第 28 天：統計——資料摘要、抽樣不確定性與推論

01 120 分鐘

描述統計：平均值、離散程度與誠實的摘要

僅在記錄預測與觀察證據之後允許求助。

前置要求

完成第 27 天數學模組；具備基礎算術能力以及第 01 天的預測—回顧流程。

學習目標

在小資料集上手工計算並解釋平均值、中位數、母體/樣本變異數與標準差，並說明兩個平均值相同的資料集仍可能不同。

操作指引

統計首先從把資料壓縮為少數幾個數字開始。平均值刻畫中心位置；變異數與標準差度量圍繞中心的離散程度。母體變異數把偏差平方和除以 n；樣本變異數除以 n−1（貝塞爾校正），以避免系統性低估離散程度。先用紙筆預測每個數值，再執行 python3 核對——核對必須發生在預測記錄之後。

自主實作與記錄

對於 data = [2, 4, 4, 4, 5, 5, 7, 9]：計算平均值、中位數、偏差平方和、母體變異數、母體標準差與樣本變異數（除數為 n−1）。然後對 data2 = [4, 5, 5, 5, 5, 5, 5, 6] 計算平均值與母體標準差，說明什麼保持不變、什麼發生變化。

標記為已完成

獨立掌握標準

手算結果與 Python 核對全部一致，並能說明平均值相等並不意味著分布相同。

權威來源資源與權利證明

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

NIST/SEMATECH e-Handbook of Statistical Methods

作者 / 機構: NIST (U.S. National Institute of Standards and Technology) and SEMATECH | 原始出處連結 ↗

對應章節 / 時間戳: Section 1.3.5 (Quantitative Techniques): measures of location and scale

驗證版本與日期: Current online edition (Verified: 2026-10-30)

入選理由:

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

來源前置條件:

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

安全回退方案（離線彈性）:

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

對比候選資源與選型分析 (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 / 機構: David Diez, Mine Cetinkaya-Rundel, Christopher Barr | 原始出處連結 ↗

對應章節 / 時間戳: Chapters 1-2: data basics and summary statistics

驗證版本與日期: 4th edition, 2019 (Verified: 2026-10-30)

入選理由:

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

來源前置條件:

Introductory algebra; comfort reading mathematical notation.

安全回退方案（離線彈性）:

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

對比候選資源與選型分析 (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 分鐘

抽樣不確定性與標準誤差

獨立掌握檢查（無 AI 輔助）。

前置要求

完成活動 28-01；本練習區塊不使用 AI 輔助。

學習目標

量化樣本統計量的波動：平均值的標準誤差 σ/√n 與比例的標準誤差 √(p(1−p)/n)。

操作指引

統計學中，樣本平均值本身也是隨機變數：重複抽樣它就會變化。它的標準差即標準誤差，按 1/√n 的速度縮小——n 變為四倍，誤差減半。先用手工計算下列各題，再用 python3 -c 配合 math.sqrt 與 math.comb 驗證。

自主實作與記錄

(a) 當 σ = 2 時，計算 n = 16 與 n = 64 下的 SE（平均值）。(b) 對公平硬幣的比例估計，計算 n = 4、n = 100、n = 400 下的 SE(p̂) = √(0.25/n)。(c) 用 math.comb 計算 P(10 次公平拋擲中正面 ≥ 7) = [C(10,7)+C(10,8)+C(10,9)+C(10,10)]/2¹⁰，並判斷 7 次正面是否構成硬幣有偏的有力證據。

標記為已完成

獨立掌握標準

兩個 SE 公式均正確套用，陳述 √n 減半規律，並在無輔助下算出二項尾部機率並正確解釋。

權威來源資源與權利證明

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

NIST/SEMATECH e-Handbook of Statistical Methods

作者 / 機構: NIST (U.S. National Institute of Standards and Technology) and SEMATECH | 原始出處連結 ↗

對應章節 / 時間戳: Sections 1.3.5 and 1.3.6: variability, probability distributions

驗證版本與日期: Current online edition (Verified: 2026-10-30)

入選理由:

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

來源前置條件:

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

安全回退方案（離線彈性）:

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

對比候選資源與選型分析 (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 / 機構: David Diez, Mine Cetinkaya-Rundel, Christopher Barr | 原始出處連結 ↗

對應章節 / 時間戳: Chapter 4: foundations for inference; sampling distributions

驗證版本與日期: 4th edition, 2019 (Verified: 2026-10-30)

入選理由:

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

來源前置條件:

Introductory algebra; comfort reading mathematical notation.

安全回退方案（離線彈性）:

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

對比候選資源與選型分析 (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 分鐘

統計推論與單次觀察的區別

僅在記錄預測與觀察證據之後允許求助。

前置要求

完成活動 28-02；求助前先記錄預測。

學習目標

區分統計論斷（對母體且帶不確定性量化的陳述）與玩具觀察（一次小樣本結果），並構造一個會誤導人的單次結果反例。

操作指引

推論統計會給論斷附上不確定性：一個估計值加上標準誤差或區間。單次結果不是論斷。比較兩個玩具模型在 20 個留出樣本上的表現：模型 A 得 16/20，模型 B 得 18/20。在下結論 B 更好之前，先問同等水準的模型出現這種差距有多常見——每個模型的比例標準誤差為 √(p(1−p)/20)，p = .8 時約 0.089、.9 時約 0.067——按計數單位的 SD 為 √(20p(1−p))，p = .8 時約 1.79、.9 時約 1.34——因此在明示的獨立性假設下，±2 的計數差在普通波動範圍內；但共享留出集使 A/B 的共變異數未指定，故這不是配對檢定或顯著性檢定。

自主實作與記錄

構造反例：兩名學習者各拋公平硬幣 10 次；A 得 4 次正面，B 得 7 次。(a) p̂_A 與 p̂_B 各是多少？(b) 用活動 28-02 中 n = 10 的 SE（≈0.16）判斷：能否聲稱 B 的硬幣有偏？(c) 用一句話區分觀察結果與統計論斷。

標記為已完成

獨立掌握標準

給出兩個 p̂、標準誤差比較，並寫出明確區分觀察與論斷的句子；不得虛構顯著性。

權威來源資源與權利證明

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

NIST/SEMATECH e-Handbook of Statistical Methods

作者 / 機構: NIST (U.S. National Institute of Standards and Technology) and SEMATECH | 原始出處連結 ↗

對應章節 / 時間戳: Section 1.3.5 (Quantitative Techniques): confidence in estimates

驗證版本與日期: Current online edition (Verified: 2026-10-30)

入選理由:

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

來源前置條件:

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

安全回退方案（離線彈性）:

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

對比候選資源與選型分析 (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 / 機構: David Diez, Mine Cetinkaya-Rundel, Christopher Barr | 原始出處連結 ↗

對應章節 / 時間戳: Chapter 5: inference for numerical data; point estimates and variability

驗證版本與日期: 4th edition, 2019 (Verified: 2026-10-30)

入選理由:

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

來源前置條件:

Introductory algebra; comfort reading mathematical notation.

安全回退方案（離線彈性）:

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

對比候選資源與選型分析 (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 分鐘

檢索回憶與統計複習檢查點

獨立掌握檢查（無 AI 輔助）。

前置要求

完成活動 28-01 至 28-03；若機率基礎薄弱，使用回退連結 day-25；本日統計內容的問題複習活動 28-01 至 28-03。

學習目標

以閉卷檢索收束本模組：默寫統計術語定義，獨立計算一組新資料的摘要，並說明什麼證據能把觀察升級為論斷。

操作指引

不看筆記、不用 AI：默寫平均值、中位數、變異數（母體與樣本）、標準差、標準誤差、基線的定義。然後對 data3 = [1, 3, 3, 5] 完成全部摘要計算。最後回答：把玩具觀察推廣為結論時，必須附帶哪兩個量？機率基礎中想不起來的點使用回退連結 day-25，統計遺漏複習活動 28-01 至 28-03，並記錄複習計畫。

自主實作與記錄

data3 = [1, 3, 3, 5]：計算平均值、中位數、Σ(x−μ)²、母體變異數與標準差、樣本變異數與標準差。再求解使公平硬幣比例 SE(p̂) = 0.05 的樣本量 n（解 √(0.25/n) = 0.05）。

標記為已完成

獨立掌握標準

無輔助作答數值全部正確，六個術語均憑記憶寫出，並對遺漏術語給出具體複習計畫。

權威來源資源與權利證明

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

NIST/SEMATECH e-Handbook of Statistical Methods

作者 / 機構: NIST (U.S. National Institute of Standards and Technology) and SEMATECH | 原始出處連結 ↗

對應章節 / 時間戳: Sections 1.3.5-1.3.6: summary formulas used across the day

驗證版本與日期: Current online edition (Verified: 2026-10-30)

入選理由:

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

來源前置條件:

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

安全回退方案（離線彈性）:

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

對比候選資源與選型分析 (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 / 機構: David Diez, Mine Cetinkaya-Rundel, Christopher Barr | 原始出處連結 ↗

對應章節 / 時間戳: Chapters 1-5 review: summaries through inference

驗證版本與日期: 4th edition, 2019 (Verified: 2026-10-30)

入選理由:

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

來源前置條件:

Introductory algebra; comfort reading mathematical notation.

安全回退方案（離線彈性）:

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

對比候選資源與選型分析 (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.