📚 Edexcel A-Level Statistics: Formula & Theorem Quick Reference Handbook | A-Level Edexcel 统计：公式定理速查手册
This comprehensive handbook collects all essential formulas, definitions and key theorems required for the Edexcel A-Level Statistics syllabus. Each section presents the core concepts in a bilingual paired-paragraph format, ensuring quick reference and deep understanding. Use it alongside past papers and revision notes to master statistical methods efficiently.
本手册汇总了 Edexcel A-Level 统计学大纲全部必背公式、定义与核心定理。每节均采用中英对照的配对段落排版，便于快速查阅和深入理解。配合真题和复习笔记使用，可高效掌握各类统计方法。
1. Descriptive Statistics: Measures of Centre and Spread | 描述性统计：集中趋势与离散程度
For a sample of size n with observations x₁, x₂, …, xₙ, the sample mean is given by x̄ = (Σx)/n. The sample variance, which measures dispersion, is s² = Σ(x – x̄)² / (n – 1), and the standard deviation s = √s². An equivalent computational formula is s² = (Σx² – (Σx)²/n) / (n – 1). The first quartile Q₁ is the median of the lower half of the data, the third quartile Q₃ is the median of the upper half; the interquartile range IQR = Q₃ – Q₁.
对于容量为 n 的样本 x₁, x₂, …, xₙ，样本均值定义为 x̄ = (Σx)/n。衡量离散程度的样本方差为 s² = Σ(x – x̄)² / (n – 1)，标准差 s = √s²。等价的简便计算公式为 s² = (Σx² – (Σx)²/n) / (n – 1)。第一四分位数 Q₁ 是数据下半部分的中位数，第三四分位数 Q₃ 是上半部分的中位数；四分位距 IQR = Q₃ – Q₁。
Outlier boundaries are conventionally defined using the 1.5×IQR rule: a value is an outlier if it falls below Q₁ – 1.5×IQR or above Q₃ + 1.5×IQR. For grouped data with frequency f and class midpoint m, the mean is estimated as Σfm / Σf, and the variance uses Σfm² adjusted by the mean.
离群值的判定通常遵循 1.5×IQR 法则：若某数值小于 Q₁ – 1.5×IQR 或大于 Q₃ + 1.5×IQR，则被视为离群值。对于分组数据，以频数 f 和组中值 m，均值估计为 Σfm / Σf，方差则通过 Σfm² 并减去均值的平方求得。
For a population of size N with mean μ, the variance is σ² = Σ(x – μ)² / N. The population standard deviation is σ = √σ². When working with census data, use N in the denominator rather than n-1.
对于个体数目为 N 的总体，均值为 μ，方差为 σ² = Σ(x – μ)² / N，总体标准差 σ = √σ²。处理普查数据时，分母使用 N 而非 n-1。
2. Probability Rules and Conditional Probability | 概率法则与条件概率
For any two events A and B, the addition rule states: 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). The complement rule says P(A′) = 1 – P(A).
对于任意两个事件 A 和 B，加法法则为：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。如果 A 与 B 互斥，则 P(A ∩ B) = 0，从而 P(A ∪ B) = P(A) + P(B)。补事件法则为 P(A′) = 1 – P(A)。
Conditional probability is defined as P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0. Rearranging gives the multiplication rule: P(A ∩ B) = P(A) P(B | A) = P(B) P(A | B). Events A and B are independent exactly when P(A ∩ B) = P(A) P(B) or equivalently P(A | B) = P(A).
条件概率定义为 P(A | B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。移项可得乘法法则：P(A ∩ B) = P(A) P(B | A) = P(B) P(A | B)。当 P(A ∩ B) = P(A) P(B) 或等价地 P(A | B) = P(A) 时，事件 A 与 B 独立。
Bayes’ theorem calculates reverse conditional probabilities: P(A | B) = [P(B | A) P(A)] / P(B). The denominator P(B) can be found via the law of total probability: P(B) = Σ P(B | Aᵢ) P(Aᵢ) for a partition {Aᵢ}.
贝叶斯定理用于计算反向条件概率：P(A | B) = [P(B | A) P(A)] / P(B)。分母 P(B) 可通过全概率公式求出：对于样本空间的一个分割 {Aᵢ}，有 P(B) = Σ P(B | Aᵢ) P(Aᵢ)。
3. Discrete Random Variables: Expectation and Variance | 离散随机变量：期望与方差
A discrete random variable X takes values x with probabilities P(X = x). Its expected value (mean) is E(X) = Σ x · P(X = x). The variance is Var(X) = E[(X – μ)²] = Σ (x – μ)² P(X = x) = E(X²) – [E(X)]². The shortcut is particularly useful: Var(X) = Σ x² P(X = x) – μ².
离散随机变量 X 以概率 P(X = x) 取值 x。其期望值（均值）为 E(X) = Σ x · P(X = x)。方差 Var(X) = E[(X – μ)²] = Σ (x – μ)² P(X = x) = E(X²) – [E(X)]²。通常使用简便公式 Var(X) = Σ x² P(X = x) – μ²。
For any constants a and b, of linear functions we have E(aX + b) = a E(X) + b and Var(aX + b) = a² Var(X). If X and Y are independent random variables, then E(X + Y) = E(X) + E(Y) and Var(X + Y) = Var(X) + Var(Y); the same holds for X – Y, giving Var(X – Y) = Var(X) + Var(Y).
对于任意常数 a 和 b，线性函数的期望与方差满足 E(aX + b) = a E(X) + b，Var(aX + b) = a² Var(X)。若 X 与 Y 独立，则 E(X + Y) = E(X) + E(Y)，Var(X + Y) = Var(X) + Var(Y)；类似地，Var(X – Y) = Var(X) + Var(Y)。
4. Binomial Distribution | 二项分布
If X ~ B(n, p), meaning n independent trials each with success probability p, the probability mass function is P(X = k) = ⁿCₖ pᵏ (1 – p)ⁿ⁻ᵏ for k = 0,1,…,n, where ⁿCₖ = n! / (k!(n – k)!). The distribution is fully specified by n and p.
若 X ~ B(n, p)，即 n 次独立试验、每次成功概率为 p，其概率质量函数为 P(X = k) = ⁿCₖ pᵏ (1 – p)ⁿ⁻ᵏ，k = 0,1,…,n，其中 ⁿCₖ = n! / (k!(n – k)!)。该分布完全由参数 n 与 p 决定。
The mean and variance of the binomial distribution are given by
E(X) = np and Var(X) = np(1 – p).
二项分布的均值与方差为
E(X) = np，Var(X) = np(1 – p)。
The binomial setting requires four conditions: a fixed number of trials, each trial is independent, only two possible outcomes (success/failure), and the probability of success p remains constant across trials.
使用二项分布需满足四个条件：试验次数固定、各次试验独立、每次试验只有两种可能结果（成功/失败）、每次成功的概率 p 保持不变。
5. Poisson Distribution | 泊松分布
A discrete random variable X follows a Poisson distribution with parameter λ > 0, written X ~ Po(λ), if it models the count of independent events occurring at a constant average rate within a fixed interval. The probability mass function is P(X = k) = (e⁻λ λᵏ) / k!, for k = 0,1,2,….
若离散随机变量 X 表示在固定区间内、以恒定平均发生率独立发生的事件次数，且 X ~ Po(λ)（λ > 0），则概率质量函数为 P(X = k) = (e⁻λ λᵏ) / k!，k = 0,1,2,…。
E(X) = λ and Var(X) = λ.
E(X) = λ，Var(X) = λ。
When a binomial random variable X ~ B(n, p) has large n and small p such that np remains moderate, the Poisson distribution provides a good approximation: X ≈ Po(np). A typical rule of thumb is n ≥ 50 and np < 5 or n ≥ 20 and p ≤ 0.1. In such cases, P(X = k) is computed using λ = np.
当二项随机变量 X ~ B(n, p) 满足 n 很大、p 很小且 np 大小适中时，可用泊松分布进行近似：X ≈ Po(np)。常用经验规则为 n ≥ 50 且 np < 5，或 n ≥ 20 且 p ≤ 0.1。此时取 λ = np，按泊松公式计算概率。
6. Normal Distribution and Approximations | 正态分布及其近似
If X follows a normal distribution with mean μ and variance σ², denoted X ~ N(μ, σ²), its probability density is symmetric and bell‑shaped. To find probabilities, standardise to the standard normal Z ~ N(0,1) using
Z = (X – μ) / σ.
若 X 服从均值为 μ、方差为 σ² 的正态分布，记作 X ~ N(μ, σ²)，其概率密度呈对称钟形。计算概率时通过标准化转化为标准正态 Z ~ N(0,1)：
Z = (X – μ) / σ。
For a random sample of size n from a normal population, the sample mean X̄ is also normally distributed: X̄ ~ N(μ, σ²/n). This result is exact if the population is normal; for large n (usually n ≥ 30) the Central Limit Theorem extends approximate normality even when the population is not normal.
对取自正态总体的容量为 n 的随机样本，样本均值 X̄ 也服从正态分布：X̄ ~ N(μ, σ²/n)。若总体为正态，该结论严格成立；当样本量较大（通常 n ≥ 30）时，即使总体非正态，中心极限定理也能保证 X̄ 的近似正态性。
The normal approximation to the binomial, X ~ B(n, p), applies when np > 5 and n(1 – p) > 5. Then X ≈ N(np, np(1 – p)), and a continuity correction is necessary: replace a discrete integer k by the interval (k – 0.5, k + 0.5) when calculating probabilities. Similarly, a Poisson X ~ Po(λ) with λ > 15 can be approximated by N(λ, λ), again applying a continuity correction.
二项分布的正态近似：当 np > 5 且 n(1 – p) > 5 时，X ~ B(n, p) 可近似为 N(np, np(1 – p))，并必须进行连续性校正——将离散整数 k 替换为区间 (k – 0.5, k + 0.5) 来计算概率。类似地，若 λ > 15，泊松分布 X ~ Po(λ) 可用 N(λ, λ) 近似，同样需要连续性校正。
If X ~ N(μₓ, σₓ²) and Y ~ N(μᵧ, σᵧ²) are independent, then X ± Y ~ N(μₓ ± μᵧ, σₓ² + σᵧ²).
若 X ~ N(μₓ, σₓ²) 与 Y ~ N(μᵧ, σᵧ²) 独立，则 X ± Y ~ N(μₓ ± μᵧ, σₓ² + σᵧ²)。
7. Sampling and the Central Limit Theorem | 抽样与中心极限定理
The Central Limit Theorem (CLT) states that for a random sample of size n drawn from any population with mean μ and finite variance σ², the sampling distribution of the sample mean X̄ tends to a normal distribution as n increases. Specifically, for sufficiently large n (often n ≥ 30), X̄ ~ N(μ, σ²/n) approximately, regardless of the population shape.
中心极限定理 (CLT) 指出：从均值为 μ、方差 σ² 有限且非特定的总体中抽取容量为 n 的随机样本，当 n 增大时，样本均值 X̄ 的抽样分布趋近于正态分布。具体而言，对于足够大的 n（常取 n ≥ 30），无论总体分布形状如何，均有 X̄ 近似服从 N(μ, σ²/n)。
For a sample proportion p̂ = X/n derived from a binomial count, the CLT yields that when np ≥ 5 and n(1 – p) ≥ 5, p̂ follows an approximate normal distribution: p̂ ~ N(p, p(1 – p) / n). This underpins confidence intervals and hypothesis tests for proportions.
对于来自二项计数变量的样本比例 p̂ = X/n，中心极限定理表明当 np ≥ 5 且 n(1 – p) ≥ 5 时，p̂ 近似服从正态分布：p̂ ~ N(p, p(1 – p) / n)。该近似是比例值的置信区间和假设检验的基础。
The standard error of a statistic is a measure of its sampling variability. For the sample mean, SE(X̄) = σ / √n (estimated by s / √n when σ is unknown). For a sample proportion, SE(p̂) = √[p(1 – p)/n] and is estimated by √[p̂(1 – p̂)/n].
统计量的标准误差衡量其抽样变异性。样本均值的标准误差为 SE(X̄) = σ / √n（当 σ 未知时用 s / √n 估计）。样本比例的标准误差为 SE(p̂) = √[p(1 – p)/n]，并以 √[p̂(1 – p̂)/n] 作为估计。
8. Confidence Intervals | 置信区间
A confidence interval provides a range of plausible values for an unknown population parameter. For a population mean with known variance σ², the 100(1 – α)% confidence interval is
x̄ ± zα/2 · σ / √n,
where zα/2 is the critical value from the standard normal distribution (e.g., 1.96 for 95% confidence). When σ is unknown and is estimated by the sample standard deviation s, and the population is normally distributed (or n is large), use the t‑distribution with n – 1 degrees of freedom:
x̄ ± tn-1, α/2 · s / √n.
置信区间为未知总体参数提供一个合理的取值范围。对于已知总体方差 σ² 的总体均值，100(1 – α)% 置信区间为
x̄ ± zα/2 · σ / √n，
其中 zα/2 是标准正态分布的临界值（例如 95% 置信水平下为 1.96）。当 σ 未知、用样本标准差 s 估计且总体正态（或 n 很大）时，使用自由度为 n – 1 的 t 分布：
x̄ ± tn-1, α/2 · s / √n。
For a population proportion p, a large-sample confidence interval uses the normal approximation:
p̂ ± zα/2 · √(p̂(1 – p̂) / n).
This formula is valid when both np̂ > 5 and n(1 – p̂) > 5. The margin of error is the half‑width of the interval: zα/2 · SE.
对于总体比例 p，大样本置信区间使用正态近似：
p̂ ± zα/2 · √(p̂(1 – p̂) / n)。
当 np̂ > 5 且 n(1 – p̂) > 5 时该公式有效。误差界限（margin of error）即区间半宽：zα/2 · SE。
9. Hypothesis Testing: Framework and Test Statistics | 假设检验：框架与检验统计量
A hypothesis test evaluates two competing statements: the null hypothesis H₀ (assumed true) and the alternative hypothesis H₁ (one‑ or two‑tailed). Testing is conducted at a chosen significance level α (commonly 5% or 1%). The decision rule compares the test statistic to a critical value, or equivalently compares the p‑value to α: reject H₀ if p‑value < α.
假设检验用于评判两个对立的陈述：原假设 H₀（假定为真）与备择假设 H₁（单尾或双尾）。检验在选定的显著性水平 α（常用 5% 或 1%）下进行。决策规则将检验统计量与临界值进行比较，或等价地将 p 值与 α 比较：若 p‑value < α，则拒绝 H₀。
Test statistic for a population mean with known σ:
Z = (x̄ – μ₀) / (σ / √n).
With unknown σ, use the t‑statistic:
t = (x̄ – μ₀) / (s / √n), df = n – 1.
已知 σ 时总体均值的检验统计量：
Z = (x̄ – μ₀) / (σ / √n)。
σ 未知时使用 t 统计量：
t = (x̄ – μ₀) / (s / √n)，df = n – 1。
For a binomial proportion, the test statistic under H₀: p = p₀ is
Z = (p̂ – p₀) / √(p₀(1 – p₀) / n),
provided np₀ ≥ 5 and n(1 – p₀) ≥ 5. For small binomial samples, exact p‑values are calculated directly from the binomial distribution.
对于二项比例，在 H₀: p = p₀ 下的检验统计量为
Z = (p̂ – p₀) / √(p₀(1 – p
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