Crude and Standardised Rates: O Level Statistics 4040 and IGCSE 0479 Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs. Use them alongside the official past papers from Cambridge.

Study the method first: crude and standardised rates study guide. Give every rate per thousand unless a question says otherwise.

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Questions

1. A town has a population of 84 000. In one year there were 1 302 births. Calculate the crude birth rate. [2]

2. A district has a population of 125 000 and a crude death rate of 8.4 per thousand. Calculate the number of deaths in the district that year. [2]

3. In a city, the death rate for people aged 65 and over is 45 per thousand. There are 6 400 people in this age group. Calculate the expected number of deaths in this age group. [2]

4. Explain why a standardised death rate is a better measure than a crude death rate for comparing the health of two regions. [2]

5. The table gives the population and the number of deaths in one year in two regions, P and Q, with a standard population.

Age group	Standard population	Region P population	Region P deaths	Region Q population	Region Q deaths
Under 20	3 000	6 000	12	8 000	20
20–59	5 000	9 000	36	10 000	50
60 and over	2 000	5 000	140	2 000	62

(a) Calculate the crude death rate for each region. [3]

(b) Calculate the death rate for each age group in each region. [3]

(c) Calculate the standardised death rate for each region. [4]

(d) Which region appears healthier? Explain why the crude and standardised rates lead to different conclusions. [3]

6. A factory reports 18 accidents in a year among its 1 200 workers. A second factory reports 11 accidents among 500 workers. State one reason why comparing these crude accident rates might not show which factory is safer, and name the information you would need to standardise them. [3]

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Answers

1. 1302 ÷ 84 000 × 1000 [1] = 15.5 per thousand [1].

2. 8.4 × 125 000 ÷ 1000 [1] = 1 050 deaths [1].

3. 45 × 6 400 ÷ 1000 [1] = 288 deaths [1].

4. Death rates depend strongly on age, so a crude rate is affected by how many old or young people a region has [1]. A standardised rate applies each region’s age-specific rates to the same standard population, so the regions are compared as if they had the same age structure [1].

5. (a) Region P: total deaths 12 + 36 + 140 = 188, population 20 000 [1]; 188 ÷ 20 000 × 1000 = 9.4 per thousand [1]. Region Q: 20 + 50 + 62 = 132 deaths out of 20 000; 6.6 per thousand [1].

(b) Region P: 12 ÷ 6 000 × 1000 = 2, 36 ÷ 9 000 × 1000 = 4, 140 ÷ 5 000 × 1000 = 28 [1]. Region Q: 20 ÷ 8 000 × 1000 = 2.5, 50 ÷ 10 000 × 1000 = 5, 62 ÷ 2 000 × 1000 = 31 [1]. All six correct, with “per thousand” [1].

(c) Region P: (3000 × 2 + 5000 × 4 + 2000 × 28) ÷ 10 000 [1] = (6 000 + 20 000 + 56 000) ÷ 10 000 = 8.2 per thousand [1]. Region Q: (3000 × 2.5 + 5000 × 5 + 2000 × 31) ÷ 10 000 [1] = (7 500 + 25 000 + 62 000) ÷ 10 000 = 9.45 per thousand [1].

(d) Region P appears healthier, because its standardised death rate is lower (8.2 against 9.45) [1]. Region P has the higher crude rate because a quarter of its population is aged 60 or over (5 000 of 20 000), compared with a tenth of Region Q’s (2 000 of 20 000) [1]. Region Q’s death rate is higher in every age group, which the standardised rate shows once the age structures are made the same [1].

6. Crude rates: 18 ÷ 1200 × 1000 = 15 per thousand and 11 ÷ 500 × 1000 = 22 per thousand. The factories may employ different mixes of workers (for example, different proportions in high-risk jobs, or of newly trained staff), and accident risk varies between these groups [1]. To standardise you would need, for each factory, the number of workers and accidents in each group [1], and a standard population of workers across the same groups [1].

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Where marks are usually lost

• Giving a standardised rate as the plain average of the age-specific rates. In 5(c) that gives (2 + 4 + 28) ÷ 3 = 11.3 for Region P, which is wrong.
• Using the region’s own population as the weights in 5(c) instead of the standard population.
• In 5(d), naming the healthier region without explaining why the crude rate pointed the other way. The explanation (age structure) carries the marks.
• Losing the “× 1000”: a rate of 0.0155 instead of 15.5 per thousand.

Next steps

• Next topic: index numbers practice
• Time series practice
• O Level Statistics 4040 hub · IGCSE Statistics 0479 hub