Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 4 - Sampling variation, inference and uncertainty

Sampling variation, inference and uncertainty

Use repeated samples to understand sampling variation and make cautious evidence-based inferences about populations.

Updated 2026-07-26 - 12 min read

Sampling variation, inference and uncertainty is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Recognise sampling variation

Different random samples from the same population usually produce different statistics because the selected individuals vary. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by recognise sampling variation and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Some difference is expected by chance; its size and the sampling method determine whether concern is warranted.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Two random samples give mean heights 161 cm and 163 cm. What is the best initial interpretation?

Step 1 - identify the governing idea: Different random samples from the same population usually produce different statistics because the selected individuals vary.

Step 2 - apply it to this evidence: Different random members produce slightly different sample means.

Result: Ordinary sampling variation may explain the difference

The relationship is visible in the working: Different random members produce slightly different sample means. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • One mean must be false — It does not agree with the required relationship: Different random members produce slightly different sample means.
  • The population mean changed instantly — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Both samples are biased — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

What usually happens to sample-to-sample variability when random sample size increases?

Step 1 - identify the governing idea: Different random samples from the same population usually produce different statistics because the selected individuals vary.

Step 2 - apply it to this evidence: Larger random samples are usually more stable representations of the population.

Result: It tends to decrease

The relationship is visible in the working: Larger random samples are usually more stable representations of the population. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • It always becomes zero — It does not agree with the required relationship: Larger random samples are usually more stable representations of the population.
  • It always increases — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It becomes bias — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Which action helps observe sampling variation?

Step 1 - identify the governing idea: Different random samples from the same population usually produce different statistics because the selected individuals vary.

Step 2 - apply it to this evidence: Repeated samples reveal the distribution of sample statistics.

Result: Take many random samples using the same method

The relationship is visible in the working: Repeated samples reveal the distribution of sample statistics. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Change the question each time — It does not agree with the required relationship: Repeated samples reveal the distribution of sample statistics.
  • Keep only the largest result — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Survey the same convenient group — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Evaluate sample evidence

Judge an inference using sample selection, sample size, variability and how well the measured variable represents the population question. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by evaluate sample evidence and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Size can reduce random variation but cannot repair systematic exclusion or leading measurement.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

An online poll of 10,000 website visitors predicts all Australians' views. What remains a major concern?

Step 1 - identify the governing idea: Judge an inference using sample selection, sample size, variability and how well the measured variable represents the population question.

Step 2 - apply it to this evidence: Visitors chose to participate and may differ systematically from the population.

Result: Self-selection bias

The relationship is visible in the working: Visitors chose to participate and may differ systematically from the population. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The sample is too small — It does not agree with the required relationship: Visitors chose to participate and may differ systematically from the population.
  • There are too many responses — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • A mean cannot be calculated — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Which sample gives stronger evidence about a school's average commute?

Step 1 - identify the governing idea: Judge an inference using sample selection, sample size, variability and how well the measured variable represents the population question.

Step 2 - apply it to this evidence: It is larger and covers the school's year-level structure.

Result: A random sample of 200 across year levels

The relationship is visible in the working: It is larger and covers the school's year-level structure. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The first 15 staff arrivals — It does not agree with the required relationship: It is larger and covers the school's year-level structure.
  • Ten students from one sports team — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Five volunteers — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

A sample result changes sharply after one extreme value is removed. What does this show?

Step 1 - identify the governing idea: Judge an inference using sample selection, sample size, variability and how well the measured variable represents the population question.

Step 2 - apply it to this evidence: The extreme observation strongly influenced the chosen summary.

Result: The statistic is sensitive to that value

The relationship is visible in the working: The extreme observation strongly influenced the chosen summary. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The value must be deleted — It does not agree with the required relationship: The extreme observation strongly influenced the chosen summary.
  • The sample now proves the population claim — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Sampling variation has ended — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Make cautious population inferences

State what the sample suggests, quantify relevant sample evidence and acknowledge sampling and measurement uncertainty. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by make cautious population inferences and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: A sample estimate is evidence about a population, not an exact census result; conclusions should use calibrated language.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

In a random sample, 62% prefer option A. Which conclusion is most defensible?

Step 1 - identify the governing idea: State what the sample suggests, quantify relevant sample evidence and acknowledge sampling and measurement uncertainty.

Step 2 - apply it to this evidence: The sample supports a cautious estimate while retaining uncertainty.

Result: The sample suggests option A may be preferred in the population

The relationship is visible in the working: The sample supports a cautious estimate while retaining uncertainty. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Exactly 62% of the population prefer A — It does not agree with the required relationship: The sample supports a cautious estimate while retaining uncertainty.
  • Every subgroup prefers A — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Option A will always win — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Two representative samples give 48% and 51%. What is supported?

Step 1 - identify the governing idea: State what the sample suggests, quantify relevant sample evidence and acknowledge sampling and measurement uncertainty.

Step 2 - apply it to this evidence: Small sample differences around 50% do not establish a stable majority without more evidence.

Result: The results are close enough that a clear majority is uncertain

The relationship is visible in the working: Small sample differences around 50% do not establish a stable majority without more evidence. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The second proves a majority — It does not agree with the required relationship: Small sample differences around 50% do not establish a stable majority without more evidence.
  • The first proves a minority — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Average them and claim certainty — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Which report best communicates uncertainty?

Step 1 - identify the governing idea: State what the sample suggests, quantify relevant sample evidence and acknowledge sampling and measurement uncertainty.

Step 2 - apply it to this evidence: It gives the estimate and information needed to judge its limits.

Result: Estimate 54%, noting sampling method and sample size

The relationship is visible in the working: It gives the estimate and information needed to judge its limits. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The true value is 54% — It does not agree with the required relationship: It gives the estimate and information needed to judge its limits.
  • Most people agree, with no data — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The survey cannot tell us anything — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. What usually happens to sample-to-sample variability when random sample size increases?
  2. Which sample gives stronger evidence about a school's average commute?
  3. Two representative samples give 48% and 51%. What is supported?

Answers

  1. It tends to decrease — Larger random samples are usually more stable representations of the population.
  2. A random sample of 200 across year levels — It is larger and covers the school's year-level structure.
  3. The results are close enough that a clear majority is uncertain — Small sample differences around 50% do not establish a stable majority without more evidence.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources