Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 4 - Data collection, sampling and distributions

Data collection, sampling and distributions

Plan data collection, choose samples and describe distributions using appropriate displays and summary measures.

Updated 2026-07-26 - 12 min read

Data collection, sampling and distributions 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.

Plan valid data collection

Define the population, variable and measurement method, then choose a sample process that reduces systematic bias. 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 plan valid data collection 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: Convenience can overrepresent one group; sampling should give relevant population members a fair chance of selection.

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

To estimate all Year 8 students' travel time, which sample is most representative?

Step 1 - identify the governing idea: Define the population, variable and measurement method, then choose a sample process that reduces systematic bias.

Step 2 - apply it to this evidence: Sampling across all classes reduces class-specific selection bias.

Result: Randomly select students from every Year 8 class

The relationship is visible in the working: Sampling across all classes reduces class-specific selection bias. 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:

  • Survey only the cycling club — It does not agree with the required relationship: Sampling across all classes reduces class-specific selection bias.
  • Ask the first 20 arrivals — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Survey only one class — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Which is a numerical variable?

Step 1 - identify the governing idea: Define the population, variable and measurement method, then choose a sample process that reduces systematic bias.

Step 2 - apply it to this evidence: Minutes are measured as numbers with meaningful differences.

Result: Minutes spent reading yesterday

The relationship is visible in the working: Minutes are measured as numbers with meaningful differences. 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:

  • Favourite book genre — It does not agree with the required relationship: Minutes are measured as numbers with meaningful differences.
  • School house colour — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Preferred author — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A survey asks, 'Don't you agree homework is excessive?' What is the problem?

Step 1 - identify the governing idea: Define the population, variable and measurement method, then choose a sample process that reduces systematic bias.

Step 2 - apply it to this evidence: The question signals a preferred answer and may bias responses.

Result: The wording is leading

The relationship is visible in the working: The question signals a preferred answer and may bias responses. 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 random — It does not agree with the required relationship: The question signals a preferred answer and may bias responses.
  • The variable is continuous — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • There are too many response options — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Choose displays and summary measures

Match the display and centre/spread measures to the variable, distribution shape and investigation 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 choose displays and summary measures 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: Outliers and skew can pull the mean, so median and interquartile information may better describe a typical value.

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

A data set is strongly right-skewed by one very large value. Which centre is usually more resistant?

Step 1 - identify the governing idea: Match the display and centre/spread measures to the variable, distribution shape and investigation question.

Step 2 - apply it to this evidence: The median depends on order and is less affected by an extreme value.

Result: Median

The relationship is visible in the working: The median depends on order and is less affected by an extreme value. 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:

  • Mean — It does not agree with the required relationship: The median depends on order and is less affected by an extreme value.
  • Range — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Maximum — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Which display is suitable for a distribution of test scores grouped into intervals?

Step 1 - identify the governing idea: Match the display and centre/spread measures to the variable, distribution shape and investigation question.

Step 2 - apply it to this evidence: A histogram shows frequencies for continuous numerical intervals.

Result: Histogram

The relationship is visible in the working: A histogram shows frequencies for continuous numerical intervals. 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:

  • Pie chart only — It does not agree with the required relationship: A histogram shows frequencies for continuous numerical intervals.
  • Pictograph — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Network diagram — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Scores are 4, 5, 5, 6, 20. What is the median?

Step 1 - identify the governing idea: Match the display and centre/spread measures to the variable, distribution shape and investigation question.

Step 2 - apply it to this evidence: The ordered middle value is 5.

Result: 5

The relationship is visible in the working: The ordered middle value is 5. 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:

  • 8 — It does not agree with the required relationship: The ordered middle value is 5.
  • 20 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Describe and compare distributions

Compare distributions using centre, spread, shape and unusual features, supported by values read from consistent scales. 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 describe and compare distributions 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: One extreme does not describe the whole distribution; compare typical values and variability.

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

Group A has median 14 and IQR 3; Group B has median 12 and IQR 8. Which description is supported?

Step 1 - identify the governing idea: Compare distributions using centre, spread, shape and unusual features, supported by values read from consistent scales.

Step 2 - apply it to this evidence: A has the larger median and smaller interquartile range.

Result: A is typically higher and less variable

The relationship is visible in the working: A has the larger median and smaller interquartile range. 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:

  • B is typically higher and less variable — It does not agree with the required relationship: A has the larger median and smaller interquartile range.
  • A has a larger range for certain — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The groups are identical — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Two boxplots use different horizontal scales. What must be done before comparison?

Step 1 - identify the governing idea: Compare distributions using centre, spread, shape and unusual features, supported by values read from consistent scales.

Step 2 - apply it to this evidence: Visual box lengths are not comparable until interpreted using their axes.

Result: Read values from each scale

The relationship is visible in the working: Visual box lengths are not comparable until interpreted using their axes. 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:

  • Compare only colours — It does not agree with the required relationship: Visual box lengths are not comparable until interpreted using their axes.
  • Ignore the medians — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Assume equal units — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A distribution has most values near 10 and one value at 60. What feature should be reported?

Step 1 - identify the governing idea: Compare distributions using centre, spread, shape and unusual features, supported by values read from consistent scales.

Step 2 - apply it to this evidence: The isolated value is far from the main cluster.

Result: A possible high outlier

The relationship is visible in the working: The isolated value is far from the main cluster. 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:

  • Perfect symmetry — It does not agree with the required relationship: The isolated value is far from the main cluster.
  • No variability — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • A low outlier — 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. Which is a numerical variable?
  2. Which display is suitable for a distribution of test scores grouped into intervals?
  3. Two boxplots use different horizontal scales. What must be done before comparison?

Answers

  1. Minutes spent reading yesterday — Minutes are measured as numbers with meaningful differences.
  2. Histogram — A histogram shows frequencies for continuous numerical intervals.
  3. Read values from each scale — Visual box lengths are not comparable until interpreted using their axes.

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