106i - Random and Non-Random Sampling

Lesson 106 — Explicit Instruction: Random and Non-Random Sampling Techniques

Strand: Statistics | Descriptor: AC9M8ST02 | Duration: 45 minutes

Learning Intentions

• To distinguish random from non-random sampling techniques.
• To identify simple random, systematic, stratified, convenience and self-selected sampling in real examples.

Success Criteria

I can:

1. Define random sampling and explain why it reduces bias.
2. Identify simple random, systematic and stratified sampling from a description.
3. Identify convenience and self-selected (voluntary response) sampling from a description.
4. Explain, for a given non-random sample, why it risks being unrepresentative.

Warmup

(6 minutes — equal chance or not? pairs, mini whiteboards)

For each method of choosing students from your class of , decide: does every student have an equal chance of being chosen?

1. Names written on identical slips, drawn from a hat.
2. The teacher picks the students sitting closest to the door.
3. Every third name on the class roll, starting from a randomly chosen point.
4. The first students to volunteer.

Answers: 1. Yes — every slip has the same chance of being drawn; 2. No — only students near the door can be chosen; 3. Yes, in a sense — once the random starting point is chosen, every student ends up with an equal chance overall (explained fully in Activity 1); 4. No — only students willing to volunteer can be chosen, which is not the same as everyone having an equal chance.

The key idea for today: a sample is random only if every member of the population has a known, fair chance of being selected — not just if the selection “feels” fair or unbiased.

Activities

Activity 1 — Explicit Instruction: Five Sampling Techniques (13 min)

I do — three random techniques:

Technique	How it works	Real example
Simple random sample	Every member of the population has an equal chance; selection is by lottery-style draw or a random number generator	Choosing students for a leadership panel by drawing names from a hat
Systematic sample	List the whole population, choose a random starting point, then select every th member	Selecting every th person on the electoral roll for jury duty summons
Stratified sample	Split the population into groups (strata) that matter for the question — e.g. year level, state — then randomly sample from each group, usually in proportion to its size	The ABS Labour Force Survey samples households from every state and territory in proportion to population, so small states are not swamped or ignored

Why stratified sampling matters — worked example. A school has Year 7s, Year 8s and Year 9s ( total). A survey needs a sample of students, proportional by year level:

Check: ✓. Within each year level, the , and students are then chosen randomly.

I do — two non-random techniques:

Technique	How it works	Real example
Convenience sample	Whoever is easiest to reach is sampled	A researcher surveys shoppers at one shopping centre on one afternoon
Self-selected (voluntary response) sample	People choose themselves to take part	An online news site’s “vote in our poll” button, or a call-in radio survey

We do — classify together:

1. A council randomly selects addresses from its full ratepayer database using a computer.
2. A radio station asks listeners to text in their opinion on a new law.
3. A researcher interviews the first people who walk past on a particular street corner.
4. A school selects every th name from an alphabetical roll of all students, after a random start.
5. A survey selects students proportionally from each year level, then randomly within each level.

(Answers: 1. Simple random; 2. Self-selected — only motivated listeners respond; 3. Convenience — whoever happens to be walking by; 4. Systematic; 5. Stratified.)

Activity 2 — Guided Practice: Identify and Critique (13 min)

Pairs. For each scenario, name the technique and state whether it is random or non-random, then explain one risk if it is non-random.

1. A teacher chooses the first five students to finish a quiz for extra feedback.
2. A market researcher uses a computer to randomly select phone numbers to call.
3. A newspaper prints a coupon readers can mail in with their opinion on a proposed tax change.
4. A wildlife survey selects one site to sample from each of five different habitat types in a national park, then randomly places counting points within each site.
5. A gym owner asks the members currently at the gym on a Tuesday morning what new equipment they would like.

Circulating prompts:

Prompt	Purpose
Could someone in the population have zero chance of being chosen?	Tests the “equal chance” definition directly.
Who is more likely to respond, and does that group differ from the whole population?	Surfaces the bias risk in self-selected samples.
If this used a random number generator instead, what would change?	Contrasts the method actually used with a genuinely random alternative.

(Answers: 1. Non-random — a form of convenience/judgement sample; students who finish first may differ systematically (e.g. faster, more confident) from the class as a whole; 2. Random — simple random sample; 3. Non-random — self-selected; people with strong opinions (especially opposed to the tax) are more likely to respond, skewing results; 4. Stratified (by habitat type) with random selection within strata — random; 5. Non-random — convenience; a Tuesday morning crowd (e.g. retirees, shift workers) may not represent all members, such as those who only attend evenings or weekends.)

Activity 3 — Inquiry: Design a Stratified Sample for Your School (7 min)

Pairs.

Your school wants student opinions on a new bell-time schedule and needs a sample of students, fairly representing the whole school.

1. Suggest a stratified sampling plan (choose your own realistic year-level population sizes if you don’t know your school’s exact numbers).
2. Calculate how many students should be sampled from each stratum, showing your working.
3. Explain how you would then choose the actual students within each stratum, and why that step must also be random.

Discussion target: stratifying without randomising within each stratum still risks bias — e.g. always picking the first names alphabetically in Year 8 is not a fair chance for every Year 8 student. Both steps — stratifying and randomising within strata — are needed.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Define random sampling in your own words.
2. Name the sampling technique: “Every 12th customer entering a store is asked to complete a short survey, starting from a randomly chosen customer.”
3. Name the sampling technique, and explain why it risks bias: “A social media app asks users to tap a button if they want to share feedback.”
4. A population of has in Group A and in Group B. A stratified sample of is needed, proportional to group size. Calculate the sample size for each group.
5. Reasoning. Explain why a convenience sample can still be large and yet unrepresentative.

Answers: 1. A sampling method where every member of the population has a known, equal (or fair, calculable) chance of being selected; 2. Systematic sampling; 3. Self-selected (voluntary response) sampling — only users motivated enough to tap the button respond, likely those with strong (often negative) opinions, so the sample may not reflect typical users; 4. Group A: ; Group B: ; 5. Size does not fix bias — if every member of a large sample is drawn from the same easy-to-reach group (e.g. all from one location or time), the sample can still systematically miss whole sections of the population, however many people are included.

Common Misconceptions

Misconception	How to pre-empt it
Believing “random” means “haphazard” or “without a plan”.	Emphasise random sampling requires a deliberate, calculable, equal-chance method — the opposite of careless.
Assuming a large sample is automatically random or representative.	Exit Q5; convenience and self-selected samples can be large and still biased.
Thinking stratifying alone is enough, without random selection within each stratum.	Activity 3’s discussion target.
Confusing systematic sampling with convenience sampling.	Systematic sampling still gives every population member a calculable chance via the fixed interval and random start; convenience sampling does not.
Believing self-selected samples are fine “if enough people respond”.	The self-selection bias (strong opinions over-represented) does not shrink with more responses — it can even grow.
Treating “stratified” as just “sampling from different groups”, without the proportional and random-within-group requirements.	Worked proportional calculation in Activity 1; Activity 3’s calculation task.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A school of students is stratified into three year levels of , and students for a sample of . How many should be sampled from the smallest year level?

Answer

.

E2 (AMC Junior style). A radio poll asking “Do you support the new stadium?” receives “yes” votes and “no” votes from callers. A separate simple random phone sample of people finds support. Explain why these two results might genuinely differ, even though the poll has far more responses.

Answer

The radio poll is self-selected — people with strong feelings (often opponents of a proposal, or organised campaigns) are far more likely to call in, while the random sample gives every person in the population a fair chance regardless of how strongly they feel, making it more likely to reflect the true population opinion despite its much smaller size.

E3 (Challenge). A researcher wants a systematic sample of every th name from a list of people, to select people. Explain how to choose the random starting point, and list the first three selected positions if the random start is position .

Answer

Choose a random whole number from to as the starting position (since ), then add repeatedly. Starting at position : positions

E4 (Challenge). A stratified sample proportionally represents age groups in a city, but within each age group, only people who answer their phone between am and pm on weekdays are ever reached. Explain why this sample is not fully random despite the stratification.

Answer

Stratifying by age fixes the proportions across groups, but within each group, people unavailable during weekday business hours (e.g. those working full-time) have zero chance of being reached — so the within-stratum selection is not actually random, undermining the overall design.

Homework

1. Define, in your own words: simple random sample, systematic sample, stratified sample, convenience sample, self-selected sample.
2. Name the technique: “A quality-control worker checks every th item off a factory production line.”
3. Name the technique, and state whether it is random or non-random: “A survey is handed only to people leaving a gym at 6 am.”
4. A population of is split into three groups of , and . Calculate a proportional stratified sample of from each group.
5. Explain why an online poll on a news website’s homepage is a self-selected sample, and describe one group of people likely to be under-represented.
6. Design a systematic sampling plan to select names from an alphabetical class list of students. State the interval and describe how you would choose the random start.
7. Reasoning. Explain why a stratified sample can be more accurate than a simple random sample of the same size, when the population naturally splits into very different groups.
8. Challenge. A council wants to sample opinions from a city of residents made up of four suburbs with populations , , and . For a stratified sample of , calculate how many residents should be sampled from each suburb, and explain why simply sampling people from each suburb (a quarter each) would not be an appropriate alternative.

Answers: Q2 — systematic sampling. Q3 — convenience sampling (and arguably self-selected among gym-goers); non-random — it misses everyone who is not a 6 am gym-goer. Q4 — ; ; . Q5 — only people who visit the site and choose to click vote, likely more engaged or opinionated users; people who do not use that news site, or do not feel strongly enough to click, are under-represented. Q6 — interval ; choose a random starting position from to , then select every th name after that. Q7 — a simple random sample could by chance under- or over-represent a smaller group, especially if that group behaves very differently; stratifying guarantees every group is represented in proportion to its true size, reducing this risk. Q8 — ; ; ; ; sampling an equal from each suburb would over-represent the smallest suburb and under-represent the largest relative to their true population share, biasing the overall city-wide estimate.