📚 A-Level WJEC Statistics: Formula & Theorem Quick Reference | A-Level WJEC 统计：公式定理速查手册
This handbook compiles the key formulae, theorems and distributions required for the WJEC A-Level Statistics specification. Each entry is presented with its standard notation, followed by a concise explanation to support fast revision and efficient problem-solving in the examination.
本手册汇集了 WJEC A-Level 统计课程中必须掌握的核心公式、定理和分布。每个条目都采用标准符号表达，并附有简明解释，旨在帮助考生快速回顾知识，高效应对考试中的计算与推理。
1. Basic Probability Rules | 基本概率法则
The addition rule for mutually exclusive events is P(A ∪ B) = P(A) + P(B). For non-exclusive events, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The complement rule states that P(A′) = 1 − P(A).
互斥事件的加法法则为 P(A ∪ B) = P(A) + P(B)。对于非互斥事件，有 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。互补法则指出 P(A′) = 1 − P(A)。
Conditional probability is defined as P(A | B) = P(A ∩ B) / P(B) for P(B) > 0. Rearranged, the multiplication rule becomes P(A ∩ B) = P(A) P(B | A) = P(B) P(A | B). Two events are independent if P(A ∩ B) = P(A) P(B) or equivalently P(A | B) = P(A).
条件概率定义为 P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。乘法法则可变形为 P(A ∩ B) = P(A) P(B | A) = P(B) P(A | B)。如果满足 P(A ∩ B) = P(A) P(B) 或等价地 P(A | B) = P(A)，则两事件独立。
2. Discrete Probability Distributions | 离散概率分布
A discrete random variable X takes values xᵢ with probabilities pᵢ. The probability mass function must satisfy Σ pᵢ = 1. The expected value is E(X) = Σ xᵢ pᵢ and the variance is Var(X) = Σ (xᵢ − μ)² pᵢ = E(X²) − [E(X)]².
离散型随机变量 X 取值 xᵢ 的概率为 pᵢ。概率质量函数必须满足 Σ pᵢ = 1。期望值为 E(X) = Σ xᵢ pᵢ，方差为 Var(X) = Σ (xᵢ − μ)² pᵢ = E(X²) − [E(X)]²。
For a linear transformation Y = aX + b, we have E(Y) = aE(X) + b and Var(Y) = a² Var(X). The standard deviation remains scaled by |a|.
对于线性变换 Y = aX + b，有 E(Y) = aE(X) + b，Var(Y) = a² Var(X)。标准差按 |a| 倍缩放。
3. Binomial Distribution | 二项分布
If X ~ B(n, p), then the probability of exactly r successes is P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, where ⁿCᵣ = n! / [r! (n − r)!]. The mean is E(X) = np and the variance Var(X) = np(1 − p).
若 X ~ B(n, p)，则恰好得到 r 次成功的概率为 P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，其中 ⁿCᵣ = n! / [r! (n − r)!]。均值为 E(X) = np，方差为 Var(X) = np(1 − p)。
Conditions for a binomial distribution: fixed number of trials n, only two outcomes (success/failure), constant probability p, and trials are independent. The distribution is symmetric when p = 0.5, skewed otherwise.
二项分布的条件为：试验次数 n 固定；每次试验只有两种结果（成功/失败）；概率 p 保持不变；各次试验相互独立。当 p = 0.5 时分布对称，否则偏斜。
4. Poisson Distribution | 泊松分布
X ~ Po(λ) describes the number of events occurring in a fixed interval, where λ is the mean rate. The probability mass function is P(X = r) = (e⁻λ λʳ) / r!. The mean and variance are both λ: E(X) = Var(X) = λ.
X ~ Po(λ) 描述固定区间内事件发生的次数，λ 为平均发生率。概率质量函数为 P(X = r) = (e⁻λ λʳ) / r!。均值和方差均等于 λ：E(X) = Var(X) = λ。
The Poisson distribution can approximate a binomial when n is large and p is small, with λ = np. Also, the sum of independent Poisson variables is Poisson: X ~ Po(λ₁) and Y ~ Po(λ₂) ⇒ X + Y ~ Po(λ₁ + λ₂).
当 n 很大、p 很小时，可用泊松分布近似二项分布，此时 λ = np。独立泊松变量之和仍服从泊松分布：若 X ~ Po(λ₁)、Y ~ Po(λ₂)，则 X + Y ~ Po(λ₁ + λ₂)。
5. Continuous Distributions – Normal | 连续分布 – 正态分布
The normal distribution X ~ N(μ, σ²) has probability density function f(x) = (1 / σ√(2π)) e^(−(x−μ)²/(2σ²)). The curve is bell-shaped, symmetric about μ, with points of inflection at μ ± σ. Total area under the curve equals 1.
正态分布 X ~ N(μ, σ²) 的概率密度函数为 f(x) = (1 / σ√(2π)) e^(−(x−μ)²/(2σ²))。曲线呈钟形，关于 μ 对称，拐点位于 μ ± σ 处。曲线下的总面积为 1。
Standardisation converts X to Z = (X − μ) / σ, giving Z ~ N(0, 1). Use standard normal tables to find probabilities. For a sample mean X̅ from a normal population, X̅ ~ N(μ, σ²/n). Central Limit Theorem extends this to large samples from any population.
标准化变换将 X 转化为 Z = (X − μ) / σ，得到 Z ~ N(0, 1)。使用标准正态分布表查找概率。若来自正态总体的样本均值为 X̅，则 X̅ ~ N(μ, σ²/n)。中心极限定理将该结论推广到任意总体的大样本情形。
6. Continuous Distributions – Student’s t | 连续分布 – 学生氏 t 分布
When population variance is unknown and sample size is small, the t-distribution with ν = n − 1 degrees of freedom is used. If X̅ and s are sample mean and standard deviation, then T = (X̅ − μ) / (s/√n) ~ t(ν).
当总体方差未知且样本量较小时，使用自由度为 ν = n − 1 的 t 分布。若样本均值为 X̅、样本标准差为 s，则 T = (X̅ − μ) / (s/√n) ~ t(ν)。
The t-curve is symmetric and bell-shaped but has heavier tails than the normal. As ν → ∞, t(ν) approaches N(0, 1). Critical values are obtained from t-tables using the desired significance level.
t 分布曲线对称且呈钟形，但尾部比正态分布更厚。当 ν → ∞ 时，t(ν) 趋近于 N(0, 1)。根据所需的显著性水平，可从 t 值表中查得临界值。
7. Estimation and Confidence Intervals | 估计与置信区间
A point estimator is unbiased if E(θ̂) = θ. For a normal population with known variance σ², the 95% confidence interval for the population mean μ is X̅ ± 1.96 × σ/√n.
点估计量若满足 E(θ̂) = θ 则称为无偏估计。对于方差 σ² 已知的正态总体，总体均值 μ 的 95% 置信区间为 X̅ ± 1.96 × σ/√n。
When σ² is unknown, replace σ with sample standard deviation s and use the t-distribution: CI = X̅ ± t_crit × s/√n. For a population proportion p from a large sample, CI = p̂ ± z_crit × √[p̂(1−p̂)/n].
当 σ² 未知时，用样本标准差 s 替代 σ，并使用 t 分布：置信区间为 X̅ ± t_crit × s/√n。由大样本估计总体比例 p 时，置信区间为 p̂ ± z_crit × √[p̂(1−p̂)/n]。
8. Hypothesis Testing | 假设检验
A hypothesis test compares a null hypothesis H₀ against an alternative H₁. The test statistic is calculated from the sample, and its value is compared with a critical region. The significance level α is the probability of a Type I error (rejecting H₀ when true).
假设检验将原假设 H₀ 与备择假设 H₁ 进行对比。通过样本计算检验统计量，并将其值与临界区域比较。显著性水平 α 是发生第一类错误（当 H₀ 为真时拒绝它）的概率。
For a one-sample z-test of mean (σ known): z = (X̅ − μ₀) / (σ/√n). For a t-test (σ unknown): t = (X̅ − μ₀) / (s/√n). The p-value approach rejects H₀ if p-value < α. Always state a conclusion in context.
单样本均值 z 检验（σ 已知）的统计量为 z = (X̅ − μ₀) / (σ/√n)。t 检验（σ 未知）使用 t = (X̅ − μ₀) / (s/√n)。p 值方法在 p < α 时拒绝 H₀。始终需要在题设背景下给出结论。
9. Correlation and Linear Regression | 相关与线性回归
Pearson’s product-moment correlation coefficient r measures linear association between two variables x and y. The formula is r = Sxy / √(Sxx Syy), where Sxx = Σ(x − x̅)², Syy = Σ(y − y̅)², Sxy = Σ(x − x̅)(y − y̅).
皮尔逊积矩相关系数 r 度量两变量 x 与 y 之间的线性关联程度。公式为 r = Sxy / √(Sxx Syy)，其中 Sxx = Σ(x − x̅)²，Syy = Σ(y − y̅)²，Sxy = Σ(x − x̅)(y − y̅)。
The least-squares regression line of y on x is y = a + bx, where b = Sxy / Sxx and a = y̅ − b x̅. The coefficient of determination r² indicates the proportion of variation in y explained by x.
y 对 x 的最小二乘回归直线为 y = a + bx，其中 b = Sxy / Sxx，a = y̅ − b x̅。决定系数 r² 表示 y 的变异中由 x 解释的比例。
10. Chi-Squared Tests | 卡方检验
The chi-squared statistic measures the discrepancy between observed (O) and expected (E) frequencies: χ² = Σ (O − E)² / E. Degrees of freedom depend on the test: goodness-of-fit (ν = categories − 1 − parameters estimated); contingency tables (ν = (r − 1)(c − 1)).
卡方统计量衡量观测频数 (O) 与期望频数 (E) 的差异程度：χ² = Σ (O − E)² / E。自由度取决于检验类型：拟合优度检验 ν = 类别数 − 1 − 估计参数的个数；列联表检验 ν = (行数 − 1)(列数 − 1)。
In a goodness-of-fit test, the expected frequencies are derived from a specified distribution. In a test for independence in a contingency table, expected frequencies are E = (row total × column total) / grand total. The test is valid when all E ≥ 5.
在拟合优度检验中，期望频数根据指定分布计算。在列联表独立性检验中，期望频数为 E = （行合计 × 列合计）/ 总合计。当所有 E ≥ 5 时检验有效。
11. Non-Parametric Tests | 非参数检验
The sign test and Wilcoxon signed-rank test are used for paired data when normality cannot be assumed. The sign test simply counts the number of positive differences and uses a binomial distribution with p = 0.5 under H₀.
符号检验和威尔科克森符号秩检验用于无法假定正态性的配对数据。符号检验仅计算正差值的个数，并在 H₀ 下运用 p = 0.5 的二项分布。
The Mann–Whitney U test compares two independent samples. The test statistic U is calculated from ranks, and its sampling distribution is used to assess whether the two populations differ in location. These tests are based on rankings rather than actual data values.
曼–惠特尼 U 检验比较两个独立样本。检验统计量 U 根据秩次计算，其抽样分布用于判断两个总体位置参数是否不同。这些检验基于数据排序，而非原始数值。
12. Quick Symbol Reference | 常用符号速查
| Symbol | Meaning | 含义 |
| μ | Population mean | 总体均值 |
| σ | Population standard deviation | 总体标准差 |
| X̅ | Sample mean | 样本均值 |
| s | Sample standard deviation | 样本标准差 |
| H₀ / H₁ | Null / Alternative hypothesis | 原假设 / 备择假设 |
| α | Significance level | 显著性水平 |
| ν | Degrees of freedom | 自由度 |
| r | Pearson correlation coefficient | 皮尔逊相关系数 |
| p̂ | Sample proportion | 样本比例 |
| χ² | Chi-squared statistic | 卡方统计量 |
Memorising these symbols and their definitions accelerates calculation and ensures correct interpretation of examination questions. Also note that tables of critical values are provided in the WJEC formula booklet, so familiarity with their layout is essential.
熟记这些符号及其定义能加快解题速度，确保对考题的正确理解。另请注意，WJEC 公式手册中提供了临界值表，因此熟悉其布局也至关重要。
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