Sampling Variation and Sample Size — AC9M8ST03

• Recognise natural variation between random samples.
• Compare sample proportions and distributions from one population.
• Explain the effect of larger sample size on variation.
• Use repeated sampling/simulation to communicate uncertainty.

Recall population, random sample, proportion, mean, median, range, relative frequency and how to compare simple data distributions.

Two random samples from the same population can contain different members, so their proportions, means, medians and ranges can differ. This is sampling variation, not automatically an error.

Repeated samples show the pattern of possible results. A single sample estimate should therefore be reported as evidence about the population, not as the exact population value.

When selection remains random and fair, larger samples usually fluctuate less from sample to sample because each estimate uses information from more population members. This does not guarantee that every larger sample is closer to the population value, and it does not fix selection bias.

Digital simulation is useful because many repeated samples can be generated quickly. At Year 8, the goal is to compare variation visually and numerically, not to introduce formal confidence intervals or hypothesis tests.

Sampling ideas have real uses: school-uniform opinion polling, heights and arm spans, weather forecasting, visitor proportions, biodiversity monitoring by First Nations Ranger Groups and other groups, and data-driven/AI decision-making.

• Different random samples should match. Natural sample-to-sample variation is expected.
• One sample proportion is the population proportion. It is an estimate based on one sample.
• A larger sample is guaranteed closer. Larger samples are usually more stable, not guaranteed perfect.
• Increasing sample size fixes bias. A biased selection method can remain biased at any size.
• Simulation proves the future. It illustrates variation under the assumptions used.

1. Compare four random-sample proportions from the same population.
2. Explain why the results differ without calling them measurement errors.
3. Compare the spread of repeated results for n=20 and n=100.
4. Identify one assumption in using historical relative frequency to predict a future proportion.

1. Calculate the proportions 27/50, 31/50 and 29/50 and compare them.
2. Explain why one sample of 50 cannot determine exact school-wide opinion.
3. Describe what you expect to happen to sample-to-sample spread when sample size increases from 25 to 250.
4. Compare two sets of repeated-sample means and identify which appears more stable.
5. Explain why removing one extreme value can change mean and range differently.
6. Describe how repeated biodiversity samples could support a cautious population claim.
7. Explain why an AI system trained on one sample should be tested on data beyond that sample.

Sample A has n=40 and reports 65% support. Sample B has n=400 and reports 58% support. A student says Sample A must be better because its percentage is higher. Explain why that reasoning is invalid, what sample size changes, and what other design information is needed before judging either estimate.

1. Why do random samples differ? They contain different randomly selected members of the population.
2. What is sampling variation? Natural variation in sample results arising from which members are selected.
3. What usually happens as random sample size increases? Estimates tend to vary less from sample to sample.
4. Does a large sample fix biased selection? No; selection bias is a design problem.
5. Why repeat samples or simulations? To see the range and pattern of plausible sample results rather than trusting one result as exact.

1. Four random samples of 50 students give support counts 24, 30, 27 and 29. Calculate proportions and explain the variation.
   Review hint: Convert to proportions, then describe sampling variation rather than labelling different results as errors.
2. Repeated samples of 25 have a visibly wider set of sample means than repeated samples of 200. Explain what this suggests and what it does not guarantee.
   Review hint: Larger samples are typically more stable but no individual sample is guaranteed closest.
3. A biodiversity team increases sample size but continues sampling only beside a road. Evaluate the change.
   Review hint: Separate random variation from systematic site-selection bias.

• I expect random samples to vary.
• I distinguish sample values from population values.
• I explain the typical effect of sample size.
• I use repeated sampling to communicate uncertainty.

Exit ticket: Why does a larger random sample usually improve stability without guaranteeing an exact population answer?

Australian Curriculum v9.0 — AC9M8ST03: compare variations in distributions and proportions from random samples of the same size and recognise the effect of sample size on variation.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8ST03: Exact.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-DAT-C-02; MAO-WM-01: Partial. NSW Stage 4 analyses simple datasets, while the national descriptor explicitly develops repeated random sampling and sample-size effects.

• AC9M8ST02 — sampling methods and distributions
• AC9M8ST04 — statistical investigations and inference

• Australian Curriculum v9.0
• Victorian Curriculum F–10 Version 2.0 Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for Variations in distributions and proportions obtained from random samples of...

Mapped skill: compare variations in distributions and proportions obtained from random samples of the same size drawn from a population and recognise the effect of sample size on this variation

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.