Year 9 Mathematics · AC9M9ST03

Comparing Numerical Data Distributions

Distributions are compared using centre, spread and shape—not one summary number alone. Histograms and comparative displays reveal skew, symmetry, modes and outliers

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Learning goalsSay it simply

Distributions are compared using centre, spread and shape—not one summary number alone. Histograms and comparative displays reveal skew, symmetry, modes and outliers.

By the end of this lesson, you should be able to:

  • construct and interpret comparative displays
  • describe symmetry, skew and bimodality
  • compare centre and spread using mean, median and range
  • explain how outliers affect summaries
Key conceptTeach from the board

Compare distributions using three lenses

Describe centre, spread and shape. A complete comparison usually addresses all three.

Mean and median respond differently to skew and outliers

The mean uses every value and is pulled by extremes; the median depends on order and is more resistant.

Shape carries information

Symmetric, positively skewed, negatively skewed and bimodal distributions can have similar centres but very different structures.

Comparative displays need common scales

Back-to-back stem-and-leaf plots or aligned histograms should use comparable scales and binning before visual comparisons are made.

Worked examplesWe do

Worked examples

AC9M9ST03 - Comparing Numerical Data Distributions
Example 1

Outlier effect Adding 100 to 4,5,5,6,6 greatly increases the mean and range but changes the median much less.

Example 2

Positive skew A long right-hand tail indicates positive skew.

Example 3

Bimodal distribution Two clear peaks may suggest two subgroups or processes.

Example 4

Compare classes Equal medians but different ranges indicate similar centre but different spread.

Curriculum examplesCopied content

Australian Curriculum v9.0 — AC9M9ST03: represent the distribution of multiple data sets for numerical variables using comparative representations; compare data distributions with consideration of centre, spread and shape, and the effect of outliers on these measures

Victoria: VC2M9ST03 — Level 9 Statistics. The mapping names direct Level 9 content where available and explicitly identifies supporting content where the Victorian structure separates an idea differently.

NSW: Stage 5 Core — Data classification, visualisation and analysis; Path — Data analysis and statistical enquiry. NSW organises Years 7–10 Mathematics through Stage 5 Core content groups and Paths rather than a one-code-per-Year-9 structure, so this lesson does not force a false one-to-one outcome.

Lesson componentAustralian CurriculumVictoriaNSW
Concept teaching + worked examplesAC9M9ST03VC2M9ST03 — Level 9 StatisticsStage 5 Core — Data classification, visualisation and analysis; Path — Data analysis and statistical enquiry
Guided + independent practiceApplies the descriptor through progressively less-scaffolded problemsBuilds the corresponding Level 9 mathematical knowledge and fluencySupports Stage 5 Core/Path application and Working mathematically
Reasoning + assessment + masteryChecks transfer, justification, interpretation and model limitsChecks Level 9 reasoning at the mapped content depthChecks relevant Stage 5 reasoning without claiming a false Year 9 equivalent
Questions and answersWith answers
What three features should I compare?
Centre, spread and shape.
Which measure of centre is more resistant to outliers?
The median.
What does bimodal mean?
The distribution has two distinct peaks or modes.
Practice and reviewReady for practice
  • A higher mean means every value is higher: Two distributions can overlap heavily.
  • Range describes typical spread perfectly: It uses only minimum and maximum and is sensitive to outliers.
  • Histogram bars represent individual values: Bars represent intervals and frequencies.
  • Same centre means same distribution: Spread and shape can differ substantially.
  1. [5 marks] Calculate and compare summary statistics for two datasets.
  2. [6 marks] Interpret comparative histograms using centre, spread, shape and outliers.
  3. [7 marks] Evaluate a claim based only on mean by using full distributions to show what information the claim omits.

Review hint: A full-mark response shows the method, keeps units and restrictions visible, interprets the result in context and checks whether the answer is reasonable.

Curriculum alignmentStart here

Distributions are compared using centre, spread and shape—not one summary number alone. Histograms and comparative displays reveal skew, symmetry, modes and outliers.

By the end of this lesson, you should be able to:

  • construct and interpret comparative displays
  • describe symmetry, skew and bimodality
  • compare centre and spread using mean, median and range
  • explain how outliers affect summaries
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