Rates of Change (Edexcel GCSE Statistics)

Revision Note

Crude Rates of Change

How do I calculate a crude rate of change?

  • Understanding rates of change in a population is important for planning purposes

    • For example, a town with a high birth rate may need to think about building more schools for the increasing number of children

  • A crude rate is a way to understand how things are changing in a population

    • It is often used to track births or deaths

      • But it can also be used to track other things, like marriages, unemployment, etc.

    • A crude rate normally tells you the number of births (or deaths, etc.) per 1000 people

  • To calculate the crude birth rate (for example) use the formula

    • crude space birth space rate equals fraction numerator number space of space births cross times 1000 over denominator total space population end fraction

      • This formula is on the exam formula sheet, so you don't need to remember it

    • To calculate a crude rate for other things, replace 'number of births' in the formula with 'number of deaths', 'number of marriages', 'number of people unemployed', etc.

  • You may be given the crude birth rate (or death rate, etc.), and asked to work out one of the other values in the formula

    • If you know any two things from the formula you can find the third one

    • Substitute in the values you know

      • and solve to find the one you want to know

  • You can also calculate a crude rate per 100 people (instead of per 1000)

    • Replace 'cross times 1000' in the formula with 'cross times 100'

    • Only do this if a question tells you to

Worked Example

Last year a particular town had a population of 75 992.

(a) Given that there were 783 deaths during the year, find the crude death rate for the town. Give your answer correct to 3 decimal places.

Use  crude space death space rate equals fraction numerator number space of space deaths cross times 1000 over denominator total space population end fraction

fraction numerator 783 cross times 1000 over denominator 75992 end fraction equals 10.303716...

Round to 3 decimal places

10.304

You could also answer '10.304 deaths per 1000 of population', but it's not necessary to get full marks

The crude birth rate for that same town last year was 11.317.

(b) Find the number of births that occurred in the town during the year.

Substitute the values you know into crude space birth space rate equals fraction numerator number space of space births cross times 1000 over denominator total space population end fraction
It will be easier to replace 'number of births' with a letter like b

11.317 equals fraction numerator b cross times 1000 over denominator 75992 end fraction equals fraction numerator 1000 b over denominator 75992 end fraction

Multiply the '75992' across

table row cell 11.317 cross times 75992 end cell equals cell 1000 b end cell row blank blank blank row cell 860001.464 end cell equals cell 1000 b end cell end table

Divide both sides by 1000

table row cell fraction numerator 860001.464 over denominator 1000 end fraction end cell equals b row blank blank blank row cell 860.001464 end cell equals b end table

Round to the nearest whole number
(Number of births has to be a whole number!)

860

Standardised Rates of Change

What is the difference between crude and standardised rates of change?

  • Crude rates can sometimes be misleading

    • Consider a town that has a high proportion of older residents

    • If the town has a high crude death rate

      • It may be because residents of the town are particularly unhealthy for some reason (the government would want to investigate this and try to fix it)

      • Or it may just be because older people die more often than younger people

    • If the town has a low crude unemployment rate

      • It may be because the town's economy is doing well

      • Or it may just be because many of the town's residents are retired and beyond employment age (a retired person would not count as 'unemployed')

  • A standardised rate of change is a way to deal with this issue

    • It allows us to compare 'like with like' when comparing rates for different populations

What is a standard population?

  • A standard population is a 'population' of 1000 people that has the same proportions in each age group as an actual population

    • It's what the real population would look like if it were 'shrunk down' to a population of just 1000

  • The standard population for a particular age group is calculated using the formula

    • standard space population space for space age space group equals fraction numerator number space in space age space group over denominator total space population end fraction cross times 1000

      • If you need this formula it will be provided for you on the exam

      • An exam question may give you any standard populations you need, rather than asking you to calculate them

    • A standard population is calculated for each age group in the total population

      • The sum of all these standard populations should equal 1000

  • For example, for a total population of 68 349 000 divided into the following 4 age groups:

Age group

Number

Standard population
(to 1 decimal place)

0-19

15 462 000

15462000 over 68349000 cross times 1000 equals 226.2

20-35

14 346 800

14346800 over 68349000 cross times 1000 equals 209.9

36-65

26 312 400

26312400 over 68349000 cross times 1000 equals 385.0

>65

12 227 800

12227800 over 68349000 cross times 1000 equals 178.9

total:

68 349 000

1000

  • So if that population were shrunk down to a town of 1000 people, 226.2 of them would be 0-19 years old, 209.9 of them would be 20-35, etc.

How do I calculate a standardised rate of change?

  • You need to know the crude rate for each age group

    • Use the crude rate of change formula

      • with the number of births (or deaths, etc.) for the age group

      • and with the 'total population' being the number of people in that age group

  • You also need to know the standard population to use for each age group

    • An exam question will specify what this should be

    • The standard populations are usually based on a much larger population group

      • For example, if you are calculating standardised rates for the different age groups in a town, the standard populations used might be those calculated for the entire country that the town is in

      • Sometimes the standard populations could be based on the population of the earth as a whole

  • Once you know the standard populations and crude rates you can calculate the standardised rates of change for each age group using the formula

    • standardised space rate equals fraction numerator crude space rate over denominator 1000 end fraction cross times standard space population

      • If you need to use this formula it will be given to you in the question, so you don't need to memorise it

  • A standardised rate of change is calculated for each age group within a population

    • The sum of the standardised rates of change for the different age groups is the standardised rate of change for the population as a whole

      • This sum is a type of weighted average

      • It tells you what the rate of change would be if the population you are looking at had the same population breakdown as the standard population

  • If you are comparing two different populations (for example two different towns or cities)

    • it might be more appropriate to compare standardised rates for each group, than to compare crude rates

      • This will be especially true if the proportions of people in each age group are different for the two populations

Examiner Tips and Tricks

  • If an exam question asks you why it might be more appropriate to use a standardised rate when comparing two populations

    • it will usually be because the proportions of the populations belonging to different age groups are different

Worked Example

In 2023 the population of Milltown was 45231. The breakdown for the population by age groups, along with the number of deaths occurring within each age group in 2023, is given in the table below.

Age group

Number

Number of deaths

0-19

4022

17

20-35

14112

87

36-65

18977

159

>65

8120

239

(a) Calculate the crude death rate for Milltown for 2023, giving your answer correct to 1 decimal place.

Use  crude space death space rate equals fraction numerator number space of space deaths cross times 1000 over denominator total space population end fraction
The total population is given in the question
The total number of deaths will be the sum of the third column in the table

fraction numerator open parentheses 17 plus 87 plus 159 plus 239 close parentheses cross times 1000 over denominator 45231 end fraction equals 11.0985...

11.1

You could also answer '11.1 deaths per 1000 of population', but it's not necessary to get full marks

A local reporter in Milltown thinks that the crude death rate is misleading. He suggests that standardising the death rate for Milltown against the population of the country as a whole would give a more realistic figure.

The formula for calculating a standardised death rate is

standardised space death space rate equals fraction numerator crude space rate over denominator 1000 end fraction cross times standard space population

(b) Calculate the standardised death rate for Milltown for 2023, using the following standard populations calculated for the country as a whole:

Age group

Standard population

0-19

226.2

20-35

209.9

36-65

385.0

>65

178.9

Give your final answer correct to 1 decimal place.
 

First calculate the crude rates for each age group using  crude space death space rate equals fraction numerator number space of space deaths cross times 1000 over denominator total space population end fraction

Age group

Crude rate

0-19

fraction numerator 17 cross times 1000 over denominator 4022 end fraction equals 4.226...

20-35

fraction numerator 87 cross times 1000 over denominator 14112 end fraction equals 6.164...

36-65

fraction numerator 159 cross times 1000 over denominator 18977 end fraction equals 8.378...

>65

fraction numerator 239 cross times 1000 over denominator 8120 end fraction equals 29.433...

Now use  standardised space death space rate equals fraction numerator crude space rate over denominator 1000 end fraction cross times standard space population to calculate the standardised rate for each age group

Age group

Standardised death rate

0-19

fraction numerator 4.226... over denominator 1000 end fraction cross times 226.2 equals 0.956...

20-35

fraction numerator 6.164... over denominator 1000 end fraction cross times 209.9 equals 1.294...

36-65

fraction numerator 8.378... over denominator 1000 end fraction cross times 385.0 equals 3.225...

>65

fraction numerator 29.433... over denominator 1000 end fraction cross times 178.9 equals 5.265...

This means that if Milltown had a population of 1000 with the same proportions in each age group as for the country as a whole, then there would have been 0.956... deaths in the 0-19 age group, 1.294... in the 20-35 age group, etc.

To find the total standardised death rate for Milltown, add those individual standardised rates together:

0.956... plus 1.294... plus 3.225... plus 5.265... equals 10.741...

Round to 1 decimal place

10.7

You could also answer '10.7 deaths per 1000 of population', but it's not necessary to get full marks

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