📚 Pre-U WJEC Statistics: Quick Reference Formula & Theorem Handbook | Pre-U WJEC 统计：公式定理速查手册
This handbook compiles the essential formulas, theorems, and key concepts required for the WJEC Pre-U Statistics syllabus. It is designed as a rapid revision aid, allowing you to check definitions, assumptions, and test procedures at a glance. Each section is presented in English followed by its Chinese translation to support bilingual learners.
本手册汇编了 WJEC Pre-U 统计课程所需的核心公式、定理和关键概念，旨在帮助快速复习。每个部分均以英文和中文配对呈现，便于双语学习者查阅。
1. Basic Probability Rules | 概率基本法则
For any events A and B, the addition rule states that P(A ∪ B) = P(A) + P(B) - P(A ∩ B). If A and B are mutually exclusive, then P(A ∩ B) = 0 and the formula simplifies to P(A ∪ B) = P(A) + P(B).
对于任意事件 A 和 B，加法法则为 P(A ∪ B) = P(A) + P(B) - P(A ∩ B)。若 A 与 B 互斥，则 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。
Conditional probability is defined as P(A|B) = P(A ∩ B) / P(B) for P(B) > 0. Rearranging gives the multiplication rule: P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(A).
条件概率定义为 P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0。变形可得乘法法则：P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(A)。
Two events are independent if and only if P(A ∩ B) = P(A) × P(B), or equivalently P(A|B) = P(A). The law of total probability for a partition {Bᵢ} states: P(A) = Σ P(A|Bᵢ) P(Bᵢ). Bayes' theorem then gives P(Bⱼ|A) = [P(A|Bⱼ) P(Bⱼ)] / Σ P(A|Bᵢ) P(Bᵢ).
两个事件独立当且仅当 P(A ∩ B) = P(A) × P(B)，或等价地 P(A|B) = P(A)。对于分割 {Bᵢ}，全概率公式为：P(A) = Σ P(A|Bᵢ) P(Bᵢ)。贝叶斯定理则给出 P(Bⱼ|A) = [P(A|Bⱼ) P(Bⱼ)] / Σ P(A|Bᵢ) P(Bᵢ)。
2. Discrete Probability Distributions | 离散概率分布
x). Expected value: E(X) = Σ x·p(x); variance: Var(X) = E(X²) - [E(X)]² = Σ (x - μ)² p(x).
离散概率质量函数 p(x) = P(X = x) 满足 0 ≤ p(x) ≤ 1 且 Σ p(x) = 1。累积分布函数为 F(x) = P(X ≤ x)。期望：E(X) = Σ x·p(x)；方差：Var(X) = E(X²) - [E(X)]² = Σ (x - μ)² p(x)。
Binomial distribution X ~ B(n, p): pmf P(X = x) = ⁿCₓ pˣ (1-p)ⁿ⁻ˣ ; variance: np(1-p). Conditions: fixed n, independent trials, constant p, binary outcomes.
二项分布 X ~ B(n, p)：概率函数 P(X = x) = ⁿCₓ pˣ (1-p)ⁿ⁻ˣ，x = 0,1,...,n。均值：np；方差：np(1-p)。条件：试验次数固定、各次独立、p 恒定、结果二元。
Poisson distribution X ~ Po(λ): pmf P(X = x) = e⁻ᴸ λˣ / x! in a fixed interval, and as approximation to binomial when n large, p small, with λ = np.
泊松分布 X ~ Po(λ)：概率函数 P(X = x) = e⁻ᴸ λˣ / x!，x = 0,1,2,...。均值 = 方差 = λ。适用于固定区间内的稀有事件，也可在 n 大、p 小时近似二项分布，令 λ = np。
3. Continuous Probability Distributions | 连续概率分布
P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. The cumulative distribution function is F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt.
对于连续随机变量 X，概率密度函数 f(x) 满足 f(x) ≥ 0 且 ∫₋∞⁺∞ f(x) dx = 1。概率为面积：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。累积分布函数为 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。
Normal distribution X ~ N(μ, σ²): pdf is f(x) = (1/(σ√2π)) exp(-(x-μ)²/(2σ²)). Standard normal Z ~ N(0,1), where Z = (X - μ)/σ. Key property: about 68% of data within μ ± σ, 95% within μ ± 2σ, 99.7% within μ ± 3σ.
正态分布 X ~ N(μ, σ²)：密度函数为 f(x) = (1/(σ√2π)) exp(-(x-μ)²/(2σ²))。标准正态 Z ~ N(0,1)，其中 Z = (X - μ)/σ。重要性质：约 68% 数据在 μ ± σ 内，95% 在 μ ± 2σ 内，99.7% 在 μ ± 3σ 内。
Mean: (a+b)/2; variance: (b-a)²/12. The exponential distribution Exp(λ) has pdf f(x) = λe⁻ᴸˣ for x ≥ 0, mean = 1/λ, variance = 1/λ².
连续均匀分布 U(a, b) 在 [a,b] 上 pdf 为常数 f(x) = 1/(b-a)。均值：(a+b)/2；方差：(b-a)²/12。指数分布 Exp(λ) 有 pdf f(x) = λe⁻ᴸˣ，x ≥ 0，均值 = 1/λ，方差 = 1/λ²。
4. Sampling Distributions & Central Limit Theorem | 抽样分布与中心极限定理
.., Xₙ be a random sample from a population with mean μ and variance σ². The sample mean X̄ has E(X̄) = μ and Var(X̄) = σ²/n. If the population is normal, X̄ ~ N(μ, σ²/n) exactly. The sample variance S² = Σ (Xᵢ - X̄)²/(n-1) is an unbiased estimator of σ².
设 X₁, X₂, ..., Xₙ 为来自均值为 μ、方差为 σ² 的总体的随机样本。样本均值 X̄ 有 E(X̄) = μ，Var(X̄) = σ²/n。若总体为正态，则 X̄ 精确服从 N(μ, σ²/n)。样本方差 S² = Σ (Xᵢ - X̄)²/(n-1) 是 σ² 的无偏估计量。
Central Limit Theorem (CLT): For large n (typically n ≥ 30), the distribution of X̄ is approximately normal regardless of the population shape, so X̄ ~ N(μ, σ²/n) approximately. The standardized statistic Z = (X̄ - μ)/(σ/√n) → N(0,1) as n → ∞. When σ is unknown, use t-distribution: T = (X̄ - μ)/(S/√n) ~ t(n-1).
中心极限定理 (CLT)：对于大样本（通常 n ≥ 30），无论总体分布形状如何，X̄ 的分布近似正态，即 X̄ 近似服从 N(μ, σ²/n)。标准化统计量 Z = (X̄ - μ)/(σ/√n) 在 n → ∞ 时收敛于 N(0,1)。当 σ 未知时，使用 t 分布：T = (X̄ - μ)/(S/√n) ~ t(n-1)。
The sampling distribution of the sample proportion p̂ = X/n (where X ~ B(n, p)) has E(p̂) = p, Var(p̂) = p(1-p)/n. By CLT, for np ≥ 5 and n(1-p) ≥ 5, p̂ is approximately normal: p̂ ~ N(p, p(1-p)/n).
样本比例 p̂ = X/n（X ~ B(n, p)）的抽样分布有 E(p̂) = p，Var(p̂) = p(1-p)/n。根据 CLT，当 np ≥ 5 且 n(1-p) ≥ 5 时，p̂ 近似正态：p̂ ~ N(p, p(1-p)/n)。
5. Point Estimation | 点估计
An estimator θ̂ is unbiased if E(θ̂) = θ. The bias is E(θ̂) - θ. For a normal population, X̄ is unbiased for μ, but the sample variance with denominator n is biased; S² = Σ (Xᵢ - X̄)²/(n-1) is unbiased for σ².
若估计量 θ̂ 满足 E(θ̂) = θ，则称为无偏估计量。偏差为 E(θ̂) - θ。对于正态总体，X̄ 是 μ 的无偏估计，而以 n 为分母的样本方差是有偏的；S² = Σ (Xᵢ - X̄)²/(n-1) 是 σ² 的无偏估计量。
bability of the observed data as a function of θ. The maximum likelihood estimator (MLE) maximises L(θ) or the log-likelihood ℓ(θ) = ln L(θ).
矩估计法将样本矩等于总体矩以求解参数估计值。似然函数 L(θ) 是观测数据作为 θ 函数的联合概率。最大似然估计量 (MLE) 最大化 L(θ) 或对数似然 ℓ(θ) = ln L(θ)。
large samples, MLEs are approximately unbiased, normally distributed, and achieve the Cramé bound. The standard error (SE) of an estimator quantifies its precision; for example, SE(X̄) = σ/√n, estimated by S/√n.
对于大样本，MLE 近似无偏、正态分布，且达到克拉美-罗下界。估计量的标准误 (SE) 衡量其精确度；例如 SE(X̄) = σ/√n，可由 S/√n 估计。
6. Confidence Intervals | 置信区间
A 100(1-α)% plausible values based on sample data. For a normal mean with known variance, the CI is X̄ ± zα/2 × (σ/√n). For unknown σ, use the t-interval: X̄ ± tα/2, n-1 × (S/√n).
参数 100(1-α)% 的置信区间给出了基于样本数据的合理取值范围。对于已知方差的正态均值，置信区间为 X̄ ± zα/2 × (σ/√n)。若 σ 未知，使用 t 区间：X̄ ± tα/2, n-1 × (S/√n)。
For a population proportion p, the approximate CI is p̂ ± zα/2 × √(p̂(1-p̂ ess-failure conditions hold. For the difference between two means, independent samples: (X̄₁ - X̄₂) ± t* × SE, where SE = √S₁²/n₁ + S₂²/n₂ or pooled version if equal variances assumed.
对于总体比例 p，近似置信区间为 p̂ ± zα/2 × √(p̂(1-p̂)/n)，前提是成功-失败条件成立。对于两个均值之差，独立样本：(X̄₁ - X̄₂) ± t* × SE，其中 SE = √S₁²/n₁ + S₂²/n₂，若假设方差相等则可使用合并版本。
A confidence interval for variance σ² of a normal population uses the chi-squared distribution: [(n-1)S²/χ²α/2, n-1, (n-1)S²/χ²1-α/2, n-1].
正态总体方差 σ² 的置信区间使用卡方分布：[(n-1)S²/χ²α/2, n-1, (n-1)S²/χ²1-α/2, n-1]。
7. Hypothesis Testing: One Sample | 假设检验：单样本
d a rejection region or p-value. Significance level α is the probability of Type I error (rejecting true H₀). Type II error β is failing to reject a false H₀. Power = 1 - β.
原假设 H₀ 陈述默认主张；备择假设 H₁ 是研究假设。检验利用检验统计量和拒绝域或 p 值评估反对 H₀ 的证据。显著性水平 α 是第 I 类错误的概率（拒绝真实的 H₀）。第 II 类错误 β 是未拒绝错误的 H₀。检验功效 = 1 - β。
One-sample z-test for mean (σ known): test statistic Z = (X̄ - μ₀)/(σ/√n). One-sample t-test (σ unknown): T = (X̄ - μ₀)/(S/√n) ~ t(n-1). Rejection rules: for two-sided test, reject H₀ if |statistic| > critical value, or p-value < α.
单样本均值 z 检验（σ 已知）：检验统计量 Z = (X̄ - μ₀)/(σ/√n)。单样本 t 检验（σ 未知）：T = (X̄ - μ₀)/(S/√n) ~ t(n-1)。拒绝规则：对于双侧检验，若 |统计量| > 临界值或 p 值 < α，则拒绝 H₀。
One-sample proportion test (normal approximation): Z = (p̂ mall samples, use the exact binomial test. Chi-squared test for variance: χ² = (n-1)S²/σ₀² ~ χ²(n-1).
单样本比例检验（正态近似）：Z = (p̂ - p₀)/√(p₀(1-p₀)/n)。小样本时使用精确二项检验。方差的卡方检验：χ² = (n-1)S²/σ₀² ~ χ²(n-1)。
8. Hypothesis Testing: Two Samples | 假设检验：双样本
ion means, independent samples: if variances are known, use the two-sample z-test: Z = [(X̄₁ - X̄₂) - (μ₁ - μ₂)₀] / √σ₁²/n₁ + σ₂²/n₂. If unknown and unequal variances (Welch's t-test): T = [(X̄₁ - X̄₂) - Δ₀] / √S₁²/n₁ + S₂²/n₂ with degrees of freedom by Welch-Satterthwaite approximation.
比较两个总体均值，独立样本：若方差已知，使用双样本 z 检验：Z = [(X̄₁ - X̄₂) - (μ₁ - μ₂)₀] / √σ₁²/n₁ + σ₂²/n₂。若方差未知且不等（Welch t 检验）：T = [(X̄₁ - X̄₂) - Δ₀] / √S₁²/n₁ + S₂²/n₂，自由度由 Welch-Satterthwaite 近似给出。
If equal variances can be assumed, use the pooled t-test: T = [(X̄₁ - X̄₂) - Δ₀] / (Sp √1/n₁ + 1/n₂), where pooled variance Sp² = [(n₁-1)S₁² + (n₂-1)S₂²] / (n₁ + n₂ - 2), df = n₁ + n₂ - 2.
若可假设方差齐性，使用合并 t 检验：T = [(X̄₁ - X̄₂) - Δ₀] / (Sp √1/n₁ + 1/n₂)，其中合并方差 Sp² = [(n₁-1)S₁² + (n₂-1)S₂²] / (n₁ + n₂ - 2)，自由度 df = n₁ + n₂ - 2。
mpute differences dᵢ = X₁ᵢ - X₂ᵢ and perform a one-sample t-test on the differences: T = d̄ / (Sd / √n) with df = n - 1. The two-proportion z-test: Z = (p̂₁ - p̂₂) / √(p̂(1-p̂)(1/n₁ + 1/n₂)), where p̂ = (x₁ + x₂)/(n₁ + n₂).
对于配对数据，计算差值 dᵢ = X₁ᵢ - X₂ᵢ，然后对差值进行单样本 t 检验：T = d̄ / (Sd / √n)，自由度 df = n - 1。双比例 z 检验：Z = (p̂₁ - p̂₂) / √(p̂(1-p̂)(1/n₁ + 1/n₂))，其中 p̂ = (x₁ + x₂)/(n₁ + n₂)。
The F-test for equality of two variances: F = S₁² / S₂² ~ F(n₁-1, n₂-1) under the assumption of normality. Reject H₀: σ₁² = σ₂² if F > Fα/2, v₁, v₂ or F < 1/Fα/2, v₂, v₁.
两个方差齐性的 F 检验：在正态性假设下，F = S₁² / S₂² ~ F(n₁-1, n₂-1)。若 F > Fα/2, v₁, v₂ 或 F < 1/Fα/2, v₂, v₁，则拒绝 H₀: σ₁² = σ₂²。
9. Chi-Squared Tests | 卡方检验
ᵢ - Eᵢ)² / Eᵢ, where Oᵢ are observed frequencies and Eᵢ are expected frequencies under H₀.
卡方检验用于评估分类数据的拟合优度或关联性。检验统计量：χ² = Σ (Oᵢ - Eᵢ)² / Eᵢ，其中 Oᵢ 为观测频数，Eᵢ 为 H₀ 下的期望频数。
categories - 1 - number of estimated parameters. Contingency table test of independence: Eij = (row i total × column j total) / grand total, df = (r-1)(c-1).
拟合优度检验：期望频数基于理论分布，自由度 = 类别数 - 1 - 估计参数个数。列联表独立性检验：Eij = (第 i 行合计 × 第 j 列合计) / 总合计，自由度 df = (r-1)(c-1)。
Validity requires all expected frequencies ≥ 5 (or no more than 20% below 5). Yates' continuity correction for 2×2 tables: χ² = Σ (|Oᵢ - Eᵢ| - 0.5)² / Eᵢ.
有效性要求所有期望频数 ≥ 5（或低于 5 的比例不超过 20%）。对于 2×2 表，使用耶茨连续性校正：χ² = Σ (|Oᵢ - Eᵢ| - 0.5)² / Eᵢ。
10. Correlation and Regression | 相关与回归
earson product-moment correlation coefficient r measures linear association: r = Σ (xᵢ - x̄)(yᵢ - ȳ) / √[Σ (xᵢ - x̄)² Σ (yᵢ - ȳ)²] = Sxy / √Sxx Syy. Range -1 ≤ r ≤ 1. Test for ρ = 0 using T = r √n-2 / √1 - r² ~ t(n-2).
皮尔逊积矩相关系数 r 衡量线性关系：r = Σ (xᵢ - x̄)(yᵢ - ȳ) / √[Σ (xᵢ - x̄)² Σ (yᵢ - ȳ)²] = Sxy / √Sxx Syy。取值范围 -1 ≤ r ≤ 1。用 T = r √n-2 / √1 - r² ~ t(n-2) 检验 ρ = 0。
Spearman's rank correlation rₛ is based on ranked data, robust to outliers. For tied ranks, use the Pearson formula on ranks. It tests monotonic association.
斯皮尔曼秩相关系数 rₛ 基于秩次数据，对异常值稳健。对于结，对秩使用皮尔逊公式。它检验单调关联。
Simple linear regression model: Y = α + βx + ε, ε ~ N(0, σ²). Least squares estimates: b = Sxy / Sxx, a = ȳ - b x̄. The fitted line is ŷ = a + bx. Residuals eᵢ = yᵢ - ŷᵢ, with Σ eᵢ = 0. Estimate of error variance: s² = Σ eᵢ² / (n-2) = (Syy - b Sxy)/(n-2).
简单线性回归模型：Y = α + βx + ε，ε ~ N(0, σ²)。最小二乘估计：b = Sxy / Sxx，a = ȳ - b x̄。拟合直线为 ŷ = a + bx。残差 eᵢ = yᵢ - ŷᵢ，满足 Σ eᵢ = 0。误差方差估计：s² = Σ eᵢ² / (n-2) = (Syy - b Sxy)/(n-2)。
Inference for slope β: SE(b) = s / √Sxx, T = (b - β₀) / SE(b) ~ t(n-2). Confidence interval: b ± tα/2, n-2 new observation at x* includes extra variability, wider than the CI for the mean response.
斜率 β 的推断：SE(b) = s / √Sxx，T = (b - β₀) / SE(b) ~ t(n-2)。置信区间：b ± tα/2, n-2 × SE(b)。在 x* 处新观测值的预测区间包含额外变异，比均值响应的置信区间更宽。
11. Nonparametric Tests | 非参数检验
Nonparametric methods avoid strong distributional assumptions. The sign test for a median compares positive and negative differences to a hypothesised value. For matched pairs, the Wilcoxon signed-rank test uses both sign and magnitude of differences.
非参数方法避免了强烈的分布假设。中位数的符号检验将正负差值与假设值比较。对于配对数据，威尔科克森符号秩检验同时利用差值的符号和大小。
The Mann-Whitney U test (Wilcoxon rank-sum test) compares two independent samples by ranking all observations together. U statistic is based on the sum of ranks in one group. The null hypothesis is that the two populations have the same distribution (shift alternative).
曼-惠特尼 U 检验（威尔科克森秩和检验）通过将所有观测合并排秩来比较两个独立样本。U 统计量基于一组的秩和。原假设为两个总体分布相同（位移备择）。
For large samples, the rank-sum statistic is approximately normal under H₀. These tests are less powerful than parametric counterparts when normality holds but are robust otherwise.
对于大样本，在 H₀ 下秩和统计量近似正态。当正态性成立时，这些检验的功效低于参数方法，但反之则更稳健。
12. Key Distributions at a Glance | 关键分布速览
| Distribution | Pmf/Pdf | Mean | Variance |
|---|---|---|---|
| Binomial B(n,p) | ⁿCₓ pˣ (1-p)ⁿ⁻ˣ | np | np(1-p) |
| Poisson Po(λ) | e⁻ᴸ λˣ / x! | λ | λ |
| Geometric G(p) | (1-p)ˣ⁻¹ p | 1/p | (1-p)/p² |
| Uniform U(a,b) | 1/(b-a) | (a+b)/2 | (b-a)²/12 |
| Normal N(μ,σ²) | (1/σ√2π) e-(x-μ)²/(2σ²) | μ | σ² |
| Exponential Exp(λ) | λ e⁻ᴸˣ | 1/λ | 1/λ² |
These distributions form the backbone of many inferential procedures. Always check model assumptions before applying any test.
这些分布构成了众多推断程序的支柱。在应用任何检验前，务必检查模型假设。
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