Comparing two distributions: shape, centre and spread

A recurring Excellence task, easy to mistake for a normal-distribution CALCULATION when it is really a qualitative COMPARISON.

The three things to compare, every time

SHAPE: is the distribution symmetrical (bell-shaped), or skewed? A cluster of values at a natural boundary (like zero, for a "time spent" variable) usually signals a skewed, non-normal shape.
CENTRE: compare the means (or medians, if the shape is skewed) — which group's typical value is higher, and by how much, IN CONTEXT.
SPREAD: compare the standard deviations (or interquartile ranges) — which group is more VARIABLE, and what that means practically (e.g. more consistent wait times vs more unpredictable ones).

Worked example Comparing Saturday and Sunday wait-time distributions Saturday wait times are roughly symmetrical with mean 12 minutes; Sunday wait times are skewed right with a cluster near 5 minutes and a long tail to 40 minutes. Compare the two distributions. SHAPE: Saturday appears roughly normal/symmetrical; Sunday is skewed right, with a cluster of short waits and a long tail of occasional long waits. CENTRE: Saturday's mean (12 min) is a fair "typical" wait; for Sunday's skewed shape, the MEDIAN is a more meaningful centre than the mean, since the long tail would pull the mean upward away from the typical experience. SPREAD: Sunday is far more VARIABLE — most customers wait only a short time, but a meaningful minority wait much longer, whereas Saturday's wait times cluster more consistently around the mean. CONCLUSION: a manager should focus improvement effort on Sunday's long-tail cases specifically, since Saturday is already fairly predictable — a conclusion the shape/centre/spread comparison makes possible that a single mean comparison would miss. ✓ The strongest answers use the RIGHT measure of centre for each shape (median for skewed, mean for symmetrical) and draw a PRACTICAL conclusion from the spread comparison, not just describing the numbers. ✕

Comparing only the means, and ignoring shape and spread

A comparison that reports only "Saturday's mean is lower than Sunday's" misses two-thirds of the instruction. Shape tells you whether the mean is even a fair summary; spread tells you about consistency/predictability, which is often the more practically useful half of the comparison.

Shape	Best measure of centre	Best measure of spread
Symmetrical / bell-shaped	Mean	Standard deviation
Skewed (long tail one side)	Median	Interquartile range

Practice for this standard covers the whole topic — flashcards and quiz.