📚 Year 13 CIE Statistics: Key Terms Memorisation Guide | Year 13 CIE 统计：词汇术语速记指南
Mastering the precise terminology of CIE A-Level Statistics is half the battle won. This guide distils the most important terms from probability, distributions, inference, and regression into bite-sized English–Chinese pairings, helping you recall definitions quickly and apply them accurately in exam contexts.
掌握 CIE A-Level 统计学的精确术语是成功的一半。本指南将概率、分布、推断和回归中最关键的概念提炼为英文–中文配对速记条目，帮助你在考试情境中快速回忆定义并准确应用。
1. Probability Basics | 概率基础
Sample space (S): The set of all possible outcomes of an experiment.
样本空间 (S)：一项试验所有可能结果的集合。
Event: A subset of the sample space; a collection of one or more outcomes.
事件：样本空间的一个子集；由一个或多个结果组成的集合。
Probability of an event A: Number of outcomes in A / Total number of outcomes in S, when outcomes are equally likely.
事件 A 的概率：当所有结果等可能时，P(A) = A 中结果个数 / S 中结果总数。
Complement of A (A' or Aᶜ): The event that A does not occur; P(A') = 1 − P(A).
A 的补集 (A' 或 Aᶜ)：A 不发生的事件；P(A') = 1 − P(A)。
Mutually exclusive events: Two events that cannot occur simultaneously; P(A ∩ B) = 0.
互斥事件：两个不可能同时发生的事件；P(A ∩ B) = 0。
Independent events: Occurrence of one does not affect the probability of the other; P(A ∩ B) = P(A) × P(B).
独立事件：一个事件的发生不影响另一个事件的概率；P(A ∩ B) = P(A) × P(B)。
Conditional probability: Probability of A given B: P(A|B) = P(A ∩ B) / P(B).
条件概率：在 B 发生的条件下 A 的概率：P(A|B) = P(A ∩ B) / P(B)。
2. Random Variables & Summary Measures | 随机变量与度量指标
Random variable (X): A numerical outcome of a random phenomenon, taking different values with certain probabilities.
随机变量 (X)：随机现象的数值结果，以特定概率取不同值。
Discrete random variable: Takes a countable number of distinct values (e.g., number of heads). Described by a probability mass function (PMF).
离散随机变量：取可数个不同值（如正面次数）。由概率质量函数 (PMF) 描述。
Continuous random variable: Takes any value in an interval; described by a probability density function (PDF); probabilities are areas under the curve.
连续随机变量：在区间内取任意值；由概率密度函数 (PDF) 描述；概率为曲线下方面积。
Expected value E(X): The long-run average of a random variable; for discrete, E(X) = ∑ xᵢ p(xᵢ).
期望值 E(X)：随机变量的长期平均值；离散情形 E(X) = ∑ xᵢ p(xᵢ)。
Variance Var(X): A measure of spread: Var(X) = E[(X − μ)²] = E(X²) − [E(X)]².
方差 Var(X)：波动性的度量：Var(X) = E[(X − μ)²] = E(X²) − [E(X)]²。
Standard deviation: σ = √Var(X), in the same units as X.
标准差：σ = √Var(X)，与 X 同单位。
Linear transformation: E(aX + b) = aE(X) + b; Var(aX + b) = a²Var(X).
线性变换：E(aX + b) = aE(X) + b；Var(aX + b) = a²Var(X)。
3. Binomial Distribution | 二项分布
Conditions: Fixed number of trials n, each trial independent, two outcomes (success/failure), constant probability of success p.
适用条件：固定试验次数 n，每次试验独立，两种结果（成功/失败），成功概率 p 恒定。
Notation: X ~ B(n, p).
记号：X ~ B(n, p)。
Probability mass function: P(X = x) = ⁿCₓ pˣ (1 − p)ⁿ⁻ˣ, for x = 0, 1, ..., n.
概率质量函数：P(X = x) = ⁿCₓ pˣ (1 − p)ⁿ⁻ˣ，x = 0, 1, ..., n。
Mean and variance: E(X) = np, Var(X) = np(1 − p).
均值与方差：E(X) = np，Var(X) = np(1 − p)。
Use of tables: Cumulative binomial probabilities P(X ≤ x) are given in statistical tables.
查表：累积二项概率 P(X ≤ x) 可从统计表中查得。
4. Poisson Distribution | 泊松分布
Conditions: Events occur independently in a fixed interval (time, area), at a constant average rate λ; no upper limit on occurrences.
适用条件：事件独立地发生在固定区间内（时间、面积），平均发生率 λ 恒定；发生次数无上限。
Notation: X ~ Po(λ).
记号：X ~ Po(λ)。
Probability mass function: P(X = x) = e-λ λˣ / x! , for x = 0, 1, 2, ...
P(X = x) = e⁻λ λˣ / x!
概率质量函数：P(X = x) = e-λ λˣ / x! ，x = 0, 1, 2, ...
Mean and variance: E(X) = λ, Var(X) = λ.
均值与方差：E(X) = λ，Var(X) = λ。
Poisson approximation to binomial: When n is large and p is small (n > 50, np < 5), B(n, p) can be approximated by Po(np).
二项分布的泊松近似：当 n 大、p 小（n > 50，np < 5）时，B(n, p) 可用 Po(np) 近似。
5. Normal Distribution | 正态分布
Continuous symmetric distribution: Bell-shaped, defined by mean μ and variance σ²; notation X ~ N(μ, σ²).
连续对称分布：钟形，由均值 μ 和方差 σ² 定义；记作 X ~ N(μ, σ²)。
Standard normal: Z ~ N(0, 1) where Z = (X − μ)/σ.
标准正态：Z ~ N(0, 1)，Z = (X − μ)/σ。
Probability as area: P(X < a) = Φ((a − μ)/σ) using standard normal tables.
概率即面积：P(X < a) = Φ((a − μ)/σ)，用标准正态表查得。
Symmetry property: Φ(−z) = 1 − Φ(z).
对称性：Φ(−z) = 1 − Φ(z)。
Normal approximation to binomial: If np > 5 and n(1−p) > 5, X ~ B(n, p) ≈ N(np, np(1−p)) with continuity correction.
二项分布的正态近似：若 np > 5 且 n(1−p) > 5，X ~ B(n, p) ≈ N(np, np(1−p)) 并需连续性校正。
6. Sampling & Central Limit Theorem | 抽样与中心极限定理
Population vs sample: The population is the entire set of items; a sample is a subset used to infer about the population.
总体与样本：总体是所有个体的集合；样本是从中抽取的子集，用于推断总体。
Statistic: A quantity calculated from a sample, e.g., sample mean x̄.
统计量：由样本计算出的量，如样本均值 x̄。
Sampling distribution: The probability distribution of a statistic.
抽样分布：统计量的概率分布。
Central Limit Theorem (CLT): For large n (usually n ≥ 30), the sample mean X̄ is approximately N(μ, σ²/n), regardless of the population distribution.
中心极限定理 (CLT)：当 n 较大（通常 n ≥ 30）时，无论总体分布如何，样本均值 X̄ 近似服从 N(μ, σ²/n)。
Unbiased estimator: An estimator with expected value equal to the true parameter; e.g., x̄ is unbiased for μ.
无偏估计量：期望值等于真实参数的估计量；如 x̄ 是 μ 的无偏估计。
7. Confidence Intervals | 置信区间
Confidence interval (CI): An interval estimate for a population parameter with a stated confidence level (e.g., 95%).
置信区间 (CI)：对总体参数的区间估计，带有指定的置信水平（如 95%）。
Confidence level: The probability that the interval contains the true parameter in repeated sampling.
置信水平：在重复抽样中，区间包含真实参数的概率。
CI for mean with known σ: x̄ ± z × (σ/√n), where z is the critical value from N(0,1).
已知 σ 时均值的 CI：x̄ ± z × (σ/√n)，其中 z 为来自 N(0,1) 的临界值。
CI for mean with unknown σ: x̄ ± t × (s/√n) using Student's t-distribution with n−1 degrees of freedom.
未知 σ 时均值的 CI：x̄ ± t × (s/√n)，使用自由度为 n−1 的 t 分布。
Margin of error: The half-width of the CI, e.g., z × σ/√n.
误差界限：置信区间的一半宽度，如 z × σ/√n。
8. Hypothesis Testing | 假设检验
Null hypothesis (H₀): A statement of no effect or no difference, assumed true unless evidence suggests otherwise.
原假设 (H₀)：声称无效应或无差异的陈述，若无充分证据则假定为真。
Alternative hypothesis (H₁): The statement we suspect might be true instead of H₀; can be one-tailed or two-tailed.
备择假设 (H₁)：我们怀疑可能为真的陈述；可以是单侧或双侧的。
Test statistic: A value calculated from sample data to decide whether to reject H₀.
检验统计量：由样本数据计算得出的值，用于决定是否拒绝 H₀。
Critical region (rejection region): The set of test statistic values that lead to rejection of H₀.
临界域（拒绝域）：导致拒绝 H₀ 的检验统计量的取值集合。
Significance level α: The probability of rejecting H₀ when it is actually true (Type I error).
显著性水平 α：当 H₀ 为真时拒绝 H₀ 的概率（第 I 类错误）。
p-value: The probability, under H₀, of obtaining a test statistic at least as extreme as the observed value.
p 值：在 H₀ 成立的条件下，得到至少与实际观测值一样极端的检验统计量的概率。
Type II error (β): Failing to reject H₀ when H₁ is true. Power = 1 − β.
第 II 类错误 (β)：当 H₁ 为真时未能拒绝 H₀。检验功效 = 1 − β。
9. Correlation & Regression | 相关与回归
Product Moment Correlation Coefficient (r): res the strength and direction of a linear relationship between two variables; −1 ≤ r ≤ 1.
皮尔逊积矩相关系数 (r)：度量两个变量间线性关系的强度和方向；−1 ≤ r ≤ 1。
Spearman's rank correlation (rₛ): A non-parametric measure based on ranks, used when data is not linear or contains outliers.
斯皮尔曼秩相关系数 (rₛ)：基于秩次的非参数度量，用于非线性或有异常值的数据。
Regression line of y on x: y = a + bx, where b = Sₓᵧ / Sₓₓ, a = ȳ − b x̄.
y 对 x 的回归线：y = a + bx，其中 b = Sₓᵧ / Sₓₓ，a = ȳ − b x̄。
Coefficient of determination (r² or
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