Pre-U Edexcel Statistics: Formula & Theorem Quick Reference Handbook | Pre-U Edexcel 统计：公式定理速查手册

📚 Pre-U Edexcel Statistics: Formula & Theorem Quick Reference Handbook | Pre-U Edexcel 统计：公式定理速查手册

This concise handbook brings together the essential formulas and theorems covered in the Pre-U Edexcel Statistics specification. Each section presents key definitions, probability models, inferential procedures, and diagnostic checks in a bilingual format, enabling quick reference and revision.

本速查手册汇总了 Pre-U Edexcel 统计课程中的核心公式与定理。每个部分以中英双语形式呈现关键定义、概率模型、推断方法以及诊断检验，便于快速查阅和复习。

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1. Probability Basics | 概率基础

For any events A and B, P(A ∪ B) = P(A) + P(B) – P(A ∩ B). If A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).

对于任意事件 A 和 B，P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若 A 与 B 互斥，则 P(A ∪ B) = P(A) + P(B)。

Conditional probability: P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

条件概率：P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。

Multiplication rule: P(A ∩ B) = P(A) P(B | A). Events A and B are independent if and only if P(A ∩ B) = P(A) P(B).

乘法法则：P(A ∩ B) = P(A) P(B | A)。事件 A 与 B 独立当且仅当 P(A ∩ B) = P(A) P(B)。

Law of total probability: P(B) = Σᵢ P(Aᵢ) P(B | Aᵢ), where {Aᵢ} forms a partition of the sample space.

全概率公式：P(B) = Σᵢ P(Aᵢ) P(B | Aᵢ)，其中 {Aᵢ} 为样本空间的一个划分。

Bayes’ theorem: P(A | B) = [P(B | A) P(A)] / P(B).

贝叶斯定理：P(A | B) = [P(B | A) P(A)] / P(B)。

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2. Random Variables & Expectation | 随机变量与期望

For a discrete random variable X, expectation: E(X) = μ = Σ x·P(X = x). Variance: Var(X) = E[(X – μ)²] = E(X²) – [E(X)]².

对于离散随机变量 X，期望值：E(X) = μ = Σ x·P(X = x)。方差：Var(X) = E[(X – μ)²] = E(X²) – [E(X)]²。

Standard deviation: σ = √Var(X).

标准差：σ = √Var(X)。

Linearity: E(aX + bY) = aE(X) + bE(Y). If X and Y are independent, Var(aX + bY) = a²Var(X) + b²Var(Y).

线性性质：E(aX + bY) = aE(X) + bE(Y)。若 X 与 Y 独立，则 Var(aX + bY) = a²Var(X) + b²Var(Y)。

Covariance: Cov(X, Y) = E[(X – μₓ)(Y – μᵧ)] = E(XY) – μₓμᵧ. Correlation: ρ = Cov(X, Y) / (σₓ σᵧ). Always -1 ≤ ρ ≤ 1.

协方差：Cov(X, Y) = E[(X – μₓ)(Y – μᵧ)] = E(XY) – μₓμᵧ。相关系数：ρ = Cov(X, Y) / (σₓ σᵧ)，取值范围 -1 ≤ ρ ≤ 1。

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3. Discrete Distributions | 离散分布

Binomial distribution X ~ B(n, p): P(X = k) = ⁿCₖ pᵏ (1-p)ⁿ⁻ᵏ, where ⁿCₖ = n!/(k!(n-k)!). Mean = np, variance = np(1-p).

二项分布 X ~ B(n, p)：P(X = k) = ⁿCₖ pᵏ (1-p)ⁿ⁻ᵏ，其中 ⁿCₖ = n!/(k!(n-k)!)。均值 np，方差 np(1-p)。

Conditions: fixed number of independent trials, constant probability of success, two outcomes.

适用条件：固定次数的独立试验，每次成功概率恒定，结果只有两种。

Poisson distribution X ~ Po(λ): P(X = k) = (λᵏ e⁻λ) / k!. Mean = λ, variance = λ.

泊松分布 X ~ Po(λ)：P(X = k) = (λᵏ e⁻λ) / k!。均值 λ，方差 λ。

Poisson approximation to binomial: if n is large and p small, (np ≈ λ), Binomial(n,p) ≈ Poisson(np).

二项分布泊松近似：当 n 大且 p 小，且 np ≈ λ，二项分布可以用泊松分布近似。

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4. Continuous Distributions | 连续分布

Normal distribution X ~ N(μ, σ²): probability density function f(x) = (1/(σ√(2π))) e^{-(x-μ)²/(2σ²)}. Standard normal Z ~ N(0,1).

正态分布 X ~ N(μ, σ²)：概率密度函数 f(x) = (1/(σ√(2π))) e^{-(x-μ)²/(2σ²)}。标准正态分布 Z ~ N(0,1)。

Student’s t-distribution with ν degrees of freedom: used when population variance unknown, t = (X̄ – μ) / (s/√n). Heavier tails.

t 分布自由度为 ν：总体方差未知时使用，t = (X̄ – μ) / (s/√n)，尾部较厚。

Chi-squared χ²(k): sum of squares of k independent N(0,1). Shape depends on degrees of freedom.

卡方分布 χ²(k)：k 个独立标准正态变量的平方和。形状取决于自由度。

F-distribution F(d₁, d₂): ratio of two independent χ² variables scaled by their degrees of freedom. Used in ANOVA and variance comparison.

F 分布 F(d₁, d₂)：两个独立卡方变量分别除以自由度的比值，用于方差分析和方差比较。

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5. Sampling Distributions & CLT | 抽样分布与中心极限定理

Sample mean X̄ from a population with mean μ and variance σ²: E(X̄) = μ, Var(X̄) = σ²/n. Standard error: SE(X̄) = σ/√n.

来自均值 μ、方差 σ² 总体的样本均值 X̄：E(X̄) = μ，Var(X̄) = σ²/n。标准误：SE(X̄) = σ/√n。

If population is normal, X̄ ~ N(μ, σ²/n). Central Limit Theorem: for large n (typically n ≥ 30), X̄ approximately ~ N(μ, σ²/n), regardless of population shape.

若总体正态，X̄ ~ N(μ, σ²/n)。中心极限定理：当 n 较大（通常 n ≥ 30），无论总体形状如何，X̄ 近似服从正态分布。

Sample variance s² = Σ(xᵢ – x̄)²/(n-1). (n-1)s²/σ² ~ χ²(n-1).

样本方差 s² = Σ(xᵢ – x̄)²/(n-1)。(n-1)s²/σ² 服从自由度为 n-1 的卡方分布。

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6. Point Estimation & Properties | 点估计与性质

An estimator θ̂ is unbiased if E(θ̂) = θ. Sample mean X̄ is unbiased for μ. Sample variance s² is unbiased for σ².

若 E(θ̂) = θ，则估计量 θ̂ 是无偏的。样本均值 X̄ 是 μ 的无偏估计，样本方差 s² 是 σ² 的无偏估计。

Consistency: an estimator is consistent if it converges in probability to the true value as sample size increases.

一致性：当样本容量增大时，估计量依概率收敛于真值，则具有一致性。

Efficiency: among unbiased estimators, the one with smaller variance is more efficient.

有效性：在无偏估计量中，方差更小者更有效。

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7. Confidence Intervals | 置信区间

A 100(1-α)% confidence interval for μ with known σ: X̄ ± z_{α/2} · σ/√n.

σ 已知时均值的 100(1-α)% 置信区间：X̄ ± z_{α/2} · σ/√n。

With unknown σ and normal population: X̄ ± t_{α/2, n-1} · s/√n.

方差未知且总体正态时：X̄ ± t_{α/2, n-1} · s/√n。

For population proportion p: p̂ ± z_{α/2} · √[p̂(1-p̂)/n], where p̂ = x/n, provided np̂ > 5 and n(1-p̂) > 5.

总体比例 p 的置信区间：p̂ ± z_{α/2} · √[p̂(1-p̂)/n]，其中 p̂ = x/n，要求 np̂ > 5 且 n(1-p̂) > 5。

Confidence interval for σ²: [(n-1)s²/χ²_{α/2, n-1}, (n-1)s²/χ²_{1-α/2, n-1}].

方差 σ² 的置信区间：[(n-1)s²/χ²_{α/2, n-1}, (n-1)s²/χ²_{1-α/2, n-1}]。

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8. Hypothesis Testing Concepts | 假设检验概念

Null hypothesis H₀: assumed true; alternative H₁. Test statistic computed from sample data. p-value: probability under H₀ of observing a value as extreme as the test statistic.

原假设 H₀：假定为真；备择假设 H₁。由样本数据计算检验统计量。p 值：在原假设成立的条件下，观察到该检验统计量或更极端值的概率。

Type I error (α): rejecting H₀ when true. Type II error (β): failing to reject H₀ when false. Power = 1 – β.

第一类错误 (α)：H₀ 为真时拒绝之。第二类错误 (β)：H₀ 为假时不拒绝。检验功效 = 1 – β。

Significance level α is set before testing. Reject H₀ if p-value < α or if |test statistic| > critical value.

显著性水平 α 在检验前设定。若 p 值 < α 或 |检验统计量| > 临界值，拒绝 H₀。

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9. Tests for Means & Proportions | 均值与比例的检验

One-sample z-test (σ known): z = (X̄ – μ₀) / (σ/√n).

单样本 z 检验（σ 已知）：z = (X̄ – μ₀) / (σ/√n)。

One-sample t-test (σ unknown): t = (X̄ – μ₀) / (s/√n), df = n-1.

单样本 t 检验（σ 未知）：t = (X̄ – μ₀) / (s/√n)，自由度 n-1。

Two-sample t-test (independent, equal variances): pooled variance s_p² = [(n₁-1)s₁² + (n₂-1)s₂²]/(n₁+n₂-2), test statistic t = (X̄₁ – X̄₂) / [s_p √(1/n₁ + 1/n₂)], df = n₁+n₂-2.

两独立样本 t 检验（等方差）：合并方差 s_p² = [(n₁-1)s₁² + (n₂-1)s₂²]/(n₁+n₂-2)，检验统计量 t = (X̄₁ – X̄₂) / [s_p √(1/n₁ + 1/n₂)]，自由度 n₁+n₂-2。

Welch’s t-test (unequal variances): t = (X̄₁ – X̄₂) / √(s₁²/n₁ + s₂²/n₂), df approx using Satterthwaite formula.

Welch t 检验（方差不相等）：t = (X̄₁ – X̄₂) / √(s₁²/n₁ + s₂²/n₂)，自由度由 Satterthwaite 近似公式给出。

Paired t-test: t = d̄ / (s_d/√n), where dᵢ = x₁ᵢ – x₂ᵢ, df = n-1.

配对 t 检验：t = d̄ / (s_d/√n)，其中 dᵢ = x₁ᵢ – x₂ᵢ，自由度 n-1。

One-sample proportion test: z = (p̂ – p₀) / √[p₀(1-p₀)/n]. Two-proportion test: z = (p̂₁ – p̂₂) / √[p̂(1-p̂)(1/n₁ + 1/n₂)], where p̂ is pooled proportion.

单样本比例检验：z = (p̂ – p₀) / √[p₀(1-p₀)/n]。两样本比例检验：z = (p̂₁ – p̂₂) / √[p̂(1-p̂)(1/n₁ + 1/n₂)]，p̂ 为合并比例。

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10. Correlation & Regression | 相关与回归

Pearson correlation coefficient r = Sₓᵧ / √(Sₓₓ Sᵧᵧ), where Sₓᵧ = Σ(xᵢ – x̄)(yᵢ – ȳ). -1 ≤ r ≤ 1.

皮尔逊相关系数 r = Sₓᵧ / √(Sₓₓ Sᵧᵧ)，其中 Sₓᵧ = Σ(xᵢ – x̄)(yᵢ – ȳ)。取值范围 -1 ≤ r ≤ 1。

Spearman’s rank correlation: use ranked data, r_rank = 1 – (6 Σ dᵢ²) / [n(n²-1)], where dᵢ = rank(xᵢ) – rank(yᵢ).

斯皮尔曼秩相关系数：基于排序数据，rₛ = 1 – (6 Σ dᵢ²) / [n(n²-1)]，其中 dᵢ 为排序差。

Simple linear regression model: y = α + βx + ε. Estimated line: ŷ = a + bx. Slope b = Sₓᵧ / Sₓₓ; intercept a = ȳ – b x̄.

简单线性回归模型：y = α + βx + ε。估计回归线：ŷ = a + bx。斜率 b = Sₓᵧ / Sₓₓ；截距 a = ȳ – b x̄。

Coefficient of determination R² = SSR / TSS = 1 – SSE/TSS, where TSS = Σ(yᵢ – ȳ)², SSR = Σ(ŷᵢ – ȳ)², SSE = Σ(yᵢ – ŷᵢ)².

决定系数 R² = SSR / TSS = 1 – SSE/TSS，其中总离差平方和 TSS = Σ(yᵢ – ȳ)²，回归平方和 SSR = Σ(ŷᵢ – ȳ)²，残差平方和 SSE = Σ(yᵢ – ŷᵢ)²。

Standard error of slope: SE(b) = s / √Sₓₓ, where s² = SSE/(n-2). Test H₀: β = 0 using t = b/SE(b) with df = n-2.

斜率的标准误：SE(b) = s / √Sₓₓ，其中 s² = SSE/(n-2)。检验 β = 0 使用 t = b/SE(b)，自由度 n-2。

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11. Analysis of Variance (ANOVA) | 方差分析

One-way ANOVA: compares means of k groups. Model: xᵢⱼ = μ + τᵢ + εᵢⱼ. Sum of squares: Total SS = SST = Σᵢ Σⱼ (xᵢⱼ – grand mean)²; Between-group SS = SSB = Σᵢ nᵢ (x̄ᵢ – grand mean)²; Within-group SS = SSW = Σᵢ Σⱼ (xᵢⱼ – x̄ᵢ)².

单因子方差分析：比较 k 个组均值。模型：xᵢⱼ = μ + τᵢ + εᵢⱼ。总平方和 SST = Σᵢ Σⱼ (xᵢⱼ – 总均值)²；组间平方和 SSB = Σᵢ nᵢ (x̄ᵢ – 总均值)²；组内平方和 SSW = Σᵢ Σⱼ (xᵢⱼ – x̄ᵢ)²。

Degrees of freedom: df_total = N-1, df_B = k-1, df_W = N-k. Mean squares: MSB = SSB/(k-1), MSW = SSW/(N-k). F-test statistic: F = MSB / MSW ~ F(k-1, N-k) under H₀.

自由度：总自由度 N-1，组间自由度 k-1，组内自由度 N-k。均方：MSB = SSB/(k-1)，MSW = SSW/(N-k)。F 检验统计量：F = MSB / MSW，在原假设下服从 F(k-1, N-k)。

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12. Chi-Squared Tests | 卡方检验

Goodness-of-fit test: H₀: data follow a specified distribution. Test statistic χ² = Σ [(Oᵢ – Eᵢ)² / Eᵢ] ~ approximately χ²(df = k-1-m), where m parameters are estimated.

拟合优度检验：H₀ 数据来自特定分布。检验统计量 χ² = Σ [(Oᵢ – Eᵢ)² / Eᵢ] 近似服从 χ²(df = k-1-m)，其中 m 为估计的参数个数。

Test of independence (contingency table): H₀: variables are independent. χ² = Σᵢⱼ (Oᵢⱼ – Eᵢⱼ)² / Eᵢⱼ, with expected frequencies Eᵢⱼ = (row total × column total) / grand total. df = (rows-1)(columns-1).

独立性检验（列联表）：H₀ 变量独立。χ² = Σᵢⱼ (Oᵢⱼ – Eᵢⱼ)² / Eᵢⱼ，期望频数 Eᵢⱼ = (行合计 × 列合计) / 总合计。自由度 = (行数-1)(列数-1)。

Conditions: all expected frequencies ≥ 5 (or at least 80% ≥ 5).

条件：所有期望频数 ≥ 5（或至少 80% ≥ 5）。

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