📚 A-Level Cambridge Statistics: Formula & Theorem Quick Reference Handbook | A-Level Cambridge 统计：公式定理速查手册
This handbook consolidates the essential formulae and theorems required for the Cambridge International A‑Level Statistics syllabus. Designed as a rapid revision tool, each section presents key results together with brief explanatory notes, ensuring you can quickly locate and apply the correct statistical method in examinations. The content is aligned with the CAIE 9709 Statistics 1 & 2 components, covering probability, distributions, inference, and bivariate data analysis.
本手册整理了剑桥国际 A‑Level 统计课程必备的公式和定理。作为快速复习工具，每一个章节都列出了核心结果并附有简要说明，确保你在考试中能迅速定位并应用正确的统计方法。内容对标 CAIE 9709 统计 1 和统计 2 模块，覆盖概率、分布、推断以及双变量数据分析。
1. Probability Fundamentals | 概率基础
For any events A and B in a sample space S, probability measures the likelihood of occurrence on a scale from 0 to 1. The addition rule connects union and intersection: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, this simplifies to P(A ∪ B) = P(A) + P(B) because A ∩ B = ∅.
对于样本空间 S 中的事件 A 和 B，概率以 0 到 1 之间的数值衡量事件发生的可能性。加法法则将并集与交集联系在一起：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。对于互斥事件，因为 A ∩ B = ∅，该公式简化为 P(A ∪ B) = P(A) + P(B)。
Conditional probability is defined as P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0. Multiplication rule follows directly: P(A ∩ B) = P(B) × 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), or equivalently P(A | B) = P(A) when P(B) > 0.
条件概率定义为 P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。乘法法则直接由此得出：P(A ∩ B) = P(B) × P(A | B) = P(A) × P(B | A)。事件 A 与 B 相互独立的充要条件是 P(A ∩ B) = P(A) × P(B)，或者等价地当 P(B) > 0 时 P(A | B) = P(A)。
2. Discrete Probability Distributions | 离散概率分布
A discrete random variable X takes countable values x₁, x₂, … with probability mass function P(X = x). The expectation (mean) is E(X) = Σ x·P(X = x), and variance Var(X) = E(X²) − [E(X)]² = Σ x²·P(X = x) − μ². For a linear transformation Y = aX + b, we have E(Y) = aE(X) + b and Var(Y) = a² Var(X).
离散随机变量 X 取可数个值 x₁, x₂, …，其概率质量函数为 P(X = x)。期望值（均值）为 E(X) = Σ x·P(X = x)，方差为 Var(X) = E(X²) − [E(X)]² = Σ x²·P(X = x) − μ²。对于线性变换 Y = aX + b，有 E(Y) = aE(X) + b，Var(Y) = a² Var(X)。
Binomial distribution: X ~ B(n, p) describes the number of successes in n independent Bernoulli trials with success probability p. The probability mass function is P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ for r = 0,1,…,n. Mean E(X) = np, variance Var(X) = np(1−p).
二项分布：X ~ B(n, p) 描述在成功概率为 p 的 n 次独立伯努利试验中成功的次数。概率质量函数为 P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ，r = 0,1,…,n。期望 E(X) = np，方差 Var(X) = np(1−p)。
Poisson distribution: X ~ Po(λ) models the number of events occurring in a fixed interval when events happen independently at a constant average rate λ. Probability function: P(X = r) = (e⁻λ λʳ) / r! for r = 0,1,2,… Both mean and variance equal λ. The Poisson can approximate a binomial when n is large and p is small; set λ ≈ np.
泊松分布：X ~ Po(λ) 用于建模在固定区间内事件发生的次数，假设事件以恒定平均率 λ 独立发生。概率函数：P(X = r) = (e⁻λ λʳ) / r!，r = 0,1,2,…。均值和方差都等于 λ。当 n 很大且 p 很小时，可以用泊松分布近似二项分布；取 λ ≈ np。
3. Continuous Probability Distributions | 连续概率分布
A continuous random variable X is described by its probability density function f(x). The probability of X lying in the interval [a, b] is the area under the curve: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. The total area under f(x) equals 1. Cumulative distribution function F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt.
连续随机变量 X 由其概率密度函数 f(x) 描述。X 落在区间 [a, b] 内的概率为曲线下面积：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。曲线下总面积为 1。累积分布函数为 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。
Normal distribution: X ~ N(μ, σ²) has density f(x) = (1/√(2πσ²)) exp(−(x−μ)²/(2σ²)). The standard normal Z = (X − μ)/σ follows N(0, 1). Phi(z) = P(Z ≤ z) is tabled. For any normal variable, P(a < X < b) = Phi((b−μ)/σ) − Phi((a−μ)/σ). The distribution is symmetric, so Phi(−z) = 1 − Phi(z).
正态分布：X ~ N(μ, σ²) 的密度函数为 f(x) = (1/√(2πσ²)) exp(−(x−μ)²/(2σ²))。标准正态 Z = (X − μ)/σ 服从 N(0, 1)。Φ(z) = P(Z ≤ z) 可查表获得。对于任意正态变量，P(a < X < b) = Φ((b−μ)/σ) − Φ((a−μ)/σ)。分布具有对称性，故 Φ(−z) = 1 − Φ(z)。
Rectangular (uniform) distribution: X ~ R(a, b) over the interval [a, b] has constant density f(x) = 1/(b−a). Mean = (a+b)/2, variance = (b−a)²/12.
矩形（均匀）分布：X ~ R(a, b) 在区间 [a, b] 上具有常密度 f(x) = 1/(b−a)。均值为 (a+b)/2，方差为 (b−a)²/12。
4. Sampling and the Central Limit Theorem | 抽样与中心极限定理
A statistic is a function of sample values X₁, X₂, …, Xₙ drawn from a population. The sample mean X̄ = (1/n) ΣXᵢ is an unbiased estimator of the population mean μ, with E(X̄) = μ and Var(X̄) = σ²/n. The Central Limit Theorem (CLT) states that for a large sample size n (typically n ≥ 30), X̄ is approximately normally distributed with mean μ and variance σ²/n, regardless of the shape of the population distribution. Hence (X̄ − μ)/(σ/√n) ≈ N(0, 1).
统计量是从总体抽取的样本值 X₁, X₂, …, Xₙ 的函数。样本均值 X̄ = (1/n) ΣXᵢ 是总体均值 μ 的无偏估计，满足 E(X̄) = μ，Var(X̄) = σ²/n。中心极限定理指出，当样本量 n 足够大（通常 n ≥ 30）时，无论总体分布形状如何，样本均值 X̄ 近似服从正态分布，均值为 μ，方差为 σ²/n。因此 (X̄ − μ)/(σ/√n) ≈ N(0, 1)。
When the population variance σ² is unknown, we estimate it using the sample variance s² = (1/(n−1)) Σ(Xᵢ − X̄)². The sample mean standardised with s follows a t‑distribution with n−1 degrees of freedom: (X̄ − μ)/(s/√n) ~ tₙ₋₁. This result holds exactly for samples from a normal population and provides the basis for confidence intervals and hypothesis tests when σ is unknown.
当总体方差 σ² 未知时，我们使用样本方差 s² = (1/(n−1)) Σ(Xᵢ − X̄)² 来估计。用 s 标准化的样本均值服从自由度为 n−1 的 t 分布：(X̄ − μ)/(s/√n) ~ tₙ₋₁。该结论对于来自正态总体的样本精确成立，并提供了 σ 未知时构造置信区间和假设检验的基础。
5. Estimation and Confidence Intervals | 估计与置信区间
A point estimate gives a single value for a population parameter. A symmetric 100(1−α)% confidence interval provides an interval estimate, derived from a sampling distribution. For a normal population with known variance, the confidence interval for μ is: X̄ ± z_{\u03b1/₂} × σ/√n, where z_{\u03b1/₂} is the critical value from N(0, 1). When σ is unknown, replace σ with s and use the t‑distribution: X̄ ± tₙ₋₁,ₐ/₂ × s/√n.
点估计给出总体参数的单一数值。一个对称的 100(1−α)% 置信区间提供了区间估计，从抽样分布导出。对于已知方差的正态总体，μ 的置信区间为：X̄ ± z_{\u03b1/₂} × σ/√n，其中 z_{\u03b1/₂} 来自 N(0, 1) 的临界值。当 σ 未知时，用 s 代替 σ 并使用 t 分布：X̄ ± tₙ₋₁,ₐ/₂ × s/√n。
For a proportion p from a binomial sample, the approximate confidence interval is p̂ ± z_{\u03b1/₂} × √(p̂(1−p̂)/n), where p̂ = x/n. This interval is valid under large-sample conditions (np̂ > 5 and n(1−p̂) > 5). For the difference between two independent means with known population variances, use (X̄₁ − X̄₂) ± z × √(σ₁²/n₁ + σ₂²/n₂).
对于来自二项样本的比例 p，近似置信区间为 p̂ ± z_{\u03b1/₂} × √(p̂(1−p̂)/n)，其中 p̂ = x/n。该区间在大样本条件（np̂ > 5 且 n(1−p̂) > 5）下有效。对于两个独立均值的差值且总体方差已知时，使用 (X̄₁ − X̄₂) ± z × √(σ₁²/n₁ + σ₂²/n₂)。
6. Hypothesis Testing Framework | 假设检验框架
A hypothesis test assesses the strength of evidence against a null hypothesis H₀ in favour of an alternative H₁. The test statistic (e.g., Z, t, χ²) measures the discrepancy between the observed data and H₀. The p‑value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming H₀ is true. Reject H₀ if p‑value < significance level α (commonly 0.05).
假设检验评估反对零假设 H₀ 并支持备择假设 H₁ 的证据强度。检验统计量（如 Z、t、χ²）衡量观测数据与 H₀ 之间的偏离程度。p 值是在 H₀ 为真的条件下，获得至少与观测值同样极端的检验统计量的概率。若 p 值 < 显著性水平 α（通常为 0.05），则拒绝 H₀。
One‑sample tests for the mean: when σ is known, use z = (X̄ − μ₀)/(σ/√n); when σ unknown, use t = (X̄ − μ₀)/(s/√n) with n−1 degrees of freedom. For a two‑tailed test, reject H₀ if |z| > critical value or equivalently if p‑value < α. The probability of a Type I error (rejecting H₀ when true) is exactly α; the probability of a Type II error (not rejecting H₀ when false) is denoted β.
单样本均值检验：当 σ 已知时，使用 z = (X̄ − μ₀)/(σ/√n)；当 σ 未知时，使用 t = (X̄ − μ₀)/(s/√n)，自由度为 n−1。对于双尾检验，若 |z| > 临界值，或等价地 p 值 < α，则拒绝 H₀。第一类错误（H₀ 为真时拒绝它）的概率恰好为 α；第二类错误（H₀ 为假时不拒绝它）的概率记为 β。
7. Chi‑squared Tests | 卡方检验
The chi‑squared statistic measures the difference between observed frequencies Oᵢ and expected frequencies Eᵢ: χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ. Two main applications appear in the syllabus: goodness‑of‑fit tests, which assess whether sample data follow a specified distribution; and tests for independence in contingency tables, which examine association between two categorical variables. Degrees of freedom for goodness‑of‑fit are k−1−p, where k is the number of categories and p is the number of estimated parameters.
卡方统计量衡量观测频数 Oᵢ 与期望频数 Eᵢ 之间的差异：χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ。课程中主要有两类应用：拟合优度检验，评估样本数据是否服从某个指定分布；以及列联表中的独立性检验，考察两个分类变量之间的关联性。拟合优度的自由度是 k−1−p，其中 k 为类别数，p 为估计参数个数。
For a ʳ × ᶜ contingency table, the expected frequency for each cell is E = (row total × column total) / grand total. The test statistic χ² = Σ(O−E)²/E has degrees of freedom (r−1)(c−1). The test is valid when all expected frequencies are at least 5. If the calculated χ² exceeds the critical value from χ² tables (or p‑value < α), we reject the hypothesis of independence.
对于 r × c 列联表，每个格子的期望频数为 E = (行总和 × 列总和) / 总合计。检验统计量 χ² = Σ(O−E)²/E 的自由度为 (r−1)(c−1)。当所有期望频数均不小于 5 时该检验有效。若计算所得 χ² 超过卡方表中的临界值（或 p 值 < α），则拒绝独立性假设。
8. Correlation and Linear Regression | 相关与线性回归
Bivariate data (xᵢ, yᵢ) can be summarised using the product moment correlation coefficient. The sample Pearson correlation r is given by r = Sₓᵧ / √(Sₓₓ Sᵧᵧ), where Sₓₓ = Σ(x−x̄)², Sᵧᵧ = Σ(y−ȳ)², Sₓᵧ = Σ(x−x̄)(y−ȳ). The value of r lies between −1 and 1, measuring the strength and direction of linear association. Spearman’s rank correlation coefficient ρₛ provides a non‑parametric alternative based on ranked data.
双变量数据 (xᵢ, yᵢ) 可用积矩相关系数进行概括。样本皮尔逊相关系数 r 由下式给出：r = Sₓᵧ / √(Sₓₓ Sᵧᵧ)，其中 Sₓₓ = Σ(x−x̄)²，Sᵧᵧ = Σ(y−ȳ)²，Sₓᵧ = Σ(x−x̄)(y−ȳ)。r 的值介于 −1 和 1 之间，衡量线性关系的强度和方向。斯皮尔曼等级相关系数 ρₛ 提供了基于排序数据的非参数替代方法。
The least‑squares regression line of y on x is y = a + bx, where b = Sₓᵧ / Sₓₓ and a = ȳ − bx̄. This line minimises the sum of squared vertical residuals Σ(yᵢ − ŷᵢ)². The coefficient of determination r² represents the proportion of the total variation in y that is explained by the linear regression on x. Caution: regression of y on x is not symmetric; a different line arises from regressing x on y.
y 对 x 的最小二乘回归直线为 y = a + bx，其中 b = Sₓᵧ / Sₓₓ，a = ȳ − bx̄。该直线最小化残差平方和 Σ(yᵢ − ŷᵢ)²。判定系数 r² 表示 y 的总变异中被 x 的线性回归所解释的比例。注意：y 对 x 的回归并非对称；x 对 y 的回归会产生不同的直线。
9. Expectation Algebra and Properties of Estimators | 期望代数与估计量性质
For any random variables X and Y, expectation is linear: E(aX + bY) = aE(X) + bE(Y), regardless of dependence. Variance is not linear in general: Var(aX ± bY) = a²Var(X) + b²Var(Y) if X and Y are independent. If they are dependent, Var(aX ± bY) = a²Var(X) + b²Var(Y) ± 2ab Cov(X, Y).
对于任意随机变量 X 和 Y，期望运算是线性的：E(aX + bY) = aE(X) + bE(Y)，与独立性无关。方差一般不是线性的：若 X 和 Y 独立，Var(aX ± bY) = a²Var(X) + b²Var(Y)。若它们相关，则 Var(aX ± bY) = a²Var(X) + b²Var(Y) ± 2ab Cov(X, Y)。
An estimator is unbiased if its expectation equals the parameter it estimates. The sample mean is unbiased for μ; the sample variance s² = Σ(x−x̄)²/(n−1) is unbiased for σ², whereas the divisor n would yield a biased estimator. Efficiency refers to the variance of an estimator; a more efficient estimator has a smaller variance. Consistent estimators converge in probability to the true parameter as sample size increases.
如果估计量的期望等于其所估计的参数，则称其为无偏估计量。样本均值是 μ 的无偏估计；样本方差 s² = Σ(x−x̄)²/(n−1) 是 σ² 的无偏估计，而除数为 n 则会产生有偏估计量。效率指估计量的方差；效率更高的估计量具有更小的方差。相合估计量随着样本量增大以概率收敛于真实参数。
10. Common Exam Pitfalls and Reminders | 常见考试陷阱与提醒
When calculating probabilities, always check whether events are mutually exclusive or independent—misapplying the multiplication or addition rules is a frequent error. With normal approximations to binomial or Poisson, include a continuity correction: for an integer value k, adjust the interval to k±0.5 before standardising. In hypothesis testing, always state the conclusion in context, linking rejection or non‑rejection of H₀ to the original problem.
计算概率时，务必检查事件是否互斥或独立——错误使用乘法或加法法则是常见错误。在用正态分布近似二项或泊松分布时，需要进行连续性修正：对于整数值 k，先将区间调整为 k±0.5 再进行标准化。在假设检验中，一定要结合情境给出结论，将拒绝或不拒绝 H₀ 与原问题相关联。
Ensure the scale of measurement is clear: correlation coefficient r has no units and is unaffected by linear coding of x or y. The regression coefficient b, however, changes with scaling. Always report degrees of freedom for t‑ and χ²‑ tests, and confirm that conditions are satisfied (random sampling, normality where required, expected frequencies > 5). Finally, remember that a small p‑value indicates evidence against H₀; a large p‑value indicates insufficient evidence to reject H₀, not proof that H₀ is true.
确保你清楚变量的测量尺度：相关系数 r 没有单位且不受 x 或 y 的线性编码影响。但回归系数 b 会随尺度改变。进行 t 检验和卡方检验时，务必报告自由度，并确认条件是否满足（随机抽样、必要时满足正态性、期望频数 > 5）。最后，记住小的 p 值表示反对 H₀ 的证据；大的 p 值则表示没有充分证据拒绝 H₀，而不代表 H₀ 是正确的。
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