Estimation and Petersen Capture-Recapture
You cannot count every fish in a lake, so statisticians estimate. Learn to estimate a population from a sample and to use the Petersen capture-recapture formula, with its assumptions.
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Counting the uncountable 🎣
How many fish are in a lake? You cannot catch and count every one. Instead, statisticians **estimate**: they study a **sample** and use it to work out a value for the whole **population**. One neat method is **capture-recapture**: catch some, mark them, let them go, then catch a second sample and see how many are marked. This module builds that method, its formula and the assumptions it depends on.
The words you need 🔑
Estimation has a precise vocabulary. Get these exact.
Match each term to its meaning 🔗
- Population
- Sample
- Estimate
- Recapture
- The whole group you want to find out about
- A smaller part of the population you actually study
- A value from a sample that stands for the whole population
- Catching a second sample to count how many are already marked
Bigger samples 📈
What generally happens to a population estimate as the sample size increases?
- It generally becomes more reliable
- It always becomes less accurate
- It stays exactly the same
- It becomes impossible to calculate
The Petersen formula 🧮
For capture-recapture, the estimated population size is: **N = (first sample x second sample) / number marked in the second sample** In words: multiply the size of the two samples, then divide by how many marked individuals turned up in the second catch. On this Higher-tier topic the formula is **not given** - you must memorise it. Now try one.
Estimate the fish 🐟
An interactive activity.
How much can you trust it? ⚖️
An estimate is only as good as the sample it came from. Sample size drives how much you can rely on the result.
Estimate the beetles 🪲
An interactive activity.
What must be true? ✅
Select the TWO assumptions the capture-recapture method relies on.
- The population is closed, with no births, deaths or migration between the samples
- The marks do not come off or affect survival
- The population must be very small
- Marked individuals deliberately avoid being recaptured
Why each assumption matters 🧷
- Closed population
- Marks stay on
- Full mixing
- Enough marked
- No births, deaths or migration change the total between samples
- A marked individual can still be recognised when recaptured
- Marked individuals spread back randomly before the second catch
- A large enough sample makes the estimate reliable
Scale up a proportion 👥
An interactive activity.
Is the estimate any good? 🧭
An interactive activity.
Your turn: estimate 🖊️
An interactive activity.
The grade-9 habit 🌟
Two things win the marks here. First, **memorise and apply the Petersen formula** cleanly - N = (first sample x second sample) divided by the number marked in the second catch - and always **show your working** so method marks are safe even if the arithmetic slips. Second, be ready to **criticise an estimate**: name the assumption that is broken (a population that is not closed, marks that wear off, poor mixing) or explain that a **small sample** makes the result unreliable. Estimation questions reward judgement, not just a number.