Topic 1.9 Comparing Distributions of a Quantitative Variable: compare two or more distributions of a quantitative variable by shape, center, spread, and unusual features, in context, using comparative language.

What this topic is asking

The College Board (Topic 1.9) wants you to compare two or more distributions of a quantitative variable, covering shape, center, spread, and unusual features, using explicitly comparative language ("higher than," "more variable than") and always in context. This is one of the most common free-response tasks in the whole course.

The comparison must be comparative

This is the rule that catches students out more than any other. The College Board's scoring guidelines repeatedly withhold credit when a response describes Group A fully, then describes Group B fully, without ever saying which is larger or more spread out. The fix is mechanical: every sentence should contain a comparison word and name both groups.

Compare like with like, across SOCS

Run the SOCS checklist, but each item becomes a comparison:

• Shape: "Group A is roughly symmetric, whereas Group B is skewed right."
• Center: "Group A has a higher median ($75$) than Group B ($68$)," so a typical A is larger.
• Spread: "Group A has a larger IQR (more variable) than Group B."
• Unusual features: "Group B has a high outlier at $65$, while Group A has none."

Always compare the same measure across groups: median to median and IQR to IQR. Mixing a mean with a median, or a range with an IQR, is not a valid comparison. When the groups have outliers or skew, prefer the resistant pair (median and IQR), exactly as in Topic 1.7.

Parallel boxplots are the natural tool

Parallel (side-by-side) boxplots on a shared axis are the standard display for comparison because they line up the five-number summaries so differences in center, spread, and skew are visible at a glance. The position of the median lines compares centers; the box widths compare spreads (IQR); the whisker lengths and plotted outlier points compare skew and unusual features. Reading a comparison straight off parallel boxplots is a core exam skill, and the same logic applies to back-to-back stemplots or overlaid histograms. Whatever the display, the discipline is identical: line up the same feature in both groups and state the difference comparatively in context.

Writing a full-credit comparison

Because comparison questions carry several marks, it is worth knowing exactly how markers award them: typically one point per SOCS component, given only if the statement is genuinely comparative and in context. So a reliable structure is four sentences, one each for shape, center, spread, and unusual features, every sentence naming both groups, the relevant measure, and a comparison word, with units attached. For example: "The commute times in City X are roughly symmetric while those in City Y are skewed right (shape); City X has a higher median commute of $25$ minutes than City Y's $20$ minutes (center); City X is more variable, with an IQR of $14$ minutes against City Y's $8$ minutes (spread); and City Y has a high outlier at $65$ minutes whereas City X has none (unusual features)." That paragraph would earn full points because it compares every component, quantifies where possible, and stays in context. The commonest ways to lose marks are to describe the groups separately, to omit a component (often spread or unusual features), or to forget the context and units, so a quick self-check against those three pitfalls before moving on is time well spent.

Try this

Q1. Rewrite "Group A has median $50$. Group B has median $40$." as a proper comparison. [1 point]

• Cue. "Group A has a higher median ($50$) than Group B ($40$), so a typical A value is larger."

Q2. When comparing two skewed distributions, which center and spread should you use, and why? [2 points]

• Cue. Median and IQR, because they are resistant to the skew and any outliers, giving a fair comparison of typical value and spread.