Lesson 105 — Problem Solving and Consolidation: Data Collection Techniques
Strand: Statistics | Descriptor: AC9M8ST01 | Duration: 45 minutes
Learning Intentions
- To select, justify and critique data collection techniques for real, multi-part problems.
- To consolidate census, sample, experiment and observation, and their practical and ethical trade-offs.
Success Criteria
I can:
- Identify the population and variable involved in a real scenario.
- Recommend a data collection technique, justifying it with cost, time, accuracy and ethics.
- Critique a flawed data collection plan and propose a specific improvement.
- Communicate a full recommendation clearly to a real audience (e.g. a council).
Warmup
(6 minutes — spot the flaw, pairs)
Each plan below has a serious flaw. Name it.
- “To find out if students enjoy the canteen menu, we will only interview the five students who complained loudest last week.”
- “To test if a new pain medicine works, we will give it to every patient in the hospital and see if they feel better, with no comparison group.”
- “To count the exact number of fish in the harbour, we sent one boat out for one hour.”
Answers: 1. The sample is not representative — it deliberately selects the most negative opinions, not a fair cross-section (this previews Lesson 106’s non-random sampling ideas); 2. No comparison group means any improvement cannot be linked to the medicine specifically — patients might have improved anyway; this is observation dressed up as an experiment; 3. Claiming an “exact number” from one hour’s data is really a rough sample-based estimate, not a genuine census — the language overstates the method’s accuracy.
Today’s focus: applying everything from Lessons 103–104 to real, layered problems where more than one technique is needed.
Activities
Activity 1 — Quick Consolidation: Classify and Justify (10 min)
Pairs, rapid-fire. For each, state the technique (or combination) and one practical or ethical reason for your choice.
- A council wants to know the exact number of registered dogs in its area (it already has registration records for all of them).
- A drink company wants to compare reactions to two new can designs among a group of
shoppers. - A marine biologist counts whale sightings from a clifftop over a migration season.
- A phone company samples
customers nationally to estimate overall satisfaction. - A school trials a new lunchtime timetable with one year level before deciding whether to roll it out to the whole school.
- A researcher wants to know community attitudes to a new law but only interviews people at one shopping centre on a weekday morning.
(Answers: 1. Census — records already cover the whole population, so no further collection is even needed beyond checking them; 2. Experiment on a sample — the design is deliberately varied, and only
Activity 2 — Main Task: the Riverbend Council Investigation (14 min)
Pairs. A single scenario with three linked sub-problems.
Riverbend Council is deciding whether to build a new skate park. They need to answer three questions before deciding:
(a) Roughly how many young people in the area would use a skate park?
(b) Would a temporary “pop-up” skate ramp actually increase how many young people visit the local park, compared with how many visit now?
(c) Exactly how many pedestrians currently cross the pedestrian bridge near the proposed site each day (needed for a safety assessment)?
For each sub-problem: recommend a data collection technique (or combination), and justify it using cost, time, accuracy and any ethical considerations.
Socratic scaffolding (Polya cycle) for the hardest part, (b):
| Prompt | Purpose |
|---|---|
| Understand the problem. What exactly is being compared in (b)? | Park visits before the pop-up ramp versus visits after — a change caused by something the council controls. |
| Is this a census, sample, experiment or observation — or more than one? | It is an experiment (the ramp is a deliberate intervention) applied to counts that are effectively a census of visitors during the study period, since every visitor can be counted. |
| Devise a plan. What might make the “before vs after” comparison unreliable? | Other things change over time too — season, weather, school holidays — so an increase might not be caused by the ramp. |
| How could you make the comparison fairer? | Compare with a similar park without a ramp over the same period, or choose “before” and “after” periods with similar weather and school terms. |
| Carry out the plan. Write your final recommendation in one or two sentences. | Should name the technique(s) and the fairness safeguard. |
| Look back. Is your recommendation practical for a real council with a limited budget? | Weigh the cost of running two comparison sites against the value of a more reliable answer. |
Reference recommendations (teacher, one valid approach each):
(a) Sample — survey a representative sample of young people across the area (e.g. through schools), since a census of “everyone who might use it” is impossible before it exists; accuracy is good enough for planning purposes and far cheaper and faster than trying to ask everyone.
(b) Experiment, ideally with a comparison park, counting visitors (a near-census of visits during the trial) before and after — balances practicality against the risk of misattributing a change in visits to the ramp when something else caused it.
(c) Census + observation — an automatic pedestrian counter recording every crossing gives an exact count, which matters here because the number feeds directly into a safety assessment where undercounting could be dangerous.
Activity 3 — Peer Critique and Refine (9 min)
Swap your Riverbend plan with another pair. Audit against this checklist, then return it for one repair.
| Check | ✓ / ✗ / ? |
|---|---|
| Population and variable are clearly identified for each sub-problem | |
| Technique choice is justified with at least two of: cost, time, accuracy, ethics | |
| Any experiment includes a safeguard against other causes (e.g. a comparison group or period) | |
| Any claim of “exact” numbers is backed by a genuine census or full observation, not an inflated sample | |
| At least one ethical consideration is mentioned somewhere in the plan |
Expected common faults, for the debrief: treating (b) as pure observation and ignoring the need for a fair comparison; recommending a census for (a) despite it being impossible to identify “everyone who might use” a park that does not yet exist; forgetting to mention any ethical or practical limitation.
Checks for Understanding
(6 minutes — exit ticket, collected)
- A council wants to know how many households in its area recycle correctly. It already inspects every household’s bin as part of routine collection. Is this a census or a sample? Justify.
- A school wants to know if a new study app improves test scores, so it gives the app to one class and compares them with a similar class that does not use it. Name the technique, and state one thing that would make the comparison fairer.
- Identify the flaw: “To find the average commute time for our city, we asked
people waiting at a train station at 8 am.” - Reasoning. Explain why “we counted every fish caught in the harbour last year” is not the same as “we counted every fish in the harbour”.
Answers: 1. Census — routine inspection already covers every household, so no separate sampling step is needed; 2. Experiment (on a sample of two classes); fairer if the two classes are otherwise similar (same year level, similar prior results) and studied over the same period, to rule out other causes of any difference; 3. The sample only includes train commuters at a busy morning time, missing people who drive, cycle, work from home, or travel at other times — not representative of “the city’s” commute times as a whole; 4. The first only counts fish that were caught (a sample, and a biased one — easier-to-catch fish are over-represented), while the second would require locating every fish in the harbour, which is not practically possible — the two numbers describe very different things.
Common Misconceptions
| Misconception | How to pre-empt it |
|---|---|
| Assuming “we asked a lot of people” means the sample is automatically fair. | Warmup Q1 and Activity 1 Q6 — size does not fix a biased selection. |
| Treating a before/after comparison as proof of cause, with no safeguard against other explanations. | The Riverbend (b) Socratic scaffolding explicitly requires a fairness safeguard. |
| Confusing “we recorded everyone who did X” with “we recorded everyone in the population”. | Exit Q4 — caught fish versus all fish in the harbour. |
| Believing every real problem needs only one technique. | The Riverbend task deliberately needs three different recommendations. |
| Skipping ethical considerations because a scenario “seems harmless”. | Peer audit checklist explicitly requires at least one ethical mention. |
| Recommending a census simply because it “sounds more thorough”. | Sub-problem (a)‘s impossibility of a true census is the direct counter-example. |
Enrichment — Competition-Style Problems
E1 (Kangaroo style). A council claims ”
Answer
Only residents who chose to visit the library and fill in a form are counted — a small, self-selected sample, not the whole population, and likely biased toward people with strong opinions (usually in favour, since opponents may not bother). The
E2 (AMC Junior style). A school of
Answer
E.g. if the exact number feeds into a binding decision, such as safety funding tied precisely to walking numbers, or if walking rates vary a lot between year levels or areas and a small sample might miss that variation.
E3 (Challenge). A researcher wants to test whether a fertiliser increases crop yield, but only has one field. Explain a design that still allows a fair experiment despite having just one field.
Answer
Split the single field into matched plots (similar soil, sunlight, drainage), apply the fertiliser to some plots and none to others, and compare yields — this creates an internal comparison group without needing a second field.
E4 (Challenge). Explain why “the exact number of pedestrians crossing a bridge, recorded by an automatic sensor for one random Tuesday” is not the same as “the exact number of pedestrians crossing the bridge on average”.
Answer
One Tuesday is a single observation, not a census over time — pedestrian numbers likely vary by day of the week, weather and season, so one day’s exact count is still only a sample in time, even though it is exact for that day.
Homework
- A hospital already records every patient’s length of stay as part of standard admin. Is this a census or sample of patients? Justify.
- A company wants to know if a new packaging design increases sales. Describe an experiment they could run, including one safeguard against other explanations for a sales change.
- Identify the flaw: “To find out how Year 8 students feel about homework, we asked the students in the top maths class.”
- A council wants an exact count of cyclists using a new bike path in its first month. Recommend a technique and justify it.
- Explain why “everyone who entered our shop today” is a census of today’s shoppers but only a sample of all possible shoppers.
- Design a data collection plan for: “Does a four-day school week improve student wellbeing?” Name the technique(s) and one ethical consideration.
- Reasoning. Explain why comparing “before” and “after” data is not, by itself, enough to prove that a change caused an improvement.
- Challenge. A national park wants to estimate the total number of a rare bird species without disturbing them. Propose a technique, and explain why a census (physically finding every bird) would be impractical and possibly harmful.
Answers: Q1 — census, since routine records already cover every patient. Q3 — the top maths class is not representative of all Year 8 students; opinions on homework likely vary by ability group, so the sample is biased. Q4 — an automatic counter on the path gives a genuine census of path use, appropriate here because “exact” figures were specifically requested and a permanent sensor makes full coverage practical. Q5 — the shop’s population of interest might be “everyone who could ever shop there”, of which today’s shoppers are only a sample in time, even though today’s count is complete in itself. Q7 — other factors could change between the two time points (season, unrelated events, natural variation), so an improvement might not be caused by the change being studied; a comparison group or controlled conditions are needed to be confident about cause. Q8 — a sample-based technique such as observation from a distance (e.g. call counts, motion-sensor cameras, or counting in a representative area and scaling up) avoids the stress, disturbance and practical impossibility of physically locating and counting every individual bird across a large, often inaccessible habitat.