📚 Year 12 OCR Statistics: Formula & Theorem Quick Reference | Year 12 OCR 统计：公式定理速查手册
This concise handbook collects every essential formula, theorem and decision rule needed for the Year 12 OCR Statistics specification. Use it for quick revision, exam preparation and as a clear reference while working through past papers.
这本速查手册汇集了 Year 12 OCR 统计学所需的所有关键公式、定理与决策规则。可用于快速复习、备考和刷历年真题时作为清晰的参考资料。
1. Probability Laws | 概率法则
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
The addition rule applies to any two events A and B. It avoids double counting the intersection.
加法法则适用于任意两个事件 A 与 B，避免了重复计算交集部分。
P(A ∩ B) = P(A) · P(B | A) = P(B) · P(A | B)
Multiplication law introduces conditional probability. If events are independent, P(B | A) = P(B) and the formula simplifies to P(A ∩ B) = P(A)·P(B).
乘法法则引入了条件概率。若事件独立，P(B | A) = P(B)，公式简化为 P(A ∩ B) = P(A)·P(B)。
P(A | B) = P(A ∩ B) / P(B), P(B) > 0
The conditional probability of A given B is defined as the intersection divided by the probability of the conditioning event.
给定 B 时 A 的条件概率定义为交集概率除以条件事件的概率。
P(A′) = 1 − P(A)
The complement rule. The probability that an event does not happen is one minus the probability that it does happen.
补集法则：事件不发生概率等于 1 减去该事件发生的概率。
2. Counting Principles | 计数原理
n! = n × (n−1) × ⋯ × 2 × 1
The number of ways to arrange n distinct objects in a line is n! (factorial).
将 n 个不同物体排成一排的总数为 n!（阶乘）。
ⁿPᵣ = n! / (n−r)!
Permutations when order matters. The number of ways to select and arrange r objects from n distinct objects is nPr.
考虑顺序的排列：从 n 个不同物体中选出并排列 r 个的方法数为 ⁿPᵣ。
ⁿCᵣ = n! / [r!(n−r)!]
Combinations when order does not matter. The number of ways to choose r objects from n is ⁿCᵣ, also written as C(n,r) or binom{n}{r}.
不考虑顺序的组合：从 n 个物体中选出 r 个的方法数为 ⁿCᵣ，也记作 C(n,r)。
ⁿC₀ = 1, ⁿCₙ = 1, ⁿCᵣ = ⁿCₙ₋ᵣ
Useful symmetry properties of the binomial coefficient. They speed up calculations and are frequently tested in problems involving the binomial expansion.
二项式系数的有用对称性，可加快计算，在涉及二项展开的问题中常被考察。
3. Discrete Random Variables | 离散随机变量
E(X) = μ = Σ x · P(X = x)
The expected value (mean) of a discrete random variable X is the probability-weighted average of all possible values. The sum is taken over all x in the support of X.
离散随机变量 X 的期望（均值）是所有可能取值的概率加权平均值，求和取遍 X 支撑集中的所有 x。
Var(X) = σ² = E(X²) − [E(X)]²
A computationally efficient form of the variance. It is always non‑negative and its square root is the standard deviation σ.
方差的计算简便形式。方差恒为非负数，其平方根为标准差 σ。
Var(X) = Σ (x − μ)² · P(X = x)
The definition of variance as the expected squared deviation from the mean. Both forms must give the same numerical result.
方差的定义：与均值之差的平方的期望值。两种形式给出的数值结果必须一致。
E(aX + b) = aE(X) + b
Var(aX + b) = a² Var(X)
Linear transformations of a random variable. The constant b shifts the mean but does not affect the variance. The constant a scales both the mean and the spread (variance is scaled by a²).
随机变量的线性变换。常数 b 平移均值但不影响方差；常数 a 同时缩放均值与离散程度（方差按 a² 缩放）。
4. Binomial Distribution | 二项分布
X ~ B(n, p)
The notation for a binomial random variable: n independent trials, constant probability of success p.
二项随机变量的记号：n 次独立试验，每次成功概率恒为 p。
P(X = k) = ⁿCₖ · pᵏ · (1−p)ⁿ⁻ᵏ, k = 0,1,…,n
The probability mass function for the binomial distribution. Each trial yields either success (probability p) or failure (probability 1−p), and trials are independent.
二项分布的概率质量函数。每次试验只有成功（概率 p）或失败（概率 1−p）两种结果，且试验间独立。
E(X) = np
Var(X) = np(1−p)
Mean and variance formulas. Often used to find p or n from summary statistics, or to check whether a binomial model is appropriate.
均值与方差公式。常用于根据摘要统计量求 p 或 n，也用来检验二项模型是否合适。
Conditions for a binomial model: fixed number of trials, two outcomes per trial, constant probability of success, independent trials.
二项模型的条件：试验次数固定，每次试验有两种结果，成功概率不变，试验间独立。
5. Poisson Distribution | 泊松分布
X ~ Po(λ)
A Poisson random variable models the number of events occurring in a fixed interval of time or space, given a known average rate λ.
泊松随机变量用于建模在固定时间或空间区间内发生的事件数，已知平均率 λ。
P(X = r) = e⁻λ · λʳ / r!, r = 0,1,2,…
The Poisson probability mass function. The parameter λ is both the mean and the variance of the distribution.
泊松概率质量函数。参数 λ 既是该分布的均值也是其方差。
E(X) = λ, Var(X) = λ
Unlike the binomial, the Poisson variance equals its mean. This property is a quick diagnostic check for Poisson data.
与二项分布不同，泊松分布的方差等于均值。这一性质是判别数据是否符合泊松分布的快速诊断方法。
Conditions for a Poisson model: events occur singly and randomly, the rate λ is constant, events in non‑overlapping intervals are independent.
泊松模型的条件：事件单独且随机发生，率 λ 恒定，不相交区间内的事件相互独立。
Po(λ) can approximate B(n, p) when n is large and p is small, with λ ≈ np.
The Poisson approximation to the binomial is useful when n > 50 and p < 0.1. It simplifies calculations by turning a two‑parameter problem into a one‑parameter model.
当 n 很大且 p 很小时，泊松分布可近似二项分布，取 λ ≈ np。通常 n > 50 且 p < 0.1 时使用，它将双参数问题化为单参数模型，简化计算。
6. Continuous Random Variables | 连续随机变量
P(X = x) = 0
For a continuous random variable, the probability of any exact single value is zero. All calculations use intervals.
对于连续随机变量，任何单一精确值的概率均为零，所有计算都基于区间。
P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx
The probability density function (pdf) f(x) must be non‑negative and the total area under the curve must equal 1. The probability between a and b is the area under the pdf.
概率密度函数 f(x) 必须非负，且曲线下总面积等于 1。a 到 b 之间的概率即为 pdf 在该区间下的面积。
F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt
The cumulative distribution function (cdf) gives the probability that X is no greater than x. It is a non‑decreasing function with limits 0 and 1.
累积分布函数给出 X ≤ x 的概率，它是一个非降函数，极限为 0 和 1。
E(X) = ∫ x f(x) dx, Var(X) = ∫ (x−μ)² f(x) dx = E(X²) − μ²
Moments of a continuous distribution are defined by integration. The same linear transformation rules as for discrete variables apply.
连续分布的矩由积分定义。离散变量下成立的线性变换规则在此同样适用。
7. Normal Distribution | 正态分布
X ~ N(μ, σ²)
The normal distribution is the most important continuous model. It is symmetric, bell‑shaped, and fully described by its mean μ and variance σ².
正态分布是最重要的连续模型，呈对称的钟形，完全由其均值 μ 和方差 σ² 刻画。
Z = (X − μ) / σ → Z ~ N(0, 1)
Standardisation converts any normal observation into a standard normal z‑score. This allows the use of standard normal tables which give Φ(z) = P(Z ≤ z).
标准化将任意正态观测值转换为标准正态 Z 值，从而可使用标准正态表，其给出 Φ(z) = P(Z ≤ z)。
P(X ≤ x) = Φ((x − μ) / σ)
Every normal probability calculation reduces to finding a z‑value and looking up Φ(z). For the inverse problem, use the percentage point table to find z and then X = μ + zσ.
所有正态概率计算都归结为求 z 值并查表 Φ(z)。反向问题则使用百分点表求出 z，再由 X = μ + zσ 得到观测值。
68% within 1σ, 95% within 1.96σ, 99.7% within 3σ
Empirical rule approximations. The 95 % interval μ ± 1.96σ is especially important for hypothesis testing and confidence intervals in later topics.
经验法则近似值。95% 区间 μ ± 1.96σ 在后续假设检验与置信区间中尤为重要。
8. Sampling & The Central Limit Theorem | 抽样与中心极限定理
Sample mean X̄ = ΣXᵢ / n
When a random sample of size n is drawn from a population with mean μ and variance σ², the sample mean is an unbiased estimator of μ, so E(X̄) = μ.
从均值为 μ、方差为 σ² 的总体中抽取大小为 n 的随机样本时，样本均值是 μ 的无偏估计量，即 E(X̄) = μ。
Var(X̄) = σ² / n, SE(X̄) = σ / √n
The variability of the sample mean decreases as the sample size increases. The standard error of the mean measures how precisely the sample mean estimates μ.
样本均值的变异随样本容量增大而减小。均值的标准误衡量样本均值估计 μ 的精确程度。
If Xᵢ ~ N(μ, σ²), then X̄ ~ N(μ, σ²/n) exactly.
When the population is normal, the sample mean also follows a normal distribution exactly for any sample size.
若总体为正态，则无论样本大小，样本均值精确服从正态分布 N(μ, σ²/n)。
Central Limit Theorem: For large n (n ≥ 30), X̄ is approximately N(μ, σ²/n) regardless of the population shape.
The CLT justifies the use of normal methods for the sample mean even when the original data are non‑normal. This underpins large‑sample hypothesis tests and confidence intervals.
中心极限定理：对大样本（n ≥ 30），无论总体分布形态如何，X̄ 近似服从 N(μ, σ²/n)。这为原始数据非正态时使用正态方法提供了理论依据，是大样本检验和置信区间的基础。
9. Hypothesis Testing for Proportions (Binomial) | 二项分布假设检验
Framework: null hypothesis H₀: p = p₀ vs. alternative H₁: p > p₀, p < p₀ or p ≠ p₀. The test statistic is the observed number of successes X, and the decision is based on the binomial distribution B(n, p₀).
检验框架：原假设 H₀: p = p₀，备择假设 H₁: p > p₀、p < p₀ 或 p ≠ p₀。检验统计量为观测成功次数 X，决策基于二项分布 B(n, p₀)。
p-value = P(X ≥ x | p₀) for upper‑tail test
For a one‑tailed test, the p‑value is the probability of observing a result at least as extreme as the sample, under H₀. If p‑value ≤ significance level α, reject H₀.
单尾检验的 p 值是在 H₀ 下观测到至少与样本一样极端的结果的概率。若 p‑值 ≤ 显著性水平 α，则拒绝 H₀。
For a two‑tailed test: p‑value = 2 × min{P(X ≥ x), P(X ≤ x)} or find the critical region such that the probability in each tail is ≤ α/2.
双尾检验：p‑值 = 2 × min{P(X ≥ x), P(X ≤ x)} 或求出两侧尾部概率 ≤ α/2 的临界域。
Conditions: n fixed, p constant, independent trials, and n large enough so that the minimum of np₀ and n(1−p₀) permits normal approximation if used.
The binomial test must satisfy the underlying model conditions. When the normal approximation is used, apply a continuity correction of ±0.5 to the observed number of successes when calculating z.
二项检验必须满足模型条件。若使用正态近似，计算 z 时需对观测成功次数进行 ±0.5 的连续性校正。
10. Hypothesis Testing for Rates (Poisson) | 泊松分布假设检验
Hypotheses are stated in terms of the rate parameter λ: H₀: λ = λ₀ vs. H₁: λ > λ₀, λ < λ₀ or λ ≠ λ₀. The test statistic is the observed count X ~ Po(λ₀) under H₀.
假设用率参数 λ 表述：H₀: λ = λ₀，备择为一侧或两侧。检验统计量为观测计数 X，在 H₀ 下 X ~ Po(λ₀)。
p-value = P(X ≥ x | λ₀) for H₁: λ > λ₀
Use the Poisson cumulative probability tables or the formula P(X ≥ x) = 1 − P(X ≤ x−1). Compare with α and conclude in the usual way.
使用泊松累积概率表或公式 P(X ≥ x) = 1 − P(X ≤ x−1) 求 p‑值。与 α 比较后按常规方式得出结论。
For a two‑sided test, double the smaller one‑tail probability or construct a critical region with half α in each tail.
双尾检验可将较小的单尾概率加倍，或构建双侧各含 α/2 概率的拒绝域。
Normal approximation: For λ₀ > 10, X ~ N(λ₀, λ₀) approximately. Use Z = (X − λ₀) / √λ₀ and apply continuity correction.
When λ₀ is large enough (＞10), the Poisson distribution can be approximated by a normal distribution with mean and variance λ₀. The continuity correction adds or subtracts 0.5 as appropriate to improve accuracy.
当 λ₀ 足够大（＞10）时，泊松分布可用均值与方差均为 λ₀ 的正态分布近似。适当加减 0.5 的连续性校正可提高精度。
The same model conditions apply: events must occur randomly, singly, and at a constant average rate. Non‑overlapping intervals must be independent.
模型适用条件不变：事件随机且独立发生，平均发生率恒定，不相交区间内事件独立。
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