Mr Kidman's Classroom
Revise the key ideas
Population estimation
- Estimate a proportion — If 36 of 120 sampled pupils cycle, estimate the population cycling proportion as 0.3. In a population of 800 this predicts about 240 cyclists. The estimate depends on representative selection, accurate responses and an appropriate date; it is not an exact census total.
- Estimate centre — Use a representative sample mean to estimate a population mean. Approximately half a comparable population may lie above the sample median, but ties and sampling variation affect this statement. An estimate summarises a population characteristic, not each individual's value.
- Size and replication — Larger random samples generally give less variable estimates; repeating comparable samples reveals variability. Independent repetitions should use consistent methods. A large biased sample can still give a consistently wrong answer; improving coverage and reducing bias matter alongside sample size.
- State limitations — Distinguish an estimate from a certainty, explain the source and method and identify likely error or bias. If a sample covers only one shift, extend the frame to other shifts before generalising about all production. Give specific improvements related to the actual weakness.
Higher — capture–recapture
- Petersen estimate — Capture and mark M individuals, release and allow them to mix, then capture C individuals and count R marked recaptures. Estimate population N ≈ MC/R. With M=40, C=50 and R=10, estimate N=200; use total second catch, not just unmarked individuals.
- Capture assumptions — Assume a closed population, retained recognisable marks, mixing and comparable capture probabilities. Marking must not change survival or capture behaviour. If marked animals become easier to catch, R can be too large and N underestimated; immigration can invalidate the closed-population assumption.
Closed means no births, deaths, arrivals or departures during the study.
- Small recaptures — A small R makes the estimate unstable because a change of one recapture changes the denominator substantially. R=0 gives no finite estimate from this formula; use a revised sampling plan rather than inventing a total. Ethical animal methods need expert oversight.
Higher — quality assurance
- Sample summaries — Means from repeated comparable samples usually vary less than individual observations from the same population because averaging balances high and low values. Use the distribution of the monitored sample statistic (a summary value, such as the mean, calculated for each sample), not the SD of individual items, when setting its limits.
- Control charts — Plot a sample mean, median or range against sample number/time, with a target and warning/action lines. Keep sample size and method consistent. A control chart checks process behaviour; specification limits (acceptable values for individual products) serve a different purpose.
- Warning and action limits — In the stated mean-chart model warning lines are target ±2 SD of sample means and action lines target ±3 SD of sample means. About 1 in 20 fall outside warning lines and almost all lie inside action lines under the approximate normal model (a symmetric, bell-shaped distribution). Random variation can occasionally produce a warning.
Sample-mean control limits. Original illustrative diagram; numerical datasets are fictional worked examples. Enlarge diagram - Respond to evidence — An observation outside action limits calls for stopping/investigating the process and correcting an identified cause. Between warning and action lines, take another sample and investigate repeated warning signals under the stated procedure. Within limits normally continue monitoring, while watching patterns rather than assuming permanent perfection.
Higher — quality assurance
- Read the control chart — Suppose the target sample mean is 100, warning lines 96 and 104, and action lines 94 and 106. A sample mean of 105 is between warning and action: follow the stated procedure by taking another sample and checking a repeated warning. A mean of 107 crosses the action line and calls for stopping and investigation. These lines concern sample means, not every individual item's permitted dimensions. A process can be statistically stable yet fail a customer's specification.
