Cumulative frequency graphs

Aled is a gardener.

In April, he records how many minutes he spends working in each of his customers’ gardens.

He draws a cumulative frequency diagram to display the findings.

Find the median time that Aled spends working in the garden for these customers.

For how many customers did Aled spend between 10 and 30 minutes working in their

garden?

On 1st June last year, 200 customers used cash to pay at Shop Lil. The cumulative frequency diagram represents the time each of these 200 customers waited to be given change at the checkout.

How many of these customers waited between 30 and 50 seconds for their

change?

Use the graph to estimate the median time these 200 customers waited for their change.

Circle your answer.

24 seconds 32 seconds 38 seconds 80 seconds 100 seconds

Calculate the fraction of these 200 customers who waited 40 seconds or longer for their change.

Give your answer in its simplest form.

On 1st June this year, the manager at Shop Lil drew a box-and-whisker plot of the times 200 customers waited for their change at the checkout.

Based on the results of these 200 customers, the manager made the following statements. Complete the statements.

“On 1st June this year, 50% of our customers were given their change in

........ seconds or less.”

“On 1st June this year, the interquartile range of the times taken to give

customers their change was........ seconds.”

Consider the 50 customers waiting the longest times to get their change on 1st June last year and this year. Has the speed of giving change at the checkout improved since last year?

Yes No

You must give a reason for your answer.

Giovanni has a takeaway pizza van. He sells whole pizzas and slices of pizza from his van.

For the last 3 days, he has timed how long it takes to complete the food order for each of his customers. Giovanni recorded his results in the table below.

Complete the cumulative frequency table and the cumulative frequency diagram.

Time, t (minutes)	Frequency	Cumulative frequency
3 t 5	8	8
5 t 7	24	32
7 t 9	74	106
9 t 11	40	..............................
11 t 13	14	..............................

Cumulative frequency

Use your cumulative frequency diagram to give the best estimates for the answers to each of the following questions.

Find the median time taken to complete a food order.

The median time is_........ minutes.

Giovanni is concerned that food orders are taking too long to complete. He says,

“Only 25% of the food orders are completed in under_........ minutes.”

Use one of the five values below to complete Giovanni’s statement.

6·4 6·6 7·2 8 9·6

For the last 3 days: • Giovanni spent £180 on ingredients • he spent £220 on the running costs for the pizza van • he received a total of £700 from the food orders.

Calculate Giovanni’s percentage profit.

Giovanni has a takeaway pizza van. He sells whole pizzas and slices of pizza from his van.

For the last 3 days, he has timed how long it takes to complete the food order for each of his customers. Giovanni recorded his results in the table below.

Complete the cumulative frequency table and the cumulative frequency diagram.

Time, t (minutes)	Frequency	Cumulative frequency
3 t 5	8	8
5 t 7	24	32
7 t 9	74	106
9 t 11	40	..............................
11 t 13	14	..............................

Cumulative frequency

Use your cumulative frequency diagram to give the best estimates for the answers to each of the following questions.

Find the median time taken to complete a food order.

The median time is_........ minutes.

Giovanni is concerned that food orders are taking too long to complete. He says,

“Only 25% of the food orders are completed in under_........ minutes.”

Use one of the five values below to complete Giovanni’s statement.

6·4 6·6 7·2 8 9·6

For the last 3 days: • Giovanni spent £180 on ingredients • he spent £220 on the running costs for the pizza van • he received a total of £700 from the food orders.

Calculate Giovanni’s percentage profit.

Deva Design Service employs 60 people. Employees were asked to log on to their computer at 9 a.m. on Thursday.

The number of employees logged on was recorded every 10 seconds from 9 a.m. The results are displayed in the cumulative frequency diagram shown below.

Use the cumulative frequency diagram to estimate the median time taken by the

employees to log on.

Deva Design Service has a policy that states the following:

‘90% of employees should be logged on to their computer by 9:01 a.m.’

Show that this policy was met on Thursday.

You must show all your working.

The shortest yacht has a length of 3 m. The longest yacht has a length of 22 m.

Use the information above to complete a box-and-whisker diagram on the graph paper

The lengths of the 68 yachts in Clwyd Marina were measured.

For these yachts: • the lower quartile of their lengths is 10 m • 25% have lengths greater than 18 m • the median length is 11·6 m.

Calculate how many of the yachts in Clwyd Marina have lengths greater

than 10 m.

Use the cumulative frequency diagram to give the best estimates for the answers to the following questions.

What is the median length of these ray fish?

How many of these ray fish have lengths greater than 72 cm?

Draw a box-and-whisker plot of this data on the grid below.

How many of the sample of 60 guppies have a length greater than or equal to

2. 4 cm?

Calculate the mass of the carp in November 2021.

Use the cumulative frequency diagram to give the best estimates for the answers to the following questions.

What is the median length of these ray fish?

How many of these ray fish have lengths greater than 72 cm?

Draw a box-and-whisker plot of this data on the grid below.

Calculate the mass of the carp in November 2021.

Bethan works as an office manager at a medical centre.

Last Monday, 60 patients each had an appointment with a doctor. Bethan recorded how long each patient’s appointment lasted. Her results are given in the table below.

Length of time, t (minutes)	0 t 4	4 t 8	8 t 12	12 t 16	16 t 20	20 t 24
Number of patients	4	24	18	6	2	6

Complete the following cumulative frequency table.

Time, t (minutes)	t 0	t 4	t 8	t 12	t 16	t 20	t 24
Cumulative frequency	0	4	28				60

Each patient is given 10 minutes for their appointment. Use your graph to give the best estimate for the number of appointments that lasted longer than 10 minutes.

Give your answer correct to the nearest whole number of appointments.

Of the patients seen last Monday, what percentage spent longer than 20 minutes with

the doctor?

The median length of the appointments last Tuesday was 11.5 minutes. (e) How much shorter was the median length of the appointments on Monday?

Give your answer correct to the nearest minute.

Bethan is considering changing the time given for each patient’s appointment to 12 minutes. She would set a target of 80% of patient appointments taking less than or equal to 12 minutes. Would this target have been met last Monday?

You must show all your working.

Bethan works as an office manager at a medical centre.

Last Monday, 60 patients each had an appointment with a doctor. Bethan recorded how long each patient’s appointment lasted. Her results are given in the table below.

Length of time, t (minutes)	0 t 4	4 t 8	8 t 12	12 t 16	16 t 20	20 t 24
Number of patients	4	24	18	6	2	6

Complete the following cumulative frequency table.

Time, t (minutes)	t 0	t 4	t 8	t 12	t 16	t 20	t 24
Cumulative frequency	0	4	28				60

Each patient is given 10 minutes for their appointment. Use your graph to give the best estimate for the number of appointments that lasted longer than 10 minutes.

Give your answer correct to the nearest whole number of appointments.

Of the patients seen last Monday, what percentage spent longer than 20 minutes with

the doctor?

The median length of the appointments last Tuesday was 11.5 minutes. (e) How much shorter was the median length of the appointments on Monday?

Give your answer correct to the nearest minute.

Bethan is considering changing the time given for each patient’s appointment to 12 minutes. She would set a target of 80% of patient appointments taking less than or equal to 12 minutes. Would this target have been met last Monday?

You must show all your working.

On Wednesday, there were no cars in a car park when it opened at 7:00 a.m.

From 7:00 a.m., the number of cars entering the car park was recorded.

The cumulative frequency diagram has been drawn from the results.

How many cars had entered the car park by 7:35 a.m.?

The first car entered the car park at 7:21 a.m. The last car entered the car park at 7:44 a.m.

Draw a box-and-whisker plot using this information and the data from the cumulative

frequency diagram.

On Wednesday, there were no cars in a car park when it opened at 7:00 a.m.

From 7:00 a.m., the number of cars entering the car park was recorded.

The cumulative frequency diagram has been drawn from the results.

How many cars had entered the car park by 7:35 a.m.?

The first car entered the car park at 7:21 a.m. The last car entered the car park at 7:44 a.m.

Draw a box-and-whisker plot using this information and the data from the cumulative

frequency diagram.

On 1st July every year, Trefor estate agents record the time from when a phone rings to when it is answered.

The time taken to answer the phone is recorded in seconds.

Trefor displays the data for their agents to see. The displays for 1st July 2018 and 1st July 2019 are shown below.

1st July 2018

Cumulative frequency

Use the diagrams on the previous page to answer the following questions.

What is the range of times taken to answer the phone for 1st July 2018?

Circle your answer.

101 seconds 80 seconds 78 seconds 106 seconds 104 seconds

What is the maximum possible range of times taken to answer the phone for 1st July 2019?

Circle your answer.

86 seconds 106 seconds 80 seconds 56 seconds 83 seconds

The manager of Trefor estate agents claims that there has been an improvement in the median time taken to answer the phone from 1st July 2018 to 1st July 2019. Is this true?

Yes No

You must show all your working.

Complete the following statements.

‘On 1st July 2018, 75% of the phone calls were answered within........

seconds.’

‘On 1st July 2019, 75% of the phone calls were answered within........

seconds.’

On 1st July every year, Trefor estate agents record the time from when a phone rings to when it is answered.

The time taken to answer the phone is recorded in seconds.

Trefor displays the data for their agents to see. The displays for 1st July 2018 and 1st July 2019 are shown below.

1st July 2018

Cumulative frequency

Use the diagrams on the previous page to answer the following questions.

What is the range of times taken to answer the phone for 1st July 2018?

Circle your answer.

101 seconds 80 seconds 78 seconds 106 seconds 104 seconds

What is the maximum possible range of times taken to answer the phone for 1st July 2019?

Circle your answer.

86 seconds 106 seconds 80 seconds 56 seconds 83 seconds

The manager of Trefor estate agents claims that there has been an improvement in the median time taken to answer the phone from 1st July 2018 to 1st July 2019. Is this true?

Yes No

You must show all your working.

Complete the following statements.

‘On 1st July 2018, 75% of the phone calls were answered within........

seconds.’

‘On 1st July 2019, 75% of the phone calls were answered within........

seconds.’

A group of pupils sat a mathematics test.

The teacher grouped their marks using the intervals 1 to 20, 21 to 40, and so on. She then drew the following cumulative frequency diagram to display the results.

Phoebe is one of the pupils who sat the test. Phoebe says, ‘The cumulative frequency diagram shows that the median mark was 70.’

Explain why the median mark may not be 70.

Consider the pupils who had a mark of 80 or less.

How many of these pupils would have needed to score more than 80 for Phoebe’s estimate of the median to be 80?

Circle your answer.

10 12 18 5 20

Which is the modal group?

Circle your answer.

20 to 25 minutes 25 to 30 minutes 30 to 35 minutes

35 to 40 minutes 40 to 45 minutes

Is it certain that the last runner’s finish time was 45 minutes?

You must give a reason for your answer.

Yes No

The organisers hoped that 80% of the runners would finish the race within 30 minutes.

Complete the following two statements.

‘_........ % of runners finished the race within 30 minutes.’

‘80% of runners finished the race within_........ minutes.’

Last year, the median finish time was 26 minutes. By how many minutes was the median time better this year?

You must show all your working.

Which is the modal group?

Circle your answer.

20 to 25 minutes 25 to 30 minutes 30 to 35 minutes

35 to 40 minutes 40 to 45 minutes

Is it certain that the last runner’s finish time was 45 minutes?

You must give a reason for your answer.

Yes No

The organisers hoped that 80% of the runners would finish the race within 30 minutes.

Complete the following two statements.

‘_........ % of runners finished the race within 30 minutes.’

‘80% of runners finished the race within_........ minutes.’

Last year, the median finish time was 26 minutes. By how many minutes was the median time better this year?

You must show all your working.

Meirion is a window cleaner. From Monday to Friday, he records how long he spends cleaning windows for each of his customers.

He draws a cumulative frequency diagram to display the findings.

Use Meirion’s cumulative frequency diagram to find the median and interquartile

range of the times he spends cleaning windows for each of his customers.

Interquartile range_........ minutes

Meirion looks back at his raw data. He finds that the median is actually 17 minutes 30 seconds. Why is there a difference between the median from his cumulative frequency

diagram and the actual median from his raw data?

Meirion is looking at the time it took to clean individual customers’ windows. Find the number of customers whose windows took between 10 and 15 minutes to clean.

Meirion thinks that for approximately 80% of his customers, he cleaned their windows in less than 20 minutes. Is Meirion correct?

You must show all your working.

Meirion is a window cleaner. From Monday to Friday, he records how long he spends cleaning windows for each of his customers.

He draws a cumulative frequency diagram to display the findings.

Use Meirion’s cumulative frequency diagram to find the median and interquartile

range of the times he spends cleaning windows for each of his customers.

Interquartile range_........ minutes

Meirion looks back at his raw data. He finds that the median is actually 17 minutes 30 seconds. Why is there a difference between the median from his cumulative frequency

diagram and the actual median from his raw data?

Meirion is looking at the time it took to clean individual customers’ windows. Find the number of customers whose windows took between 10 and 15 minutes to clean.

Meirion thinks that for approximately 80% of his customers, he cleaned their windows in less than 20 minutes. Is Meirion correct?

You must show all your working.

140 girls were asked how long they spent revising for their GCSE examinations. The cumulative frequency diagram shows the results.

Cumulative frequency

Estimate the median time the girls spent revising.

Circle your answer.

35 hours 40 hours 48 hours 52 hours 70 hours

Calculate the number of girls who spent between 40 and 50 hours revising.

Circle your answer.

0 girls 5 girls 10 girls 15 girls 20 girls

Circle either TRUE or FALSE for each of the following statements.

140 boys were asked how long they spent revising for their GCSE examinations. The cumulative frequency diagram below shows the results.

Cumulative frequency

Trefor makes two statements.

Are both Trefor’s statements correct?

Show calculations and give reasons to support your answers.

Statement 1:_........

Statement 2:_........

140 girls were asked how long they spent revising for their GCSE examinations. The cumulative frequency diagram shows the results.

Cumulative frequency

Estimate the median time the girls spent revising.

Circle your answer.

35 hours 40 hours 48 hours 52 hours 70 hours

Calculate the number of girls who spent between 40 and 50 hours revising.

Circle your answer.

0 girls 5 girls 10 girls 15 girls 20 girls

Circle either TRUE or FALSE for each of the following statements.

140 boys were asked how long they spent revising for their GCSE examinations. The cumulative frequency diagram below shows the results.

Cumulative frequency

Trefor makes two statements.

Are both Trefor’s statements correct?

Show calculations and give reasons to support your answers.

Statement 1:_........

Statement 2:_........