← Back to course

Comparing Distributions: What a Complete Comparison Requires

You’ll be able to

Four things, in context, comparatively

Almost every AP Statistics free-response set contains a comparison, and it is scored on four elements: shape, center, spread and unusual features. Two things are required beyond naming them. The description must be in context — the variable and the units, not "the data" — and it must be comparative, using words like higher, more variable, more strongly skewed. Writing a complete paragraph about group A and then a complete paragraph about group B answers a different question and does not earn the comparison.

Why "comparative" is a scoring rule and not a style note

A reader is checking whether you have connected the two distributions. "The median for the treatment group is 42 minutes" and "the median for the control group is 35 minutes" are two facts; "the treatment group's median is about 7 minutes higher than the control's" is a comparison. The information is identical and only the second earns the point, because only the second demonstrates that you looked at both together. The cheapest fix in the whole course is to insert higher than, less variable than, more skewed than into sentences you were already writing.

Let the shape choose the summaries

For a roughly symmetric distribution with no outliers, report the mean and standard deviation. For a skewed distribution or one with outliers, report the median and IQR, because they are resistant — a single extreme value moves the mean and standard deviation substantially and barely moves the median and IQR. Reporting a mean for a strongly skewed income distribution is not wrong arithmetic; it is a poor choice of summary, and questions are written specifically to test whether you notice.

On the exam

A reliable checklist: shape, center, spread, unusual — in context, comparatively. Four elements, two requirements. Say the variable name and units at least once. If the distributions are skewed, say so and switch to median and IQR in the same breath.

Worked example

Two boxplots show commuting times in minutes for two cities. City A: min 8, Q1 18, median 25, Q3 34, max 71. City B: min 12, Q1 22, median 28, Q3 33, max 44. Write a complete comparison.

  1. 1.Shape: City A's box has a long upper whisker (34 to 71) against a short lower one, indicating right skew; City B is closer to symmetric.
  2. 2.Center: compare the medians comparatively — 25 minutes for A against 28 for B, so B is about 3 minutes higher.
  3. 3.Spread: compare IQRs — A is 34 − 18 = 16 minutes, B is 33 − 22 = 11 minutes, so A is more variable in the middle half.
  4. 4.Unusual features: A's maximum of 71 is far above Q3 + 1.5(IQR) = 34 + 24 = 58, so it is an outlier; B has none by that rule since 33 + 16.5 = 49.5 exceeds its maximum of 44.
Answer: City A's commute times are right-skewed while City B's are roughly symmetric. City B has the higher median commute, about 28 minutes against about 25 for City A. City A is more variable, with an IQR of 16 minutes against City B's 11 minutes. City A also has a high outlier at 71 minutes, above the 58-minute upper fence, while City B has none. Because City A is skewed with an outlier, median and IQR are the appropriate summaries for both groups.
Checkpoint

A student writes: "Group 1 has a median of 40 and an IQR of 12. Group 2 has a median of 55 and an IQR of 9." On a comparison question, this response would —

Tip

The 1.5 × IQR rule gives fences at Q1 − 1.5(IQR) and Q3 + 1.5(IQR). Anything beyond them is an outlier by that convention. Compute the fence and state it, rather than asserting that a value "looks like" an outlier — the arithmetic is the evidence.

Checkpoint

A distribution of household incomes is strongly right-skewed. The most appropriate pair of summary statistics is —

Answer the 2 checkpoints as you read.

Sign in to save your progress