Histograms

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.

The mass of some parcels are recorded. The table and the histogram each show part of this information.

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.

The histogram summarises the heights, h cm, of some plants in a garden centre.

Show that there are 80 plants with a height in the interval 0 < h ⩽ 100.

The value, in pounds, of each plant depends on the plant’s height. The table below shows this information.

Use this information to find the total value of the plants represented in the histogram.

A farmer grows pumpkins. The farmer records the masses, m kilograms, of 80 of their pumpkins. The table shows the results.

Complete the cumulative frequency table.

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.

A group of students record the time taken to travel to school. All students in the group took less than an hour to travel to school. Some of their results are recorded on this histogram.

16 students took less than 5 minutes to travel to school.

How many students took less than 20 minutes to travel to school?

9 students took between 30 and 45 minutes to travel to school.

Add these students’ results to the histogram.

The histogram shows the heights of some plants out of a total of 80 plants.

Complete the histogram to show the plants with heights between 0 cm and 20 cm.

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.

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.

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.

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.

80 cyclists take part in a race. A summary of their times is shown in the table.

Complete the cumulative frequency table.

Draw the cumulative frequency graph to show the information.

Reece makes two comments about the times taken to complete the race.

For each comment, decide if Reece is right or wrong and give a reason for your answer.

of the 80 cyclists took more than 30 minutes to complete the race.

Reece is........ because........

The longest time that any of the 80 cyclists took to complete the race must have been 50 minutes.

Reece is........ because........

The height, h cm, of each member of a tennis club is recorded. The histogram shows some of the results.

40% of the members have a height in the interval 160 ⩽ h < 170. 30% of the members have a height in the interval 170 ⩽ h < 180. 100% of the members have a height in the interval 140 ⩽ h < 200.

Complete the histogram for the intervals 170 ⩽ h < 180 and 180 ⩽ h < 200.

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.

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 histogram shows information about the weights of some of the parcels handled by a delivery company in one month.

Zoe says

There are fewer parcels weighing between 450 g and 700 g than parcels weighing between 300 g and 450 g.

Is Zoe correct? Show how you decide.

The delivery company delivered 6500 parcels weighing between 50 g and 100 g.

Complete the histogram to show this information.

Zoe uses the histogram to calculate the number of parcels weighing between 200 g and 250 g.

Explain why Zoe’s answer is unlikely to be reliable.

30 students completed a puzzle and their times were recorded. All of the students completed the puzzle in less than 50 seconds. The histogram shows information about some of their times.

Complete the histogram for those completing the puzzle in less than 10 seconds.

Ceri records the time taken, t minutes, to travel to school for a sample of 168 students at her

Draw a histogram to represent this information.

Ceri says

The longest time that any of these students took to travel to school was 80 minutes.

Is she correct? Give a reason for your answer.

Ceri also claims that 25% of all of the students at this Academy took more than 30 minutes to travel to school.

Show how Ceri might have worked out her claim.

State one assumption that Ceri has made in making her claim.

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) ?

The cumulative frequency graph shows the speeds, in miles per hour (mph), of vehicles passing a 40 mph speed limit sign on a road.

A speed camera will be installed if more than 30% of vehicles go over the speed limit of 40 mph.

Use information from the graph to decide if a speed camera should be installed.

Ana records the amount of money spent by 140 customers in her shop on one day.

Draw a histogram to represent this information.

Ana wants to offer a discount to the customers who spend the most money in her shop.

She wants to give the discount to approximately 25% of her customers.

Suggest a suitable amount of money for Ana to use on her voucher. Justify your decision.