 people each try to solve a puzzle. The table summarises their recorded times.
| Recorded time (t minutes) | Frequency |
| 0 < t ⩽ 5 | 12 |
| 5 < t ⩽ 15 | 19 |
| 15 < t ⩽ 30 | 18 |
| 30 < t ⩽ 50 | 11 |
Draw a histogram to show this information.
Those people who failed to solve the puzzle within 50 minutes were given a recorded time of 50 minutes.
Nina uses mid-interval values to estimate the mean recorded time of the 60 people.
Explain why Nina’s answer is likely to be an under-estimate for the mean of the actual time taken by the 60 people.
Also involves Percentages
Charlie weighs many apples.
The weights of the apples are summarised below.
• heaviest apple = 75 g
• range = 50 g
• median = 60 g
• lower quartile = 45 g
• 50% of the apples weigh between 45 g and 65 g
• mean = 63 g
Write down the interquartile range for the weights of the apples.
Write down the percentage of the apples that weigh between 45 g and 60 g.
Draw a box plot to show the distribution of the weights of the apples.
Charlie eats two of the apples. The apples that they eat weigh 58 g and 66 g.
Charlie says
The mean weight of all the apples was 63 g. I ate one apple that weighed less than the mean and another apple that weighed more than the mean. Therefore, the mean of the remaining apples will still be 63 g.
Is Charlie correct? Explain your reasoning.
Here is a list of numbers.
6 9 2 12 3 8 3
Write down the mode.
Work out the range.
This list represents four numbers.
127 x x + 1 2 x
The mean of the four numbers is 180.
Work out the numbers. You must show your working.
127
........
Also involves Fractions
Some students were asked how they travel to school. Each student gave one answer. The pie chart shows the proportion who travel by bus, by train or walk.
All of the remaining students travel to school either by bike or by car. The ratio of the number who travel by bike to the number who travel by car is 2: 3.
Complete the pie chart.
You must show your working.
Which way of travelling to school is the mode?
In a dance competition, four judges award marks to each dancer. Each judge can award 1, 2, 3, 4 or 5 marks.
The four judges’ median mark, m, is put into the formula
S = 10 m - 5
to get the dancer’s score, S.
Sam is awarded marks of 4, 3, 1 and 4. Work out Sam’s score.
Taylor gets a score of 40. Taylor says
The judges must have awarded marks of 4, 4, 5 and 5 because the median is 4.5 and 4.5 × 10 - 5 = 40.
Why is Taylor not correct? Show working to support your reason.
Also involves The four operations · Estimation and approximation
Morgan is playing a computer game. They can score 0, 1, 2 or 3 points on each turn. They record their scores for 100 turns. The table shows the relative frequencies of their scores.
| Score | 0 | 1 | 2 | 3 |
| Relative frequency | 0.08 | 0.42 | 0.38 |
Complete the table.
Morgan says
I scored more than 160 points in total in my 100 turns.
Is Morgan correct? Show how you decide.
Here is a list of numbers.
6 9 2 3 9 1
Work out the range of the numbers.
Work out the mean of the numbers.
Also involves Fractions
A sports team played the same number of matches in 2019 and 2020. The two pie charts summarise their results.
What fraction of the matches did the team win in 2019?
Did the team’s results improve in 2020? Explain how you know.
Also involves Box plots and stem and leaf diagrams
The box plot shows the distribution of the lengths, in cm, of 60 full-grown mice owned by a pet
Find the range.
Work out the number of these mice that have a length of at least 7.2 cm.
Sam owns 5 full-grown mice. Sam picks the third longest mouse and measures its length. Sam then looks at the box plot.
Sam says
This mouse is 7.35 cm long. Therefore, the mice I own are longer than the full-grown mice owned by the pet shop.
Give a mathematical reason to support Sam’s conclusion.
Give a mathematical reason why Sam’s conclusion may be unreliable.
This is Nadia’s bar chart to show the number of goals scored by four teams during November.
Blake says
Nadia’s bar chart shows that United scored twice as many goals as City.
Is Blake correct? Give a reason for your answer.
Give one way in which Nadia can improve her bar chart.
Kareem says
Out of these four teams, United achieved the highest mean number of goals per game during November.
What assumption has Kareem made?
A teacher asks nine of his pupils how many pets they have at home.
Here are the results.
1 1 1 2 3 4 5 7 111
Work out the range of the nine results.
The median of the nine results is 3. The mean is 15.
Write down the mode.
The teacher wants to use a sensible average to summarise the results.
Which average should he use and why?
Also involves Measures of dispersion
Ali and Beth take it in turns to play a computer game. On each turn, the player achieves a score out of 50. Ali and Beth play the computer game many times and record their scores.
Ali’s scores are summarised below.
• median = 31
• highest score = 38
• range = 23
• lower quartile = 24
• interquartile range = 11
Draw a box plot to show the distribution of Ali’s scores.
This box plot shows the distribution of Beth’s scores.
Find the interquartile range of Beth’s scores.
Kareem says
Beth was more consistent than Ali because Beth had a lower median score.
Is his statement correct? Explain your reasoning.
The masses, m kg, of some parcels are shown below.
4 15 14 11 12 3 1 18 13 2 16 10
Jack constructs this grouped frequency table to record the masses.
Explain why Jack’s table is unsuitable to record the masses.
The histogram summarises the times taken, in minutes, by some students to complete a
Show that 70 students took between 45 and 70 minutes to complete the race.
Calculate an estimate of the mean time taken to complete the race. Show your working.
Also involves Rounding
Plaza United are playing a football match away from home.
379 supporters are going to the match by coach. Each coach seats 45 people.
What is the smallest number of coaches that will be needed?
In their last 50 matches, Plaza United have drawn 10 matches, lost 5 and won the rest.
Sam claims
The probability that Plaza United will win this match is 0.7.
Show calculations to support Sam’s claim.
Give one reason why Sam’s claim may not be reliable.
30 students each own one pet. The pie chart shows the proportion of each type of pet owned by the 30 students.
Which type of pet is the mode?
Use the information in the pie chart to complete this bar chart.
The cumulative frequency graph summarises the annual salary, p (£ thousands), of the 60 workers in a factory.
Use the graph to estimate the median annual salary.
(a) £........ thousands
Complete this cumulative frequency table.
Use the information in the cumulative frequency table to calculate an estimate of the mean annual salary.
(c) £........ thousands
Explain why your estimate of the median is more reliable than your estimate of the mean.
The box plot shows the distribution of the marks scored by some boys in a test.
Find the interquartile range.
The marks for some girls in the same test are summarised below.
• median = 58
• lowest mark = 32
• range = 44
• upper quartile = 66
• interquartile range = 12
Draw a box plot to show the distribution of the marks scored by the girls.
Eleanor says
The boys did better, on average, in the test as they had a bigger interquartile range.
Is her statement correct? Explain your reasoning.
Mr and Mrs Wilde have five children who are all different ages.
• The mean age is 6.4.
• The range is 9.
• The median is 6.
• The oldest child is 12.
Work out the ages of the children. Write their ages from youngest to oldest.
youngest oldest
Luke is an office receptionist. Each day, for 60 days, he records the number of people visiting the office.
Calculate an estimate of the mean number of people visiting the office.
Luke says the range is 40.
Explain why he may be wrong.
Also involves Percentages
The cumulative frequency graph shows the distribution of the heights of members of a rowing
Find the median.
Find the interquartile range.
Calculate the percentage of the members who are at least 180 cm tall.
The histogram summarises the heights of the 153 members of a swimming club.
Which club has the greater median height? You must show all your working.
These are the heights, in metres, of the players in a netball team.
1.30 1.13 1.20 1.23 1.22 1.24 1.15
Find the median height of the 7 players.
Work out the range of the heights of the 7 players.
The sum of the heights of the 7 players is 8.47 m.
Calculate the mean height of the 7 players.
The tallest player is replaced by a substitute. The median height of the players is unchanged. The mean height of the players becomes smaller.
Write down a possible height for the substitute.
The police record the speed of vehicles passing a speed checkpoint. The speeds are recorded in the table below.
Calculate an estimate of the mean speed of the vehicles.
Explain why it is not possible to use the information from this table to calculate the exact value of the mean speed.
Also involves Box plots and stem and leaf diagrams
The box plot shows the distribution of the runs scored by David in some cricket matches.
Another player, Peter, has
• a median score of 26
• a highest score of 39
• a lowest score of 8
• a lower quartile of 14
• an inter-quartile range of 18.
Show the distribution of Peter’s scores as a box plot on the diagram above.
Decide whether David or Peter best satisfies each of these questions. Give a reason for each of your decisions.
Who scored more runs on average?
Whose scores were more consistent?
Also involves Cumulative frequency graphs
The cumulative frequency graph shows information about the marks scored by a group of 80 students in a test.
Find the interquartile range.
The ratio of the number of students passing the test compared to failing the test is 4: 1. Find the minimum mark needed to pass the test.
The police record the speed of vehicles passing a speed checkpoint. The speeds are recorded in the table below.
Calculate an estimate of the mean speed of the vehicles.
Explain why it is not possible to use the information from this table to calculate the exact value of the mean speed.
Here is a list of numbers.
2 8 5 12 6
From this list, write down
the odd number,
the cube number.
Using the same list of numbers, work out
the median,
the range.
The table below shows the weight, w kg, of the bags that people took on a plane.
Calculate an estimate of the mean weight of the 60 bags.
Two groups of students go on a water sport holiday. Each student chooses one activity.
Students in Group A choose from Diving, Swimming, Paddleboarding and Kayaking. Their choices are to be shown in a pie chart.
Complete this table for Group A.
| Activity | Number of students | Angle of sector |
| Diving | 5 | 60 |
| Swimming | 120 | |
| Paddleboarding | ||
| Kayaking | 9 | 108 |
Complete the pie chart for Group A.
One student in Group A changes activity. There is now a new modal activity for Group A.
Write down the student’s original activity and new activity.
original activity........
The choices made by Group B are shown in this pie chart.
A teacher thinks more students chose Diving in Group B than in Group A.
Give a reason why the teacher may be wrong.
Also involves Forming and solving equations
Ping chooses four numbers.
The mode of these four numbers is 8, the range is 7 and the mean is 11.
Find Ping’s four numbers.
…………, …………, …………, …………
Ping chooses four numbers.
The mode of these four numbers is 8, the range is 7 and the mean is 11.
Find Ping’s four numbers.
…………, …………, …………, …………
State the median salary.
Find the interquartile range.
Make two different comparisons between the distribution of the salaries at Bexbridge Biscuits and the salaries at Camford Cookies.
One morning, eight buses arrive at a bus stop. The number of minutes late for each bus is shown below.
0 7 2 6 9 2 0 7
In the afternoon, two more buses arrive at the bus stop.
The median number of minutes late of all ten buses is 3.5. The mode number of minutes late of all ten buses is 0.
How many minutes late were the two afternoon buses?
The average mass of a man is 84 kg and of a woman is 70 kg.
A lift can safely carry 630 kg.
To find how many people the lift can safely carry, Dan divides the safe total mass by the average mass of a person.
630 ÷ 77 = 8.18...
How has the average mass of a person, 77 kg, been worked out?
Dan decides that his answer shows the lift can safely carry 8 people.
Explain why he is wrong and give an example, with working, to support your answer.
Also involves The four operations
A company makes sweets. The sweets are put into packets.
Here are some facts.
Calculate the mean number of sweets in one packet.
Sweets are made on 288 days each year.
Calculate the number of sweets made each year. Give your answer in standard form.
The company has 152 machines making the sweets. Each machine operates for 15 hours each day.
Calculate the number of sweets made by one machine each hour. Give your answer as an ordinary number correct to the nearest 10.
State one assumption you have made in part (c)(i).
A shop records the time taken by its customers to complete a purchase on its website. The results from one day are summarised in this table.
Calculate an estimate of the mean time taken.
Explain why it is not possible to use the information from this table to calculate the exact value of the mean time taken.
Also involves Fractions
The histogram summarises a health score for a group of people.
Estimate the fraction of the group who had a score of 45 or more.
What assumption did you make in answering part (a) ?
Iqrah carries out a survey of 200 families in the north of England on their weekly spending on food. The cumulative frequency diagram summarises the results.
Find
the median,
the interquartile range.
Iqrah says
15% of these families spent over £120.
Is her statement correct? State the evidence you have used in making your decision.
In a survey of 200 families in the south of England, the median weekly amount spent on food was £84 and the interquartile range was £28.
Make two comparisons between the weekly amounts spent on food in the north of England and the south of England. State the evidence you have used in making your comparisons.
A shop records the time taken by its customers to complete a purchase on its website. The results from one day are summarised in this table.
Calculate an estimate of the mean time taken.
Explain why it is not possible to use the information from this table to calculate the exact value of the mean time taken.
Also involves Using a calculator
A company makes sweets. The sweets are put into packets.
Here are some facts.
Calculate the mean number of sweets in one packet.
Sweets are made on 288 days each year.
Calculate the number of sweets made each year. Give your answer in standard form.
The company has 152 machines making the sweets. Each machine operates for 15 hours each day.
Calculate the number of sweets made by one machine each hour. Give your answer as an ordinary number correct to the nearest 10.
State one assumption you have made in part (c)(i).
The table shows the average number of customers per day entering a shop.
| 2015 | 2016 | |||||||
| Months | Jan- Mar | Apr- Jun | July- Sep | Oct- Dec | Jan- Mar | Apr- Jun | July- Sep | Oct- Dec |
| Average number of customers per day | 119 | 264 | 368 | 172 | 130 | 304 | 381 | 192 |
Complete the time series graph below.
Make two different comments comparing the number of customers entering the shop in 2015 and 2016.
Also involves Forming and solving equations
Jenny played four games of golf. For these games her modal score was 76 and her mean score was 75. Her range of scores was 10.
What were her scores for the four games?
The table shows the average number of customers per day entering a shop.
| 2015 | 2016 | |||||||
| Months | Jan- Mar | Apr- Jun | July- Sep | Oct- Dec | Jan- Mar | Apr- Jun | July- Sep | Oct- Dec |
| Average number of customers per day | 119 | 264 | 368 | 172 | 130 | 304 | 381 | 192 |
Complete the time series graph below.
Make two different comments comparing the number of customers entering the shop in 2015 and 2016.
Jenny played four games of golf. For these games her modal score was 76 and her mean score was 75. Her range of scores was 10.
What were her scores for the four games?
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