📚 AS Eduqas Statistics: Formula and Theorem Quick Reference Handbook | AS Eduqas 统计：公式定理速查手册
This handbook distils the core formulas, theorems, and statistical tests required for the AS Eduqas Statistics specification. Use it as a fast reference when revising or working through problems. Each entry is presented in a compact bilingual format, covering descriptive measures, probability, distributions, inference, and relationships between variables.
本手册浓缩了 AS Eduqas 统计课程所需的核心公式、定理和检验方法，可供复习或解题时快速查阅。每个条目都以简洁的双语形式呈现，内容涵盖描述性统计量、概率、分布、推断以及变量间关系。
1. Sample Mean, Variance and Standard Deviation | 样本均值、方差与标准差
For a sample of n observations x₁, x₂, …, xₙ, the sample mean is x̄ = (Σx)/n.
对于 n 个观测值 x₁, x₂, …, xₙ，样本均值为 x̄ = (Σx)/n。
The sum of squares Sxx = Σ(x − x̄)² = Σx² − (Σx)²/n.
离差平方和 Sxx = Σ(x − x̄)² = Σx² − (Σx)²/n。
Sample variance: s² = Sxx/(n − 1) = [Σx² − (Σx)²/n] / (n − 1).
样本方差：s² = Sxx/(n − 1) = [Σx² − (Σx)²/n] / (n − 1)。
Sample standard deviation: s = √s².
样本标准差：s = √s²。
2. Quartiles, IQR and Outliers | 四分位数、四分位距与离群值
The lower quartile Q₁ is the median of the lower half of the data; the upper quartile Q₃ is the median of the upper half.
下四分位数 Q₁ 是数据下半部分的中位数；上四分位数 Q₃ 是数据上半部分的中位数。
Interquartile range: IQR = Q₃ − Q₁.
四分位距：IQR = Q₃ − Q₁。
observation is a possible outlier if it lies below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR.
若观测值低于 Q₁ − 1.5 × IQR 或高于 Q₃ + 1.5 × IQR，则可能为离群值。
The five‑number summary consists of minimum, Q₁, median, Q₃, maximum, often displayed in a box plot.
五数综合由最小值、Q₁、中位数、Q₃、最大值组成，常用箱线图表示。
3. Basic Probability Rules | 概率基本法则
For any event A, 0 ≤ P(A) ≤ 1.
对任意事件 A，有 0 ≤ P(A) ≤ 1。
Complement rule: P(A') = 1 − P(A).
补集法则：P(A') = 1 − P(A)。
Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
若 A 与 B 互斥，则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。
4. Conditional Probability and Independence | 条件概率与独立性
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)，其中 P(B) > 0。
Multiplication rule: P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(A).
乘法法则：P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(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).
事件 A 与 B 独立当且仅当 P(A ∩ B) = P(A) × P(B)，或等价地 P(A|B) = P(A)。
Law of total probability: If B₁, B₂, …, Bₖ partition the sample space, then P(A) = Σ P(A|Bᵢ) P(Bᵢ).
全概率公式：若 B₁, B₂, …, Bₖ 构成样本空间的一个划分，则 P(A) = Σ P(A|Bᵢ) P(Bᵢ)。
Bayes' theorem: P(Bⱼ|A) = [P(A|Bⱼ) P(Bⱼ)] / Σ P(A|Bᵢ) P(Bᵢ).
贝叶斯定理：P(Bⱼ|A) = [P(A|Bⱼ) P(Bⱼ)] / Σ P(A|Bᵢ) P(Bᵢ)。
5. Discrete Random Variables | 离散随机变量
lues xᵢ and probabilities pᵢ = P(X = xᵢ), the expected value (mean) is E(X) = μ = Σ xᵢ pᵢ.
对于离散随机变量 X，取值 xᵢ，概率 pᵢ = P(X = xᵢ)，期望值（均值）为 E(X) = μ = Σ xᵢ pᵢ。
Variance: Var(X) = E[(X − μ)²] = Σ (xᵢ − μ)² pᵢ = E(X²) − [E(X)]², where E(X²) = Σ xᵢ² pᵢ.
方差：Var(X) = E[(X − μ)²] = Σ (xᵢ − μ)² pᵢ = E(X²) − [E(X)]²，其中 E(X²) = Σ xᵢ² pᵢ。
Linear transformations: E(aX + b) = a E(X) + b; Var(aX + b) = a² Var(X).
线性变换：E(aX + b) = a E(X) + b；Var(aX + b) = a² Var(X)。
The standard deviation of X is σ = √Var(X).
X 的标准差为 σ = √Var(X)。
6. Binomial Distribution | 二项分布
If X ~ B(n, p), where n is the number of independent trials and p is the probability of success in each trial, then the probability of exactly r successes is:
若 X ~ B(n, p)，其中 n 为独立试验次数，p 为每次试验成功的概率，则恰好 r 次成功的概率为：
P(X = r) = C(n, r) pr (1 − p)n − r, r = 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)。
Cumulative probabilities P(X ≤ r) are obtained from tables or technology.
累积概率 P(X ≤ r) 可通过查表或使用技术工具获得。
>5), a normal approximation may be used: X ≈ N(np, np(1−p)) with a continuity correction.
当 np 和 n(1−p) 较大（通常大于 5）时，可使用正态近似：X ≈ N(np, np(1−p))，并应用连续性校正。
7. The Normal Distribution | 正态分布
If X ~ N(μ, σ²), its probability density function is symmetric and bell‑shaped centred at μ.
若 X ~ N(μ, σ²)，其概率密度函数为关于 μ 对称的钟形曲线。
The standard normal variable Z = (X − μ) / σ follows N(0, 1).
标准正态变量 Z = (X − μ) / σ 服从 N(0, 1)。
For a given value x, the standardised score (z‑score) is z = (x − μ) / σ.
对给定值 x，标准化分数（z 值）为 z = (x − μ) / σ。
Probabilities such as P(Z < z) are read from the standard normal table. The 95% central interval is μ ± 1.96σ, and the 90% interval is μ ± 1.645σ.
通过标准正态表可查得 P(Z < z) 等概率。95% 中心区间为 μ ± 1.96σ，90% 中心区间为 μ ± 1.645σ。
Inverse normal: given a probability P(Z < z) = p, find z from the table; then x = μ + zσ.
逆向正态：给定概率 P(Z < z) = p，查表得 z；再由 x = μ + zσ 回求 x 值。
8. Confidence Interval for a Population Mean (σ known) | 总体均值的置信区间（σ 已知）
drawn from a normally distributed population with known standard deviation σ, a (1−α)×100% confidence interval for the population mean μ is:
当从已知标准差 σ 的正态总体中抽取容量为 n 的随机样本时，总体均值 μ 的 (1−α)×100% 置信区间为：
x̄ ± z* × (σ / √n)
the critical value from N(0,1) such that P(−z* < Z < z*) = 1 − α. Common critical values:
其中 z* 为标准正态分布的临界值，满足 P(−z* < Z < z*) = 1 − α。常用临界值：
| Confidence level | 置信水平 | z* |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
The interval widens as the confidence level increases or as the sample size decreases.
置信水平越高或样本量越小，区间越宽。
9. Hypothesis Test for a Population Mean (σ known) | （σ 已知）总体均值的假设检验
Null hypothesis H₀: μ = μ₀. Alternative H₁ can be two‑tailed (μ ≠ μ₀) or one‑tailed (μ < μ₀ or μ > μ₀).
原假设 H₀: μ = μ₀。备择假设 H₁ 可为双侧 (μ ≠ μ₀) 或单侧 (μ < μ₀ 或 μ > μ₀)。
Test statistic: Z = (x̄ − μ₀) / (σ / √n).
检验统计量：Z = (x̄ − μ₀) / (σ / √n)。
e the observed Z with the critical value(s) from N(0,1) at significance level α. For a two‑tailed test at 5%, critical values are ±1.96.
将观测 Z 值与显著性水平 α 下的标准正态临界值比较。双侧 5% 检验的临界值为 ±1.96。
Alternatively, compute the p‑value = P(Z ≤ zobs) for a lower‑tail test, P(Z ≥ zobs) for an upper‑tail test, or 2 × P(Z ≥ |zobs|) for a two‑tailed test. Reject H₀ if p
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