114i - Selecting Ethical and Fair Sampling Methods

Lesson 114 — Selecting Ethical and Fair Sampling Methods

Strand: Statistics | Descriptor: AC9M8ST04 | Duration: 45 minutes

Block note. Stage 2 (Plan, continued) of the investigation. Today the class finalises how the screen-time sample will be selected and collected, ready for Lesson 115.

Learning Intentions

• To compare sampling methods for fairness and practicality.
• To apply ethical principles — consent, privacy, and freedom from bias — to a real data collection plan.

Success Criteria

I can:

1. Describe and distinguish simple random, systematic, stratified and convenience sampling.
2. Identify a source of bias in a sampling method and explain its likely effect.
3. Explain why consent, anonymity and non-leading questions matter in data collection.
4. Choose and justify a sampling method for the class investigation.

Warmup

(5 minutes — spot the problem, pairs)

Each plan claims to sample “Year 8 students fairly”. What is wrong with each?

1. Survey every student who follows the school’s Instagram account.
2. Stand at the canteen at lunch and ask whoever walks past first.
3. Ask only the students in your own friendship group.
4. Hand the survey to the front row of your own class only.

Answers: 1. Only students who use Instagram are included — excludes everyone else, a biased subgroup; 2. Canteen visitors may differ from students who bring lunch from home; 3. Friends often share similar habits, so the sample is not representative; 4. One row of one class is far too small and specific to represent all Year 8 students.

The pattern across all four: each method makes it easy to collect data, but easy and fair are not the same thing.

Activities

Activity 1 — Explicit Instruction: Sampling Methods and Ethics (15 min)

Four sampling methods, compared:

Method	How it works	Strength	Risk
Simple random	Every member of the population has an equal chance of selection (e.g. draw names from a hat, use a random number generator on a class list)	Least biased if done correctly	Can, by chance, still under-represent a group in a small sample
Systematic	Select every th person from an ordered list (e.g. every th name on the school roll)	Easy to apply consistently	Can align accidentally with a pattern in the list (e.g. class order)
Stratified	Divide the population into groups (strata) — e.g. by class — then randomly sample from each group in proportion to its size	Guarantees fair representation of known subgroups	Requires knowing the subgroup sizes in advance
Convenience	Sample whoever is easiest to reach (e.g. your own class, people walking past)	Fast, simple	Usually biased — the “easy to reach” group is rarely representative

I do — choosing a method for our investigation.

Our population is Year 8 students across classes of . A convenience sample of just my own class would answer for students, not all . A stratified random sample — randomly selecting, say, students from each of the classes ( total) — guarantees every class is represented, and within each class every student has an equal chance of selection. This is the method we will use.

Ethical principles for data collection — the four we must apply:

Principle	What it requires	Why it matters here
Consent	Participants (or their guardians, for minors) understand what is being asked and agree to take part	Students must know their data is for a maths investigation and may opt out
Anonymity / privacy	Individual responses cannot be traced back to a named student	Screen-time habits are personal; no student should fear judgement for their answer
Non-leading questions	Wording does not push participants toward an answer	”Don’t you agree you spend too much time on your phone?” is leading; a neutral yes/no test is not
Right to decline	No student is forced or pressured to answer	Some students may not wish to disclose this information, and that choice must be respected

We do — spot the ethical issue together:

1. The survey asks students to write their name “so we can follow up if needed.”
2. A teacher reads results aloud by student name to “make it more engaging.”
3. The question is worded: “How many hours do you waste on your phone each day?”

(Answers: 1. Breaks anonymity — names are not needed for a proportion investigation and should not be collected; 2. Breaks confidentiality and could embarrass students, discouraging honest answers in future; 3. “Waste” is a loaded, judgemental word — leading rather than neutral.)

Activity 2 — Guided Practice: Designing Our Data Collection (14 min)

Pairs, converging on one class-agreed method.

Using the stratified random sampling method introduced above, design the full collection plan:

1. How will the students per class be randomly selected? (e.g. numbering the class roll and using a random number generator or dice)
2. How will anonymity be protected? (e.g. no names collected; responses submitted in a way that cannot be traced to an individual)
3. What will the exact wording of the question be, using the precise variable definition from Lesson 113?
4. How will students be told about the investigation and given the chance to decline?
5. How will the data be recorded and stored, so it stays anonymous and secure?

Circulating prompts:

Prompt	Purpose
If a selected student is absent, what will you do?	Forces a fair, pre-decided replacement rule (e.g. next name on the randomised list), not an ad hoc convenience choice.
Could a reader trace any answer back to a specific student?	The core anonymity test.
Does your wording match Lesson 113’s precise variable definition exactly?	Consistency between the plan and the question.
Is any student pressured — by a teacher’s presence, by peers watching — to answer a certain way?	Freedom from social pressure, a subtler form of bias.

Activity 3 — Inquiry: Auditing a Flawed Plan (6 min)

Pairs.

A different school runs the same investigation this way: “We put a link to the survey on the school app. Any student who wants to can fill it in over the week.”

1. Which sampling method is this closest to?
2. Who is likely to respond, and who is likely to be missing?
3. Name the specific type of bias this produces.
4. Suggest one change that would make it fairer.

Answers: 1. Convenience / voluntary response sampling; 2. Students who check the app often, and who feel strongly about the topic (perhaps very high or very low screen-time users), are more likely to respond; students who are indifferent or do not use the app regularly are missing; 3. Voluntary response bias (a form of self-selection bias) — those who opt in are not representative of everyone; 4. Instead, randomly select students from a full class list (stratified across classes) and approach them directly, rather than waiting for volunteers.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Name the four sampling methods studied today.
2. Which method guarantees every class is represented in the sample? Explain why.
3. State two ethical principles that must be applied when collecting the screen-time data.
4. Identify the flaw: “A survey is left on the staffroom table for teachers to fill in about their own screen use, for a whole-school investigation into student habits.”
5. Reasoning. Explain why “convenient” and “fair” are not the same thing, using an example from today.

Answers: 1. Simple random, systematic, stratified, convenience; 2. Stratified — it deliberately samples from every class (stratum) rather than relying on chance to include them all; 3. Any two of: consent, anonymity/privacy, non-leading questions, right to decline; 4. The survey is about teachers, not students — it cannot answer a question about student habits at all; a completely wrong population; 5. E.g. asking your own class (convenience) is fast and easy, but it only represents of the students and likely shares similar habits as friends, so it is not fair as a representation of the whole Year 8 population.

Common Misconceptions

Misconception	How to pre-empt it
Believing any large sample is automatically fair, regardless of how it was selected.	E4’s flawed-plan inquiry — size never fixes a biased selection method.
Treating “anonymous” and “confidential” as the same thing.	Anonymous means no names are ever collected; confidential means names are collected but kept private — anonymity is the stronger, preferred protection here.
Thinking ethics is only about being “nice”, not about the validity of the data.	Frame consent and non-leading wording as also protecting data quality, not just participants’ feelings.
Assuming stratified sampling requires surveying an equal number from every group regardless of group size.	Note that strata are sampled in proportion to their size when group sizes differ — our seven classes are equal here, which simplifies it, but this will not always be the case.
Believing a systematic sample (every th name) is the same as a simple random sample.	Contrast: systematic is not random once the starting point is fixed, and can hide bias if the list itself has a pattern.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A school of students is divided into year levels of each. A stratified sample of students is needed, in proportion to year level size. How many students should be selected from each year level?

Answer

E2 (AMC Junior style). A researcher numbers a population list to and selects every th name, starting from a randomly chosen number between and . How many people are selected in total?

Answer

people.

E3 (Challenge). A survey about exercise habits is only sent to students enrolled in the school’s sports teams. Explain the direction of the bias this would produce in an estimate of “the proportion of all students who exercise regularly.”

Answer

The estimate would be biased upward — sports-team members are, by definition, already active, so the sample proportion meeting an exercise guideline would likely be much higher than the true proportion across all students, including those not on any team.

E4 (Challenge). A stratified sample of students (six from each of seven classes) is compared with a simple random sample of students drawn from the whole Year 8 list without regard to class. Both are valid random methods. Explain one advantage the stratified method still has.

Answer

A simple random sample of from could, by chance, draw very unevenly from the seven classes — even leaving one class completely unsampled. The stratified method removes this particular risk by guaranteeing every class contributes exactly six students, so representation across classes is certain rather than left to chance.

E5 (Challenge). A question is worded: “Most experts agree screens harm sleep — do you limit your screen time before bed?” Identify every ethical problem with this single question.

Answer

It is a leading question — the opening claim (“most experts agree…”) primes the respondent toward answering “yes”. It also mixes two things: it does not clearly ask for a testable yes/no fact about the respondent’s own behaviour, but invites a self-justifying answer shaped by the stated opinion. A neutral rewrite would simply ask: “In the hour before you go to sleep, do you use a screen — yes or no?”

Homework

1. Describe, in your own words, the difference between simple random and stratified sampling.
2. A population of is split into two groups: Year 8 and Year 9 students. For a stratified sample of , how many should come from each group?
3. Explain why a systematic sample (every th name on a list ordered by class) could be biased if the list happens to group students by class in blocks of exactly .
4. Identify the ethical issue and rewrite it appropriately: “Please write your name and enter to win a prize if you answer honestly about your screen time.”
5. A student collects data by asking only their friends in the school chat group. Name the sampling method this resembles and the type of bias it introduces.
6. State two reasons anonymity is important in the class screen-time investigation specifically.
7. Reasoning. Explain why offering students the right to decline can still produce a fair sample, even though not everyone selected will respond.
8. Reasoning. A convenience sample of students is compared with a stratified random sample of . Explain why the smaller stratified sample can still be the more trustworthy one.
9. Challenge. Design a systematic sampling plan to select students fairly from the full Year 8 roll of (ordered by student ID number), including the random starting point rule.

Answers: Q2 — Year 8: ; Year 9: . Q3 — every selected student would come from the same position within their class group (e.g. always the 1st, 11th, 21st name), which could accidentally align with some other pattern (such as alphabetical or ability grouping), producing an unrepresentative sample. Q4 — collecting names removes anonymity and the prize incentive could pressure students to answer in a way they think sounds “better” rather than honestly; rewrite: no names collected, and no incentive tied to a particular type of answer — e.g. “Your response is anonymous and will only be used for our class maths investigation.” Q5 — convenience (and voluntary response) sampling; friendship-group bias, since friends often share similar habits and are not representative of the whole population. Q6 — screen time is a personal habit some students may not want judged, and anonymity encourages honest answers rather than answers shaped by fear of embarrassment or social pressure. Q7 — as long as the selection of who is invited to participate was random and fair, allowing declines does not reintroduce bias by itself — though a very low response rate could still create new bias if those who decline differ systematically from those who respond, which is why response rates should also be monitored. Q8 — the convenience sample’s method of selection is unfair to begin with (e.g. drawn from wherever data was easy to get), so no amount of extra size fixes that bias, whereas the smaller stratified sample was selected to fairly represent every subgroup and remains a trustworthy estimate despite being smaller. Q9 — e.g. order the students by ID number; use a random number generator to choose a starting point between and (since ); then select every th student from that starting point, wrapping around the list if needed, until students are selected.