📚 CIE A-Level Statistics: Formula & Theorem Quick Reference | CIE A-Level 统计学：公式定理速查手册
Welcome to your all-in-one revision companion for CIE A-Level Statistics. This guide brings together the essential formulas, definitions, and theorems you need to master topics in Papers 5 and 6. Each section pairs a concise English statement with its Chinese equivalent, helping you internalise concepts from multiple angles. Whether you are tackling measures of central tendency, probability distributions, or hypothesis tests, you will find logically grouped content, worked examples, and handy tables to speed up your revision.
欢迎来到 CIE A-Level 统计学一站式复习手册。本文汇集了你在 Paper 5 和 Paper 6 中必须掌握的核心公式、定义与定理。每一部分都采用简明的英文表述与对应的中文解读，帮助你从不同角度理解概念。无论你在复习集中量数、概率分布还是假设检验，这里都有按逻辑分组的内容、示例和速查表，让你的复习更高效。
1. Measures of Central Tendency and Spread | 集中趋势与离散程度的度量
For ungrouped data, the mean is x̄ = (Σxᵢ)/n. For grouped data, use midpoints m, giving x̄ = (Σfᵢmᵢ)/Σfᵢ.
对于未分组数据，均值 x̄ = (Σxᵢ)/n。对于分组数据，使用组中值 m，公式为 x̄ = (Σfᵢmᵢ)/Σfᵢ。
The median is the middle value. For continuous grouped data, median = L + ((n/2 – F)/f)·w, where L is the lower boundary, F is the cumulative frequency before the median class, f is the class frequency, and w is the class width.
中位数是居中的数值。对于连续分组数据，中位数 = L + ((n/2 – F)/f)·w，其中 L 是下界，F 是中位数组之前的累积频数，f 是该组频数，w 是组距。
Variance σ² = (Σ(xᵢ – x̄)²)/n for a population, or s² = (Σ(xᵢ – x̄)²)/(n – 1) for a sample. The standard deviation is the square root of the variance.
总体方差 σ² = (Σ(xᵢ – x̄)²)/n，样本方差 s² = (Σ(xᵢ – x̄)²)/(n – 1)。标准差是方差的平方根。
2. Basic Probability Rules | 基本概率法则
For any event A, 0 ≤ P(A) ≤ 1. The complement rule is P(A′) = 1 – P(A).
对于任意事件 A，有 0 ≤ P(A) ≤ 1。互补事件的概率 P(A′) = 1 – P(A)。
The addition rule for mutually exclusive events: P(A ∪ B) = P(A) + P(B). If not mutually exclusive, P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
互斥事件的加法法则：P(A ∪ B) = P(A) + P(B)。若事件不互斥，则 P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。
Multiplication rule: P(A ∩ B) = P(A)·P(B|A). For independent events, P(A ∩ B) = P(A)·P(B).
乘法法则：P(A ∩ B) = P(A)·P(B|A)。独立事件满足 P(A ∩ B) = P(A)·P(B)。
3. Permutations and Combinations | 排列与组合
Factorial n! = n·(n–1)·(n–2)·…·1. Permutations: order matters, ⁿPᵣ = n!/(n–r)! .
阶乘 n! = n·(n–1)·(n–2)·…·1。排列（顺序重要）公式：ⁿPᵣ = n!/(n–r)!。
Combinations: order does not matter, ⁿCᵣ = n!/(r!(n–r)!). Also written as (n choose r).
组合（顺序不重要）公式：ⁿCᵣ = n!/(r!(n–r)!)，也记作 C(n,r)。
Distinguish between selections and arrangements. For example, the number of ways to arrange 5 people in 3 seats is ⁵P₃ = 60, while choosing 3 people from 5 is ⁵C₃ = 10.
区分选择与排列。例如，5 个人坐 3 个座位有 ⁵P₃ = 60 种排列方式；从 5 人中选 3 人则有 ⁵C₃ = 10 种组合。
4. Discrete Random Variables | 离散随机变量
A discrete random variable X has a probability mass function P(X = x) where ΣP(X = x) = 1. The expected value E(X) = Σx·P(X = x).
离散随机变量 X 的概率质量函数满足 ΣP(X = x) = 1。期望值 E(X) = Σx·P(X = x)。
The variance Var(X) = E[(X – μ)²] = Σ(x – μ)²·P(X = x), or equivalently Var(X) = E(X²) – [E(X)]².
方差 Var(X) = E[(X – μ)²] = Σ(x – μ)²·P(X = x)，等价于 Var(X) = E(X²) – [E(X)]²。
For linear transformations, E(aX + b) = aE(X) + b, and Var(aX + b) = a²Var(X).
线性变换中，E(aX + b) = aE(X) + b，Var(aX + b) = a²Var(X)。
Use a probability distribution table to organise values, probabilities, and calculations of E(X) and Var(X).
使用概率分布表整理取值、概率，并计算 E(X) 和 Var(X)。
5. Binomial Distribution | 二项分布
If X ~ B(n, p), the probability of exactly x successes is P(X = x) = ⁿCₓ·pˣ·(1 – p)ⁿ⁻ˣ.
若 X ~ B(n, p)，恰好 x 次成功的概率为 P(X = x) = ⁿCₓ·pˣ·(1 – p)ⁿ⁻ˣ。
Mean: μ = np. Variance: σ² = np(1 – p). The distribution is symmetric when p = 0.5, skewed otherwise.
均值 μ = np，方差 σ² = np(1 – p)。当 p = 0.5 时分布对称，否则偏斜。
Cumulative probabilities P(X ≤ x) are often found using tables or the calculator binomialcdf function.
累积概率 P(X ≤ x) 常使用统计表或计算器的 binomialcdf 函数获取。
6. Geometric Distribution | 几何分布
Let X ~ Geo(p) count the number of trials up to and including the first success. Then P(X = x) = (1 – p)ˣ⁻¹·p, for x = 1, 2, 3, …
设 X ~ Geo(p) 表示直到首次成功为止的试验次数。则 P(X = x) = (1 – p)ˣ⁻¹·p，x = 1, 2, 3, …
Mean: E(X) = 1/p. Variance: Var(X) = (1 – p)/p². The geometric distribution has the memoryless property: P(X > s + t | X > s) = P(X > t).
均值 E(X) = 1/p，方差 Var(X) = (1 – p)/p²。几何分布具有无记忆性：P(X > s + t | X > s) = P(X > t)。
For P(X > k), use (1 – p)ᵏ, which is the probability that the first k trials are all failures.
计算 P(X > k) 用 (1 – p)ᵏ，即前 k 次全部失败的概率。
7. Normal Distribution | 正态分布
If X ~ N(μ, σ²), the probability density function is symmetric about μ. The total area under the curve is 1.
若 X ~ N(μ, σ²)，概率密度函数关于 μ 对称，曲线下总面积为 1。
Transform to the standard normal Z ~ N(0, 1) using Z = (X – μ)/σ. Then use standard normal tables to find Φ(z) = P(Z ≤ z).
通过 Z = (X – μ)/σ 转换为标准正态分布 Z ~ N(0, 1)，再查标准正态表求 Φ(z) = P(Z ≤ z)。
Key probabilities: P(μ – σ < X < μ + σ) ≈ 0.6827, P(μ – 2σ < X < μ + 2σ) ≈ 0.9545, P(μ – 3σ < X < μ + 3σ) ≈ 0.9973.
常用概率：P(μ – σ < X < μ + σ) ≈ 0.6827，P(μ – 2σ < X < μ + 2σ) ≈ 0.9545，P(μ – 3σ < X < μ + 3σ) ≈ 0.9973。
Inverse normal: given probability p, find z such that Φ(z) = p, then X = μ + zσ.
逆正态：给定概率 p，求 z 使得 Φ(z) = p，那么 X = μ + zσ。
8. Sampling and the Central Limit Theorem | 抽样与中心极限定理
The sample mean X̄ is an unbiased estimator of μ, with Var(X̄) = σ²/n for random samples of size n.
样本均值 X̄ 是 μ 的无偏估计量，对于大小为 n 的随机样本，Var(X̄) = σ²/n。
Central Limit Theorem: If X₁, X₂, …, Xₙ are independent and identically distributed with mean μ and variance σ², then the distribution of X̄ approximates N(μ, σ²/n) as n → ∞, regardless of the population shape.
中心极限定理：若 X₁, X₂, …, Xₙ 独立同分布，均值为 μ，方差为 σ²，则 n → ∞ 时 X̄ 的分布近似于 N(μ, σ²/n)，无论总体分布形状如何。
This justifies using normal-based confidence intervals and hypothesis tests when n is large (usually n ≥ 30).
这为大样本（通常 n ≥ 30）下基于正态的置信区间和假设检验提供了依据。
9. Confidence Intervals | 置信区间
For a population mean with known variance: x̄ ± z·(σ/√n). Common z-values: 90%: 1.645, 95%: 1.960, 99%: 2.576.
已知方差的总体均值置信区间：x̄ ± z·(σ/√n)。常用 z 值：90% 对应 1.645，95% 对应 1.960，99% 对应 2.576。
For unknown variance, use sample standard deviation s and the t-distribution: x̄ ± t·(s/√n) with n – 1 degrees of freedom.
方差未知时，使用样本标准差 s 和 t 分布：x̄ ± t·(s/√n)，自由度为 n – 1。
The confidence interval width decreases as sample size increases. Interpretation: if we repeatedly sample, the true mean lies within the interval in the stated percentage of samples.
置信区间宽度随样本量增大而减小。解释：若重复抽样，在理想比例下真值会落在此区间内。
10. Hypothesis Testing | 假设检验
Set null hypothesis H₀: μ = μ₀, against alternative H₁: μ < μ₀, μ > μ₀, or μ ≠ μ₀. Choose significance level α (commonly 5%).
设定原假设 H₀: μ = μ₀，备择假设 H₁: μ < μ₀、μ > μ₀ 或 μ ≠ μ₀。选择显著性水平 α（通常 5%）。
Test statistic: Z = (x̄ – μ₀)/(σ/√n) for known σ, or t = (x̄ – μ₀)/(s/√n) for unknown σ. Compare with critical value or use p-value.
检验统计量：已知 σ 用 Z = (x̄ – μ₀)/(σ/√n)，未知 σ 用 t = (x̄ – μ₀)/(s/√n)。与临界值比较或使用 p 值。
Decision rule: reject H₀ if the test statistic falls in the rejection region (e.g., |Z| > 1.96 for two-tailed α = 0.05).
决策规则：若检验统计量落入拒绝域（如双尾 α = 0.05 时 |Z| > 1.96），则拒绝 H₀。
The p-value is the probability of observing a test statistic at least as extreme as the one obtained, assuming H₀ is true. Reject H₀ if p < α.
p 值是在 H₀ 成立时，观察到比当前更极端检验统计量的概率。若 p < α，拒绝 H₀。
Type I error: rejecting H₀ when true. Type II error: failing to reject H₀ when false. The power of a test is 1 – P(Type II error).
第一类错误：H₀ 真时拒绝它。第二类错误：H₀ 假时未拒绝它。检验的功效 = 1 – P(第二类错误)。
11. Correlation and Regression (Continuum) | 相关与回归（进阶）
Pearson’s product-moment correlation coefficient r measures linear strength: r = Sxy /√(Sxx·Syy), where Sxx = Σ(x – x̄)², Syy = Σ(y – ȳ)², Sxy = Σ(x – x̄)(y – ȳ).
皮尔逊积矩相关系数 r 度量线性强度：r = Sxy /√(Sxx·Syy)，其中 Sxx = Σ(x – x̄)²，Syy = Σ(y – ȳ)²，Sxy = Σ(x – x̄)(y – ȳ)。
Least squares regression line: y = a + bx, where b = Sxy/Sxx and a = ȳ – bx̄.
最小二乘回归直线：y = a + bx，其中 b = Sxy/Sxx，a = ȳ – bx̄。
Interpretation: b is the change in y per unit increase in x. Extrapolation beyond the data range is unreliable.
解释：b 表示 x 每增加一个单位时 y 的变化量。超出数据范围的外推不可靠。
12. Key Quick-Reference Table | 核心速查表
| Concept / 概念 | Formula / 公式 |
|---|---|
| Mean (ungrouped) / 均值（未分组） | x̄ = Σx/n |
| Sample variance / 样本方差 | s² = Σ(x – x̄)²/(n – 1) |
| P(A|B) / 条件概率 | P(A|B) = P(A ∩ B)/P(B) |
| Binomial probability / 二项概率 | P(X = x) = ⁿCₓ pˣ (1 – p)ⁿ⁻ˣ |
| Geometric mean / 几何分布均值 | E(X) = 1/p |
| Standard normal Z / 标准正态 Z | Z = (X – μ)/σ |
| CLT for sample mean / 样本均值 CLT | X̄ ~ N(μ, σ²/n) approx. |
| 95% CI for μ / 95% 置信区间 | x̄ ± 1.96σ/√n |
| Correlation r / 相关系数 r | r = Sxy/√(SxxSyy) |
| Regression slope / 回归斜率 | b = Sxy/Sxx |
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