ST21: Index Numbers & Population Estimation
Covers index numbers including RPI, CPI and GDP, weighted index numbers, crude and standardised rates for population comparisons, and the capture-recapture method for estimating population size.
Index Numbers Population Estimation
Index Numbers & Population Estimation
Covers index numbers including RPI, CPI and GDP, weighted index numbers, crude and standardised rates for population comparisons, and the capture-recapture method for estimating population size.
Key Fact: An index number measures how a quantity changes relative to a base period; the base value is always 100
Key Fact: Index number = (current value ÷ base value) × 100; an index of 120 means a 20% increase from the base period
Key Fact: RPI (Retail Prices Index) measures the change in cost of a basket of goods and services including mortgage interest payments
Key Fact: CPI (Consumer Prices Index) is similar to RPI but excludes mortgage interest and housing costs; CPI is used for the government’s inflation target
Key Fact: GDP (Gross Domestic Product) measures the total value of goods and services produced in a country; GDP per capita = GDP ÷ population
Key Fact: A price relative = (current price ÷ base price) × 100; it shows the change in price for a single item
Key Fact: Weighted index numbers use weights reflecting the importance of each item; overall index = ∑(weight × price relative) ÷ ∑weights
Key Fact: Crude rate = (number of events ÷ total population) × a scaling factor (e.g. per 1000); it does not account for population structure
Key Fact: Standardised rates adjust for differences in age structure between populations, allowing fair comparisons; direct standardisation applies a standard population’s age distribution
Key Fact: Capture-recapture: population estimate = (first sample × second sample) ÷ number of marked recaptures; N = (n₁ × n₂) ÷ m
Key Fact: Capture-recapture assumptions: the population is closed (no births, deaths, migration), marks are not lost, and every individual has an equal chance of being captured
Key Fact: Chain base index: each period’s index is calculated using the previous period as the base, making it easier to see period-to-period changes
📋 Key Vocabulary and Concepts
For Index Numbers & Population Estimation, you must know:
- Index number: A measure of relative change compared to a base value of 100
- RPI: Retail Prices Index: measures inflation using a basket of goods including housing costs such as mortgage interest
- CPI: Consumer Prices Index: measures inflation excluding housing costs; used for the government’s inflation target
- Weighted index: An index that accounts for the relative importance of different items by assigning weights before averaging
- Crude rate: The number of events per unit of population without adjusting for age or other structural differences
- Capture-recapture: A method to estimate population size: N = (first sample × second sample) ÷ number of recaptures
❓ Practice Questions
Q: In the base year 2020, a basket of goods cost £200. In 2024 the same basket costs £260. Calculate the index number for 2024 with 2020 as the base.
Q: A weighted price index is calculated from three items. Item A: price relative 120, weight 4. Item B: price relative 105, weight 3. Item C: price relative 135, weight 1. Find the overall weighted index.
Q: In a capture-recapture study, 48 rabbits are caught, tagged and released. Later, 60 rabbits are caught and 12 have tags. Estimate the total rabbit population.
Q: Town A has a crude death rate of 12 per 1000. Town B has a crude death rate of 8 per 1000. Why might it be misleading to conclude Town A is less healthy?
Q: The CPI in January was 112.4 and in February was 113.6. Calculate the monthly rate of inflation.
✅ Answers
- Index = (260 ÷ 200) × 100 = 130. Prices have risen by 30% since the base year.
- Weighted index = (4×120 + 3×105 + 1×135) ÷ (4+3+1) = (480 + 315 + 135) ÷ 8 = 930 ÷ 8 = 116.25
- N = (48 × 60) ÷ 12 = 2880 ÷ 12 = 240. The estimated population is 240 rabbits.
- Town A may have an older population structure, which naturally leads to more deaths. Crude rates do not account for age differences. Standardised rates would allow a fairer comparison.
- Inflation rate = ((113.6 − 112.4) ÷ 112.4) × 100 = (1.2 ÷ 112.4) × 100 ≈ 1.07%
🎯 Exam Tips
- Always state the base year when giving an index number, and show the calculation as (current/base) × 100
- For weighted index questions, set out a clear table with columns for item, price relative, and weight before calculating
- When discussing RPI vs CPI, mention that CPI excludes mortgage interest payments and is the government’s preferred measure
- In capture-recapture questions, list the assumptions and explain how violating each one would affect the estimate
- For crude vs standardised rate questions, always explain that crude rates can be misleading due to different population structures
📝 Exam Technique
GCSE Statistics Exam Tips — Index Numbers & Population Estimation:
1. For Index Numbers & Population Estimation questions, show every step of your working clearly — method marks count even if the final answer is wrong
2. Check your answer makes sense in context (estimation, units, reasonableness)
3. Use correct mathematical notation and state formulae before substituting values
4. If a Index Numbers & Population Estimation question asks you to 'prove' or 'show', write a logical chain of reasoning with a conclusion line
5. For problem-solving, identify the topic first, then recall the relevant method
1. For Index Numbers & Population Estimation questions, show every step of your working clearly — method marks count even if the final answer is wrong
2. Check your answer makes sense in context (estimation, units, reasonableness)
3. Use correct mathematical notation and state formulae before substituting values
4. If a Index Numbers & Population Estimation question asks you to 'prove' or 'show', write a logical chain of reasoning with a conclusion line
5. For problem-solving, identify the topic first, then recall the relevant method
⚠️ Common Errors
✗ Saying an index of 130 means prices are 130% higher ✓ An index of 130 means a 30% increase from the base period (index of 100)
✗ Using a simple average instead of a weighted average when items have different importance ✓ Multiply each price relative by its weight, sum, then divide by the sum of weights
✗ Forgetting to check capture-recapture assumptions before calculating ✓ Always state assumptions: closed population, no lost marks, equal chance of capture
✗ Confusing CPI and RPI ✓ CPI excludes housing costs like mortgage interest; RPI includes them. CPI is used for the inflation target.
✍️ Model Answer
Full-Mark Response
A health researcher compares two towns. Town X has a crude death rate of 14 per 1000. Town Y has a crude death rate of 9 per 1000. After standardising for age, Town X has a standardised rate of 10 per 1000 and Town Y has a standardised rate of 11 per 1000. Explain these results.
The crude rates suggest Town X has a higher death rate (14 vs 9 per 1000), which might lead to the conclusion that Town X is less healthy. However, after standardising for age, Town Y actually has the higher standardised death rate (11 vs 10 per 1000). This reversal happens because Town X likely has a larger proportion of older residents. Older people are more likely to die, so Town X’s crude rate is inflated by its age structure. Once we adjust for this by applying the same standard age distribution to both towns, the true picture emerges: Town Y has slightly worse health outcomes when age is accounted for. This demonstrates why standardised rates are essential for fair comparisons between populations with different age structures.
📊 AO Deep Dive
Assessment Objective Analysis
AO1 (Knowledge & Understanding): Demonstrate knowledge and understanding of index numbers & population estimation, including data collection, presentation and calculation techniques relevant to Edexcel 1ST0 & AQA 8382.
AO2 (Application): Apply knowledge and understanding of index numbers & population estimation to interpret data, reason statistically and draw conclusions in context.
AO3 (Evaluation): Evaluate statistical methods and conclusions, assessing appropriateness, reliability, validity and bias through the statistical enquiry cycle.
📝 Exam Questions by Topic
- AQA Past Papers - Topic Questions -- AQA
- Edexcel Past Papers - Topic Questions -- Edexcel
- CCEA Past Papers - Topic Questions -- CCEA
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