Comparing two groups is where statistics starts to answer real questions: Is route A faster than route B? Do two brands of battery last equally long? You now have tools for center, spread and outliers. A good comparison uses all of them, in context.
Key idea
To compare distributions, describe shape, center and spread (and any outliers) for each group, using comparison words ("higher," "more variable") and the units of the data. Use median and IQR when a group is skewed or has outliers; mean and standard deviation when both are roughly symmetric.
Why it works
Two groups can share a center and still behave very differently. If both commute routes have a median of 25 minutes but one varies from 20 to 30 and the other from 10 to 60, only the first lets you plan when to leave. Spread tells you how much an individual can differ from the center, and overlap tells you how often a member of the "lower" group beats a member of the "higher" one.
Worked example: A complete comparison
Use the box plots of 15 commutes on each route: Route A has median 25, $Q_1 = 22$, $Q_3 = 29$ (IQR 7), range 18 to 34. Route B has median 27, $Q_1 = 20$, $Q_3 = 38$ (IQR 18), range 15 to 47. Compare them.
- Shape: A is roughly symmetric; B is skewed right (its upper whisker and box half are longer).
- Center: B's median is 2 minutes longer (27 versus 25 minutes).
- Spread: B is far more variable, with an IQR of 18 minutes versus 7.
- Overlap: the boxes overlap a lot, so on many days B is faster. The big difference is predictability, not typical time.
Answer: Route A is slightly faster on a typical day and much more consistent.
Worked example: Using mean and standard deviation
Class 1 has quiz scores with mean 78 and SD 5; Class 2 has mean 74 and SD 12. Both are roughly symmetric. Compare.
- Center: Class 1 scored 4 points higher on average.
- Spread: Class 2 is more than twice as variable (SD 12 versus 5).
- So Class 2 likely has both the lowest scores and some of the highest. A 4-point gap is small compared with Class 2's spread.
Answer: Class 1: higher and more consistent; Class 2: lower on average but much more spread out.
Common mistake
Comparing only the centers ("B is higher") and stopping. That ignores spread and overlap, which are often the bigger story. Check your comparison for four words: shape, center, spread, outliers, each with numbers and units.