为什么小样本容易得出惊人的错误结论？
Why do small samples easily produce surprising false conclusions?
小数定律指人们误以为小样本也能像大样本一样稳定反映真实比例。样本越小，随机波动越大，越容易碰到极端结果，于是把噪声当成了规律。
The law of small numbers is the bias of expecting small samples to mirror true proportions. Smaller samples have larger random swings, so extremes look like patterns when they are just noise.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-04
来源 · Source
Shatz,《The law of small numbers: When statistics and psychology go to war》,Significance,2023 (DOI: 10.1093/jrssig/qmad044)
Shatz, 'The law of small numbers: When statistics and psychology go to war', Significance, 2023, DOI: 10.1093/jrssig/qmad044
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.1093/jrssig/qmad044
#统计陷阱 #小数定律
为什么图表把纵轴不从零开始会误导人？
Why is starting a chart axis above zero misleading?
纵轴截断会放大差距的视觉比例，让小幅变化看起来很剧烈。诚实的数据新闻通常从零开始，或用标注说明截断。
Truncating the vertical axis exaggerates differences, making small changes look dramatic. Honest data journalism starts at zero or clearly marks the break.
难度 进阶 · 适合 大众
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：本题原挂出处与答案内容不符，已撤回；具体书目待补，暂不展示来源。
本题原挂出处与答案内容不符，已撤回；具体书目待补。
Previous citation withdrawn as unsupported for this answer; a specific source is pending.
#数据新闻 #图表误导
为什么新闻里说平均收入很高，很多人却觉得没那么多？
Why do reported average incomes feel too high to most people?
少数人收入极高会拉高平均数，而中位数更能代表普通人的水平。当分布很不均匀时，平均数往往大于一半人的实际收入。
A few very high earners pull the mean up, while the median better reflects a typical person. With skewed income, the mean exceeds what more than half the people actually earn.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
《概率论与数理统计》
Probability Theory and Mathematical Statistics (《概率论与数理统计》)
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787312018381（豆瓣书目条目: 概率论与数理统计）
#统计陷阱 #平均数与中位数
为什么大量四舍五入后再相加，总额会悄悄偏差？
Why does summing many rounded numbers drift from the true total?
每次四舍五入都引入约半个单位的随机误差，成百上千次后会累积。财务系统常用四舍五入到偶数或保留更多小数来减小这种偏差。
Each rounding adds about half a unit of error, and over thousands of values these accumulate. Accounting systems use round-half-to-even or extra decimal places to reduce drift.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 2 条独立证据
来源 · Source
美国数学会(AMS)《数值分析》
American Mathematical Society (AMS), Numerical Analysis
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780538733519 (2011) · 核验依据 2 ISBN 9780387987934 (1999)
#统计陷阱 #舍入误差
为什么看人均数据有时会误判一个地方的生活水平？
Why can per-capita data mislead you about living standards?
人均是把总量除以人数，掩盖了内部的贫富差距。一个少数富人拉高人均、多数人并不富裕的地区，人均数字就失真了。
Per-capita divides totals by population and hides internal inequality. Where a few rich people inflate the average, most residents may be far less well off than the figure suggests.
难度 进阶 · 适合 大众
CC BY 4.0 · 来源待核验 · Source pending review
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出处备注：本题原挂出处与答案内容不符，已撤回；具体书目待补，暂不展示来源。
本题原挂出处与答案内容不符，已撤回；具体书目待补。
Previous citation withdrawn as unsupported for this answer; a specific source is pending.
#统计陷阱 #人均
为什么同一道题，问法不同调查结果就会变？
Why can the same question get different survey results depending on wording?
诱导性措辞、选项顺序和选项范围都会改变人们的回答，这叫问卷偏差。严谨的民调会把问题写成中立、对称的句式。
Leading wording, option order and the range of choices all shift responses; this is questionnaire bias. Rigorous polls use neutral, balanced phrasing.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-04
来源 · Source
Constantin,《Gender Beliefs Measurement. How a Slightly Different Wording of the Same Question Changes the Story》,Social Change Review,2012 (DOI: 10.2478/scr-2013-0010)
Constantin, 'Gender Beliefs Measurement. How a Slightly Different Wording of the Same Question Changes the Story', Social Change Review, 2012, DOI: 10.2478/scr-2013-0010
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.2478/scr-2013-0010
#统计陷阱 #问卷偏差
冠军次年常回落、垫底球队多反弹——「向均值回归」为什么并不是一种真实的因果力量?
Champions often slump the next season while bottom teams rebound — why is regression to the mean not a genuine causal force?
因为一切测量都掺着随机波动,极端成绩通常是真实实力与好运气叠加的产物。下一次测量时运气重新洗牌,随机成分回到平常水准,观测值便自动向平均靠拢——不需要任何机制施加「拉回」,这只是条件期望的数学性质。高尔顿1886年研究遗传身高时发现该现象并创造了regression一词;若误当成因果,就会推出表扬有害、惩罚有益的荒唐结论。
Every measurement carries random noise, so an extreme result usually blends true ability with good luck. On remeasurement the luck reshuffles, the random component returns to its ordinary level, and the observation drifts back toward the average — no mechanism pulls it, for this is simply a property of conditional expectation. Galton noticed the effect while studying hereditary height in 1886 and coined the word regression; read as causation, it absurdly implies that praise harms and punishment helps.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-04
来源 · Source
Galton,《Regression Towards Mediocrity in Hereditary Stature.》,Journal of the Anthropological Institute,1886 (DOI: 10.2307/2841583)
F. Galton, 'Regression Towards Mediocrity in Hereditary Stature.', Journal of the Anthropological Institute, 1886, DOI: 10.2307/2841583
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.2307/2841583
#向均值回归 #随机波动 #相关与因果
某种疗法在男性组和女性组里都胜过对照，可把两组合并之后它反而显得更差——这种分组与合计结论相悖的现象叫什么？通常是哪个隐藏因素在作祟？
A treatment beats the control both among men and among women, yet pooling the two groups makes it look worse — what is this reversal between subgroup and aggregate conclusions called, and which hidden factor is usually at work?
这叫辛普森悖论：在每个分层内部成立的趋势，合并数据后可能整体反转。症结在于各层样本规模悬殊，且层的归属本身与治疗分配相关——性别作为混杂因素既影响分组构成又影响痊愈概率，合计时权重失衡压倒了组内优势，把方向整个扭了过来。因此看到合计率先别下结论，应排查混杂并按层加权或标准化，这才是绕开此类陷阱的正路。
This is Simpson's paradox: a trend that holds within every stratum can flip once the data are pooled. The culprit is wildly unequal group sizes whose composition correlates with treatment assignment — sex acts as a confounder shaping both who lands in which arm and how likely recovery is, so merging the arms lets lopsided weights drown out the within-group advantage and reverse the direction. Aggregate rates therefore deserve suspicion first: hunt down confounders, then weight or standardize by stratum.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-04
来源 · Source
Simpson,《The Interpretation of Interaction in Contingency Tables》,Journal of the Royal Statistical Society Series B 第13卷,1951 (DOI: 10.1111/j.2517-6161.1951.tb00088.x)
E. H. Simpson, 'The Interpretation of Interaction in Contingency Tables', Journal of the Royal Statistical Society Series B, 1951, DOI: 10.1111/j.2517-6161.1951.tb00088.x
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.1111/j.2517-6161.1951.tb00088.x
#辛普森悖论 #混杂变量 #分层分析
二战时盟军统计返航轰炸机的弹孔分布，打算哪里中弹多就加固哪里，统计学家瓦尔德为什么建议加固相反的部位？
During World War II, analysts tallied bullet holes on bombers that returned home, planning to reinforce the most-hit areas — why did statistician Abraham Wald urge reinforcing the opposite ones?
因为样本全是幸存者：飞机能返航，本身就说明其弹孔部位并不致命，所以机翼与机身布满弹孔；发动机附近几乎看不到弹孔，不是炮火打不准，而是中弹的飞机大多坠毁了。瓦尔德主张反向推断：返航机上没有弹孔的部位才是真正的致命处。这就是幸存者偏差——只用「活下来」的数据做判断，结论会被系统性扭曲，基金业绩榜与成功学案例中也屡见不鲜。
Because the sample contains only survivors. A plane that made it back proves its hit locations were survivable, so wings and fuselage show dense hole patterns; engines look almost untouched not because gunfire missed, but because engine hits usually brought the plane down. Wald argued in reverse: the clean areas on returning aircraft are the truly lethal weak points. This is survivorship bias — judging from data that lived on distorts conclusions, and it recurs in fund rankings and success stories alike.
难度 入门 · 适合 大众
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录含书名，但缺版次 / 出版社 / 年份中的一部分，需编辑补全后才能做独立比对。
Wald, A.，《A Method of Estimating Plane Vulnerability Based on Damage of Survivors》，Statistical Research Group, Columbia University,1944
A. Wald, A Method of Estimating Plane Vulnerability Based on Damage of Survivors, Statistical Research Group, Columbia University, 1944
#幸存者偏差 #选择偏差 #缺失数据