Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 4 - Comparing distributions and statistical investigations

Comparing distributions and statistical investigations

Represent and compare multiple numerical distributions, choose displays and conduct evidence-based investigations.

Updated 2026-07-26 - 11 min read

Comparing distributions and statistical investigations is part of the Year 9 curriculum because students must do more than carry out a familiar calculation. They need to choose a relationship, represent it accurately, explain the result and decide whether it makes sense. The sections below develop those decisions through explicit rules, worked examples, misconception repair and transfer.

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.

Compare centre, spread and shape

One summary cannot describe every distribution. Median and IQR often suit skewed data; mean and standard deviation/range may suit reasonably symmetric data at this level.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by compare centre, spread and shape 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: Interpret centre with spread, shape and the meaning of higher or lower values.

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

Group A median 12, IQR 3; B median 13, IQR 9. Best comparison?

Step 1 - identify the governing idea: A defensible comparison names the group, relevant centre, spread, shape and any outliers in context.

Step 2 - apply it to this evidence: The median difference is 1 while IQR triples.

Result: B is slightly higher at centre but much more variable

The relationship is visible in the working: The median difference is 1 while IQR triples. 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 always better — It does not agree with the required relationship: The median difference is 1 while IQR triples.
  • A has higher centre — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • They are identical — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

A distribution has a long right tail. Best shape description?

Step 1 - identify the governing idea: A defensible comparison names the group, relevant centre, spread, shape and any outliers in context.

Step 2 - apply it to this evidence: Large values extend the right tail.

Result: Right-skewed

The relationship is visible in the working: Large values extend the right tail. 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:

  • Left-skewed — It does not agree with the required relationship: Large values extend the right tail.
  • Symmetric — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Uniform because values vary — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Why mention an outlier in a comparison?

Step 1 - identify the governing idea: A defensible comparison names the group, relevant centre, spread, shape and any outliers in context.

Step 2 - apply it to this evidence: Its impact and context need evaluation.

Result: It can affect summaries and may represent important variation or error

The relationship is visible in the working: Its impact and context need evaluation. 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 must always be deleted — It does not agree with the required relationship: Its impact and context need evaluation.
  • It proves the sample is invalid — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It changes every median — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Choose and interpret displays

Side-by-side box plots support several-group centre/spread comparisons; histograms show shape; scatterplots show relationships between two numerical variables.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by choose and interpret displays 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: Choose by data type, comparison and features, then justify the 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 2.1

Best display to compare distributions for three classes?

Step 1 - identify the governing idea: Display choice should make the investigation question and relevant distribution features visible.

Step 2 - apply it to this evidence: They align median and spread summaries across groups.

Result: Side-by-side box plots

The relationship is visible in the working: They align median and spread summaries across groups. 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:

  • Three unrelated pie charts — It does not agree with the required relationship: They align median and spread summaries across groups.
  • A single pictograph — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • A map without data — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Best display for relationship between height and arm span?

Step 1 - identify the governing idea: Display choice should make the investigation question and relevant distribution features visible.

Step 2 - apply it to this evidence: Both variables are numerical and paired.

Result: Scatterplot

The relationship is visible in the working: Both variables are numerical and paired. 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 — It does not agree with the required relationship: Both variables are numerical and paired.
  • One box plot — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Frequency table with no pairs — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Why use a histogram for one large numerical data set?

Step 1 - identify the governing idea: Display choice should make the investigation question and relevant distribution features visible.

Step 2 - apply it to this evidence: Binned frequencies show peaks, gaps and skew.

Result: It reveals distribution shape across intervals

The relationship is visible in the working: Binned frequencies show peaks, gaps and skew. 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 lists every name — It does not agree with the required relationship: Binned frequencies show peaks, gaps and skew.
  • It proves causation — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It removes outliers — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Plan, conduct and report an investigation

Variables, population, sampling, display and summaries should be chosen before seeing results. Conclusions must answer the question without exceeding the sample evidence.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by plan, conduct and report an investigation 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 conclusion must interpret the relevant pattern and discuss evidence strength and limitations.

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

Which question is statistical?

Step 1 - identify the governing idea: A statistical investigation cycles through question, plan, data, analysis, conclusion and evaluation.

Step 2 - apply it to this evidence: It anticipates variation across individuals.

Result: How does travel time vary among Year 9 students?

The relationship is visible in the working: It anticipates variation across individuals. 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:

  • What is 7+5? — It does not agree with the required relationship: It anticipates variation across individuals.
  • What is one student's name? — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Is 10 greater than 8? — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A study compares reaction time before and after practice on same students. Useful design?

Step 1 - identify the governing idea: A statistical investigation cycles through question, plan, data, analysis, conclusion and evaluation.

Step 2 - apply it to this evidence: Pairing controls between-person differences.

Result: Paired measurements for each student

The relationship is visible in the working: Pairing controls between-person 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:

  • Two unrelated convenience groups only — It does not agree with the required relationship: Pairing controls between-person differences.
  • One measurement total — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Ask students to guess — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Sample evidence is small and variable. Best conclusion language?

Step 1 - identify the governing idea: A statistical investigation cycles through question, plan, data, analysis, conclusion and evaluation.

Step 2 - apply it to this evidence: The claim is calibrated to evidence strength.

Result: The sample suggests a pattern, but more representative data are needed

The relationship is visible in the working: The claim is calibrated to evidence strength. 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 result proves the rule for everyone — It does not agree with the required relationship: The claim is calibrated to evidence strength.
  • No conclusion of any kind is possible — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Delete variability — 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. A distribution has a long right tail. Best shape description?
  2. Best display for relationship between height and arm span?
  3. A study compares reaction time before and after practice on same students. Useful design?

Answers

  1. Right-skewed — Large values extend the right tail.
  2. Scatterplot — Both variables are numerical and paired.
  3. Paired measurements for each student — Pairing controls between-person differences.

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.

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