起源故事 · 三个「中心」的分工
Origin Story · A Division of Labor Among the Three Centers
人类很早就想用一个数概括一堆数据。最朴素的是算术平均（均值） —— 全部加起来除以个数，公平地把总量摊给每个人，
却也因此被任何一个极端值「绑架」。于是统计学家请来中位数：把数据排好队，只认正中间那一个，
多富多穷的尾巴都撼不动它。再加上众数 —— 谁出现得最多就是它，专治那些根本没法做加减的类别数据（最畅销的颜色、最常见的故障码）。
三者各司其职，「平均」从来不只有一种。
People have long wanted to summarize a pile of numbers with a single value. The simplest answer is the arithmetic mean — sum everything and divide by the count,
spreading the total evenly across the group. But that same property leaves it hostage to any single extreme observation. So statisticians brought in the median:
line the data up in order and take the middle one. No matter how rich or poor the tails get, the median barely budges.
Add the mode — whichever value occurs most often — and you finally have a tool for categorical data that can't be added or subtracted at all (best-selling color, most common defect code).
Three measures, three jobs. "Average" was never one thing.
1 拖动圆点 · 看三条中心线打架 Drag the Dots · Watch the Three Centerlines Argue
均值 ≈ 中位数mean ≈ median2 现实里该用哪个中心 Which Center to Use in the Real World
收入 / 房价 → 中位数：少数富豪和豪宅把均值拉爆，政府报告一律用「中位数收入 / 中位数房价」才不失真。
Income / housing prices → median: a handful of billionaires and trophy homes blow up the mean, so government reports use median income and median home price to stay honest.
交付工期 → 中位数 + 均值并看：均值算总产能，中位数看「一半订单多久能到」，差距大说明有长尾延误。
Lead time → median alongside mean: the mean tells you total throughput; the median tells you when half the orders actually arrive. A wide gap reveals a long-tail delay problem.
类别数据 → 众数：最畅销的尺码、最常见的故障码无法做加减，只能数「谁出现最多」。
Categorical data → mode: best-selling size, most common defect code — you can't add or average a category, you can only count which one appears most often.
对称无异常 → 均值：身高、测量误差这类对称分布，三者几乎重合，均值信息量最足、最好算。
Symmetric, no outliers → mean: for symmetric distributions like adult height or measurement error, all three nearly coincide — the mean is the most informative and the easiest to compute.
一句话In One Line
「平均」是个被滥用的词。均值把每个值都算进去，所以一个极端值就能把它拽到没人代表的位置；
中位数只关心排序的中点，天生抗极端值；众数则适用于连加减都做不了的类别数据。
看到一份报告说「平均」时，先问一句：这分布对称吗？有没有长尾？
数据一偏，均值和中位数就会分家 —— 它们的差距本身，就是「有没有极端值」的信号。
"Average" is one of the most abused words in statistics. The mean counts every value, so a single outlier can drag it to a spot that represents nobody;
the median only cares about the sorted midpoint and is robust to extremes by design; the mode is the only one that works on categorical data you can't even add up.
Whenever a report says "average", the first question to ask is: is the distribution symmetric, or is there a long tail?
The moment data is skewed, the mean and median split apart — and the size of that gap is itself the signal that outliers are present.
常见误用Common Mistakes
偏态数据用均值代表「典型值」。收入、房价这类长尾分布改用中位数，否则会高估大多数人的水平。
Using the mean as the "typical value" of skewed data. For long-tailed distributions like income or housing prices, switch to the median — otherwise you overstate where most people actually sit.
对类别数据求均值（比如给颜色编号再算平均）。类别 / 名义数据只能用众数。
Averaging categorical data (assigning numbers to colors and then taking a mean). Nominal data only supports the mode — never the mean or median.
只报一个中心、不看分布形状。均值与中位数并列，差得越远越说明有极端值或偏态。
Reporting a single center without looking at the shape. Always show the mean and median together — the wider the gap, the stronger the evidence of outliers or skew.