📚 Year 12 WJEC Statistics: Formula and Theorem Quick Reference Handbook | WJEC 统计 Year 12 公式定理速查手册
This quick-reference handbook summarises every essential formula, theorem and concept you need to master for the Year 12 WJEC Statistics specification. Each section pairs concise English statements with their Chinese equivalents, helping you revise efficiently whether you prefer studying in English, Chinese, or both. Keep this guide open while solving past‑paper questions; it is designed to match the exact notation, language and scope expected in WJEC AS‑level Statistics examinations.
这本速查手册汇总了 Year 12 WJEC 统计课程中每一个核心公式、定理与概念。每个小节都将精简的英文表述与对应的中文释义交替呈现，无论你习惯用英文、中文还是双语境复习，都能高效巩固。刷历年真题时请把本指南放在手边——它的符号、用语和范围完全贴合 WJEC AS 统计考试的要求。
1. Measures of Location & Spread | 位置与离散程度的度量
For raw data x₁, x₂, …, xₙ, the sample mean is x̄ = Σxᵢ / n; the population mean is written μ.
对于原始数据 x₁, x₂, …, xₙ，样本均值 x̄ = Σxᵢ / n；总体均值记为 μ。
The median is the middle value when data are ordered. If n is even, take the mean of the two central values. The mode is the most frequent data value.
中位数是排序后位于中间的值。若 n 为偶数，则取中间两个值的平均数。众数是出现次数最多的数据值。
Range = max – min. Interquartile range IQR = Q₃ – Q₁. Q₁ and Q₃ are the lower and upper quartiles.
极差 = 最大值 – 最小值。四分位距 IQR = Q₃ – Q₁，其中 Q₁ 和 Q₃ 分别为下四分位数与上四分位数。
Population variance σ² = Σ(xᵢ – μ)² / N. Sample variance s² = Σ(xᵢ – x̄)² / (n – 1). The corresponding standard deviations are σ and s.
总体方差 σ² = Σ(xᵢ – μ)² / N。样本方差 s² = Σ(xᵢ – x̄)² / (n – 1)。对应的标准差分别为 σ 与 s。
2. Basic Probability Rules | 基本概率法则
Probability of an event A: 0 ≤ P(A) ≤ 1. P(not A) = 1 – P(A). The sample space S has P(S) = 1.
事件 A 的概率满足 0 ≤ P(A) ≤ 1。P(非 A) = 1 – P(A)。样本空间 S 的概率 P(S) = 1。
Addition rule for mutually exclusive events: P(A ∪ B) = P(A) + P(B). For any two events, 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)。对于任意两事件，P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。
Multiplication rule for independent events: P(A ∩ B) = P(A) × P(B). If A and B are not independent, P(A ∩ B) = P(A) × P(B | A).
独立事件的乘法法则：P(A ∩ B) = P(A) × P(B)。若 A 与 B 不独立，P(A ∩ B) = P(A) × P(B | A)。
Conditional probability: P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.
条件概率：P(A | B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。
3. Conditional Probability & Tree Diagrams | 条件概率与树状图
Tree diagrams display sequential events. Multiply probabilities along branches; add probabilities of branches meeting at a terminal node for a combined outcome.
树状图用于呈现顺序事件。沿分支依次相乘概率；将汇聚于同一终点的分支概率相加，即得联合结果的概率。
When sampling without replacement, probabilities on the second set of branches depend on the first outcome, meaning events are not independent.
在不放回抽样中，第二层分支的概率依赖于第一阶段的结果，此时事件不独立。
Use the conditional probability formula to reverse the direction of conditioning: P(B | A) = [P(A | B) P(B)] / P(A) (this is Bayes’ theorem in its simplest form for the AS syllabus, though you will often apply it via a tree diagram without quoting the formula explicitly).
可利用条件概率公式逆转条件方向：P(B | A) = [P(A | B) P(B)] / P(A)（这是 AS 阶段贝叶斯定理的简化形式，不过你通常通过树状图直接求解而无需显式引用该公式）。
4. Discrete Random Variables | 离散随机变量
A discrete random variable X takes values x₁, x₂, … with probabilities p(x) = P(X = x). The sum of all probabilities is 1.
离散随机变量 X 取值为 x₁, x₂, …，概率为 p(x) = P(X = x)。所有概率之和等于 1。
Expected value (mean): E(X) = μ = Σ x·P(X = x). For a function g(X), E[g(X)] = Σ g(x)·P(X = x).
期望值（均值）：E(X) = μ = Σ x·P(X = x)。对于函数 g(X)，E[g(X)] = Σ g(x)·P(X = x)。
Variance: Var(X) = E[(X – μ)²] = Σ (x – μ)²·P(X = x) = E(X²) – [E(X)]². Standard deviation σ = √Var(X).
方差：Var(X) = E[(X – μ)²] = Σ (x – μ)²·P(X = x) = E(X²) – [E(X)]²。标准差 σ = √Var(X)。
For a linear transformation Y = aX + b: E(Y) = aE(X) + b, Var(Y) = a² Var(X).
对于线性变换 Y = aX + b：E(Y) = aE(X) + b，Var(Y) = a² Var(X)。
5. Binomial Distribution | 二项分布
Conditions: fixed number of trials n, each independent, two outcomes (success/failure), constant success probability p. We write X ~ B(n, p).
条件：试验次数 n 固定，各次独立，每次仅有成功与失败两种结果，成功概率 p 恒定。记作 X ~ B(n, p)。
Probability mass function: P(X = r) = C(n, r) · pʳ · (1–p)ⁿ⁻ʳ, where C(n, r) = n! / [r!(n–r)!].
概率质量函数：P(X = r) = C(n, r) · pʳ · (1–p)ⁿ⁻ʳ，其中 C(n, r) = n! / [r!(n–r)!]。
Mean: E(X) = np. Variance: Var(X) = np(1–p). Standard deviation: √[np(1–p)].
均值：E(X) = np。方差：Var(X) = np(1–p)。标准差：√[np(1–p)]。
Cumulative probability P(X ≤ k) can be found using binomial tables; for P(X ≥ k) use the complement 1 – P(X ≤ k–1).
累积概率 P(X ≤ k) 可通过二项分布表查找；P(X ≥ k) 则利用补集 1 – P(X ≤ k–1) 计算。
6. Poisson Distribution | 泊松分布
The Poisson model counts the number of events occurring independently at a constant average rate λ in a fixed interval. We write X ~ Po(λ).
泊松模型描述在固定区间内，以恒定平均速率 λ 独立发生的随机事件次数。记作 X ~ Po(λ)。
Probability function: P(X = r) = e⁻λ λʳ / r!, r = 0, 1, 2, …
概率函数：P(X = r) = e⁻λ λʳ / r!，r = 0, 1, 2, …
Mean: E(X) = λ. Variance: Var(X) = λ. Therefore standard deviation = √λ.
均值：E(X) = λ。方差：Var(X) = λ。因此标准差 = √λ。
For λ > 10 the Poisson distribution can be approximated by a normal distribution N(λ, λ), provided continuity correction is applied.
当 λ > 10 时，泊松分布可用正态分布 N(λ, λ) 近似，但必须应用连续性校正。
7. Normal Distribution | 正态分布
A continuous random variable X with mean μ and variance σ² follows a normal distribution: X ~ N(μ, σ²). Its probability density function is bell‑shaped and symmetric about μ.
若连续随机变量 X 的均值为 μ、方差为 σ²，则 X ~ N(μ, σ²)。其概率密度函数呈钟形，关于 μ 对称。
Standardisation: Z = (X – μ) / σ, so Z ~ N(0, 1). Standard normal tables give Φ(z) = P(Z < z).
标准化：Z = (X – μ) / σ，于是 Z ~ N(0, 1)。标准正态分布表给出 Φ(z) = P(Z < z)。
To find P(a < X < b), standardise both endpoints: P(a < X < b) = Φ((b–μ)/σ) – Φ((a–μ)/σ).
求 P(a < X < b) 时，将两端点标准化：P(a < X < b) = Φ((b–μ)/σ) – Φ((a–μ)/σ)。
Inverse normal: given a left‑tail probability p, find z such that Φ(z) = p, then X = μ + zσ.
逆正态：给定左尾概率 p，求出满足 Φ(z) = p 的 z 值，再代回 X = μ + zσ。
8. Sampling & the Central Limit Theorem | 抽样与中心极限定理
A parameter describes a population; a statistic is a value calculated from a sample. The sampling distribution of a statistic is its probability distribution over all possible samples of the same size.
参数描述总体；统计量则由样本计算得到。一个统计量的抽样分布是指，在相同样本容量下所有可能样本得到的统计值所构成的概率分布。
For a sample mean x̄ from a population with mean μ and variance σ²: E(x̄) = μ, Var(x̄) = σ²/n, so standard error SE = σ/√n.
对于从均值为 μ、方差为 σ² 的总体中抽出的样本均值 x̄：E(x̄) = μ，Var(x̄) = σ²/n，标准误 SE = σ/√n。
Central Limit Theorem: when n is sufficiently large (usually n ≥ 30), the distribution of x̄ is approximately normal, regardless of the population’s shape, i.e. x̄ ~ N(μ, σ²/n).
中心极限定理：当样本容量 n 足够大（通常 n ≥ 30），无论总体分布形状如何，x̄ 的分布近似正态，即 x̄ ~ N(μ, σ²/n)。
9. Confidence Intervals | 置信区间
A 100(1–α)% confidence interval for the population mean μ when σ is known: x̄ ± z* × (σ/√n), where z* is the critical value from N(0,1), e.g. z* = 1.96 for a 95% interval.
当 σ 已知时，总体均值 μ 的 100(1–α)% 置信区间为：x̄ ± z* × (σ/√n)，其中 z* 为标准正态临界值，例如 95% 置信水平下 z* = 1.96。
If σ is unknown and n is large, replace σ with the sample standard deviation s (or use the t‑distribution if n is small; however, WJEC Year 12 expects large‑sample z‑procedures or straightforward application of given σ).
若 σ 未知而 n 较大，可用样本标准差 s 替代 σ（若 n 较小则需使用 t 分布；但 WJEC Year 12 通常要求大样本下的 z 方法或直接使用给定的 σ）。
Confidence interval for a population proportion p: p̂ ± z* × √[p̂(1–p̂)/n]. Conditions: np̂ > 5 and n(1–p̂) > 5 for the normal approximation to be valid.
总体比例 p 的置信区间：p̂ ± z* × √[p̂(1–p̂)/n]。使用正态近似需满足条件：np̂ > 5 且 n(1–p̂) > 5。
10. Hypothesis Testing | 假设检验
Steps: (1) State null hypothesis H₀ and alternative H₁ (one‑tailed or two‑tailed). (2) Choose significance level α (e.g., 0.05). (3) Calculate the test statistic. (4) Compare with critical value or find p‑value. (5) Reject H₀ if the test statistic falls in the critical region (or p < α), otherwise do not reject H₀.
步骤：(1) 提出原假设 H₀ 与备择假设 H₁（单尾或双尾）。(2) 选定显著性水平 α（如 0.05）。(3) 计算检验统计量。(4) 与临界值比较或计算 p 值。(5) 若检验统计量落入拒绝域（或 p < α）则拒绝 H₀，否则不拒绝 H₀。
For a binomial test with H₀: p = p₀, find P(X ≥ observed) or P(X ≤ observed) using the binomial distribution, and compare with α.
对于原假设为 p = p₀ 的二项检验，利用二项分布求出 P(X ≥ 观测值) 或 P(X ≤ 观测值)，并与 α 比较。
For testing a mean when σ is known, use the z‑test: z = (x̄ – μ₀) / (σ/√n). The critical values are ±z* for two‑tailed or z* for one‑tailed tests.
当 σ 已知时对均值进行检验，采用 z 检验：z = (x̄ – μ₀) / (σ/√n)。双尾检验临界值为 ±z*，单尾为 z*。
Type I error: rejecting a true H₀ (probability = α). Type II error: failing to reject a false H₀ (probability = β).
第 I 类错误：H₀ 为真却被拒绝（概率等于 α）。第 II 类错误：H₀ 不成立却未被拒绝（概率为 β）。
11. Correlation & Regression | 相关与回归
Pearson product‑moment correlation coefficient: r = Sxy / √(Sxx · Syy), where Sxx = Σ(x – x̄)², Syy = Σ(y – ȳ)², and Sxy = Σ(x – x̄)(y – ȳ).
皮尔逊积矩相关系数：r = Sxy / √(Sxx · Syy)，其中 Sxx = Σ(x – x̄)²，Syy = Σ(y – ȳ)²，Sxy = Σ(x – x̄)(y – ȳ)。
r measures the strength and direction of a linear relationship. –1 ≤ r ≤ 1. A value close to 1 or –1 indicates strong linear association; r = 0 suggests no linear correlation.
r 衡量线性关系的强弱与方向，满足 –1 ≤ r ≤ 1。接近 1 或 –1 的值表示强线性相关；r = 0 表明没有线性相关。
Spearman’s rank correlation coefficient rₛ = 1 – [6 Σdᵢ²] / [n(n² – 1)], where dᵢ is the difference between the ranks of corresponding x and y. This measures monotonic association.
斯皮尔曼等级相关系数 rₛ = 1 – [6 Σdᵢ²] / [n(n² – 1)]，其中 dᵢ 为对应 x 与 y 的秩次之差。它衡量单调关联强度。
Least‑squares regression line of y on x: y = a + bx, where b = Sxy / Sxx and a = ȳ – b x̄. The line passes through (x̄, ȳ).
y 对 x 的最小二乘回归线：y = a + bx，其中 b = Sxy / Sxx，a = ȳ – b x̄。该直线经过点 (x̄, ȳ)。
Interpretation: b represents the estimated change in y for a one‑unit increase in x. Use the line for interpolation only within the range of the data; extrapolation may be unreliable.
解释：b 表示 x 每增加一个单位时 y 的估计变化量。此直线仅适用于原始数据范围内的内插；外推可能不可靠。
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