拟合数据点画直线时，为什么偏偏要“最小化误差的平方”而不是绝对值？
When fitting a line to data points, why minimize the squared errors rather than the absolute ones?
最小二乘有代数上的理由：对参数求导令导数为零，平方和得到关于参数的正规方程组，是线性的、可解析求解；而绝对值之和在数据点处不可导，通常要用迭代算法。几何上看，最小二乘解等于把观测向量正交投影到数据矩阵的列空间，与线性代数完美衔接。它还给大误差更大权重，这在正态误差假设下对应最大似然估计，因而有统计上的最优性。
Least squares is algebraically blessed: differentiating the squared-error sum and setting derivatives to zero yields the normal equations, which are linear in the parameters and solvable in closed form, whereas a sum of absolute errors is non-differentiable at data points and needs iterative methods. Geometrically, the least-squares fit is the orthogonal projection of the observation vector onto the column space of the data matrix, linking it directly to linear algebra. Giving large errors more weight also matches maximum-likelihood estimation under normal errors, giving it statistical optimality.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#最小二乘 #回归 #正交投影
贝叶斯方法如何更新我们对数据的看法？
How do Bayesian methods update our view of the data?
贝叶斯方法从一个“先验”信念出发，结合新观察到的证据，按贝叶斯公式计算出“后验”信念。它把概率理解为对不确定性的主观置信度，并随证据不断调整。与只依赖频率的经典统计不同，贝叶斯框架天然适合逐步学习和个性化推断。
Bayesian methods start from a prior belief, combine it with newly observed evidence, and use Bayes' formula to compute a posterior belief. It treats probability as a degree of subjective confidence that is updated as evidence arrives. Unlike classical statistics that rely only on frequencies, the Bayesian framework naturally supports step-by-step learning and personalized inference.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
吴军,《数学之美》第3版,人民邮电出版社,2020
Wu Jun, Beauty of Mathematics, 3rd ed., Posts & Telecom Press, 2020
独立核验依据 · Independent verification: 核验依据 2 ISBN 9787115537973（豆瓣书目条目: 数学之美）
#贝叶斯 #先验 #后验
信息论里的“熵”是什么意思？
What does entropy mean in information theory?
香农熵衡量一个随机来源所产生信息的不确定性或平均信息量，事件越不可预测，熵越高。它用概率的对数来定义，是数据压缩和通信的理论极限。熵为零表示结果确定无疑，熵最大表示所有结果等可能。
Shannon entropy measures the uncertainty or average information content of a random source; the less predictable the event, the higher the entropy. Defined via logarithms of probabilities, it is the theoretical limit for data compression and communication. Zero entropy means a certain outcome, and maximum entropy means all outcomes are equally likely.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
吴军,《数学之美》第3版,人民邮电出版社,2020
Wu Jun, Beauty of Mathematics, 3rd ed., Posts & Telecom Press, 2020
独立核验依据 · Independent verification: 核验依据 2 ISBN 9787115537973（豆瓣书目条目: 数学之美）
#香农熵 #信息论 #不确定性
回归中的多重共线性是什么问题？
What problem is multicollinearity in regression?
多重共线性指回归的多个自变量之间高度相关，导致模型难以区分各自独立的影响。结果是个别变量的系数估计变得不稳定、标准误变大，符号甚至可能与直觉相反。常用剔除变量、合并指标或正则化来缓解，但要注意它主要损害解释而非预测精度。
Multicollinearity occurs when several independent variables in a regression are highly correlated, making it hard for the model to separate their individual effects. The result is unstable coefficient estimates with inflated standard errors, and signs may even reverse against intuition. Dropping variables, combining indicators, or regularization help, but note it mainly hurts interpretability rather than prediction accuracy.
难度 专业 · 适合 高中(15-18)
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）
#多重共线性 #回归 #变量相关
采样偏差会怎样危害统计结论？
How can sampling bias undermine statistical conclusions?
采样偏差指样本不能代表总体，例如只调查上网人群会忽略不使用网络的人，使结论偏离真实情况。即使样本量大，偏差也不会因数量多而消失，反而会被放大。可靠的推断要求抽样随机且有代表性，否则再精巧的模型也无济于事。
Sampling bias means the sample fails to represent the population, for example surveying only internet users ignores those offline, skewing conclusions. Even a large sample size does not remove bias; it can amplify it. Reliable inference requires random, representative sampling, or even the most elaborate model is useless.
难度 专业 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
吴军,《数学之美》第3版,人民邮电出版社,2020
Wu Jun, Beauty of Mathematics, 3rd ed., Posts & Telecom Press, 2020
独立核验依据 · Independent verification: 核验依据 2 ISBN 9787115537973（豆瓣书目条目: 数学之美）
#采样偏差 #代表性 #随机抽样
中心极限定理说的是什么？
What does the Central Limit Theorem say?
中心极限定理指出，只要样本量足够大，无论总体本身是什么分布，样本平均数的分布都会近似服从正态分布。这正是为什么正态分布在统计中如此普遍，也让我们可以用样本均值去推断总体。它是许多统计检验的理论基石。
The Central Limit Theorem states that, with a large enough sample size, the distribution of sample means approximates a normal distribution regardless of the population's original shape. This is why the normal distribution is so ubiquitous in statistics and lets us infer populations from sample means. It underpins many statistical tests.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
吴军,《数学之美》第3版,人民邮电出版社,2020
Wu Jun, Beauty of Mathematics, 3rd ed., Posts & Telecom Press, 2020
独立核验依据 · Independent verification: 核验依据 2 ISBN 9787115537973（豆瓣书目条目: 数学之美）
#中心极限定理 #样本均值 #正态分布
相关系数高就说明一个变量导致另一个变量变化吗？
Does a high correlation mean one variable causes the other to change?
不，相关不等于因果。两个变量可能都受第三个隐藏变量影响，或者只是巧合，也可能因果方向相反。要证明因果关系，通常需要控制实验、随机化或严谨的因果推断方法。例如冰淇淋销量和溺水人数的相关系数很高，但真正原因是夏天高温同时推高了两者。
No, correlation does not imply causation. Two variables may both be driven by a hidden third variable, may be coincidental, or the causal direction may be reversed. Establishing causation usually requires controlled experiments, randomization, or rigorous causal inference. For example, ice-cream sales and drownings are highly correlated, but the real cause is hot summer weather driving up both.
难度 专业 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
吴军,《数学之美》第3版,人民邮电出版社,2020
Wu Jun, Beauty of Mathematics, 3rd ed., Posts & Telecom Press, 2020
独立核验依据 · Independent verification: 核验依据 2 ISBN 9787115537973（豆瓣书目条目: 数学之美）
#相关非因果 #混淆变量 #统计误区
统计学里的 p 值是什么意思？
What does the p-value mean in statistics?
p 值是在“原假设成立”的前提下，观察到当前结果（或更极端结果）的概率。p 值很小（通常小于 0.05）说明原假设下出现这种数据很不寻常，从而提供反对原假设的证据。p 值小不等于效应大，也不等于原假设为假的概率，误读它会导致许多错误结论。
The p-value is the probability of observing the current result, or something more extreme, assuming the null hypothesis is true. A small p-value (typically below 0.05) means such data would be unusual under the null, giving evidence against it. A small p-value does not mean a large effect or the probability that the null is false; misreading it causes many errors.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Casella, G. 与 Berger, R. L., 《统计推断》(第2版), Duxbury
Casella, G. & Berger, R. L., 'Statistical Inference' (2nd ed.), Duxbury
独立核验依据 · Independent verification: 核验依据 1 ISBN 9788131503942 (2002)
#p值 #假设检验 #显著性
什么是辛普森悖论？
What is Simpson's paradox?
辛普森悖论指在各分组中趋势一致的方向，在合并后的总体数据中却发生反转的现象。它通常源于各组样本量或比例很不均衡，掩盖了隐藏的结构。分析数据时必须警惕“只看总体”而忽略分组结构。
Simpson's paradox occurs when a trend appearing in several separate groups reverses when the groups are combined. It usually arises because the groups have very unequal sizes or proportions, hiding an underlying structure. One must beware of looking only at aggregates and ignoring group structure.
难度 专业 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Pearl、Glymour 与 Jewell《Causal Inference in Statistics: A Primer》（Wiley）
Pearl, Glymour and Jewell, Causal Inference in Statistics: A Primer, Wiley
独立核验依据 · Independent verification: 核验依据 1 ISBN 978-1-119-18687-8 (2016, Wiley)
#辛普森悖论 #统计悖论 #分组效应
回归模型中的“过拟合”是什么？
What is overfitting in a regression model?
过拟合指模型在训练数据上表现极好，却在新数据上表现很差，因为它记住了噪声而非真实规律。比如用很高次多项式去拟合少量数据点，曲线会穿过每一个点却失去预测能力。常用交叉验证、简化模型或正则化来防止过拟合。
Overfitting occurs when a model performs extremely well on training data but poorly on new data, because it has memorized noise rather than the true pattern. For example, fitting a very high-degree polynomial to a few points makes the curve pass through every point yet lose predictive power. Cross-validation, simpler models, or regularization are common ways to prevent it.
难度 专业 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
《深度学习》Goodfellow、Bengio 与 Courville
Deep Learning by Goodfellow Bengio and Courville
独立核验依据 · Independent verification: 核验依据 1 ISBN 978-0-262-03561-3 (MIT Press)
#过拟合 #正则化 #交叉验证
主成分分析（PCA）在做什么？
What does principal component analysis (PCA) do?
PCA 是一种降维方法：在尽量保留数据总变异的前提下，把多个相关变量旋转压缩成少数几个彼此不相关的新变量（主成分）。第一主成分指向数据方差最大的方向，后续成分依次正交且方差递减。它常用于去噪、可视化和特征提取。
PCA is a dimensionality-reduction method that, while preserving as much total variation as possible, rotates and compresses many correlated variables into a few uncorrelated new ones (principal components). The first component points along the direction of greatest variance, and later ones are orthogonal with decreasing variance. It is used for denoising, visualization, and feature extraction.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
吴军,《数学之美》第3版,人民邮电出版社,2020
Wu Jun, Beauty of Mathematics, 3rd ed., Posts & Telecom Press, 2020
独立核验依据 · Independent verification: 核验依据 2 ISBN 9787115537973（豆瓣书目条目: 数学之美）
#主成分分析 #降维 #方差
切比雪夫不等式有什么用？
What is Chebyshev's inequality useful for?
切比雪夫不等式指出：无论数据分布形状如何，落在均值 k 个标准差之外的概率不超过 1/k²。它不需要假设正态分布，因此是一个很通用的“安全网”界限。例如至少 75% 的数据落在均值两个标准差内，与具体分布无关。
Chebyshev's inequality says that, regardless of the data's distribution shape, the probability of lying more than k standard deviations from the mean is at most 1/k². It needs no normal-distribution assumption, making it a very general safety-net bound. For example, at least 75% of data lie within two standard deviations of the mean, independent of the distribution.
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）
#切比雪夫 #不等式 #标准差
什么是散点图？
What is a scatter plot?
散点图用平面上的一个个点表示两个变量的成对观测值，横轴一个变量、纵轴另一个。通过观察点的分布走向，可以直观判断两变量之间是否存在相关关系及其强弱。它是探索性数据分析中最常用的图形之一。
A scatter plot places points on a plane for paired observations of two variables, one on the horizontal axis and one on the vertical. By looking at the spread and trend of the points, you can judge whether and how strongly the two variables are related. It is among the most common plots in exploratory data analysis.
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）
#散点图 #相关 #数据可视化
什么是正态分布？
What is the normal distribution?
正态分布又称高斯分布，是一条中间高、两边对称下降的钟形曲线，由均值和标准差两个参数决定。许多自然现象（如身高、测量误差）近似服从正态分布。它的重要性质是约 68% 的数据落在均值一个标准差内，约 95% 落在两个标准差内。
The normal distribution, also called the Gaussian distribution, is a bell-shaped curve that is high in the middle and falls off symmetrically on both sides, defined by its mean and standard deviation. Many natural phenomena, such as heights and measurement errors, are approximately normal. A key property is that about 68% of data lie within one standard deviation of the mean, and about 95% within two.
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）
#正态分布 #钟形曲线 #高斯
直方图和条形图有什么不同？
How does a histogram differ from a bar chart?
直方图用来展示连续数值数据的分布，横轴是按区间划分的数值，柱子之间通常没有间隙。条形图则用来比较不同类别的数量，横轴是离散的类别，柱子之间有空隙。简言之，直方图看“分布形状”，条形图看“各类对比”。
A histogram shows the distribution of continuous numerical data, with the horizontal axis divided into numeric intervals and bars usually touching. A bar chart compares counts across discrete categories, with gaps between bars. In short, a histogram shows distribution shape, while a bar chart shows category comparisons.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
吴军,《数学之美》第3版,人民邮电出版社,2020
Wu Jun, Beauty of Mathematics, 3rd ed., Posts & Telecom Press, 2020
独立核验依据 · Independent verification: 核验依据 2 ISBN 9787115537973（豆瓣书目条目: 数学之美）
#直方图 #条形图 #数据分布
为什么民意调查几百上千人，就能“代表”整个国家？误差从哪来？
How can a poll of a few hundred or thousand people “represent” an entire country, and where does its error come from?
只要样本是随机抽取的，统计上就能用样本推算总体，但要附上误差范围。通常报告“置信区间”：比如 95% 置信区间是 52%±3%，意思是如果反复重新抽样，约 95% 的区间会覆盖真实比例。误差主要来自抽样随机性，而非样本大小本身，因此关键是抽样是否随机、样本是否足够。
As long as the sample is drawn randomly, statistics lets you estimate the population from it — but with a margin of error attached. Usually a confidence interval is reported: a 95% confidence interval of 52%±3% means that if sampling were repeated, about 95% of such intervals would cover the true proportion. Error comes mostly from sampling randomness, so what matters is random selection and adequate size, not just bigness.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#抽样 #置信区间 #误差范围
方差和标准差有什么区别？
What is the difference between variance and standard deviation?
方差衡量数据偏离平均值的平方的平均距离，而标准差是方差的正平方根。标准差和原始数据使用相同的单位，因此比方差更容易直观理解。两者都描述数据的离散程度，数值越大说明数据越分散。
Variance measures the average of the squared distances from the mean, while standard deviation is the positive square root of the variance. Because standard deviation uses the same units as the original data, it is more intuitively interpretable than variance. Both describe how spread out the data are; larger values mean more dispersion.
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）
#方差 #标准差 #离散程度
标准差在日常生活里有什么直观意义？
What is the intuitive meaning of standard deviation in daily life?
标准差告诉我们一组数据围绕平均值波动的“典型幅度”。标准差小，说明大家的水平很接近；标准差大，说明个体差异明显。例如两个班级数学平均分都是 80，但一个班标准差 5、另一个 15，后者成绩更参差不齐。
Standard deviation tells us the typical amount by which data fluctuate around the mean. A small standard deviation means values are close together; a large one means individuals differ greatly. For example, two classes may both average 80 in math, but one with standard deviation 5 and another with 15 shows the latter is far more uneven.
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）
#标准差 #波动 #离散程度
为什么样本方差要除以 n−1 而不是 n？
Why does sample variance divide by n−1 instead of n?
用样本去估计总体方差时，如果除以样本量 n，会系统性地低估真实方差，这称为偏差。除以 n−1（即自由度）可以对这种低估进行修正，得到无偏估计，这一做法被称为贝塞尔校正。当样本量很大时，除以 n 还是 n−1 差别很小。
When using a sample to estimate the population variance, dividing by n systematically underestimates the true variance; this is called bias. Dividing by n−1 (the degrees of freedom) corrects for this underestimation and yields an unbiased estimate, known as Bessel's correction. When the sample is large, the difference between using n and n−1 is negligible.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Casella, G. 与 Berger, R. L., 《统计推断》(第2版), Duxbury
Casella, G. & Berger, R. L., 'Statistical Inference' (2nd ed.), Duxbury
独立核验依据 · Independent verification: 核验依据 1 ISBN 9788131503942 (2002)
#样本方差 #贝塞尔校正 #自由度
什么是幸存者偏差？
What is survivorship bias?
幸存者偏差指我们只观察到“幸存”下来的样本，而忽略了已经消失的样本，从而得出错误结论。例如二战时统计返航飞机弹孔位置来加固装甲，却忽略了被击落飞机才是关键。正确做法要考虑那些“没被看到”的数据。
Survivorship bias means we only observe the samples that survived while ignoring those that disappeared, leading to wrong conclusions. For example, during WWII, plotting bullet holes on returning aircraft to add armor missed the fact that downed planes were the crucial evidence. The correct approach accounts for the data you cannot see.
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：此条未指向带版次的可引用著作（通识性内容、机构文件或未展开的英文书名），本库不提供独立书目出处；如需精确引用，需由编辑补全具体篇目与版次。
Wald 战时统计研究
Wald's wartime statistics
#幸存者偏差 #样本偏差 #决策误区
什么是相关系数？
What is the correlation coefficient?
相关系数（常用皮尔逊 r）衡量两个变量之间线性关系的强弱和方向，取值范围在 −1 到 1 之间。r 接近 1 表示强正相关，接近 −1 表示强负相关，接近 0 表示几乎没有线性关系。它只衡量线性关联，无法捕捉曲线关系。
The correlation coefficient, commonly Pearson's r, measures the strength and direction of the linear relationship between two variables, ranging from −1 to 1. An r near 1 indicates strong positive correlation, near −1 strong negative correlation, and near 0 little linear relationship. It captures only linear association, not curved relationships.
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）
#相关系数 #皮尔逊 #线性关系
什么是箱线图？
What is a box plot?
箱线图用一个小箱子展示数据的中位数、上下四分位数，并用须线标出大致范围，还能直观显示异常值。它能在一张图里比较多个组的分布形状和离散程度。相比只看平均数，箱线图更能暴露数据的偏斜和离群点。
A box plot uses a box to show the median and the upper and lower quartiles, with whiskers marking the rough range and outliers often shown individually. It lets you compare the shape and spread of several groups in one picture. Compared with looking only at means, it better reveals skewness and outliers.
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）
#箱线图 #四分位数 #异常值
什么是百分位数？
What is a percentile?
百分位数把排序后的数据分成 100 等份，第 p 百分位数表示有 p% 的数据小于或等于它。例如第 90 百分位数意味着你比 90% 的人高（或成绩更好）。常见的“中位数”其实就是第 50 百分位数。
A percentile divides sorted data into 100 equal parts; the p-th percentile is the value below which p% of the data fall. For example, the 90th percentile means you are above 90% of people in height or score. The familiar median is simply the 50th percentile.
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）
#百分位数 #分位数 #排名
什么是离群值？
What is an outlier?
离群值是明显偏离其余数据的数据点，可能来自测量错误、罕见事件或真实但极端的情形。它会影响平均数和标准差，却对中位数影响较小。处理时要先判断其来源，再决定保留、修正还是剔除。
An outlier is a data point that lies markedly far from the rest, possibly from measurement error, a rare event, or a genuine but extreme case. It strongly affects the mean and standard deviation but hardly the median. One should first judge its source before deciding to keep, correct, or remove it.
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）
#离群值 #异常点 #数据清洗
经验累积分布函数（ECDF）是什么，它怎样用样本画出“理论分布的形状”？
What is the empirical cumulative distribution function (ECDF), and how does it draw the 'shape of the theoretical distribution' from a sample?
ECDF 在任意实数 x 处取值为“样本中 ≤ x 的比例”，随样本量增大逐步逼近真实累积分布函数（格利文科-坎特尔利定理）。它不假设分布形态，就能直观对比样本与正态、指数等理论曲线，是探索性数据分析的基本工具。
The ECDF at any real x equals the fraction of sample values ≤ x, and as the sample grows it converges to the true CDF (Glivenko–Cantelli theorem). Assuming no parametric form, it lets you visually compare a sample with normal, exponential, or other theoretical curves—a basic tool of exploratory data analysis.
难度 进阶 · 适合 初中(12-15)
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。
Wasserman, L., 《全部统计学》, Springer
Wasserman, L., 'All of Statistics', Springer
#经验累积分布 #ECDF #格利文科-坎特尔利 #探索性分析
中位数和平均数有什么不同？
How does the median differ from the mean?
中位数是把数据按大小排序后位于正中间的那个数，而平均数是所有数的和除以个数。当数据中存在极大或极小的异常值时，中位数比平均数更能代表“典型”水平。例如收入数据中少数富豪会大幅拉高平均收入，此时中位数更贴近多数人的实际情况。
The median is the middle value when the data are sorted from smallest to largest, while the mean is the sum divided by the count. When extreme outliers exist, the median represents the typical level better than the mean. For instance, a few very wealthy people can inflate average income, so the median is closer to what most people actually earn.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
吴军,《数学之美》第3版,人民邮电出版社,2020
Wu Jun, Beauty of Mathematics, 3rd ed., Posts & Telecom Press, 2020
独立核验依据 · Independent verification: 核验依据 2 ISBN 9787115537973（豆瓣书目条目: 数学之美）
#中位数 #平均数 #异常值
什么是折线图？
What is a line chart?
折线图把数据点按某种顺序（通常是时间）用线段依次连接，用来展示数值随时间变化的趋势。它适合表现增长、下降或周期性波动，例如气温变化或股价走势。与散点图不同，折线图强调“顺序和变化”而非两两关系。
A line chart connects data points in sequence, usually over time, with line segments to show how a value trends. It suits growth, decline, or periodic fluctuation, such as temperature or stock prices. Unlike a scatter plot, it emphasizes order and change rather than pairwise relations.
难度 入门 · 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）
#折线图 #趋势 #时间序列
什么是数据标准化（z-score），为什么要让不同单位的数据可比？
What is data standardization (z-score), and why make differently-scaled data comparable?
z-score 把一个数值减去均值再除以标准差：z = (x−μ)/σ，表示它离平均水平多少个“标准差”。这样摄氏温度、身高等不同单位的数据都能变成同一尺度，便于比较、画正态图，也是很多机器学习算法（如 PCA）的预处理步骤。
A z-score subtracts the mean and divides by the standard deviation: z = (x−μ)/σ, telling how many standard deviations a value lies from the average. It puts data in different units (temperature, height) onto one scale, aiding comparison and plots, and is standard preprocessing for methods like PCA.
难度 入门 · 适合 初中(12-15)
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。
Wasserman, L., 《全部统计学》, Springer
Wasserman, L., 'All of Statistics', Springer
#标准化 #z分数 #标准差 #可比性
百分数（%）表示什么？
What does a percentage (%) represent?
百分数表示“每一百中有多少”，是分母为 100 的比例写法。比如 25% 就是 25/100，也就是四分之一。它方便我们比较不同基数的比例，例如比较两个班级及格率时不用关心总人数。
A percentage shows how many per hundred, writing a ratio with denominator 100. For example, 25% is 25/100, or one quarter. It makes comparing proportions with different bases easy, such as comparing pass rates of two classes without caring about their sizes.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
吴军,《数学之美》第3版,人民邮电出版社,2020
Wu Jun, Beauty of Mathematics, 3rd ed., Posts & Telecom Press, 2020
独立核验依据 · Independent verification: 核验依据 2 ISBN 9787115537973（豆瓣书目条目: 数学之美）
#百分数 #比例 #比较
什么是混杂变量（confounder），它怎样让“相关”看起来像“因果”？
What is a confounder, and how can it make a correlation look like causation?
混杂变量是同时影响自变量和因变量的“第三方因素”，会制造虚假关联：例如冰淇淋销量与溺水人数都随气温上升，二者相关却无因果。随机实验或回归中控制混杂变量，才能把真正的因果效应从相关中剥离出来。
A confounder is a third factor that influences both variables and manufactures a spurious association—ice-cream sales and drownings both rise with temperature, yet neither causes the other. Randomized experiments or regression that controls for confounders is needed to separate true causal effect from mere correlation.
难度 入门 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Pearl, J. 与 Mackenzie, D., 《为什么之书：因果的新科学》, Basic Books
Pearl, J. & Mackenzie, D., 'The Book of Why: The New Science of Cause and Effect', Basic Books
独立核验依据 · Independent verification: 核验依据 1 ARCHIVE 标识 judea-pearl-dana-mackenzie-the-book-of-why-th
#混杂变量 #相关非因果 #控制变量
什么是平均数（均值）？
What is the average (mean)?
平均数是把一组数的所有数值相加，再除以数值的个数。它反映这组数据的整体集中水平，但容易被极端值拉高或拉低。例如三个数 2、3、10 的平均数是 5，但中位数只有 3。
The mean is found by adding up all the values in a set and then dividing by how many values there are. It describes the central tendency of the data, but it can be pulled upward or downward by extreme values. For example, the mean of 2, 3, and 10 is 5, while the median is only 3.
难度 入门 · 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）
#平均数 #集中趋势 #统计
什么是数据的“范围”（极差）？
What is the range of data?
极差是一组数据中最大值减去最小值得到的差距，用来粗略描述数据的跨度。它计算简单，但只依赖两个极端值，容易受离群值影响。要了解整体分布，通常还要配合平均数、标准差等指标。
The range is the difference between the maximum and minimum values in a dataset, giving a rough sense of its spread. It is simple to compute but depends only on the two extremes, so it is sensitive to outliers. To understand the full distribution, it is usually paired with the mean, standard deviation, and other measures.
难度 入门 · 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）
#极差 #范围 #离散程度