起源故事 · 从二项到正态近似
Origin Story · From Binomial to Normal Approximation
掷硬币正面几次、一批零件合格几个 —— 这类"成功计数"服从二项分布。当样本足够多时,
De Moivre 与 Laplace 发现二项分布会逐渐逼近一条钟形曲线(正态近似),于是比例的检验可以借用 z 统计量,变得异常简洁。
这套数学今天换了个时髦名字 —— A/B 测试:网站、App、广告每天都在比两个版本的转化率,背后跑的正是百年前的比例检验。
How many heads in n coin tosses; how many good parts in a batch — those "success counts" follow the binomial distribution. With enough trials,
De Moivre and Laplace noticed the binomial drifts toward a bell curve (the normal approximation), which lets us test proportions with a tidy z statistic.
Today that math wears a trendier label — A/B testing. Every site, app, and ad system pits two conversion rates against each other; under the hood it is the century-old proportion test running at scale.
1 两个比例的抽样分布:重叠还是分开? Two Sampling Distributions: Overlapping or Separated?
观察重叠2 转化方块:每版本转化的比例直观看 Conversion Grid: Each Variant's Hit Rate at a Glance
每片网格代表一次访问,实心=转化、空心=流失。直觉上两版差几片格子;但要不要相信,得看下面的 z 值有没有超过临界线。 Each tile is one visit — filled = converted, empty = lost. Eyeballed, the two rows differ by a few tiles; whether to trust that gap depends on whether the z statistic clears the critical line.
3 现实里的比例检验 Proportion Tests in the Real World
A/B 测试:新旧落地页转化率比较,互联网产品最常见的决策依据。
A/B testing: new vs. old landing-page conversion — the everyday decision tool of digital product teams.
合格率对比:改善前后一次合格率有没有真提升,而非批次波动。
First-pass-yield lift: did the improvement really raise FPY, or is it just batch-to-batch noise?
缺陷率改善:两条产线或两种工艺的不良比例对比,验证改进有效。
Defect-rate reduction: compare two lines or two processes — confirm the change actually moved the needle.
点击率 CTR:两版广告创意的点击率谁高,投放预算的分配凭证。
Ad CTR: which of two creatives wins on click-through — the evidence that decides budget allocation.
一句话In One Line
比例检验把"两个百分比的差"翻译成用标准误衡量的 z 值:差得多 + 样本大 + 比例本身波动小 → z 大 → 可信。
它解释了 A/B 测试的两条铁律:差异不大就别急着下结论,样本不够也别急着下结论。
一个 12% vs 15% 的"胜出",在 200 个样本里可能毫无意义,在 5000 个样本里却铁证如山 —— 数字相同,证据强度天壤之别。
The proportion test translates "the gap between two percentages" into a z value measured in standard errors: bigger gap + bigger sample + smaller intrinsic noise → larger z → more credible.
It nails the two iron rules of A/B testing: don't call it when the gap is small, and don't call it when the sample is thin.
A 12% vs. 15% "win" can mean nothing at n = 200 and be ironclad at n = 5000 — same numbers, wildly different evidence.
常见误用Common Mistakes
看到差异就停测下结论(偷看 peeking)。先按样本量算够再看结果,避免早停假阳性。
Stopping the test the moment a gap appears (peeking). Pre-compute the required sample size and wait — early stops inflate false positives.
极端比例 / 极小样本仍用正态近似。np 或 n(1−p) 过小时改用精确二项检验。
Forcing the normal approximation on extreme p or tiny n. When np or n(1−p) drops below ~5, switch to an exact binomial (or Fisher's exact) test.
把统计显著当成业务显著。0.1% 的提升可能显著但不值得上线,要看效应量。
Treating statistical significance as business significance. A 0.1% lift can be highly significant and still not worth shipping — always check the effect size.