Comparisons of the Distributions for One Quantitative Variable

Comparing two distributions of the same quantitative variable takes three disciplines. Language: both groups in one sentence, with a directional word, values, units, and context ("the median commute on Route A is 8 minutes shorter than on Route B"). Coverage: shape, outliers, center, and spread all compared, not just the one feature that happens to match or differ. And displays: parallel boxplots, back-to-back stemplots, and stacked dotplots share one scale so centers, spreads, and overlap can be read directly, while the z-score $z = \frac{x - \mu}{\sigma}$ lets a single value from one distribution stand next to a value from another by converting both to standard deviations from their own means.

The failures are predictable. Two tidy paragraphs, one per group, with no comparative word between them; or one feature compared (usually center) and the groups declared "basically the same" while their spreads differ by a factor of three. Boxplot misreads carry over: a bigger box read as more data, matching plots read as matching shapes, "every A beats every B" claimed while the plots visibly overlap. And raw values compared across different scales, a 92 called better than an 82 when one class's test was far easier, which is precisely the comparison z-scores exist to fix.