Comparing Distributions of a Quantitative Variable

Unit 1 · Exploring One-Variable Data

What AP Stats asks here

Comparing distributions is a recurring free-response prompt and the most common rubric tripwire in Unit 1. AP graders require comparison language — 'Group A is more variable than Group B' — not a list of separate descriptions. Cover all four SOCS dimensions, tie everything to the variable's context, and lead with the most distinguishing piece.

Comparison checklist

Compare each SOCS piece

\text{Shape} \to \text{Outliers} \to \text{Center} \to \text{Spread}

Comparison phrase

\text{higher than, more variable than, less spread out than}

End with context

\text{tie numbers to the named variable and units}

AP Tip: Lead with the more distinguishing SOCS piece. If centers are within a few points but spreads differ by a factor of two, the spread is the headline.

Caution: Listing 'Group A median 75, IQR 10; Group B median 60, IQR 25' loses comparison credit on AP. The rubric wants 'A is higher and less variable than B' explicitly.

Type 1

Two-group SOCS comparison

Given side-by-side boxplots or a five-number summary table, write (or pick) a complete comparison addressing each SOCS dimension with comparison language.

Example 1

Monthly textbook spending ($) for two majors: Engineering: Min 80, Q1 120, Median 180, Q3 240, Max 350 English: Min 40, Q1 70, Median 95, Q3 130, Max 200 Which choice best compares the distributions? (A) Engineering and English have different summary statistics, but the distributions are otherwise the same. (B) Engineering students spend roughly twice as much on textbooks (median

180 vs

95) and show greater variability (IQR

120 vs

60); both distributions are slightly right-skewed with no obvious outliers. (C) Engineering: median

180, IQR

120. English: median

95, IQR

60. (D) English students spend more because their distribution starts lower.

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Type 2

Comparison language critique

AP rubrics require explicit comparison phrases. A student who lists statistics for each group separately receives partial credit at best.

Example 2

A student writes: "Group A has a median of 75. Group A's IQR is 10. Group B has a median of 60. Group B's IQR is 25." Which rewritten answer best fixes the rubric problem? (A) "Group A median is greater. Group A IQR is less." (B) "Group A has a higher median (75) than Group B (60), and Group A's scores are less variable (IQR 10 vs 25), meaning Group A performs better and more consistently." (C) "Group A and Group B both have medians and IQRs." (D) "Group A is normal; Group B is skewed."

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Type 3

Same center, different spread

When centers are essentially identical, the spread tells the story. Interpret what that means in context — consistency vs variability.

Example 3

Two classes both have median test score 50, but distribution P has IQR 5 and distribution Q has IQR 30 (scores out of 100). Which choice best describes the practical difference? (A) The two distributions are essentially the same because their medians match. (B) P scores are tightly clustered around 50 (middle half roughly 47.5 to 52.5); Q scores spread widely (middle half roughly 35 to 65). Same typical score, very different consistency. (C) Q is better because more students score above 50. (D) The IQR difference is meaningless without the means.

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