📚 Year 13 CCEA Statistics: Core Knowledge Review | Year 13 CCEA 统计：核心知识点梳理
This article summarises the core topics covered in Year 13 of the CCEA GCE Statistics specification. These topics extend AS-level foundations and introduce advanced techniques in statistical inference, probability modelling, and data analysis. A firm grasp of each area is essential for success in A2 examinations and for building the analytical skills required in higher education.
本文梳理 CCEA GCE 统计课程 Year 13 阶段的核心知识点。这些内容在 AS 基础上进一步深入，涵盖统计推断、概率建模和数据分析的高级方法。牢固掌握各模块是 A2 考试成功的关键，也为高等教育所需的分析能力打下坚实基础。
1. Probability and Conditional Probability | 概率与条件概率
Probability measures the likelihood of events on a scale from 0 to 1. The addition rule for mutually exclusive events is P(A ∪ B) = P(A) + P(B); for non‑mutually exclusive events, P(A ∪ B) = P(A) + P(B) – P(A ∩ B). Conditional probability captures how the probability of A changes when we know B has occurred.
概率衡量事件发生的可能性，值域为 0 到 1。互斥事件的加法规则为 P(A ∪ B) = P(A) + P(B)；对于非互斥事件，P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。条件概率用于刻画已知 B 发生时 A 的概率如何变化。
P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0
条件概率公式：P(A|B) = P(A ∩ B) / P(B), 其中 P(B) > 0
Bayes’ theorem reverses the conditioning: P(A|B) = [P(B|A) × P(A)] / P(B). It is useful for updating probabilities in light of new evidence. Two events are independent if P(A ∩ B) = P(A) × P(B) or equivalently P(A|B) = P(A).
贝叶斯定理可以逆转条件：P(A|B) = [P(B|A) × P(A)] / P(B)，常用于根据新证据更新概率。若 P(A ∩ B) = P(A) × P(B) 或等价地 P(A|B) = P(A)，则两事件独立。
2. Discrete Random Variables & Expectation | 离散随机变量与期望
A discrete random variable X takes a countable set of values, each with a probability mass function (PMF) P(X = x). The cumulative distribution function (CDF) is F(x) = P(X ≤ x). The expected value E(X) = Σ x · P(X = x) gives the long‑run average, while variance Var(X) = E[(X – μ)²] = E(X²) – [E(X)]² measures spread.
离散随机变量 X 取可数个值，每个值对应概率质量函数 P(X = x)。累积分布函数为 F(x) = P(X ≤ x)。期望 E(X) = Σ x · P(X = x) 表示长期平均值，方差 Var(X) = E[(X – μ)²] = E(X²) – [E(X)]² 度量离散程度。
- Linearity of expectation: E(aX + b) = aE(X) + b, and for independent X and Y, E(XY) = E(X)E(Y) but Var(X + Y) = Var(X) + Var(Y).
- 期望的线性性质：E(aX + b) = aE(X) + b；若 X 与 Y 独立，则 E(XY) = E(X)E(Y) 且 Var(X + Y) = Var(X) + Var(Y)。
3. Binomial and Poisson Distributions | 二项分布与泊松分布
The binomial distribution, X ~ B(n, p), models the number of successes in n independent trials each with success probability p. Conditions: fixed n, independent trials, constant p, two outcomes. Its mean is np, variance np(1 – p). The Poisson distribution, X ~ Po(λ), models rare events occurring randomly in a fixed interval; λ is both the mean and the variance.
二项分布 X ~ B(n, p) 描述 n 次独立试验中成功的次数，每次成功概率为 p。条件：固定 n、独立试验、恒定 p、两种结果。均值 np，方差 np(1 – p)。泊松分布 X ~ Po(λ) 描述固定区间内随机发生的稀有事件，λ 既是均值也是方差。
When n is large and p is small, the Poisson distribution can approximate the binomial with λ = np. Typically the approximation is valid when n ≥ 50 and np < 5.
当 n 较大、p 较小时，可用 λ = np 的泊松分布近似二项分布。一般 n ≥ 50 且 np < 5 时近似效果较好。
4. Continuous Random Variables & Probability Density Functions | 连续随机变量与概率密度函数
A continuous random variable has an uncountable range. Probability is given by the area under the probability density function (PDF) f(x). The total area must be 1: ∫ f(x) dx = 1 over the domain. The cumulative distribution function is F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt. The median m satisfies F(m) = 0.5.
连续随机变量的取值范围不可数，概率由概率密度函数 f(x) 下的面积给出。全域积分 ∫ f(x) dx = 1。累积分布函数 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。中位数 m 满足 F(m) = 0.5。
The expected value E(X) = ∫ x f(x) dx, and Var(X) = ∫ (x – μ)² f(x) dx = ∫ x² f(x) dx – μ². Key continuous distributions include the uniform and exponential distributions, which are used to model constant likelihood or waiting times.
期望 E(X) = ∫ x f(x) dx，方差 Var(X) = ∫ (x – μ)² f(x) dx = ∫ x² f(x) dx – μ²。重要的连续分布包括均匀分布和指数分布，分别用于等可能模型和等待时间模型。
5. Normal Distribution & Normal Approximations | 正态分布与正态近似
The normal distribution X ~ N(μ, σ²) is symmetric and bell‑shaped. Probabilities are found by standardising: Z = (X – μ)/σ ~ N(0, 1). Use statistical tables or technology to obtain Φ(z) = P(Z ≤ z).
正态分布 X ~ N(μ, σ²) 呈对称钟形。通过标准化 Z = (X – μ)/σ ~ N(0, 1) 求概率，借助统计表或技术工具查询 Φ(z) = P(Z ≤ z)。
Normal approximations are used for the binomial (when np ≥ 5 and nq ≥ 5) and for the Poisson (when λ > 15). A continuity correction of ±0.5 is applied when a discrete distribution is approximated by the continuous normal.
正态近似用于二项分布（条件 np ≥ 5 且 nq ≥ 5）和泊松分布（条件 λ > 15）。由于用连续分布近似离散分布，需进行 ±0.5 的连续性校正。
Binomial approx: X ~ N(np, np(1–p)), with correction
二项近似：X ~ N(np, np(1–p))，并做连续性校正
6. Sampling Distributions & Central Limit Theorem | 抽样分布与中心极限定理
The sample mean X̄ is a random variable. If the population is N(μ, σ²), then X̄ ~ N(μ, σ²/n). The Central Limit Theorem (CLT) states that for a large sample (n ≥ 30), X̄ is approximately normal regardless of the population shape. For a proportion p̂ = X/n, its sampling distribution is approximately N(p, p(1–p)/n) provided np ≥ 10 and n(1–p) ≥ 10.
样本均值 X̄ 是一个随机变量。若总体服从 N(μ, σ²)，则 X̄ ~ N(μ, σ²/n)。中心极限定理指出，当样本量足够大 (n ≥ 30)，无论总体分布如何，X̄ 均近似服从正态分布。对于样本比例 p̂ = X/n，其抽样分布近似为 N(p, p(1–p)/n)，条件为 np ≥ 10 且 n(1–p) ≥ 10。
7. Estimation: Point Estimates & Confidence Intervals | 估计：点估计与置信区间
A point estimate gives a single plausible value for a population parameter, while a confidence interval provides a range of plausible values with a specified confidence level (1 – α). For μ with known σ: X̄ ± zₐ/₂ × σ/√n. When σ is unknown, use the t‑distribution with n – 1 degrees of freedom.
点估计给出总体参数的单一合理值，置信区间则以指定置信水平 (1 – α) 提供一个合理范围。总体方差已知时，μ 的置信区间为 X̄ ± zₐ/₂ × σ/√n；方差未知时，使用自由度为 n – 1 的 t 分布构造区间。
CI for μ (σ unknown): X̄ ± t* × s/√n
μ 的置信区间（σ 未知）：X̄ ± t* × s/√n
For a proportion p, the confidence interval is p̂ ± z* × √[p̂(1 – p̂)/n]. Sample size determination can be performed by rearranging the margin of error formula.
总体比例 p 的置信区间为 p̂ ± z* × √[p̂(1 – p̂)/n]。如需确定样本量，可通过调整误差边际公式进行计算。
8. Hypothesis Testing: z-tests and t-tests | 假设检验：z 检验与 t 检验
A hypothesis test evaluates claims about a population parameter. Set up the null hypothesis H₀ and the alternative H₁. The significance level α is the probability of a Type I error (rejecting H₀ when true). The p‑value is the probability of observing a test statistic at least as extreme as the one obtained, assuming H₀ is true. If p‑value < α, reject H₀.
假设检验用于评估关于总体参数的声明。设立原假设 H₀ 和备择假设 H₁。显著性水平 α 为发生第一类错误的概率（当 H₀ 为真时拒绝 H₀）。p 值是在 H₀ 成立的条件下，观察到至少与当前一样极端的检验统计量的概率。若 p 值 < α，则拒绝 H₀。
One‑sample z‑test for μ (σ known), one‑sample t‑test (σ unknown), two‑sample t‑tests (independent or paired) and a z‑test for a proportion are common. Check assumptions: normality, independence, and homogeneity of variance for two‑sample tests.
常见检验包括：单样本 z 检验（σ 已知）、单样本 t 检验（σ 未知）、两样本 t 检验（独立或配对）以及比例 z 检验。需检查假设条件：正态性、独立性，两样本检验还需方差齐性。
The power of a test is 1 – β, where β is the probability of a Type II error (failing to reject a false H₀).
检验功效为 1 – β，其中 β 是第二类错误的概率（H₀ 为假但未拒绝）。
9. Chi-Squared Tests | 卡方检验
The chi‑squared distribution is used for goodness‑of‑fit tests and tests of independence. For a goodness‑of‑fit test, compare observed frequencies Oᵢ with expected frequencies Eᵢ under a specified model: X² = Σ (Oᵢ – Eᵢ)² / Eᵢ. Degrees of freedom = number of categories – 1 – number of estimated parameters. The null hypothesis is that the data follow the specified distribution.
卡方分布用于拟合优度检验和独立性检验。拟合优度检验比较观测频数 Oᵢ 与期望频数 Eᵢ，统计量 X² = Σ (Oᵢ – Eᵢ)² / Eᵢ。自由度 = 类别数 – 1 – 估计的参数量。原假设为数据服从指定分布。
For a test of independence in a two‑way contingency table, Eᵢⱼ = (row total × column total) / grand total. Degrees of freedom = (r – 1)(c – 1). Yates’ correction for continuity may be applied to a 2×2 table when expected frequencies are small.
对于独立性检验（两向列联表），期望频数 Eᵢⱼ = (行合计 × 列合计) / 总计。自由度 = (行数 – 1)(列数 – 1)。当 2×2 表中期望频数较小时，有时会使用 Yates 连续性校正。
10. Correlation and Linear Regression | 相关与线性回归
The product‑moment correlation coefficient r measures the strength and direction of a linear relationship between two variables. Its formula involves standardised sums of products: r = Sₓᵧ / √(Sₓₓ Sᵧᵧ). The null hypothesis ρ = 0 can be tested using a t‑test. The coefficient of determination r² gives the proportion of variation in y explained by x.
积矩相关系数 r 衡量两变量间线性关系的强度与方向，其计算公式涉及标准化乘积之和：r = Sₓᵧ / √(Sₓₓ Sᵧᵧ)。对 ρ = 0 的原假设可用 t 检验。决定系数 r² 表示 y 的变异中被 x 解释的比例。
The least‑squares regression line is y = a + bx, where b = Sₓᵧ / Sₓₓ and a = ȳ – bx̄. Residual analysis helps verify linearity, constant variance, and normality. Confidence intervals for the slope and prediction intervals for a new observation are typical inferential tasks.
最小二乘回归线为 y = a + bx，其中 b = Sₓᵧ / Sₓₓ，a = ȳ – bx̄。残差分析用于检验线性、方差齐性和正态性假设。对斜率的置信区间和新观测的预测区间是常见的推断任务。
11. Probability Generating Functions | 概率生成函数
For a non‑negative integer discrete random variable X, the probability generating function (PGF) is G(t) = E(tˣ) = Σ tˣ P(X = x). The PGF uniquely identifies the distribution, and its derivatives at t = 1 recover moments: G'(1) = E(X), G”(1) = E[X(X – 1)].
对于非负整数值离散随机变量 X，概率生成函数定义为 G(t) = E(tˣ) = Σ tˣ P(X = x)。PGF 唯一确定分布，其在 t = 1 处的导数可提取矩：G'(1) = E(X), G”(1) = E[X(X – 1)]。
The variance can be obtained via Var(X) = G”(1) + G'(1) – [G'(1)]². If X and Y are independent, the PGF of their sum is the product of their PGFs. Recognising standard PGFs, such as for the binomial ( (q + pt)ⁿ ) or Poisson ( e^{λ(t–1)} ), enables quick identification of distributions of sums.
方差可通过 Var(X) = G”(1) + G'(1) – [G'(1)]² 得到。若 X 与 Y 独立，则二者之和的 PGF 为各自 PGF 的乘积。熟记标准分布的 PGF，例如二项分布的 (q + pt)ⁿ 和泊松分布的 e^{λ(t–1)}，可快速识别独立变量和的分布。
Published by TutorHao | Statistics Revision Series | aleveler.com
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