Measures of location

Johan sells tickets for a walking tour, a boat tour and a bus tour.

The pictogram shows the number of people who chose each type of tour one Saturday morning in June.

Write down the modal type of tour.

How many more people chose the boat tour than the bus tour on this morning?

Johan thinks that the number of people choosing each type of tour will change in July compared to June.

He wants to investigate the number of people choosing each type of tour.

Write down a hypothesis that he could use for this investigation.

Suggest two things that Johan should do when collecting his data to ensure that he can compare his results for July with the results for June.

There is an award for board and card games.

Matthew collected a list of the 9 games that were recommended by the awards panel in the critics prize category for 2022

Here is a list of the ratings given to these games by players. A higher rating represents a better game.

8.0 7.79 7.52 7.0 6.6 7.2 6.9 7.59 6.9

(Source: www.spiel‑des‑jahres.de and boardgamegeek.com)

Work out the mean of the ratings.

Work out the range of the ratings.

Matthew also collected the ratings awarded to the games that were recommended in 2021

The table gives information about these ratings.

Matthew concludes that the players liked the games recommended by the panel in 2022 more than they liked the games recommended by the panel in 2021

Do the statistics support this conclusion? Give a reason for your answer.

Give another conclusion based on these statistics.

A company offers a number of activities for its employees. Yusra wants to investigate what employees at the company think about the activities.

Here is her plan for data collection, for calculations and for diagrams.

Discuss whether Yusra’s plans for data collection, for calculations and for diagrams are appropriate.

Passengers for airline flights have their suitcases weighed when they check in.

LocalHop offers flights to and from islands in a small plane.

Here is a list of the weights, in kilograms, of each of the suitcases for a flight to one of the islands on Monday.

16.8 8.9 15.7 16.7 15.4 4.1 19.2 17.4 16.4 15.3 14.9

Mikolaj thinks that it is more appropriate to use the median rather than the mean as an average for these weights.

Explain why Mikolaj may be correct.

Calculate the interquartile range for these weights.

For the return flight on Sunday the interquartile range for the weights of the suitcases is 1.6 kilograms.

Compare these interquartile ranges in context.

Tsena is researching the ages of Oscar winners.

She has collected data from the internet about the ages of the winners of the Best Actor and Best Actress categories from 2000 to 2022

Data from the internet is an example of secondary data.

Explain what is meant by secondary data.

Here are the ages of the winners of the Best Actor category from 2000 to 2022

40 36 47 29 43 37 38 45 50 48 60 50

39 55 44 32 41 41 59 37 83 53 54 (Source: www.openintro.org and www.oscars.org)

The stem and leaf diagram gives the ages of the winners of the Best Actress category from 2000 to 2022

Complete the stem and leaf diagram with the ages of the winners of the Best Actor category. You should also complete the key.

(Source: www.openintro.org and www.oscars.org)

Work out the median age for the winners of the Best Actress category from 2000 to 2022

Michelle thinks it is more appropriate to use the interquartile range than the range as a measure of spread for these sets of data.

Give a reason why Michelle may think this.

Work out the interquartile range for the winners of the Best Actress category from 2000 to 2022

The table gives information about the ages of the winners of the Best Actor and Best Actress categories before 2000 It also gives information about the ages of the winners of the Best Actor category from 2000 to 2022

Compare the distribution of ages for the winners of the Best Actor category with the distribution of ages for the winners of the Best Actress category. You may use the blank column in the table to record your answers to parts (c) and (e) to help you. You should interpret your comparisons in context.

Tsena is researching the ages of Oscar winners.

She has collected data from the internet about the ages of the winners of the Best Actor and Best Actress categories from 2000 to 2022

Data from the internet is an example of secondary data.

Explain what is meant by secondary data.

Here are the ages of the winners of the Best Actor category from 2000 to 2022

40 36 47 29 43 37 38 45 50 48 60 50

39 55 44 32 41 41 59 37 83 53 54 (Source: www.openintro.org and www.oscars.org)

The stem and leaf diagram gives the ages of the winners of the Best Actress category from 2000 to 2022

Complete the stem and leaf diagram with the ages of the winners of the Best Actor category. You should also complete the key.

(Source: www.openintro.org and www.oscars.org)

Work out the median age for the winners of the Best Actress category from 2000 to 2022

Michelle thinks it is more appropriate to use the interquartile range than the range as a measure of spread for these sets of data.

Give a reason why Michelle may think this.

Work out the interquartile range for the winners of the Best Actress category from 2000 to 2022

The table gives information about the ages of the winners of the Best Actor and Best Actress categories before 2000 It also gives information about the ages of the winners of the Best Actor category from 2000 to 2022

Compare the distribution of ages for the winners of the Best Actor category with the distribution of ages for the winners of the Best Actress category. You may use the blank column in the table to record your answers to parts (c) and (e) to help you. You should interpret your comparisons in context.

Agatha is investigating the times taken in the 2022 Olympic Women’s Skiathlon race.

There are two parts to the Skiathlon race. There is a 7.5 km classic stage and a 7.5 km free stage.

Agatha thinks there may be a relationship between the time taken by each skier to complete the classic stage and the time taken by each skier to complete the free stage. She suggests calculating a correlation coefficient to check.

Discuss the appropriateness of this suggestion. You should consider the type of data in your answer.

Agatha found the times taken for the event by all the skiers.

The incomplete histogram and incomplete grouped frequency table give information about the times taken, in minutes, to complete the 7.5 km classic stage of the race.

Use the information in the histogram to complete the table.

Use the information in the table to complete the histogram.

Agatha also drew this histogram for the overall times for the race.

Agatha concludes that the median is in the 50 t ≤ 52 minute time interval.

Does the histogram support Agatha’s conclusion? Show calculations to support your answer.

A ski manufacturer produces wax used on skis.

It has made a new type of wax and wants to find out if using this wax improves the skiers’ times.

Agatha thinks a matched pairs approach should be used to find out if the wax improves the time taken to complete the race.

Agatha suggests matching skiers based on the type of ski that they use and their times in the race. One skier in each pair would use their original wax and the other skier would use the new wax.

Comment on the appropriateness of this suggestion. Give a reason for your answer.

David collected data on the times, in hours, taken for men to complete the Ironman Canada 2022 race and the Ironman UK 2022 race.

These graphs and summary statistics were produced for this data using statistical software.

Mean
13.53 Median
13.49 Number of male competitors 992

Standard deviation 1.839 Skewness 0.0653

Mean
13.98 Median
14.12 Number of male competitors 1128

Standard deviation 1.489 Skewness –0.282

(Source: www.coachcox.co.uk)

Show that there are no outliers in the times for the Ironman Canada 2022 men’s results.

Compare, in context, the distribution of the Ironman Canada 2022 men’s results with the distribution of the Ironman UK 2022 men’s results.

Dan competed in the Ironman Canada 2022 race.

Here is information about the performance of all male competitors in the swim, bike and run stages of the race, and some of Dan’s standardised scores.

Dan took 5.64 hours to complete the run.

Use standardised scores to compare Dan’s performances relative to the other competitors in each stage of the race. Give reasons for your comparisons.

The table shows information about the number of films that each of 12 students watched during one week.

Work out the mean number of films watched per student.

The mean number of films watched by twelve pensioners in the same week is 1.5

Alina claims that the students watched more television in this week than the pensioners.

Explain why Alina’s claim may be incorrect.

For each of the three diagrams below, state whether the mode, median or mean would be the most appropriate average to use to summarise the data.

Tim recorded the number of sheep on each of eight farms. The results are shown below.

44 41 72 74 68 80 68 97

Work out the median.

Tim recorded the number of sheep on the eight farms incorrectly. There are 2 more sheep that need to be included on every farm.

Work out the new median.

Tim also recorded the number of cows and the number of pigs on each of the eight farms. The mean number of cows is 48 The mean number of pigs is 70

Compare the number of cows with the number of pigs for these eight farms.

Tim plans to compare the two sets of data using the interquartile range.

Which average would be the most appropriate to use with the interquartile range?

Leya has collected data about 15 butterflies.

The table gives information about the length of time, in days, each butterfly lived.

Length of time lived (d days)	Frequency
0 d 3	7
3 d 6	4
6 d 9	3
9 d 12	0
12 d 15	0
15 d 18	1

Leya uses the table to calculate an estimate for the mean length of time these butterflies lived. Her estimate for the mean is 4.5 days.

Without performing a calculation, explain why Leya’s value for the mean is an estimate.

Explain why using the mean to represent the average length of time these butterflies lived may not be appropriate.

The population pyramid shows the age distribution of the world population in the year 1960

Each percentage is given correct to one decimal place.

10% 8% 6% 4% 2% 0% 2% 4% 6% 8% 10%

Work out the percentage of males aged 65 or over in 1960

Kiran is carrying out research into how the population of Finland is changing. She uses the data for Finland in 2023

	Finland 2023
Age (y years)	Percentage of the population (%)
0 y 15	14.91
15 y 30	17.13
30 y 45	19.52
45 y 60	18.39
60 y 75	18.88
75 y	11.17

Use linear interpolation to work out an estimate of the median age in Finland in 2023

In 1960, the median age in Finland was 27 years correct to the nearest year. Kiran claims that Finland’s population is getting older.

Explain whether or not Kiran’s claim is correct. You should justify your answer.

On a website, Kiran reads that the median age in Finland in 2023 was 42.8 years. (Source: www.worldometers.info)

Suggest a reason why the median of 42.8 from the website and a median found using linear interpolation would be different.

The table gives information about the median house price in a town called Bulkington, in England, between the years 2015 and 2020

The table also gives, correct to one decimal place, some of the chain base index numbers for these prices.

Calculate the geometric mean of the chain base index numbers from 2016 to 2020 and interpret your result. You must show your working.

Here are some summary statistics for house prices in 2020 for all of England and Wales.

Calculate the skew of this distribution and interpret your result.

A road safety officer records the speed of each vehicle travelling faster than the speed limit of 30 mph as they pass a site.

Here are the results.

Speed of vehicles (vmph)	Frequency
30 v 35	25
35 v 40	18
40 v 45	p
45 v 50	14
50 v 55	3

An estimate for the mean of these data was calculated to be 40.5 mph.

Find the missing value p in the table.

The government recommends that a speed camera should be installed if at least 20% of vehicles travel faster than the speed limit at a site. (Source: www.ageas.co.uk)

Explain why the data collected by the road safety officer cannot be used as evidence that a speed camera should be installed at this site.

A fjord is a deep and narrow part of a sea with steep land on three sides.

Emily is investigating the length of fjords in Norway. She collects some data from the internet and puts the data into a grouped frequency table.

The grouped frequency table below shows information about the results she collected.

Length of fjord (lkm)	Frequency
0 l 50	199
50 l 100	17
100 l 150	12
150 l 200	3
200 l 250	1

(Source: https://en.wikipedia.org/wiki/List_of_Norwegian_fjords)

Work out the number of fjords that have a length of at least 100 km.

Calculate an estimate of the mean length of these fjords. Give your answer to 1 decimal place.

Explain why your answer to part (b)(i) is only an estimate.

How could Emily have improved the accuracy of her answer to part (b)(i)?

Emily plans to use a frequency polygon to represent the lengths of the fjords.

Discuss whether or not a frequency polygon would be an appropriate diagram to use.

Describe and interpret the trend shown by the graph.

Discuss any seasonal variation shown by the graph and interpret one in context. Do not do any calculations.

Work out the mean seasonal variation for Quarter 1 Give your answer correct to 1 decimal place.

Interpret your answer to part (c)(i) in context.

Use your answer to part (c)(i) and the time series graph to predict the number of visitors to Canada from the UK in Quarter 1 of 2020 You must show your working.

Discuss the reliability of using your answer to part (c)(i) to predict the number of visitors to Canada from the UK in Quarter 1 of 2025 Give a reason for your answer.

Jenny is investigating how many days per week people use a gym.

She asks the 40 people in her fitness group how often they use the gym each week. Jenny draws this bar chart for her data.

One of these people is chosen at random.

Find the probability that this person uses the gym exactly 2 days per week.

What is the modal number of days to use the gym each week?

Jenny thinks that there are a lot of people in her fitness group who are exercising less than 2 days per week as there is a total of 10 people who used the gym on 0 days or 1 day per week.

Explain why Jenny might not be correct.

Ben is researching information about the number of British swimming medals won at the Olympics.

Here are his results, giving the number of British swimming medals won at the Olympics from 1900 to 2016

(Source: www.teamgb.com)

Fill in the tally chart for Ben’s results and complete the frequency column.

Suggest a suitable diagram that could be used for Ben’s results.

Write down the mode or modes.

Work out the median.

Ben wants to use an average to summarise the data.

Which of the mode or the median would be more appropriate? Give a reason for your answer.

Tachi collects data on the heights, in metres, of a sample of Egyptian pyramids.

Here is her data.

136.4 101.1 104 62.6 138.8 65.5 93.5 (Source: www.rankred.com)

Tachi picks one of these pyramids at random.

Find the probability that this pyramid will have a height of more than 100 m.

The mean height of a sample of Mexican pyramids is 53.5 m.

Tachi says, “On average these Egyptian pyramids are twice as high as the Mexican pyramids.”

Is Tachi correct? You must show working to support your answer.

The range of heights for the Mexican pyramids is 45 m. The lowest height of the Mexican pyramids is 30 m.

Work out the greatest height of the Mexican pyramids.

Claire is planning an investigation into the length of time that a learner has to wait for a driving test. She wants to find out about how waiting time varies in different regions of the UK.

Here is her plan for data collection, for calculations and for diagrams.

Discuss whether Claire’s plans for data collection, for calculations and for diagrams are appropriate.

Zack is comparing the players from the England Rugby Union team with the players from the Welsh Rugby Union team. The players are listed as Rugby Union Forwards or Rugby Union Backs.

Here are the weights, in kilograms, of the England Rugby Union team players who are listed as Backs.

(Source: www.englandrugby.com/england/senior‑men#squad)

The back‑to‑back stem and leaf diagram gives the weights, in kilograms, of the Welsh Rugby Union Backs.

Complete the back‑to‑back stem and leaf diagram with the weights, in kilograms, of the England Rugby Union Backs.

(Source: www.wru.wales/fixtures‑and‑teams/teams/wales/)

Work out the median weight for the Welsh Rugby Union Backs.

Work out the interquartile range for the Welsh Rugby Union Backs.

The table gives information about the weights for the England Rugby Union Backs.

Zack thinks that the Welsh Rugby Union Backs are heavier and have less consistent weights than the England Rugby Union Backs.

Do the statistics support these conclusions? You must give reasons for your answer.

Amy wants to use the median and interquartile range statistics in a news article for a sports magazine. The article will compare the players on the two teams who are Backs.

Comment on the appropriateness of using the median and the interquartile range in the article.

Give a limitation of using Zack’s statistics to compare all the players on the two teams.

The Consumer Price Index (CPI) is a measure of the rate of change of prices in everyday life.

The table shows the annual average CPI from 2017 to 2021 with 2015 as the base year.

Give an interpretation of the number 108.9 in the table.

Find the percentage increase in consumer prices from 2017 to 2021 You must show your working. Give your answer correct to one decimal place.

A representative sample of consumer product prices is included in the calculations for CPI. In the non‑alcoholic beverages category there are two classes of product included.

The table gives the weightings for these two classes and the price index for one of the

The price index for non‑alcoholic beverages in November 2022 was 124.2 (Source: www.ons.gov.uk)

Calculate the price index for mineral waters, soft drinks and juices in November 2022 Give your answer correct to one decimal place.

The weights in the non‑alcoholic beverage category were different in 2021 and in 2022

The table gives these weights.

Explain what this means for the relative importance in the CPI of ‘Coffee, tea and cocoa’ and ‘Mineral waters, soft drinks and juices’ in 2021 compared with 2022

The rate of inflation is the change in prices for goods and services over time.

David wants to compare annual inflation in Germany from 2017 to 2021 with annual inflation in the UK from 2017 to 2021

The table gives the annual inflation for the UK from 2017 to 2021

(Source: worlddata.info)

From 2017 to 2021 the average annual inflation for Germany was 1.66%.

By calculating an appropriate geometric mean, compare the average annual inflation for Germany from 2017 to 2021 with the average annual inflation for the UK from 2017 to 2021 You must show your working.

David asked 15 of his friends about the number of pets they each have. Here is the data he collected.

0 0 0 0 0 1 1 2 2 2 4 4 4 4 8

Circle the word in the list below that describes this type of data.

continuous qualitative discrete grouped

Write down the modal number of pets.

Find the median number of pets.

State which average, the mode or the median, best represents these data. Give a reason for your answer.

Find the interquartile range of the number of pets.

Wanda asked some of her friends about the number of pets they each have. The table below is a summary of the data she collected.

Compare the distribution of the numbers of pets for David with the distribution of the numbers of pets for Wanda. Give two comparisons and interpret each of your comparisons.

Wanda recorded the highest number of pets as 15 She says that this must be an outlier and concludes that it should be removed from her data.

Give one reason why Wanda’s conclusion may be appropriate.

Give one reason why Wanda’s conclusion may not be appropriate.

Kyle is investigating the heights and the weights of professional basketball players.

He found the weight, in kilograms, of some professional basketball players from 1950 to 1959

Circle the word in the list below that describes weight, in kilograms, as a type of data.

discrete continuous ordinal categorical

The incomplete histogram and incomplete grouped frequency table give information about the weights, in kilograms, of the professional basketball players from 1950 to 1959

Weight (w kilograms)
Frequency

65 w 70	5
70 w 75	15
75 w 80	61
80 w 85	81
85 w 90
90 w 95
95 w 100	35
100 w 105	14
105 w 110	9
110 w 115	1

Use the information in the histogram to complete the table.

Use the information in the table to complete the histogram.

Kyle also drew a histogram for the weights of professional basketball players from 2000 to 2009 This histogram was negatively skewed.

Interpret the negative skew of the weights of professional basketball players from 2000 to 2009

Kyle also collected data about the heights of professional basketball players from 1950 to 1959 and the heights of professional basketball players from 2000 to 2009

The grouped frequency table below gives information about the heights of professional

The estimate of the mean height for professional basketball players from 1950 to 1959 is calculated to be 190.9 cm to one decimal place.

Calculate an estimate of the mean height of basketball players from 2000 to 2009

Comment on how the mean height of professional basketball players has changed between the two sets of data.

Norbert asked each of the students in his class to name their favourite fruit from Apple, Banana, Orange or Pear.

The results are shown below.

Banana Orange Apple Banana Pear Apple

Apple Banana Orange Pear Apple Banana Apple

Apple Apple Orange Apple Pear Banana Banana

Fill in the tally chart for this information and complete the frequency column.

How many students are in the class?

One of the students is chosen at random.

Find the probability that this student’s favourite fruit is Orange.

Compare the number of students whose favourite fruit is Apple to the number of students whose favourite fruit is Pear.

Norbert decides to find the favourite fruit that is the mode.

Explain why the mode is an appropriate average for Norbert to find for this type of data.

Give one advantage of the tally chart over the raw data.

Norbert wants to draw a diagram to represent his results.

Rose is investigating the number of brothers and sisters that students in her secondary school have.

To investigate this she asks 10 students in Year 8 and 10 students in Year 11 how many brothers and sisters they each have.

Assess Rose’s method for her data collection.

The vertical line graph shows the data that she collected.

How many students have 2 or more brothers and sisters?

Write down the mode.

Rose uses her vertical line graph to conclude that no student in her school has 5 or more brothers or sisters.

Assess whether or not Rose’s conclusion is appropriate.

Connie is going to write a report on the difference in total rainfall between London and Aberdeen in 2019

She collects secondary data to investigate this.

What should Connie include in her report?

source of the data her telephone number her age
name of her school

Describe one way that she could obtain this secondary data.

The table shows the total rainfall, in cm, for each month in 2019 in London.

The mean monthly rainfall in Aberdeen in 2019 is 6.2 cm.

Connie considers the data in the table and concludes that the mean monthly rainfall for Aberdeen in 2019 is greater than the mean monthly rainfall in London in 2019

Is Connie correct? You must show how you get your answer.

The cumulative frequency step polygon shows information about the number of goals scored in each of 28 matches played by the German women’s national football team.

(Source: www.worldfootball.net/teams/deutschland‑frauen‑team/)

Give a reason why a cumulative frequency step polygon is used to represent this information rather than a cumulative frequency curve.

Find the mode of the number of goals scored.

Find the number of these matches where

exactly 6 goals were scored,

more than 6 goals were scored.

In 24 matches fewer than n goals were scored.

Find the value of n

Klara tries to calculate the interquartile range of the number of goals scored.

She gets an answer of 14

Explain how you know that her answer is incorrect.

Jane is a car mechanic at MPG Autos.

Jane recorded the time, in minutes, that she took to change a tyre on each of 9 cars.

Here are her results.

25 32 19 27 17 24 35 29 15

Explain whether or not her results have a mode.

Find the median of her results.

Show that the range of her results is 20 minutes.

The range of the times that Jane took to change the oil in each of these 9 cars is 28 minutes.

Are the times that Jane took to change a tyre on each of these 9 cars more consistent than the times that she took to change the oil in each of these 9 cars? Give a reason for your answer.

Dylan investigates the ages of 150 people at a music festival. The youngest person at the festival is 21 years old and the oldest person is 68 years old.

Dylan decides to use a grouped frequency table to show his results.

Give two advantages of using grouped data rather than raw data.

Give one disadvantage of using grouped data rather than raw data.

Dylan is considering using one of the two possible frequency tables, Table A or Table B, shown below.

Dylan claims that Table B gives more detail than Table A about his results.

Assess whether or not Dylan’s claim is appropriate.

Dylan wants to work out the average age of the 150 people at the music festival. He decides to use Table B.

Calculate an estimate for the mean age of the 150 people at the music festival. Give your answer to one decimal place.

Bill is investigating how being grown in the shade and being grown in sunlight affects the heights of tree seedlings.

The following stem and leaf diagrams give information about the heights, in centimetres, of 17 tree seedlings grown in the shade and 17 tree seedlings grown in sunlight.

The seedlings were all planted at the same time.

Compare the average height of the tree seedlings grown in the shade with the average height of the tree seedlings grown in sunlight.

State clearly the value of each average you use in order to make your comparison.

Weronika works for a road traffic organisation. One day she is investigating the speeds of cars and the speeds of motorcycles along a motorway.

Here is part of the spreadsheet that Weronika used to record her results.

Speed (s miles per hour)	Percentage of cars	Percentage of motorcycles
30 s 40	3	0
40 s 50	18	two
50 s 60	42	7
60 s 70	27	124
70 s 80	6	56
80 s 90		11
Total	100	100

Give a reason why Weronika will need to clean the data.

Weronika concludes that the value of 124 in the spreadsheet must be wrong.

Explain why.

Here is the information about motorcycles from the spreadsheet with the data cleaned.

Speed (s miles per hour)	Percentage of motorcycles
30 s 40	0
40 s 50	2
50 s 60	7
60 s 70	24
70 s 80	56
80 s 90	11
Total	100

Use linear interpolation to work out an estimate of the median speed of the motorcycles.

The frequency polygon for the speeds of cars has been drawn on the grid.

On the same grid, draw the frequency polygon for the speeds of the motorcycles.

Using the two frequency polygons, compare the skew of the distribution of the speeds of the cars with the skew of the distribution of the speeds of the motorcycles. Explain what your comparison means in context.

Bill is investigating how being grown in the shade and being grown in sunlight affects the heights of tree seedlings.

The following stem and leaf diagrams give information about the heights, in centimetres, of 17 tree seedlings grown in the shade and 17 tree seedlings grown in sunlight.

The seedlings were all planted at the same time.

Compare the average height of the tree seedlings grown in the shade with the average height of the tree seedlings grown in sunlight.

State clearly the value of each average you use in order to make your comparison.

Weronika works for a road traffic organisation. One day she is investigating the speeds of cars and the speeds of motorcycles along a motorway.

Here is part of the spreadsheet that Weronika used to record her results.

Speed (s miles per hour)	Percentage of cars	Percentage of motorcycles
30 s 40	3	0
40 s 50	18	two
50 s 60	42	7
60 s 70	27	124
70 s 80	6	56
80 s 90		11
Total	100	100

Give a reason why Weronika will need to clean the data.

Weronika concludes that the value of 124 in the spreadsheet must be wrong.

Explain why.

Here is the information about motorcycles from the spreadsheet with the data cleaned.

Speed (s miles per hour)	Percentage of motorcycles
30 s 40	0
40 s 50	2
50 s 60	7
60 s 70	24
70 s 80	56
80 s 90	11
Total	100

Use linear interpolation to work out an estimate of the median speed of the motorcycles.

The frequency polygon for the speeds of cars has been drawn on the grid.

On the same grid, draw the frequency polygon for the speeds of the motorcycles.

Using the two frequency polygons, compare the skew of the distribution of the speeds of the cars with the skew of the distribution of the speeds of the motorcycles. Explain what your comparison means in context.

Suzie is investigating the profits made by two different shops, shop A and shop B. Suzie has obtained the annual percentage profits made by shop A for the years 2015 to 2019 and the annual percentage profits made by shop B for the years 2016 to 2019

The table below gives this information.

(Source: adapted from www.tescoplc.com/investors/reports)

Suzie concludes that the average annual percentage profit made by shop B over the 4 years is greater than the average annual percentage profit made by shop A over the 5 years.

By using appropriate geometric means, assess Suzie’s conclusion. You must show your working.

For a survey, Kayleigh counted the number of people in each of the 20 cars that arrived at her office car park one morning.

Here are her results.

Fill in the tally chart for this information and complete the frequency column.

Write down the mode.

At Kayleigh’s office there is a scheme to reduce the number of cars arriving at the office car park which have only 1 person in the car.

Kayleigh concludes from her survey that the scheme is not working.

Explain whether or not Kayleigh’s results support her conclusion.

John suggests that Kayleigh’s results are not suitable for her to reach a reliable conclusion.

Give one reason why John may be correct.

A garden centre has a register of 400 loyalty card members. The owner of the garden centre wants to find out information about the distribution of the amount of money spent by each loyalty card member on each visit to the garden centre.

The owner decides on the following plan.

• Record the amount of money spent by each of the first 50 customers to the garden centre on Monday morning.

• Draw a bar chart for the amounts of money spent.

• Find the mean of the amounts of money spent.

Discuss whether the manager’s plan is likely to be effective.

You should comment on both the collection of data and on the analysis of results.

Where appropriate you should suggest improvements to the plan.

The mean of the daily winter temperatures and the mean of the daily summer temperatures in the UK have been recorded for each year from 1910 to 2018

This information has been used to model the two distributions of the mean temperatures.

The graphs below give information about these models.

Write down the name of the distribution that is suggested by each of these graphs.

Comment on the difference between the means of these two distributions.

The standard deviation for the distribution of the summer temperatures is 0.8 °C, correct to one decimal place.

Using the graph for the winter temperatures, calculate an estimate for the standard deviation of the distribution of the winter temperatures, correct to one decimal place.

The maximum height of the graph above the temperature axis for the winter temperatures is less than the maximum height of the graph above the temperature axis for the summer temperatures.

Explain why.

Carol says that the standard deviation of daily temperatures in summer for the years from 1910 to 2018 will also be 0.8 °C.

Explain whether or not Carol is correct.

Greta assumes that the means of the daily temperatures each year are independent.

She works out the probability that the means of the daily summer temperatures are greater than 15.6 °C for two consecutive years.

She concludes that this probability is less than 0.001

Using the model for the daily summer temperatures, assess Greta’s conclusion.

Julie asked 20 of her friends how many days last week they each played a sport.

Here are her results.

3 4 5 3 2 2 3 3 2 5 1 0 1 0 2 3 3 2 5 3

Which word from the list below best describes Julie’s data before it is processed?

Julie draws the following tally chart for her results.

Write down two things that could be misleading or are wrong with Julie’s tally chart.

The median for Julie’s results is 3

Interpret this value in context.

Julie says the mode of her results is 7

Is Julie correct? Give a reason for your answer.

Albert groups Julie’s results and then draws the following bar chart.

Assess whether or not Albert’s way of presenting Julie’s results is appropriate. Give two reasons for your answer.

Bethany is the union representative at the large company where she works. She is investigating the salaries of the employees at the company.

Bethany takes a simple random sample of 8 employees at the company and asks each of them to tell her their salary.

Describe what you understand by the term ‘simple random sample’.

The salaries of the 8 employees are listed below.

£16 000 £3 000 £23 000 £31 000

£21 000 £24 000 £17 000 £25 000

Work out the mean.

Show that the range is £28 000

Bethany believes that one of the salaries is an outlier.

Write down the salary that is most likely to be an outlier.

Explain why you think this value is an outlier.

Bethany removes the outlier and calculates the mean of the remaining seven salaries.

Without carrying out any further calculations, state whether the mean of the remaining seven salaries is greater than, is equal to or is less than the mean of all eight salaries. Give a reason for your answer.

Bethany uses the mean of the remaining seven salaries to draw conclusions about the salaries of all of the employees at her company.

Describe two things that could affect the reliability of her conclusions.

Shona and Luke are investigating housing.

Shona wants to find out about the types of house in which people live. She plans to collect data by asking people to tell her in which type of house they each live.

Complete the sentence with an appropriate statistical word.

Shona is collecting the data herself and so this is_........ data.

Design a data collection sheet for Shona to use.

Luke has collected data about the number of people living in each of the 12 houses in his street.

Here is his data.

2 2 3 3 3 4 4 4 4 5 5 11

Luke wants to work out the average of the number of people living in each house in his street.

Which average, mean or median, would it be more appropriate for Luke to use? Give a reason for your answer.

There is one outlier in the data.

Excluding the outlier, work out the range of the number of people living in the houses in the street.

Shona has collected data about the number of people living in each of the houses in her street. She has collected data for 2014 and for 2019

The table gives the range of the number of people living in the houses in Shona’s street

Compare the range for 2014 with the range for 2019 and interpret your comparison in context.

Ruth is investigating the results of the Six Nations rugby competition.

She finds the following summary information for the Six Nations rugby competition in 2018 and in 2019

In 2018, which two teams had fewer ‘points scored for’ than ‘points scored against’?

Two teams finished in the same final position in 2019 as they had finished in 2018

Write down the names of these two teams.

Which team won 2 matches in 2018 and 3 matches in 2019?

Ruth concludes that in 2018 and in 2019 Wales had a more successful rugby team than Scotland.

Does the information in the tables support her conclusion? Give a reason for your answer.

Ruth found from a newspaper the age of each player from the England team for a 2019 Six Nations rugby match.

Here are the ages.

20 24 25 25 26 26 26 27 27 28 28 28 28 29 29

(Source: www.irishtimes.com)

Find the mode of the ages.

Ruth plans to use a stem and leaf diagram to represent the ages.

Discuss whether or not a stem and leaf diagram would be an appropriate diagram to use.

Ben is investigating how far people travel to work each day. He asks the people in his office how far, in miles, they travel to work each day.

The grouped frequency table gives information about their answers.

Distance (d miles)	Frequency
0 d 5	7
5 d 10	10
10 d 15	9
15 d 20	6
20 d 25	3

Calculate an estimate of the mean distance. Give your answer correct to one decimal place.

Ben finds a newspaper report that says that the average distance travelled to work each day in England and Wales in 2001 was 8.3 miles. (Source: www.telegraph.co.uk)

From this Ben concludes that the distance travelled to work each day in England and Wales has increased since 2001

Give two reasons why Ben’s conclusion may not be reliable.

A firm produces pottery. The firm produces 1000 plates each week.

The manager of the firm wants to take a systematic sample of 5% of the plates to check for faults.

Describe how the manager should take this sample.

Give one disadvantage of using systematic sampling.

The manager also collects information about the orders that the firm has received.

The grouped frequency table gives a summary of the costs of the 80 orders received last week.

Cost (£c)	Frequency
0 c 50	15
50 c 100	18
100 c 150	20
150 c 200	19
200 c 250	3
250 c 300	5

Write down the class interval which contains the median cost.

Use linear interpolation to work out an estimate of the median cost.

Ben is investigating how far people travel to work each day. He asks the people in his office how far, in miles, they travel to work each day.

The grouped frequency table gives information about their answers.

Distance (d miles)	Frequency
0 d 5	7
5 d 10	10
10 d 15	9
15 d 20	6
20 d 25	3

Calculate an estimate of the mean distance. Give your answer correct to one decimal place.

Ben finds a newspaper report that says that the average distance travelled to work each day in England and Wales in 2001 was 8.3 miles. (Source: www.telegraph.co.uk)

From this Ben concludes that the distance travelled to work each day in England and Wales has increased since 2001

Give two reasons why Ben’s conclusion may not be reliable.

Birgitta works in a wildlife reserve. She has been researching the size of tiger feet in order to help in the identification of animal tracks.

The table gives information about the front pad width of adult tigers.

(Source: keywordsuggest.org)

Compare in context the distribution of the front pad widths of adult male tigers with the distribution of the front pad widths of adult female tigers.

Explain when it would be more appropriate to use the median rather than the mean in describing a distribution.

Matthew is investigating how the cost of computer data storage has changed over time.

Suggest a hypothesis Matthew could use.

Matthew finds the cost per terabyte of computer data storage in each first quarter of five successive years.

The table gives cost per terabyte, in US$, for computer data storage. It also gives two of the chain base index numbers, correct to one decimal place, for this information.

(Source: jcmit.net)

Find the chain base index numbers for 2018 and 2019 and write them in the table. Give each value correct to one decimal place.

Find the geometric mean of the four chain base index numbers. You must show your working. Give your answer correct to one decimal place.

Interpret your answer.

Dalia works in a cafe. One Monday morning she recorded what type of milk each customer wanted in their coffee.

The tally chart gives the number of customers who wanted whole milk or skimmed milk or soya milk or no milk.

The list below gives the type of milk wanted by each of the other customers.

oat oat almond oat almond almond oat almond almond almond almond

Complete the tally chart.

Write down the type of milk that is the mode.

On Tuesday, Dalia again recorded what type of milk each customer wanted in their coffee.

The table below gives information about her results.

Determine whether or not both days have the same mode. You must give a reason for your answer.

Cindy works in a library. She recorded the number of books borrowed by each person in a random sample of 20 people who borrowed books from the library one Monday. The table gives information about her results.

Cindy also recorded the number of books borrowed by each person in a random sample of 20 people who borrowed books from the library one Friday. The table gives two summary statistics for her results.

She claims that the mean number of books borrowed from the library on a Monday is greater than the mean number of books borrowed from the library on a Friday. Cindy also claims that there is a smaller spread in the number of books borrowed from the library on a Monday than on a Friday.

Comment on whether each of Cindy’s claims is supported by her results. You must give reasons for your answers.

Suzanne suggests repeating the investigation with a larger number of Mondays and Fridays.

Give a reason why Suzanne’s suggestion may be appropriate.

Describe the trend in visitor numbers to the museum over these three years.

For which quarter each year does the museum have the fewest visitors?

David has drawn a right-angled triangle on the graph from 2015 Quarter 1 to 2016 Quarter 4 so that he can calculate the gradient of the trend line, correct to one decimal place.

Here is his calculation.

Interpret, in context, the gradient of 1.4

David used the trend line and mean seasonal variation (or average seasonal effect) to predict the number of visitors to the museum for Quarter 2 of 2018

Comment on the validity of a prediction found in this way.

The actual number of visitors to the museum in Quarter 2 of 2018 was 203 100

Determine how close David’s prediction was to 203 100

Richard finds information from the internet about the most popular colours for cars in the UK in 2017

Table 1 gives this information.

Explain why the information in Table 1 is secondary data.

Richard drew this pictogram to represent his information about the most popular car colours.

Complete this key for the pictogram.

Jill says that a pictogram is not the most appropriate way to display Richard’s information.

Is Jill correct? Give a reason for your answer.

Richard decides to collect some data about car colours by recording the colour of each of the 60 cars in a car park.

Table 2 gives information about his results.

The modal car colour from Table 2 is blue.

Explain whether or not blue is the expected modal car colour based on the information in Table 1

Explain why, for the information in Table 2, the mode is the most appropriate average to use.

Using the information in Table 1 and Table 2, compare the number of silver cars in the car park with the expected number of silver cars in a car park with 60 cars.

Some people were asked to complete three puzzles, Puzzle X, Puzzle Y and Puzzle Z.

The table shows some summary statistics about the times taken, in minutes, to complete Puzzle X and Puzzle Y.

The shortest time taken to complete Puzzle X was 26 minutes.

Work out the longest time taken to complete Puzzle X.

Use the information in the table to compare the median time to complete Puzzle X with the median time to complete Puzzle Y. You should interpret your comparison.

The list below gives the times taken, in minutes, for these people to complete Puzzle Z.

25 27 31 39 32 32 37 34

38 31 46 45 47 42 41

Work out the range of these times.

Which puzzle, X, Y or Z, is completed with the most consistent times by these people? Explain your answer.

Hannah wants to find the median of the 15 times taken, in minutes, to complete Puzzle Z.

She draws this stem and leaf diagram for these 15 times.

She says that as there are 15 values, the median is the 8th value, which is 34 minutes.

Explain what Hannah has done wrong. You should give the correct value of the median.

Daniel is interested in the price of houses.

He collects data on the average selling price of houses in the UK for four months in 2017

His results are shown in the table.

(Source: landregistry.data.gov.uk)

Work out the mean of the four prices in the table.

Daniel uses statistical software to work out these summary statistics.

(Source: landregistry.data.gov.uk)

Daniel writes these two conclusions.

Conclusion 1:

The average selling price for each of the other 8 months in 2017 must be less than £221 244

Conclusion 2:

The mean of the monthly average selling price for all of 2018 is greater than the mean of the monthly average selling price for all of 2017

Using the information from the tables and your answer to part (a), assess the validity of Daniel’s two conclusions.

Daniel is also interested in how the prices of detached houses have changed since he bought his house in 2011

Using the internet, he finds that the index number for the price of a detached house in 2018, using 2011 as the base year, is 132

Give an interpretation of this index number.

The price of the detached house that Daniel bought in 2011 was £260 000

Which is the correct calculation to find an estimate for the price of this house in 2018?

Daniel could not find an index number for the prices of flats in 2018, using 2011 as the base year.

He knows the price of a flat in 2011

He plans to use the index number 132 that he found for the price of detached houses in order to find an estimate for the price of this flat in 2018

Assess the suitability of Daniel’s plan.

The Forestry Commission planted Field Maple trees and Silverleaf Maple trees in region A. They measured the heights of the trees after 14 years.

The incomplete histogram and incomplete grouped frequency table give information about the heights, in metres, of the Field Maple trees in region A.

Height (h metres)	Frequency
0.8 h 1.2	2
1.2 h 1.6	4
1.6 h 2.0	13
2.0 h 2.4	25
2.4 h 2.8	10
2.8 h 3.2
3.2 h 3.6

(Source: data.gov.uk)

Use the information in the histogram to complete the table.

Use the information in the table to complete the histogram.

The histogram below gives information about the heights of the Silverleaf Maple trees after 14 years.

Identify and interpret the type of skew shown in the histogram for Silverleaf Maple trees.

The Forestry Commission also planted Field Maple trees and Silverleaf Maple trees in region B. The grouped frequency table below gives information about the heights of these trees.

		Frequency
Height (h centimetres)	Field Maple	Silverleaf Maple
0 h 80	1	1
80 h 160	14	4
160 h 240	27	32
240 h 320	21	13
Total	63	50

The estimate of the mean for Field Maple trees is calculated to be 206.3 cm to 1 decimal place.

David thinks that the estimate of the means for Field Maple trees and for Silverleaf Maple trees suggests that Field Maple trees are taller than Silverleaf Maple trees.

Is David correct? You must show your working. Give one limitation of your conclusion.

The Forestry Commission planted Field Maple trees and Silverleaf Maple trees in region A. They measured the heights of the trees after 14 years.

The incomplete histogram and incomplete grouped frequency table give information about the heights, in metres, of the Field Maple trees in region A.

Height (h metres)	Frequency
0.8 h 1.2	2
1.2 h 1.6	4
1.6 h 2.0	13
2.0 h 2.4	25
2.4 h 2.8	10
2.8 h 3.2
3.2 h 3.6

(Source: data.gov.uk)

Use the information in the histogram to complete the table.

Use the information in the table to complete the histogram.

The histogram below gives information about the heights of the Silverleaf Maple trees after 14 years.

Identify and interpret the type of skew shown in the histogram for Silverleaf Maple trees.

The Forestry Commission also planted Field Maple trees and Silverleaf Maple trees in region B. The grouped frequency table below gives information about the heights of these trees.

		Frequency
Height (h centimetres)	Field Maple	Silverleaf Maple
0 h 80	1	1
80 h 160	14	4
160 h 240	27	32
240 h 320	21	13
Total	63	50

The estimate of the mean for Field Maple trees is calculated to be 206.3 cm to 1 decimal place.

David thinks that the estimate of the means for Field Maple trees and for Silverleaf Maple trees suggests that Field Maple trees are taller than Silverleaf Maple trees.

Is David correct? You must show your working. Give one limitation of your conclusion.

Lena is researching information about the numbers of British Oscar winners.

Here are her results, giving the number of British Oscar winners for each of the years 1981 to 2010

(Source: www.theguardian.com)

Lena found the data in a newspaper.

Correctly complete the following sentence.

Because Lena found the data in a newspaper, the data is_........ data.

Fill in the tally chart for Lena’s results and complete the frequency column.

Write down the modal group.

Work out the total number of years between 1981 and 2010 for which there were fewer than 7 British Oscar winners.

Suggest a suitable diagram that could be used for Lena’s results.

Here are the weights, in kilograms, of 10 dogs.

9.2 4.3 68.2 12.3 6.5 10.6 14.2 16.0 9.1 8.5

Work out the range.

Work out the median.

The median is more appropriate than the mean to summarise the data.

Explain why.

Suresh is a secondary school student.

He wants to find out how many days off school each teacher has had in the last year.

Suresh plans to find out this information by using his school’s employee records.

Give one potential problem with Suresh’s data collection plan.

Ami and Tia work in a primary school.

Ami and Tia are investigating the numbers of days that people working for their school have been absent due to illness.

The table gives information about the number of days that each of 30 people working for the primary school have been absent due to illness in the last year.

Ami and Tia want to work out the average number of days absent due to illness.

Which of these two methods is correct, Ami’s or Tia’s? You must give a reason for your answer.

Find the median number of days absent due to illness.

Chris is carrying out an experiment to see if left handed people have a better memory than right handed people. He has two groups of people with the same number of people in each group.

Group A contains left handed people only and group B contains right handed people only.

Each person is given 15 objects to memorise and then they are tested to see how many objects they remember.

The variables for the experiment are shown in the table.

For each variable, put a tick () in the correct column of the table to show the type of variable.

Chris is not sure whether to work out the mean or the median number of objects each group remembered.

Describe a situation for which it would be more appropriate to work out the median than the mean.

Chris worked out these summary statistics for the number of objects people in group A remembered.

Lower quartile = 5 Median = 7 Upper quartile = 8

Sanjit is a member of group A.

He remembered 14 objects.

Determine whether or not the number of objects Sanjit remembered is an outlier.

The table shows information about the amount of time that each member of a group of 46 teenagers spent on social media during one day.

Time spent on social media (t minutes)	Frequency
0 t 50	1
50 t 100	4
100 t 150	8
150 t 200	17
200 t 300	16

Use linear interpolation to find an estimate of the median time spent on social media by the 46 teenagers.

The average person will spend approximately 116 minutes on social media each day.

(Source: www.socialmediatoday.com)

Compare the amount of time spent on social media by the 46 teenagers with the amount of time spent on social media by the average person.

The table below gives the mean, the standard deviation and the median for the times spent on social media during one day by a sample of sixty year olds.

Calculate the skew of the times spent on social media by the sample of sixty year olds.

Interpret your answer to part (b).

Mr Singh collected data about the number of students in each of three classes in his school and the mean mark of each class in a science test.

The table gives some information about his data.

Mr Singh plans to use one of the following two methods to work out the mean mark of all the students in the three classes.

Method 1 Work out the mean of 63, 72 and 55

Method 2 Given that Mr Singh knows the value of n, work out the weighted mean mark for the three classes.

For each of these two methods, assess whether or not the method is an appropriate way to work out the mean mark of all the students in the three classes.

Method 1

Method 2

The weighted mean mark for the three classes is 64.1 correct to one decimal place.

Calculate the value of n.

The three classes have also taken a mathematics test.

Mr Singh thinks that there is a relationship between the science marks and the mathematics marks.

He draws a scatter diagram for each of the three classes.

For each diagram, he used the science mark, x, for each student in the class as the explanatory variable and the corresponding mathematics mark, m, as the response variable.

Mr Singh then used statistical software to find the equation of the regression line for the data in each scatter diagram.

Here are the equations.

Mr Singh thinks that he has made a mistake with one of the equations.

Compare the equations of the three regression lines and explain which one is most likely to be incorrect. Explain your answer in context.

Mr Singh decides to compare the science marks, x and the history marks, y, of class A.

He uses statistical software to draw this scatter diagram and the regression line.

Find an equation of the regression line of this scatter graph in the form y = a + bx