Two Classes, Same Mean
Two sixth-grade classes take the same five-question quiz. Class A's scores: 72, 74, 75, 76, 78. Class B's scores: 60, 70, 75, 80, 90. Both have a mean of 75 and a median of 75. Asked which class did better, 21 of 28 students say "they're the same, because the averages match," and a few argue Class B is better "because it has the highest score, 90." The teacher then displays two bar graphs of the same data drawn with different vertical axis scales, and several students read the compressed one as showing "no real difference." The next unit introduces interquartile range and mean absolute deviation.
1. Quantify what "the same" misses, and justify which measure of spread you would lead with for these data.
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Compute both:
- Class A: deviations from 75 are 3, 1, 0, 1, 3 → MAD = 8/5 = 1.6. Range = 6.
- Class B: deviations 15, 5, 0, 5, 15 → MAD = 40/5 = 8. Range = 30.
Same center, very different consistency: a typical Class B score sits 8 points from the mean, a typical Class A score 1.6 points.
Which to lead with - a real judgment call. Range is immediate but rests on a single pair of values and will mislead when one outlier appears. IQR is resistant to outliers, but with n = 5 the quartiles are coarse and unstable (each quartile is essentially one data point), so IQR is a poor fit for this data set. MAD is the better lead here because it uses every value, has a natural meaning ("average distance from the mean"), and is what students need to see the difference. Save IQR for a larger data set where it is interpretable.
2. How do you respond to the "highest score" argument and to the graph confusion?
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The 90 argument: a single value is not a summary. Ask: "If you had to be randomly assigned to one of these classes for the next quiz, which would you pick?" Most choose A, because a typical score is what will happen to them. Reframe the goal as describing the whole distribution, not its best member. Note that the 90 pulls B's mean up while B's low scores pull it down - the mean coincidentally matches A's, which is precisely why a second number is needed.
The graphs: have students read the vertical axis increments and the bar baseline before comparing. Two lessons to draw out:
- A truncated axis usually exaggerates small differences.
- A compressed axis hides real differences - which is what happened here, since the spread difference is genuinely large (MAD 8 vs 1.6). Both are scale choices, and students should ask "what are the units per tick?" before drawing any conclusion. This is a Mathematical Processes moment: critique the representation, don't just consume it.
3. Sequence the next two lessons and describe the evidence you would accept that students are ready to move on.
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Sequence:
- Build the idea before the formula. Put both classes' scores on one line plot with a shared axis from 55 to 95. The spread difference is visible without any computation. Ask: "Both classes average 75. Are they the same?"
- Invent the measure. "How far is a typical score from 75?" Students propose averaging the distances; that is MAD, and they should compute it themselves for Class A (1.6) and B (8) before the term is introduced.
- Then the procedure and the notation, with a second data set (n = 12+) so IQR and quartiles are meaningful.
Evidence of readiness:
- Students describe a distribution with two numbers (center and spread) without prompting.
- They can predict the effect of a change: adding a score of 75 to Class A leaves the mean at 75 but decreases the MAD (12/6 = 2.0? - have them check: 8/6 ≈ 1.33). Predictions like this cannot be made by formula-following.
- They can say why a measure of center alone is an incomplete summary, in their own words.
Avoid introducing MAD as a formula to compute on a worksheet first; that produces correct arithmetic with the original misconception intact.