Comparing distributions and evaluating claims

A useful comparison addresses both centre and spread in context. A larger mean is not automatically better: high scores may be desirable, but long waiting times usually are not. Standard deviation compares spread about the mean; IQR compares the middle half about the median.

Example

For two delivery services, A has mean time \(28\) minutes and standard deviation \(4\) minutes; B has mean \(31\) minutes and standard deviation \(2\) minutes. A is faster on average; B’s times are more consistent around its mean. These summaries do not show that every A delivery is faster than every B delivery.

For box plots of waiting times, a lower median means a shorter typical wait; a smaller IQR means less variation in the middle half. The longest whisker does not identify the quarter with the most observations.

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.