Histograms

The headteacher of Ynysgorwen School wants to know how long it takes pupils to travel to school.

The results for a sample of 120 pupils are shown in the table below.

Time taken, t (minutes)	Frequency	Frequency density
0 t 10	34	3∙4
10 t 20	40	........................................................
20 t 35	24	........................................................
35 t 50	18	........................................................
50 t 70	4	........................................................

Complete the frequency density column.

Draw a histogram to display this data.

Frequency density

Calculate an estimate of the median travel time for the sample of pupils.

What is the greatest possible median travel time for the sample of pupils?

Anisa is comparing the ages of the populations of Cardiff and Wrexham in July 2019.

She has used published data to draw these two histograms.

In July 2019, the biggest difference between the populations of Cardiff and Wrexham was in the number of 20-to 30-year-olds. Calculate how many more 20- to 30-year-olds there were in Cardiff than there were in

Wrexham.

Anisa has used her histograms to carry out data analysis. Some of her results are shown in the table below.

Use Anisa’s histogram to calculate an estimate of the median age of the population of

Cardiff in July 2019.

The frequency table below shows the monthly rainfall in Porthislwyn for the 5-year period from January 2010 to December 2014.

Rainfall, r (mm)	Number of months	Frequency density
0 r 50	6	0∙12
50 r 75	10	0∙4
75 r 100	10	..............................................
100 r 125	7	..............................................
125 r 150	12	..............................................
150 r 200	8	..............................................
200 r 300	7	..............................................

Complete the table above and the histogram below to illustrate this data.

Frequency density

A wet month is classified as one where the monthly rainfall is greater than 60 mm. Calculate an estimate for the percentage of wet months in Porthislwyn in this 5-year period.

You must show all your working.

Madeleine is researching the effects of waves. She does this at two different beaches, Llanddawel and Abertig.

She measures the masses of pebbles in a sample taken from each beach.

Look at the histogram below. It shows the masses of the pebbles in the sample taken from Llanddawel beach.

Calculate an estimate for the percentage of pebbles in Madeleine’s sample that had a

mass of less than 70 g.

Look at the histogram below. It shows the masses of the pebbles in the sample taken from Abertig beach.

The scale on the vertical axis is missing.

120 pebbles each had a mass of less than 30 g.

Use this fact to complete the frequency density axis above.

What was the median mass of the pebbles in the sample taken from Abertig beach?

Circle your answer.

30 g 40 g 45 g 50 g 50∙5 g

Robert’s Plants is a company that grows ornamental trees to sell in its nursery.

The histogram below shows the heights of the 80 trees in the nursery.

Frequency density

Trees that can be sold are at least 50 cm tall.

How many trees are too short to be sold?

Use the histogram to give the best estimates for the answers to the following questions.

What is the median height of the 80 trees?

Circle your answer.

47∙5 cm 48 cm 50 cm 55 cm 60 cm

Calculate the inter-quartile range of the heights of the 80 trees.

PowrUp is a company that makes a certain type of battery.

To find out how long these batteries last, it tests a sample of new batteries. The lifetime of each battery in the sample is recorded. The results are shown in the histogram below.

Frequency density

Use the histogram to answer the following questions:

Calculate the total number of batteries that were in the sample.

PowrUp can advertise the lifetime of these batteries based on the results of these tests. 75% of the sample must exceed this lifetime.

Calculate an estimate of the lifetime of these batteries that PowrUp should use in its adverts.

You must show all your working.

15 patients had to wait up to 20 minutes.

Complete the scale on the frequency density axis.

Show that there were 80 patients in total who waited to see a doctor on this Tuesday.

Calculate an estimate of the inter-quartile range of the waiting times recorded.

You must show all your working.

Estimate of the inter-quartile range =_{……........ …………………} minutes

One Saturday, the waiting time for each patient was recorded. The inter-quartile range of the waiting times for this Saturday was smaller than the inter- quartile range for the Tuesday.

The management used this fact to conclude that waiting times were shorter on Saturdays than on Tuesdays. Was the management correct to come to this conclusion?

Yes No

Explain your answer.

Calculate an estimate of the percentage of passengers who were aged 60 or more.

On the graph paper opposite, draw a histogram to show the distribution of the ages of the

passengers.

Danielle is studying the growth of the population of Wales.

She used the 2001 and 2011 Census data to draw these histograms.

Use Danielle’s histograms to answer the following questions:

Circle the earliest decade in which anyone included in the histograms could have

been born.

1890–1899 1900–1909 1910–1919 1920–1929 1930–1939

From Danielle’s histogram, the population of under-50s in Wales was 1 870 000 in 2001. Calculate the increase in the population of under-50s in Wales from 2001 to 2011.

The 60 to 69-year-olds saw the biggest increase in population from 2001 to 2011. Calculate the percentage increase in the number of 60 to 69-year-olds from 2001 to 2011.

Give your answer to the nearest whole number.

The data in this table was also available to Danielle.

Explain why Danielle did not include this data in her histograms.

The number of 40 to 49-year-olds in 2011 was different to the number of 30 to 39-year- olds in 2001.

Give a full explanation for what could have caused this.

The heights of all the Year 11 girls at a school were measured.

Nia has started to draw a histogram of the results.

Frequency density

There were 24 girls in Year 11 whose heights were in the group 155 h 160 cm.

Use this information to complete Nia’s histogram.

Nia has started to do some data analysis on the heights of the Year 11 girls. She has estimated the median and the upper quartile, as shown in the table below.

Use the histogram to calculate an estimate of the lower quartile of the heights of the

Year 11 girls.

The times taken by a group of pupils to answer a numeracy question were recorded. The histogram below shows some of the results.

The remaining 16 pupils took between 25 and 35 seconds to answer the question.

Complete the histogram.

What is the greatest possible range of times taken by the pupils to answer the question?

Circle your answer.

50 seconds 4 seconds 40 seconds 45 seconds 35 seconds

Calculate the total number of pupils that were in the group.

Gareth was one of the pupils in the group. He says,

“The time I took to answer the question was 18 seconds. This means I was in the fastest 50% of the pupils.”

Explain how Gareth’s statement could be true.

You must use calculations to justify your answer.

Explain how Gareth’s statement could be false.

The Big Fish Cymru annual fishing competition is held on the west coast of Wales.

Information about last year’s competition is displayed in the Big Fish Cymru booklet. A section of this booklet is shown below.

Last year, how many of the fish caught had a mass of less than 250 g?

Last year, the final angler to catch their first fish did so after 3 hours.

How many other anglers took more than 3 hours to catch their first fish?

The number of anglers taking part this year was three times as many as took part last year.

How many anglers took part in the competition this year?

Number of anglers this year was_........

The median mass of the fish caught this year was 0.9 kg.

What is the difference, in kg, between the median mass of the fish caught this year and

the median mass of the fish caught last year?

Difference in mass is_........ kg

Approximately 10% of the anglers this year caught their first fish within 1 hour.

How does this percentage compare with last year’s percentage?

You must show all your working.

Do you think it is fair to compare last year’s competition results with this year’s competition results?

You must give a reason for your answer.