Statistics rarely deals with just one group. The real power comes from comparison. Does the new teaching method produce better results? Is this year's class stronger than last year's? This lesson teaches you the systematic approach: examine centre, spread, and shape side by side using parallel box plots and summary statistics.
Today's hook, School A: most marks 70–80, a few at 90. School B: marks spread evenly from 50–100. Both have mean = 75. Which school would you send your child to, and why?
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Recall, your gut answer first
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
Worksheets
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
School A: most marks 70–80, a few at 90. School B: marks spread evenly from 50–100. Both have mean = 75. Which school would you send your child to? Why?
Without calculating write your gut feeling. We'll revisit at the end.
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The comparison framework you need
+5 XP to read Every statistical comparison uses three dimensions.
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The comparison framework you need
+5 XP to read
Every statistical comparison uses three dimensions. Miss one and you give an incomplete answer in the HSC.
Centre
Compare medians (or means). Higher median = typically higher performance. Use same measure for both groups.
Spread
Compare IQR or range. Smaller IQR = more consistent. Larger IQR = more variable. Consistency matters for quality control and assessment.
Shape
Is the distribution symmetric, left-skewed, or right-skewed? Does one group have outliers that the other does not?
Key insight: Always compare using the same measures. Comparing the median of one group with the mean of another can be misleading.
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What you'll master
Know
Key facts
What to compare: centre, spread, shape
The tools for comparison (box plots, summary stats)
Why consistent measures matter
Understand
Concepts
Why context matters in interpretation
How spread affects decision-making
When differences are statistically meaningful
Can do
Skills
Compare distributions using parallel box plots
Compare using summary statistics
Write well-structured comparative statements
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Key terms
MedianThe middle value when data is ordered. Resistant to extreme values.
IQRInterquartile range: $Q_3 - Q_1$. Measures the spread of the middle 50% of data.
Parallel box plotsTwo or more box plots drawn on the same scale, the best visual tool for comparing groups.
SymmetricDistribution where data is evenly distributed around the centre. Mean ≈ median.
SkewedDistribution with a longer tail on one side. Positive (right) skew has a tail to the right.
OutlierA data point more than 1.5 × IQR below Q1 or above Q3.
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Comparing centre: which group is higher?
Compare the medians (or means) of the two distributions to determine typical performance:
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Comparing centre: which group is higher?
core concept
Compare the medians (or means) of the two distributions to determine typical performance:
If Median A > Median B, Group A typically scores higher
If medians are similar, typical performance is comparable
Use medians when data may be skewed or contain outliers
Example: School X median = 78, School Y median = 72.
"School X typically achieves higher results than School Y, with a median 6 marks higher."
Comparing spread: Use IQR or range:
Smaller IQR = more consistent, less variable
Larger IQR = more variable, less predictable
Example: Machine X IQR = 2 mm, Machine Y IQR = 5 mm.
"Machine X produces more consistent parts, its middle 50% of measurements vary by only 2 mm compared to 5 mm for Machine Y."
To compare two distributions, compare medians (typical value), ranges (spread), and shape (symmetry, skew). State which group is higher and by how much, a comparison needs two values and a direction, not just individual summaries.
Pause, copy the three-part comparison framework: compare centre (medians), spread (IQR or range), and shape (symmetric/skewed), and note that each comparison needs a direction ("Group A has a higher median than Group B") into your book.
Quick check: Group A has IQR = 8. Group B has IQR = 20. Which group is more consistent?
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Shape, outliers, and the full picture
We just saw that comparing two data sets begins with comparing medians to identify which group has a higher typical value.
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Shape, outliers, and the full picture
core concept
We just saw that comparing two data sets begins with comparing medians to identify which group has a higher typical value. That raises a question: two groups can have the same median but look completely different as distributions, what else should you compare? This card answers it → always report spread (range or IQR) and shape (symmetric, positively or negatively skewed) alongside centre, and identify whether outliers are present before choosing which measure to use.
Shape: Is one distribution symmetric while the other is skewed? A positively skewed distribution has its tail pointing right, the mean is pulled above the median.
Outliers: Does one group have more extreme values? Outliers affect the mean but not the median, so always check.
Full comparison framework in action:
"Group A has a higher median than Group B (78 vs 68), indicating better typical performance. However, Group A also has a larger IQR (20 vs 13), suggesting less consistency. Group B is more symmetric while Group A is slightly right-skewed. Group A has two high outliers at 95 which Group B does not."
Notice the structure: centre comparison → spread comparison → shape/outlier comment. Full marks require all three.
Shape: symmetric (median ≈ mean), positively skewed (tail right, mean > median), negatively skewed (tail left, mean < median). Outliers pull the mean but not the median, always identify outliers before choosing a measure of centre.
Pause, copy the three distribution shapes (symmetric: mean ≈ median; positive skew: tail right, mean > median; negative skew: tail left, mean < median) and the outlier rule (outliers pull the mean but not the median) into your book.
True or false: Two distributions with identical means always have identical shapes.
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Worked examples · reveal step by step
Worked examples · reveal step by step
Worked examples · reveal step by step
PROBLEM · COMPARING TWO CLASSES
Class X: min=45, Q1=60, median=72, Q3=80, max=90. Class Y: min=50, Q1=65, median=68, Q3=78, max=95. Compare centre, spread, and shape.
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Compare centre X median = 72, Y median = 68
Class X typically scores 4 marks higher than Class Y.
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Compare spread X IQR = 80 − 60 = 20; Y IQR = 78 − 65 = 13
Class Y is more consistent, its middle 50% of scores span only 13 marks vs 20 for Class X.
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Compare shape/range X range = 90 − 45 = 45; Y range = 95 − 50 = 45
Both have equal range. Y has a higher maximum (95 vs 90). Both appear roughly symmetric from their five-number summaries.
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Write the comparison statement
Class X typically scores higher (median 72 vs 68) but is less consistent (IQR 20 vs 13). Class Y is more consistent with a slightly wider overall range.
Fill the gap: Team A: mean = 80, SD = 5. Team B: mean = 75, SD = 12. Team has a higher average and Team is more consistent.
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Common errors · traps that cost marks
Common errors · traps that cost marks
Common errors · traps that cost marks
Trap 01
Comparing different measures
Comparing the median of Group A with the mean of Group B when the data is skewed. Use the same measure for both groups.
Trap 02
Ignoring spread
Saying "Group A is better because its median is higher" without noting that it is also less consistent. Incomplete comparisons lose marks.
Trap 03
Comparing without numbers
"Group A is more spread out" scores 0. Always include values: "Group A has an IQR of 20 compared to 13 for Group B."
Match each term to its meaning:
Median
IQR
Positive skew
Parallel box plots
Tail pointing right
Best tool for visual comparison
Typical value, resistant to outliers
Spread of middle 50%
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Revisit your thinking
Quick-fire practice
Quick-fire practice
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Group 1: mean = 65, SD = 8. Group 2: mean = 70, SD = 15. Write one sentence comparing centre and one comparing spread.
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From box plots: School A median = 80, IQR = 10. School B median = 78, IQR = 18. Which school is more consistent? Which typically scores higher?
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Factory A: mean = 50 mm, SD = 1 mm. Factory B: mean = 50.5 mm, SD = 3 mm. For precision engineering, which factory would you choose and why?
Top 3 list: Name THREE things you must address in a complete statistical comparison of two groups.
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Revisit your thinking
School A is more predictable, most students score 70–80, so you know what to expect. School B has wide variation, meaning some students do very well and some poorly. For a risk-averse parent, School A is preferable. For a high-achieving student who believes they can score at the top, School B offers the chance of a higher mark. The key insight: identical means can hide very different experiences. Always examine spread, not just centre.
What has changed in your thinking? What did you get right?
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Show what you have learned
Multiple choice, then short answer under exam conditions.
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Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence. That tells the system what to drill next.
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Short answer
ApplyBand 42 marks
SA 1. Factory A: mean = 50 mm, SD = 1 mm. Factory B: mean = 50.5 mm, SD = 3 mm. (a) Compare the centres of the two distributions. (b) Compare the spreads. (c) Which factory would you choose for precision engineering, and why? (2 marks)
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AnalyseBand 53 marks
SA 2. Two schools' results: School X median = 82, IQR = 8. School Y median = 80, IQR = 15. School X has two outliers at 95; School Y has no outliers. (a) Compare the distributions comprehensively. (b) A parent with a high-achieving student prefers School X. A parent with an average student prefers School Y. Explain both choices using statistics. (3 marks)
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📖 Answers (click to reveal)
Drill 1: Centre, Group 2 has higher mean (70 vs 65). Spread, Group 2 more variable (SD 15 vs 8). Drill 2: School A more consistent (IQR 10 vs 18); School A higher typical score (80 vs 78). Drill 3: Factory A, consistency matters more than small mean difference; SD 1 mm far more precise than 3 mm.
SA 1 (2 marks): (a) Factory B slightly higher mean (50.5 vs 50) [0.5]. (b) Factory A much smaller SD (1 vs 3), far more consistent [0.5]. (c) Factory A for precision, consistency is critical; the 0.5 mm mean difference is negligible compared to the variability benefit [1].
SA 2 (3 marks): (a) School X higher median (82 vs 80), more consistent (IQR 8 vs 15), two outliers at 95 [1]. (b) High-achiever prefers X: outliers show some students can reach 95, and high median suggests a stronger cohort [1]. Average student might prefer Y: with wider IQR they have more chance to stand out; or X: more consistent performance means the average student is surrounded by peers performing at a similar level [1].
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Consolidate and move on
Sit the module quiz, then close the lesson off.
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Take the full module quiz
quiz
A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.