Comparing is a sentence, not two

The graders' standard is blunt: listing facts about each group separately earns nothing, however accurate the facts. A comparison must put the two groups into the same sentence with a direction between them.

1. Not a comparison: "Route A's median commute is 22 minutes. Route B's median commute is 30 minutes."
2. A comparison: "The median commute on Route A is 8 minutes shorter than on Route B (22 vs 30 minutes)."

The second sentence has the three required parts: both groups, a directional word (shorter), and context with units (commute time, minutes). Magnitude, saying by how much, strengthens it further. "A and B are different" fails for lack of direction; "A is better" fails for lack of values and context. The test of a real comparison: a reader who has never seen the data should come away knowing which group is which, in what respect, and by roughly how much.

A full comparison of two distributions addresses the same checklist a single description does, but comparatively, feature by feature:

1. Shape: "both are roughly symmetric", or "plan A's costs are right-skewed while plan B's are symmetric".
2. Outliers and unusual features: gaps, clusters, and flagged values, in either group or both.
3. Center: which group runs higher, and by about how much.
4. Spread: which group is more variable, comparing range, IQR, or standard deviation.

Two groups can agree on center and still tell opposite stories: identical medians with a fourfold difference in IQR means one group is predictable and the other is not, and calling them "about the same" because one feature matches is the classic error. When either distribution is skewed or holds outliers, compare the resistant pair (medians and IQRs) rather than means and standard deviations, and say every value with its units and its variable name.

The standard comparison pictures put two or more distributions on a single axis: side-by-side (parallel) boxplots, back-to-back stemplots, stacked dotplots or histograms. The shared scale is what makes vertical alignment meaningful: a glance shows whose median sits higher, whose box is wider, and where the groups overlap.

Overlap is the part beginners skip. "Group A runs higher than group B" is a statement about centers; it almost never means every A value beats every B value. If B's upper whisker reaches past A's median, then a healthy share of B's values live among typical A values, and claims like "all" or "always" collapse.

Each display carries its own limits into the comparison. Parallel boxplots compare centers, spreads, and skew cleanly, but they still cannot show modes, gaps, or sample sizes, so two matching boxplots do not establish matching shapes. When counts drive the question, remember the bars: comparing raw frequencies across groups of different sizes misleads, so compare relative frequencies instead.

To compare individual values from different distributions, strip each of its units and measure it in standard deviations from its own mean:

$$z = \frac{x - \mu}{\sigma}.$$

A positive $z$ sits above the mean, a negative $z$ below, and the size of $z$ says how unusual the value is within its own distribution. When the population parameters $\mu$ and $\sigma$ are unknown, the sample's $\bar{x}$ and $s$ stand in.

A runner finishes a 5K in 20 minutes on a team whose times average 23 minutes with standard deviation 2: $z = \frac{20 - 23}{2} = -1.5$. The negative sign is good news here, 1.5 standard deviations faster than typical: interpretation always routes through the context. The same logic compares across distributions: a test score of 640 where the mean is 530 and the SD is 110 gives $z = 1.0$, which beats a 25 on a different test with mean 21 and SD 5 ($z = 0.8$), even though the raw numbers cannot be compared at all. The z-score is relative position, and relative position, not raw size, is what "did better" means across different scales.