Core answer: Mean = sum ÷ count (balance point); median = middle value after sorting (robust to outliers); mode = most frequent value. For the salaries [5k, 6k, 6k, 7k, 50k]: mean = 14.8k (misleading), median = 6k (typical), mode = 6k. That's why official statistics report MEDIAN income — one billionaire in the room raises everyone's "average" without changing anyone's life.
The three averages, computed
Data: [3, 5, 5, 7, 9, 11, 100]
- Mean = 140 ÷ 7 = 20
- Median = 7 (4th of 7 sorted values)
- Mode = 5 (appears twice)
Even count? Median = average of the two middle values: [1, 2, 3, 4] → (2+3)/2 = 2.5.
When to use which
| Situation | Use | Why |
|---|---|---|
| Symmetric data (heights, test scores) | mean | uses all information |
| Skewed data (income, house prices) | median | outliers don't drag it |
| Categorical/bestseller data | mode | mean/median meaningless for categories |
| Small samples with outliers | median | one bad value can't hijack it |
| Further math (variance, z-scores) | mean | it's the algebraic center |
Worked examples
Example 1 — The income report. 9 people earn ¥8k, one earns ¥800k: mean ¥87k vs median ¥8k. A news headline quoting the mean ("average salary in this village is ¥87k!") is technically true and practically a lie — median is the honest "typical".
Example 2 — Exam analysis. Scores [40, 65, 70, 72, 75, 78, 80, 85, 90, 95]: mean 75.0, median 76.5 — close, so the distribution is near-symmetric and either works. Mean < median (left skew) signals a tail of low scorers dragging the average.
Example 3 — Real estate. Neighborhood prices: ¥2M ×6, ¥2.5M ×2, ¥30M villa ×1. Mean ¥5.4M, median ¥2M, mode ¥2M. Buyers should anchor on median; the mean describes a house that doesn't exist.
Example 4 — Mode for business. Shoe sizes sold: mode = 42 → stock depth follows the mode, not the mean size (39.7 — a size you can't even order).
Common mistakes and myths
- "Average" always means mean — ask WHICH average; the choice often decides the story.
- Mean of rates — averaging +50% and −50% gives 0% but the money went ¥100 → ¥150 → ¥75 (−25%); rates need geometric mean.
- Ignoring the distribution — mean 70 could be everyone near 70 or half at 40, half at 100; report spread (standard deviation) alongside.
- Median worship — for symmetric data the median throws away information; use the right tool per shape.
- Sample vs population confusion — a mean of 5 surveyed friends says nothing about the city; averages inherit the biases of their samples.