Rates of Change (Edexcel GCSE Statistics: Higher): 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

  • $\mathrm{crude}\mathrm{birth}\mathrm{rate}=\frac{\mathrm{number}\mathrm{of}\mathrm{births}\times 1000}{\mathrm{total}\mathrm{population}}$

    • 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 '$\times 1000$' in the formula with '$\times 100$'

  • Only do this if a question tells you to

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

  • $\mathrm{standard}\mathrm{population}\mathrm{for}\mathrm{age}\mathrm{group}=\frac{\mathrm{number}\mathrm{in}\mathrm{age}\mathrm{group}}{\mathrm{total}\mathrm{population}}\times 1000$

    • 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:

• 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

  • $\mathrm{standardised}\mathrm{rate}=\frac{\mathrm{crude}\mathrm{rate}}{1000}\times \mathrm{standard}\mathrm{population}$

    • 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