Rates of Change (Edexcel GCSE Statistics)

Revision Note

Written by: Roger B

Reviewed by: Dan Finlay

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

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 $crude death rate = \frac{number of deaths \times 1000}{total population}$

$\frac{783 \times 1000}{75992} = 10.303716 . . .$

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 $crude birth rate = \frac{number of births \times 1000}{total population}$
It will be easier to replace 'number of births' with a letter like $b$

$11.317 = \frac{b \times 1000}{75992} = \frac{1000 b}{75992}$

Multiply the '75992' across

$\begin{array}{rcl} 11.317 \times 75992 & = & 1000 b \\ 860001.464 & = & 1000 b \end{array}$

Divide both sides by 1000

$\begin{array}{rcl} \frac{860001.464}{1000} & = & b \\ 860.001464 & = & b \end{array}$

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

Age group	Number	Standard population (to 1 decimal place)
0-19	15 462 000	$\frac{15462000}{68349000} \times 1000 = 226.2$
20-35	14 346 800	$\frac{14346800}{68349000} \times 1000 = 209.9$
36-65	26 312 400	$\frac{26312400}{68349000} \times 1000 = 385.0$
>65	12 227 800	$\frac{12227800}{68349000} \times 1000 = 178.9$
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
- $standardised rate = \frac{crude rate}{1000} \times standard population$
  - 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 $crude death rate = \frac{number of deaths \times 1000}{total population}$
The total population is given in the question
The total number of deaths will be the sum of the third column in the table

$\frac{(17 + 87 + 159 + 239) \times 1000}{45231} = 11.0985 . . .$

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

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

(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 $crude death rate = \frac{number of deaths \times 1000}{total population}$

Age group	Crude rate
0-19	$\frac{17 \times 1000}{4022} = 4.226 . . .$
20-35	$\frac{87 \times 1000}{14112} = 6.164 . . .$
36-65	$\frac{159 \times 1000}{18977} = 8.378 . . .$
>65	$\frac{239 \times 1000}{8120} = 29.433 . . .$

Now use $standardised death rate = \frac{crude rate}{1000} \times standard population$ to calculate the standardised rate for each age group

Age group	Standardised death rate
0-19	$\frac{4.226 . . .}{1000} \times 226.2 = 0.956 . . .$
20-35	$\frac{6.164 . . .}{1000} \times 209.9 = 1.294 . . .$
36-65	$\frac{8.378 . . .}{1000} \times 385.0 = 3.225 . . .$
>65	$\frac{29.433 . . .}{1000} \times 178.9 = 5.265 . . .$

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:

$0.956 . . . + 1.294 . . . + 3.225 . . . + 5.265 . . . = 10.741 . . .$

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

Last updated: 7 May 2024

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Rates of Change (Edexcel GCSE Statistics)

Revision Note

Crude Rates of Change

How do I calculate a crude rate of change?

Worked Example

Standardised Rates of Change

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

What is a standard population?

How do I calculate a standardised rate of change?

Examiner Tips and Tricks

Worked Example

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Author: Roger B

Author: Dan Finlay