Crude and Standardised Rates: O Level Statistics 4040 and IGCSE 0479 Study Guide

This guide covers crude and standardised rates: Topic 7 of Cambridge O Level Statistics (4040, syllabus for examination in 2025, 2026 and 2027) and Topic 9 of Cambridge IGCSE Statistics (0479, syllabus for examination in 2027). The two syllabuses describe the same skills; 0479 simply lists them as five separate learning outcomes. Practise straight afterwards with the crude and standardised rates practice questions.

What the syllabuses ask for

4040 Topic 7 (guidance)	0479 Topic 9 (learning outcomes)
Crude and standardised rates, and their appropriate use, “including application to death rates, fertility rates, accident rates”	9.1 Calculate a crude rate of events in a population (births, deaths, accidents)
	9.2 Understand the reason for finding a standardised rate
	9.3 Calculate standardised rates, given a standard population
	9.4 Form conclusions about one or more populations from crude and standardised rates
	9.5 Use crude rates to find the total number of events in a population or subgroup

Both syllabuses: 4040 syllabus PDF, 0479 syllabus PDF. 0479 notes that rates are defined “for example, as births per thousand of the population”.

1. Crude rate

A crude rate compares the number of events with the size of the whole population, per thousand people:

crude rate = (number of events ÷ total population) × 1000

A town of 84 000 people with 1 302 births in a year has a crude birth rate of 1302 ÷ 84 000 × 1000 = 15.5 per thousand.

It is “crude” because it ignores how the population is made up. Deaths depend strongly on age, so a town with many elderly residents will have a high crude death rate even if every age group in it is healthy.

2. Age-specific rates

To see past the age structure, work out a separate rate for each age group (or other subgroup):

age-specific rate = (events in the group ÷ people in the group) × 1000

This is the step most questions are built on. If a table gives population and deaths by age group, you will almost always need these rates first.

3. Why standardise?

Two populations can only be compared fairly if they are measured against the same age structure. A standardised rate asks: what would this population’s rate be if it had the age structure of a chosen standard population? The standard population is always given in the question.

4. Standardised rate

0479 describes a standardised rate as “a weighted average of rates in subgroups”. The weights are the standard population:

standardised rate = Σ (standard population in group × age-specific rate) ÷ Σ (standard population)

The standard population may be given as numbers of people or as percentages. The method is the same; with percentages, the divisor is 100.

Worked example

Two towns each have 40 000 residents. The standard population is given as percentages.

Age group	Standard population (%)	Town A population	Town A deaths	Town B population	Town B deaths
0–14	20	9 000	18	12 000	30
15–44	40	16 000	48	20 000	70
45–64	25	10 000	90	6 000	60
65 and over	15	5 000	250	2 000	110
Total	100	40 000	406	40 000	270

Crude death rates

• Town A: 406 ÷ 40 000 × 1000 = 10.15 per thousand
• Town B: 270 ÷ 40 000 × 1000 = 6.75 per thousand

Age-specific death rates (per thousand)

Age group	Town A	Town B
0–14	18 ÷ 9 000 × 1000 = 2.0	30 ÷ 12 000 × 1000 = 2.5
15–44	48 ÷ 16 000 × 1000 = 3.0	70 ÷ 20 000 × 1000 = 3.5
45–64	90 ÷ 10 000 × 1000 = 9.0	60 ÷ 6 000 × 1000 = 10.0
65 and over	250 ÷ 5 000 × 1000 = 50.0	110 ÷ 2 000 × 1000 = 55.0

Standardised death rates

• Town A: (20 × 2.0 + 40 × 3.0 + 25 × 9.0 + 15 × 50.0) ÷ 100 = (40 + 120 + 225 + 750) ÷ 100 = 11.35 per thousand
• Town B: (20 × 2.5 + 40 × 3.5 + 25 × 10.0 + 15 × 55.0) ÷ 100 = (50 + 140 + 250 + 825) ÷ 100 = 12.65 per thousand

Conclusion. Town A has the higher crude death rate, but Town B has the higher standardised death rate. Town A’s crude rate is high because one in eight of its residents is 65 or over (5 000 of 40 000), compared with one in twenty in Town B. Once both towns are measured against the same age structure, Town B is the less healthy place to live: its death rate is higher in every age group.

That reversal is the point of the topic, and 0479 outcome 9.4 asks you to form exactly this kind of conclusion. Name the population feature that causes it (here, the larger proportion of elderly people in Town A), not just “the populations are different”.

5. Working backwards: rate to number of events

A rate can be turned back into a number of events:

number of events = rate × population ÷ 1000

• A district of 125 000 people with a crude death rate of 8.4 per thousand has 8.4 × 125 000 ÷ 1000 = 1 050 deaths.
• If the age-specific rate for people aged 65 and over is 45 per thousand and 6 400 people are in that group, the expected number of deaths in that group is 45 × 6 400 ÷ 1000 = 288.

Other kinds of rate

The same method applies to birth rates and fertility rates (fertility rates are usually per thousand women in a stated age range, so check the denominator the question defines) and to accident rates (for example, accidents per thousand workers, grouped by job type or years of experience). Always use the definition the question gives.

Where marks are usually lost

• Dividing by the whole population when an age-specific rate is asked for, or the other way round.
• Averaging the age-specific rates without weights. The standardised rate is a weighted average; the weights are the standard population.
• Using the town’s own population as the weights. The weights must be the standard population given in the question.
• Forgetting the units: every rate here is “per thousand” (unless the question says otherwise).
• Stating which rate is higher without explaining why the crude and standardised rates disagree.

Related pages

• Crude and standardised rates: practice questions with worked answers
• Index numbers (4040 Topic 8): another weighted average, built the same way
• Time series and moving averages (4040 Topic 10)
• Cambridge O Level Statistics 4040 syllabus hub and topic checklist
• Cambridge IGCSE Statistics 0479 syllabus hub