The contest, fairly judged
Mean and median are not rivals for a crown; they are tools with different failure modes. The mean uses every value at full weight, which makes it the right summary when totals and further arithmetic matter, and a fragile one when extreme values are noise or freaks. The median uses only order, which makes it indifferent to extremes, and blind to how extreme they are. Choose by asking: what would a freak value do to my conclusion, and is that freak information or contamination?
Two datasets tell the story. Salaries (thousands): 32, 35, 38, 40, 42, 45, 48, 240. Mean 65, median 41. The mean describes a payroll (65 × 8 = 520 thousand must actually be paid); the median describes a typical employee. House prices (thousands): 180, 195, 210, 225, 240, 260, 890. Mean 314, median 225. A report quoting the mean says the typical home costs 314k in a street where six of seven homes sold under 260k.
Decision rules that hold up
Skew or outliers present: median leads, quartiles beside it. Symmetric, clean data: mean leads, SD beside it; the two are a matched pair. Totals matter (budgets, production, revenue): the mean is unavoidable, whatever the shape, because total = mean × n. Ordinal data (ratings, ranks): median and mode are meaningful; the mean of "satisfaction codes" is a number about labels. When in doubt, report both: their gap is itself a one-line skew diagnostic.
The trimmed compromise done silently: deleting the inconvenient top values and reporting the mean of the rest. Trimming is a legitimate technique when declared (a 5% trimmed mean is a defined statistic). Undeclared, it is how averages get manufactured.
The mean-median gap as a signal
For right-skewed data the mean exceeds the median; for left-skewed data the order flips; for symmetric data they coincide. Pearson's rough skew gauge, 3 × (mean − median) / SD, puts a number on it. Salaries: 3 × (65 − 41) / 70.9 ≈ 1.0: strongly right-skewed, confirming what the 240 did to the room.
Frequently asked questions
Which should I use for exam scores?
Usually the mean: scores are bounded, roughly symmetric in a well-set exam, and totals matter for grading curves. If a couple of zeros (absent students) drag the mean, report the median too and say why.
Is the median ever misleading?
Yes: it ignores magnitudes. Two classes can share a median while one has far more top scores. Pair the median with quartiles or a chart, and use the mean when the extremes carry real information.
What does it mean when the mean and median are far apart?
Skew: the tail on one side is pulling the mean toward it. The gap's direction names the skew; its size relative to the SD gauges it. Investigate the extreme values before summarizing further.