第 28 天：统计——数据摘要、抽样不确定性与推断

01 120 分钟

描述统计：均值、离散程度与诚实的摘要

仅在记录预测与观察证据之后允许求助。

前置要求

完成第 27 天数学模块；具备基础算术能力以及第 01 天的预测—回顾流程。

学习目标

在小数据集上手工计算并解释均值、中位数、总体/样本方差与标准差，并说明两个均值相同的数据集仍可能不同。

操作指引

统计首先从把数据压缩为少数几个数字开始。均值刻画中心位置；方差与标准差度量围绕中心的离散程度。总体方差把偏差平方和除以 n；样本方差除以 n−1（贝塞尔校正），以避免系统性低估离散程度。先用纸笔预测每个数值，再运行 python3 核对——核对必须发生在预测记录之后。

自主演练与记录

对于 data = [2, 4, 4, 4, 5, 5, 7, 9]：计算均值、中位数、偏差平方和、总体方差、总体标准差与样本方差（除数为 n−1）。然后对 data2 = [4, 5, 5, 5, 5, 5, 5, 6] 计算均值与总体标准差，说明什么保持不变、什么发生变化。

标记为已完成

独立掌握标准

手算结果与 Python 核对全部一致，并能说明均值相等并不意味着分布相同。

权威源资源与权利证明

linking NIST public information (U.S. government site; NIST/SEMATECH co-production)

NIST/SEMATECH e-Handbook of Statistical Methods

作者 / 机构: NIST (U.S. National Institute of Standards and Technology) and SEMATECH | 原始出处链接 ↗

对应章节 / 时间戳: Section 1.3.5 (Quantitative Techniques): measures of location and scale

验证版本与日期: Current online edition (Verified: 2026-10-30)

入选理由:

Primary engineering reference for descriptive statistics, exploratory data analysis, and uncertainty quantification with worked formulas.

来源前置条件:

High-school arithmetic; no calculus required for the descriptive sections.

安全回退方案（离线弹性）:

Every formula used in the lesson (mean, variance, standard error) is reproduced as original explanation; all exercises run fully offline.

对比候选资源与选型分析 (2)

OpenIntro Statistics (4th edition) ↗ compared_selected_for_day28 [CC BY-SA 3.0]

Undergraduate statistics text with a stronger narrative bridge from probability to inference; retained as the second linked source for inferential framing.

Seeing Theory (Brown University) ↗ compared_alternative [Free web resource (site copyright)]

Strong interactive visualizations of probability and sampling, but a browser-only interactive format is a weaker fit for an offline-capable 480-minute ledger.

linking CC BY-SA 3.0 share-alike

OpenIntro Statistics (4th edition)

作者 / 机构: David Diez, Mine Cetinkaya-Rundel, Christopher Barr | 原始出处链接 ↗

对应章节 / 时间戳: Chapters 1-2: data basics and summary statistics

验证版本与日期: 4th edition, 2019 (Verified: 2026-10-30)

入选理由:

Rigorous open textbook treatment of sampling distributions, standard error, and the difference between an inferential claim and an anecdotal observation.

来源前置条件:

Introductory algebra; comfort reading mathematical notation.

安全回退方案（离线弹性）:

All required definitions and worked computations are given in the lesson itself; the book is a further-reading link.

对比候选资源与选型分析 (2)

NIST/SEMATECH e-Handbook of Statistical Methods ↗ compared_selected_for_day28 [NIST public information (U.S. government site; NIST/SEMATECH co-production)]

Engineering handbook strength for formulas, tables, and measurement practice; OpenIntro supplies the probability-to-inference narrative.

Strong probability course, but lectures and the print book are under full copyright and the syllabus spans a semester rather than a single study block.

02 120 分钟

抽样不确定性与标准误

独立掌握检查（无 AI 辅助）。

前置要求

完成活动 28-01；本练习块不使用 AI 辅助。

学习目标

量化样本统计量的波动：均值的标准误 σ/√n 与比例的标准误 √(p(1−p)/n)。

操作指引

统计学中，样本均值本身也是随机变量：重复抽样它就会变化。它的标准差即标准误，按 1/√n 的速度缩小——n 变为四倍，误差减半。先用手工计算下列各题，再用 python3 -c 配合 math.sqrt 与 math.comb 验证。

自主演练与记录

(a) 当 σ = 2 时，计算 n = 16 与 n = 64 下的 SE（均值）。(b) 对公平硬币的比例估计，计算 n = 4、n = 100、n = 400 下的 SE(p̂) = √(0.25/n)。(c) 用 math.comb 计算 P(10 次公平抛掷中正面 ≥ 7) = [C(10,7)+C(10,8)+C(10,9)+C(10,10)]/2¹⁰，并判断 7 次正面是否构成硬币有偏的有力证据。

标记为已完成

独立掌握标准

两个 SE 公式均正确套用，陈述 √n 减半规律，并在无辅助下算出二项尾部概率并正确解释。

权威源资源与权利证明

linking NIST public information (U.S. government site; NIST/SEMATECH co-production)

NIST/SEMATECH e-Handbook of Statistical Methods

作者 / 机构: NIST (U.S. National Institute of Standards and Technology) and SEMATECH | 原始出处链接 ↗

对应章节 / 时间戳: Sections 1.3.5 and 1.3.6: variability, probability distributions

验证版本与日期: Current online edition (Verified: 2026-10-30)

入选理由:

Primary engineering reference for descriptive statistics, exploratory data analysis, and uncertainty quantification with worked formulas.

来源前置条件:

High-school arithmetic; no calculus required for the descriptive sections.

安全回退方案（离线弹性）:

Every formula used in the lesson (mean, variance, standard error) is reproduced as original explanation; all exercises run fully offline.

对比候选资源与选型分析 (2)

OpenIntro Statistics (4th edition) ↗ compared_selected_for_day28 [CC BY-SA 3.0]

Undergraduate statistics text with a stronger narrative bridge from probability to inference; retained as the second linked source for inferential framing.

Seeing Theory (Brown University) ↗ compared_alternative [Free web resource (site copyright)]

Strong interactive visualizations of probability and sampling, but a browser-only interactive format is a weaker fit for an offline-capable 480-minute ledger.

linking CC BY-SA 3.0 share-alike

OpenIntro Statistics (4th edition)

作者 / 机构: David Diez, Mine Cetinkaya-Rundel, Christopher Barr | 原始出处链接 ↗

对应章节 / 时间戳: Chapter 4: foundations for inference; sampling distributions

验证版本与日期: 4th edition, 2019 (Verified: 2026-10-30)

入选理由:

Rigorous open textbook treatment of sampling distributions, standard error, and the difference between an inferential claim and an anecdotal observation.

来源前置条件:

Introductory algebra; comfort reading mathematical notation.

安全回退方案（离线弹性）:

All required definitions and worked computations are given in the lesson itself; the book is a further-reading link.

对比候选资源与选型分析 (2)

NIST/SEMATECH e-Handbook of Statistical Methods ↗ compared_selected_for_day28 [NIST public information (U.S. government site; NIST/SEMATECH co-production)]

Engineering handbook strength for formulas, tables, and measurement practice; OpenIntro supplies the probability-to-inference narrative.

Strong probability course, but lectures and the print book are under full copyright and the syllabus spans a semester rather than a single study block.

03 120 分钟

统计推断与单次观察的区别

仅在记录预测与观察证据之后允许求助。

前置要求

完成活动 28-02；求助前先记录预测。

学习目标

区分统计论断（对总体且带不确定性量化的陈述）与玩具观察（一次小样本结果），并构造一个会误导人的单次结果反例。

操作指引

推断统计会给论断附上不确定性：一个估计值加上标准误或区间。单次结果不是论断。比较两个玩具模型在 20 个留出样本上的表现：模型 A 得 16/20，模型 B 得 18/20。在下结论 B 更好之前，先问同等水平的模型出现这种差距有多常见——每个模型的比例标准误为 √(p(1−p)/20)，p = .8 时约 0.089、.9 时约 0.067——按计数单位的 SD 为 √(20p(1−p))，p = .8 时约 1.79、.9 时约 1.34——因此在明示的独立性假设下，±2 的计数差在普通波动范围内；但共享留出集使 A/B 的协方差未指定，故这不是配对检验或显著性检验。

自主演练与记录

构造反例：两名学习者各抛公平硬币 10 次；A 得 4 次正面，B 得 7 次。(a) p̂_A 与 p̂_B 各是多少？(b) 用活动 28-02 中 n = 10 的 SE（≈0.16）判断：能否声称 B 的硬币有偏？(c) 用一句话区分观察结果与统计论断。

标记为已完成

独立掌握标准

给出两个 p̂、标准误比较，并写出明确区分观察与论断的句子；不得虚构显著性。

权威源资源与权利证明

linking NIST public information (U.S. government site; NIST/SEMATECH co-production)

NIST/SEMATECH e-Handbook of Statistical Methods

作者 / 机构: NIST (U.S. National Institute of Standards and Technology) and SEMATECH | 原始出处链接 ↗

对应章节 / 时间戳: Section 1.3.5 (Quantitative Techniques): confidence in estimates

验证版本与日期: Current online edition (Verified: 2026-10-30)

入选理由:

Primary engineering reference for descriptive statistics, exploratory data analysis, and uncertainty quantification with worked formulas.

来源前置条件:

High-school arithmetic; no calculus required for the descriptive sections.

安全回退方案（离线弹性）:

Every formula used in the lesson (mean, variance, standard error) is reproduced as original explanation; all exercises run fully offline.

对比候选资源与选型分析 (2)

OpenIntro Statistics (4th edition) ↗ compared_selected_for_day28 [CC BY-SA 3.0]

Undergraduate statistics text with a stronger narrative bridge from probability to inference; retained as the second linked source for inferential framing.

Seeing Theory (Brown University) ↗ compared_alternative [Free web resource (site copyright)]

Strong interactive visualizations of probability and sampling, but a browser-only interactive format is a weaker fit for an offline-capable 480-minute ledger.

linking CC BY-SA 3.0 share-alike

OpenIntro Statistics (4th edition)

作者 / 机构: David Diez, Mine Cetinkaya-Rundel, Christopher Barr | 原始出处链接 ↗

对应章节 / 时间戳: Chapter 5: inference for numerical data; point estimates and variability

验证版本与日期: 4th edition, 2019 (Verified: 2026-10-30)

入选理由:

Rigorous open textbook treatment of sampling distributions, standard error, and the difference between an inferential claim and an anecdotal observation.

来源前置条件:

Introductory algebra; comfort reading mathematical notation.

安全回退方案（离线弹性）:

All required definitions and worked computations are given in the lesson itself; the book is a further-reading link.

对比候选资源与选型分析 (2)

NIST/SEMATECH e-Handbook of Statistical Methods ↗ compared_selected_for_day28 [NIST public information (U.S. government site; NIST/SEMATECH co-production)]

Engineering handbook strength for formulas, tables, and measurement practice; OpenIntro supplies the probability-to-inference narrative.

Strong probability course, but lectures and the print book are under full copyright and the syllabus spans a semester rather than a single study block.

04 120 分钟

检索回忆与统计复习检查点

独立掌握检查（无 AI 辅助）。

前置要求

完成活动 28-01 至 28-03；若概率基础薄弱，使用回退链接 day-25；本日统计内容的问题复习活动 28-01 至 28-03。

学习目标

以闭卷检索收束本模块：默写统计术语定义，独立计算一组新数据的摘要，并说明什么证据能把观察升级为论断。

操作指引

不看笔记、不用 AI：默写均值、中位数、方差（总体与样本）、标准差、标准误、基线的定义。然后对 data3 = [1, 3, 3, 5] 完成全部摘要计算。最后回答：把玩具观察推广为结论时，必须附带哪两个量？概率基础中想不起来的点使用回退链接 day-25，统计遗漏复习活动 28-01 至 28-03，并记录复习计划。

自主演练与记录

data3 = [1, 3, 3, 5]：计算均值、中位数、Σ(x−μ)²、总体方差与标准差、样本方差与标准差。再求解使公平硬币比例 SE(p̂) = 0.05 的样本量 n（解 √(0.25/n) = 0.05）。

标记为已完成

独立掌握标准

无辅助作答数值全部正确，六个术语均凭记忆写出，并对遗漏术语给出具体复习计划。

权威源资源与权利证明

linking NIST public information (U.S. government site; NIST/SEMATECH co-production)

NIST/SEMATECH e-Handbook of Statistical Methods

作者 / 机构: NIST (U.S. National Institute of Standards and Technology) and SEMATECH | 原始出处链接 ↗

对应章节 / 时间戳: Sections 1.3.5-1.3.6: summary formulas used across the day

验证版本与日期: Current online edition (Verified: 2026-10-30)

入选理由:

Primary engineering reference for descriptive statistics, exploratory data analysis, and uncertainty quantification with worked formulas.

来源前置条件:

High-school arithmetic; no calculus required for the descriptive sections.

安全回退方案（离线弹性）:

Every formula used in the lesson (mean, variance, standard error) is reproduced as original explanation; all exercises run fully offline.

对比候选资源与选型分析 (2)

OpenIntro Statistics (4th edition) ↗ compared_selected_for_day28 [CC BY-SA 3.0]

Undergraduate statistics text with a stronger narrative bridge from probability to inference; retained as the second linked source for inferential framing.

Seeing Theory (Brown University) ↗ compared_alternative [Free web resource (site copyright)]

Strong interactive visualizations of probability and sampling, but a browser-only interactive format is a weaker fit for an offline-capable 480-minute ledger.

linking CC BY-SA 3.0 share-alike

OpenIntro Statistics (4th edition)

作者 / 机构: David Diez, Mine Cetinkaya-Rundel, Christopher Barr | 原始出处链接 ↗

对应章节 / 时间戳: Chapters 1-5 review: summaries through inference

验证版本与日期: 4th edition, 2019 (Verified: 2026-10-30)

入选理由:

Rigorous open textbook treatment of sampling distributions, standard error, and the difference between an inferential claim and an anecdotal observation.

来源前置条件:

Introductory algebra; comfort reading mathematical notation.

安全回退方案（离线弹性）:

All required definitions and worked computations are given in the lesson itself; the book is a further-reading link.

对比候选资源与选型分析 (2)

NIST/SEMATECH e-Handbook of Statistical Methods ↗ compared_selected_for_day28 [NIST public information (U.S. government site; NIST/SEMATECH co-production)]

Engineering handbook strength for formulas, tables, and measurement practice; OpenIntro supplies the probability-to-inference narrative.

Strong probability course, but lectures and the print book are under full copyright and the syllabus spans a semester rather than a single study block.