Index numbers & rates of change

The choropleth map gives information about the percentage change in the mileage driven in Great Britain for each region from 2020 to 2021

Percentage change in mileage driven in 2020 to 2021

(Source: https://roadtraffic.dft.gov.uk)

The table shows the number of miles driven in London in 2020 and in 2021

Show that London has been shaded correctly on the choropleth map.

Which region had the lowest percentage change in the mileage driven between 2020 and 2021?

Amy concludes that the percentage change in the mileage driven between 2020 and 2021 is the same for Wales and South West England.

Explain why Amy may not be correct.

Tom concludes that the number of motor vehicles using the roads has increased the most in Scotland between 2020 and 2021

Explain whether or not the choropleth map supports this conclusion.

The choropleth map gives information about the percentage change in the mileage driven in Great Britain for each region from 2020 to 2021

Percentage change in mileage driven in 2020 to 2021

(Source: https://roadtraffic.dft.gov.uk)

The table shows the number of miles driven in London in 2020 and in 2021

Show that London has been shaded correctly on the choropleth map.

Which region had the lowest percentage change in the mileage driven between 2020 and 2021?

Amy concludes that the percentage change in the mileage driven between 2020 and 2021 is the same for Wales and South West England.

Explain why Amy may not be correct.

Tom concludes that the number of motor vehicles using the roads has increased the most in Scotland between 2020 and 2021

Explain whether or not the choropleth map supports this conclusion.

Henry is investigating the price of some ingredients needed to make a cake.

The table shows the price index of twelve eggs and 1 kilogram of sugar in the UK in the years 2010, 2015 and 2020

The base year price is 2010 for both ingredients.

(Source: © Office for National Statistics)

Calculate the percentage change in the price of twelve eggs between 2010 and 2020

In 2010, the price of 1 kilogram of sugar in the UK was 95 p.

Calculate the price of 1 kilogram of sugar in 2020 Give your answer correct to the nearest penny.

In 2020, the total cost of all the ingredients was 450 p. Henry predicts that he will not have to pay more than 450 p for the same ingredients today.

Comment on the appropriateness of Henry’s prediction. Give a reason for your answer.

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.

The table gives the total labour force and the unemployment rate for the UK in 2017

(Source: www.economicsonline.co.uk and www.ons.gov.uk)

In 2019 the total number of unemployed people was 1.29 million.

Using the formula above, work out the unemployment rate for the UK in 2019 Give your answer correct to 2 decimal places and write your answer in the table.

Using your answer to part (a)(i), what conclusion can be drawn about the unemployment rate in the UK between 2017 and 2019?

Bob says, without doing any calculations, that the total number of people unemployed decreased from 2017 to 2018

Using the data in the table, assess Bob’s claim.

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.

The graph below shows the crude birth rate and crude death rate for Malta from 2000 to 2014

Birth rate and death rate in Malta from 2000 to 2014

Birth rate in Malta from 2000 to 2014 Death rate in Malta from 2000 to 2014

(Source: www.indexmundi.com /)

Using the information from the graph above Lottie concludes

“The total population of Malta has increased between 2000 and 2014”

Explain how the graph can be used to support Lottie’s conclusion.

Give one reason why Lottie’s conclusion might not be correct.

In 2015, the population of Malta was 445 053 In the same year there were 4398 births in Malta.

Using the formula below, work out the crude birth rate in Malta in 2015 Give your answer correct to 1 decimal place.

The time series graph shows the percentages of households in the United Kingdom with

(Source: Office for National Statistics)

Write down an estimate for the percentage of households with internet access in

2000

2018

The percentage of households with internet access in 2019 was 93%

Plot this information on the time series graph.

Write down the first year where the percentage of households with internet access was greater than 50

Using the information given in the time series graph,

compare the change in percentage of households with internet access from 2001 to 2005 with the change in percentage of households with internet access from 2015 to 2019

You must show how you get your answer.

The table gives the chain base index number for the average price of houses in the United Kingdom in January for each of the years 2016 to 2019

(Source: landregistry.data.gov.uk)

Describe what the chain base index numbers show about the average price of houses in January for the years 2016 to 2019

In January 2017, the average price of houses in the United Kingdom was £215 243

Calculate the average price of houses in the United Kingdom in January 2016 Give your answer correct to the nearest £

The table gives the populations, in thousands, and the crude birth rates for two regions of the United Kingdom in 2016

The table also gives the population, in thousands, of the United Kingdom in 2016

(Source: Office for National Statistics)

In 2016 the number of births in the United Kingdom was 774 835

Using the formula above, work out the crude birth rate for the United Kingdom in 2016 Give your answer correct to one decimal place and write your answer in the table.

Work out the number of births in Cumbria in 2016

Stephen says that without doing any calculations he knows there were more births in Norfolk than in Cumbria, in 2016

Using the information in the table, explain why Stephen is correct.

The table shows the index number for the average cost of a package holiday for 2018 and for 2019, using 2015 as base year.

(Source: Office for National Statistics)

A family spent a total of £4650 on a package holiday in 2015

Work out an estimate for the cost of the same holiday in 2018

Thomas does the following calculation to reach a conclusion about the cost of package holidays.

Is Thomas correct? Give reasons for your answer.

Changes in the cost of living in the United Kingdom are measured by the consumer price index (CPI).

The table shows the index numbers for June 2019, using 2015 as base year, of the items that contribute to the CPI.

The table also shows the weightings for how the spending by the Jones family is divided between these items.

(Source: Office for National Statistics)

Using 2015 as base year, the CPI for June 2019 was 107.9

Compare the overall change in the cost of living for the Jones family between 2015 and June 2019 with the change in the CPI between 2015 and June 2019

Explain whether or not it is possible to conclude from your calculations that the Jones family is financially better off or worse off in June 2019 as compared with 2015

The table shows the value of the gross domestic product (£ million) of the UK for each

(Source: https://www.ons.gov.uk/economy/grossdomesticproductgdp)

A country is in recession when its gross domestic product falls in two or more consecutive quarters.

The UK went into recession in 2008 Quarter 3

A country comes out of recession in the quarter in which its gross domestic product rises.

According to the table, in which year and quarter did the UK come out of this recession?

Using 2007 Quarter 1 as the base, work out the simple index number for the gross domestic product of the UK in 2008 Quarter 1 Give your answer correct to the nearest whole number.

The table below shows the simple index number for the gross domestic product of the UK for each quarter in 2010 using 2007 Quarter 1 as the base.

Calculate the value of the gross domestic product of the UK in 2010 Quarter 1

Marc says that the UK was in recession throughout 2010 because all of the simple index numbers are less than 100

Explain whether or not Marc is correct.

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.

The map below gives information about the percentage change in the number of police officers in each of the 42 different police force regions of England and Wales between 2014 and 2015

Write down the percentage change in the number of police officers in region 17

In how many of the 42 regions did the number of police officers increase?

Sadiq thinks that the map shows that there was a decrease in the overall number of police officers in England and Wales between 2014 and 2015

Explain why Sadiq may not be correct.

Write down the statistical name for the type of map used in this question.

The map below gives information about the percentage change in the number of police officers in each of the 42 different police force regions of England and Wales between 2014 and 2015

Write down the percentage change in the number of police officers in region 17

In how many of the 42 regions did the number of police officers increase?

Sadiq thinks that the map shows that there was a decrease in the overall number of police officers in England and Wales between 2014 and 2015

Explain why Sadiq may not be correct.

Write down the statistical name for the type of map used in this question.

The table gives the price index number for the average rail fare in Great Britain for each of six years, with 2012 as base year.

The table also gives some of the chain base index numbers for the same information.

(Source: Office of Rail and Road)

Find, correct to 2 decimal places, the chain base index number for 2017

Chris wanted to know the percentage increase in the cost of the average rail fare in Great Britain between 2016 and 2017

Here is his working.

Explain whether or not Chris is correct.

The geometric mean of the chain base index numbers for 2013 to 2017 is 102.18

Interpret this geometric mean in the context of the average rail fare in Great Britain.

The incomplete table gives some information about the number of births in the UK, the total population of the UK and the crude birth rate for the UK in 1960 and in 2016

Using the formula below, complete the table.

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.

Mike is investigating whether or not there is a relationship between the distance from the nearest car park to a restaurant and the annual profit made by the restaurant.

He finds out the distance, in metres, from the nearest car park to a restaurant and the annual profit, in £, made by the restaurant. He does this for twenty restaurants in a city.

Mike then draws a scatter diagram for this information.

Explain why annual profit is the response variable for this scatter diagram.

Mike’s hypothesis is that, for these restaurants, the further the restaurant is from its nearest car park the less is its annual profit.

Explain, giving a statistical reason, whether or not the scatter diagram supports Mike’s hypothesis.

Mike wants to draw a line of best fit on the scatter diagram. Using statistical software he obtains the following information about these restaurants.

Using this information, draw a line of best fit on the scatter diagram.

Interpret the value of the intercept of the line of best fit on the Annual profit axis.

Restaurant A and restaurant B are two other restaurants in the city. Restaurant A is 250 m from its nearest car park. Restaurant B is 700 m from its nearest car park. Mike uses the scatter diagram to find an estimate for the annual profit of each of these restaurants.

Explain which of these two estimates will be the more reliable estimate.

Mike finds a positive correlation between the number of tables at a restaurant and its annual profit. He concludes that as the number of tables increases this causes the annual profit to increase.

Explain whether or not this conclusion is valid.

Mike reads an article in a newspaper that says that restaurant profits for the top 100 restaurants had fallen from £345 million to £125 million in the past year. (Source: telegraph.co.uk)

Using the data in the newspaper article, calculate the percentage decrease in restaurant profits for the top 100 restaurants in the past year.

Mike is investigating whether or not there is a relationship between the distance from the nearest car park to a restaurant and the annual profit made by the restaurant.

He finds out the distance, in metres, from the nearest car park to a restaurant and the annual profit, in £, made by the restaurant. He does this for twenty restaurants in a city.

Mike then draws a scatter diagram for this information.

Explain why annual profit is the response variable for this scatter diagram.

Mike’s hypothesis is that, for these restaurants, the further the restaurant is from its nearest car park the less is its annual profit.

Explain, giving a statistical reason, whether or not the scatter diagram supports Mike’s hypothesis.

Mike wants to draw a line of best fit on the scatter diagram. Using statistical software he obtains the following information about these restaurants.

Using this information, draw a line of best fit on the scatter diagram.

Interpret the value of the intercept of the line of best fit on the Annual profit axis.

Restaurant A and restaurant B are two other restaurants in the city. Restaurant A is 250 m from its nearest car park. Restaurant B is 700 m from its nearest car park. Mike uses the scatter diagram to find an estimate for the annual profit of each of these restaurants.

Explain which of these two estimates will be the more reliable estimate.

Mike finds a positive correlation between the number of tables at a restaurant and its annual profit. He concludes that as the number of tables increases this causes the annual profit to increase.

Explain whether or not this conclusion is valid.

Mike reads an article in a newspaper that says that restaurant profits for the top 100 restaurants had fallen from £345 million to £125 million in the past year. (Source: telegraph.co.uk)

Using the data in the newspaper article, calculate the percentage decrease in restaurant profits for the top 100 restaurants in the past year.

The two population pyramids show the percentage of males and the percentage of females in each age group in the UK and in India for 2016

Each pyramid is based on the total population of that country for 2016

Percentage of population
Percentage of population

(Source: www.populationpyramid.net

Compare the percentages of males and females in the UK who are 24 years old or younger with the percentages of males and females in India who are 24 years old or younger. You should give two comparisons.

The population of the UK in 2016 was 65 648 000 and the number of live births to women aged 15 to 49 was 775 300 (Source: ons.gov.uk)

Calculate the general fertility rate for the UK in 2016

You may use

In 2016, the population of New Zealand was 4 660 833 In the same year, there were 59 430 births in New Zealand.

(Source: www.worldometers.info and www.stats.govt.nz)

Using the formula below, work out the crude birth rate for New Zealand in 2016

Give your answer correct to 1 decimal place.

The crude birth rate for Albania in 2015 was 12 The crude birth rate for Bolivia in 2015 was 24 (Source: data.worldbank.org)

Louise says, “There were twice as many births in Bolivia as in Albania in 2015”

State what must be true about the populations of Albania and Bolivia for Louise to be correct.

The table gives the crude birth rates and crude death rates, per 1000 people, for two Caribbean islands in the year 2015

(Source: www.data.worldbank.org)

Jamil concludes that the crude birth rate and the crude death rate for Saint Lucia show that the population of Saint Lucia increased in 2015

Explain how the data can be used to support Jamil’s conclusion.

Give a reason why the data might not support Jamil’s conclusion.

In 2015, the total population of Barbados was 284 217

Using the formula below, work out the number of births in Barbados in 2015

The standard population of Barbados and of Saint Lucia for three different age groups in 2015 is shown in the table.

Give an interpretation of the number 64 in the table.

Compare the crude death rate for Barbados with the crude death rate for Saint Lucia.

By referring to the information in the Standard population table, suggest a reason for your comparison.