1.9 Comparing Distributions of a Quantitative Variable

Compare shape, center, and spread

When comparing distributions, always discuss shape, center, and spread — never just one. Use comparative language: 'greater than,' 'less than,' 'about the same as.' For example, say Group A has a higher median and less variability than Group B, rather than describing each group separately without connecting them.

Choose the right graphical display

Back-to-back stemplots and parallel boxplots work well for comparing two or more groups. Side-by-side (parallel) boxplots are especially useful for comparing center (median) and spread (IQR, range) across many groups at once, while also revealing outliers.

Use numerical summaries together

Pair shape (symmetric, skewed, unimodal/bimodal) with appropriate numerical measures: use mean and standard deviation for roughly symmetric data, or median and IQR for skewed data or data with outliers, since these are more resistant to outliers. Always tie numbers back to context.

Key terms

What three features should you always compare between distributions?

Shape, center, and spread.

Why use comparative language when describing distributions?

It directly links the groups (e.g., 'higher than,' 'less variable than') instead of describing them in isolation.

Best display for comparing multiple groups' medians and IQRs

Parallel (side-by-side) boxplots.

When should you use median/IQR instead of mean/SD for comparison?

When data is skewed or has outliers, since median and IQR are resistant to these features.

Back-to-back stemplot

A display that puts two datasets on opposite sides of a shared stem for direct comparison of individual values.

Try an AP-style question

Which comparison is correct?

1. Class A has a higher typical score and more variability
2. The classes have the same variability
3. Class B has a higher score for every student
4. Class B has a higher typical score and more variability