How to compare two distributions

By Jude Wallis · Updated July 31, 2026

Compare two distributions with sentences that hold both groups at once, like "Group A's median of 42 minutes is higher than Group B's median of 35 minutes." Cover shape, center, variability, and unusual features, always in context. Two separate descriptions written in sequence is not a comparison.

AP Statistics: Unit 1 (topics 1.9 Comparisons of the Distributions for One Quantitative Variable, 1.7 Summary Statistics for One Quantitative Variable). In the Fall 2026 AP Statistics course, comparing distributions of one quantitative variable is Unit 1 topic 1.9, scored under skill 4.A (describe and compare representations of data and summary statistics).

A comparison is one sentence holding both groups

The most common way to lose points on a comparison question is to describe one distribution, then describe the other, and stop. That is two descriptions sitting next to each other, and it leaves the actual comparison for the reader to do.

A comparison sentence names both groups and a direction of difference:

Not a comparison: "Group A's median is 42 minutes. Group B's median is 35 minutes."
A comparison: "Group A's median of 42 minutes is higher than Group B's median of 35 minutes."

The second version states the same two numbers and adds three words that do the scoring work. The AP skill being tested, 4.A, is to describe and compare representations of data and summary statistics, so a response with no comparative language has not answered the question asked. Comparing distributions is topic 1.9 Comparisons of the Distributions for One Quantitative Variable.

Compare all four features, not just center

Center is the feature students remember and often the only one they write about. A full comparison covers the same four features you would use to describe a single distribution, each one comparatively.

Shape. Which is skewed and which way, or are both roughly symmetric. Number of peaks counts here too.
Center. Compare medians, or compare means, and say by how much when you can.
Variability. Compare IQRs or standard deviations, and say which group is more spread out.
Unusual features. Outliers, gaps, and clusters, said out loud for each group, including "neither group has any outliers" when that is true.

Context still applies to every sentence. Name the variable, the two groups, and the units, or a grader cannot tell what the numbers measure.

The comparative words that earn the point

You need a word or phrase that puts the two groups in a relation. These are the ones that do it cleanly:

Center: higher than, lower than, greater than, about the same as, roughly equal to, about 8 minutes longer than.
Variability: more variable than, less variable than, more spread out than, about equally variable, has a larger IQR than.
Shape: is skewed right while the other is roughly symmetric, both are skewed right but A's tail is longer.
Unusual features: has a high outlier while the other has none, both contain one low outlier.

Two phrasings look comparative but are not. "Group A is 27 and Group B is 19" lists values without relating them. "Group A is bigger" relates them without saying which feature or by how much, and bigger could mean center, spread, or sample size.

Comparing from boxplots

Side-by-side boxplots are built for comparison, and the AP course says they support comparing center, variability, outliers, and skewness or symmetry. Read them in that order.

Center: compare the median lines, which are actual values you can read off the axis.
Variability: compare the box lengths, which are the IQRs, and the whisker-to-whisker distances, which are the ranges.
Shape: a median sitting closer to $Q_1$ with a longer right whisker indicates right skew; a median near the middle of the box with even whiskers indicates rough symmetry.
Outliers: on a modified boxplot they are the separate dots or asterisks beyond the whiskers.

What boxplots cannot show is just as important. They hide gaps and clusters inside the box, they hide the number of peaks, and they hide the sample size. If a question asks which group is bimodal, a boxplot cannot answer it, and neither can a pair of five-number summaries.

Comparing from histograms, dotplots, and stemplots

Histograms, dotplots, and back-to-back stem-and-leaf plots support everything boxplots do plus the features boxplots hide: shape in detail, clusters, and gaps. The trade-off is that you have to estimate the center and the spread rather than read them off a marked line.

Two practical rules make these comparisons safe.

Check the axes match. Two histograms are only visually comparable when their horizontal axes cover the same values with the same bin width. If they do not, compare estimated numbers rather than the shapes of the bars.
Compare percentages when the group sizes differ. With 40 shoppers in one group and 120 in another, a taller bar can just mean a bigger group. Relative frequency puts both groups on the same footing.

Back-to-back stemplots avoid the axis problem entirely, since both groups share one stem. That is why they show up on exam questions with small data sets.

The trap: judging spread by how wide the box looks

This one catches strong students. When two boxplots are printed as separate figures rather than on one shared axis, the physical width of a box tells you nothing until you read the axis scale under it.

Take two figures whose axis lines are both drawn 10 centimeters long. A box that measures 2 centimeters on the page then represents an IQR of 40 minutes if its axis runs from 0 to 200, while a box measuring 3 centimeters represents an IQR of only 15 minutes if its axis runs from 0 to 50. The narrower-looking box is the more variable group. The second worked example below runs those numbers.

The fix is mechanical. Read $Q_1$ and $Q_3$ off each axis, subtract to get each IQR, and compare the two numbers. The same warning applies to histograms drawn at different widths or with different bin sizes, and to any pair of graphs where one axis has been stretched.

A template you can fill in

Write one sentence per feature, each naming both groups.

"The distribution of [variable] for [group 1] is [shape], while the distribution for [group 2] is [shape]."
"[Group 1] has a higher [median or mean] [variable] ([value] [units]) than [group 2] ([value] [units]), a difference of about [value] [units]."
"[Group 1] is [more / less] variable than [group 2]: its IQR of [value] [units] is [larger / smaller] than [group 2]'s IQR of [value] [units]."
"[Group 1] has [an outlier at (value) (units) / no outliers], while [group 2] has [...]."

Four sentences, eight numbers, both group names in every one. If a question then asks you to justify a claim from the comparison, point at the specific feature that supports it rather than repeating the whole description.

Where this sits in the AP course

Comparing distributions is topic 1.9 in Unit 1 of the Fall 2026 AP Statistics course, and topic 1.7 (1.7.E) adds the matching task for summary statistics, which can compare center, variability, shape, and outliers across two or more independent samples. Unit 1 carries 20 to 30% of the multiple-choice section.

This skill also carries forward. The same comparative sentences appear when you compare two groups in an experiment, and again in Unit 4 when a two-sample t procedure asks you to describe the data before running inference. Practice on describing distributions practice, or build two data sets side by side in the descriptive statistics sandbox. The official course description is at AP Central.

Comparing two rowing-machine sessions from raw data

A gym records how many minutes each member spent on the rowing machine at two sessions. Morning (11 members): 25, 32, 18, 27, 40, 22, 30, 20, 35, 24, 28. Evening (12 members): 16, 45, 12, 22, 10, 26, 60, 15, 33, 18, 14, 20. Compare the two distributions.

Sort the morning data: 18, 20, 22, 24, 25, 27, 28, 30, 32, 35, 40. With $n = 11$ , the median is the 6th value, 27 minutes.
Find the morning quartiles with the median-excluded convention. Lower half: 18, 20, 22, 24, 25, so $Q_1 = 22$ . Upper half: 28, 30, 32, 35, 40, so $Q_3 = 32$ . The morning five-number summary is 18, 22, 27, 32, 40 minutes.
Morning spread and outlier check: $\text{IQR} = 32 - 22 = 10$ minutes, range $= 40 - 18 = 22$ minutes. Fences: $1.5 \times 10 = 15$ , so $22 - 15 = 7$ and $32 + 15 = 47$ minutes. No morning value falls outside, so there are no outliers.
Morning shape check: the sum is 301 minutes, so $\bar{x} = 301 / 11 = 27.36$ minutes (2 decimals). The mean sits within 0.4 minutes of the median of 27, so the shape is roughly symmetric.
Sort the evening data: 10, 12, 14, 15, 16, 18, 20, 22, 26, 33, 45, 60. With $n = 12$ , the median is the average of the 6th and 7th values: $(18 + 20)/2 = 19$ minutes.
Find the evening quartiles. Lower half (6 values): 10, 12, 14, 15, 16, 18, so $Q_1 = (14 + 15)/2 = 14.5$ . Upper half: 20, 22, 26, 33, 45, 60, so $Q_3 = (26 + 33)/2 = 29.5$ . The evening five-number summary is 10, 14.5, 19, 29.5, 60 minutes.
Evening spread and outlier check: $\text{IQR} = 29.5 - 14.5 = 15$ minutes, range $= 60 - 10 = 50$ minutes. Fences: $1.5 \times 15 = 22.5$ , so $14.5 - 22.5 = -8$ and $29.5 + 22.5 = 52$ minutes. The value 60 is above 52 and is an outlier; 45 is below 52 and is not.
Evening shape check: the sum is 291 minutes, so $\bar{x} = 291 / 12 = 24.25$ minutes. The mean sits 5.25 minutes above the median of 19, which points to right skew.
Compare feature by feature: shape (roughly symmetric versus skewed right), center (27 versus 19, a gap of 8 minutes), variability (IQR 10 versus 15, range 22 versus 50), and outliers (none versus one at 60 minutes).

The morning distribution is roughly symmetric (mean 27.36 minutes against a median of 27 minutes), while the evening distribution is skewed to the right (mean 24.25 minutes against a median of 19 minutes). The morning session has a higher median rowing time, 27 minutes compared with 19 minutes for the evening, a difference of 8 minutes. Evening times are more variable: the evening IQR of 15 minutes is larger than the morning IQR of 10 minutes, and the evening range of 50 minutes is more than twice the morning range of 22 minutes. The evening session has one high outlier at 60 minutes (above its upper fence of 52 minutes), while the morning session has no outliers.

Reading spread off two boxplots drawn on different scales

Two boxplots of delivery times are printed in separate figures, each with an axis line 10 centimeters long. Figure A's axis runs from 0 to 50 minutes and its box measures 3 centimeters wide. Figure B's axis runs from 0 to 200 minutes and its box measures 2 centimeters wide. Which group has more variable delivery times?

Find the scale of figure A. The axis covers $50 - 0 = 50$ minutes across 10 centimeters, so 1 centimeter represents $50 / 10 = 5$ minutes.
Convert A's box width to minutes. The box spans $Q_1$ to $Q_3$ , so A's IQR is $3 \times 5 = 15$ minutes.
Find the scale of figure B. The axis covers $200 - 0 = 200$ minutes across 10 centimeters, so 1 centimeter represents $200 / 10 = 20$ minutes.
Convert B's box width to minutes: B's IQR is $2 \times 20 = 40$ minutes.
Compare the two IQRs in the same units: 40 minutes for B against 15 minutes for A.

Group B is the more variable group, with an IQR of 40 minutes against group A's 15 minutes, even though B's box is physically narrower on the page. Box width is only comparable when both boxplots share one axis; otherwise convert to the units first and compare the numbers.

Frequently asked questions

Is it enough to describe each distribution fully and let the reader compare?

No. Two complete descriptions written in sequence do not earn the comparison. The response has to contain sentences that relate the groups, using words like higher than, more variable than, or about the same as.

Do I still compare shape if both distributions are skewed the same way?

Yes, and saying so is already comparative: "both distributions are skewed to the right." You can add which tail is longer if the graphs support it. Skipping shape entirely costs the same as skipping center.

Which measure of spread should I compare?

Match it to the shape. If either group is skewed or has an outlier, compare IQRs, since the IQR is resistant. If both are roughly symmetric with no outliers, comparing standard deviations is fine. See standard deviation vs IQR.

What if the two groups have different sample sizes?

Compare anyway, but use relative frequency rather than counts when reading histograms, so a taller bar means a larger share and not just a larger group. Sample size does not stop you comparing medians, IQRs, shape, or outliers.

Can I compare means if one group has an outlier?

You can report both means, but base the comparison on medians, because a single extreme value pulls a mean and makes the comparison misleading. In the worked example the evening mean of 24.25 minutes looks close to the morning mean of 27.36, while the medians differ by 8 minutes. See mean vs median.