Comparing distributions and evaluating claims

A useful comparison addresses both centre and spread in context. A larger average is not automatically better: high scores may be desirable, but long waiting times usually are not. The IQR compares the spread of the middle half, while the range uses the extremes.

Example

For two delivery services, A has median time 28 minutes and IQR 8 minutes; B has median 31 minutes and IQR 4 minutes. A has the shorter typical delivery time; B’s middle half is more consistent. These summaries do not show that every A delivery is faster than every B delivery.

A lower median waiting time suggests a shorter typical wait; a smaller IQR means less variation in the middle half. Compare the same measures and units for both groups.

Check sample sizes, selection method, time period, axis units and scale before accepting a claim. A \(50\%\) pie sector from \(20\) people represents \(10\) people; a \(40\%\) sector from \(100\) represents \(40\). Percentages alone cannot compare counts.

A line graph shows change over time but does not prove its cause. An unusually high mean could arise from a few extreme observations, so inspect the distribution and a suitable alternative average. A well-labelled graph supports only conclusions its data actually establish.