Comparing Numerical Data Distributions — AC9M9ST03

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

Recall mean, median, range, stem-and-leaf plots, histograms and ordered numerical data.

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.

• 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. Calculate mean, median and range for two short datasets.
2. Describe which dataset is more spread out.
3. Identify skew from a histogram.
4. Add an outlier and compare changes in mean, median and range.

Guided method: Name the mathematical structure first, show the calculation or representation, then verify with an estimate, inverse operation, second representation or digital check.

1. Compare two datasets with equal medians but different ranges.
2. Explain how a large outlier changes mean and median.
3. Describe a supplied bimodal histogram.
4. Choose mean or median for a skewed dataset and justify.
5. Create a back-to-back stem-and-leaf comparison.
6. Write a complete centre-spread-shape comparison sentence.

Two sports teams have the same mean score, but one distribution is tightly clustered while the other is bimodal with a large range. Explain why the means alone give a poor comparison of performance.

Reasoning standard: Make a claim, show the relevant mathematical evidence, explain why it supports the conclusion and state any condition or limitation.

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.

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.

• I can compare centre, spread and shape.
• I can describe skew and bimodality.
• I can explain outlier effects.
• I can choose mean or median appropriately.
• I can use common scales for fair visual comparison.

Exit ticket: Solve one unfamiliar example and explain the key decision in words, not just symbols.

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.

Use these links to move through the Year 9 learning sequence rather than studying the code in isolation.

• AC9M9ST02 — Sampling Methods, Sample Variation and Misleading Displays
• AC9M9ST04 — Choosing and Interpreting Data Displays

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

Curriculum equivalents for Represent the distribution of multiple data sets for numerical variables...

Mapped skill: 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

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.