📚 Year 13 OCR Statistics Formula & Theorem Quick Reference | A Level 统计公式定理速查手册
This handy reference guide collects the essential formulas, theorems and definitions for OCR A Level Mathematics Statistics (Year 13). Use it to revise probability, distributions, inference and correlation before your examination. Keep the sheet by your side while working through past papers – every key result is stated concisely with its most practical form.
这份速查手册汇集了 OCR A Level 数学统计（13 年级）的核心公式、定理与定义，供你在考前复习概率、分布、推断与相关分析时使用。做题时放在手边，所有关键结论都以最实用的形式给出，一目了然。
1. Probability Basics and Conditional Probability | 概率基础与条件概率
The probability of an event A is written P(A), with 0 ≤ P(A) ≤ 1. For complementary events, P(A’) = 1 − P(A). The addition rule for mutually exclusive events A and B is P(A ∪ B) = P(A) + P(B). When events are not mutually exclusive, subtract the intersection: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
事件 A 的概率记作 P(A)，满足 0 ≤ P(A) ≤ 1。互补事件满足 P(A’) = 1 − P(A)。若 A 与 B 互斥，则加法公式为 P(A ∪ B) = P(A) + P(B)。当事件不互斥时须减去交集：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
Conditional probability is fundamental in Year 13. It is defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. Multiplication rule follows: P(A ∩ B) = P(B) × P(A|B) = P(A) × P(B|A). Two events are independent if and only if P(A ∩ B) = P(A) × P(B), equivalently P(A|B) = P(A) when P(B) > 0.
条件概率是 13 年级重点。定义式为 P(A|B) = P(A ∩ B) / P(B)，要求 P(B) > 0。乘法公式为 P(A ∩ B) = P(B) × P(A|B) = P(A) × P(B|A)。两事件独立当且仅当 P(A ∩ B) = P(A) × P(B)，亦可等价表示为 P(A|B) = P(A) (P(B) > 0 时)。
2. Discrete Random Variables and Expectation | 离散随机变量与期望
For a discrete random variable X taking values xᵢ with probabilities pᵢ = P(X = xᵢ), the expected value (mean) is E(X) = μ = Σ xᵢ pᵢ. The variance is Var(X) = E(X²) − [E(X)]², where E(X²) = Σ xᵢ² pᵢ. Expectation is linear: E(aX + b) = aE(X) + b, and Var(aX + b) = a² Var(X) for constants a, b.
对于离散随机变量 X，取值为 xᵢ 且对应概率为 pᵢ = P(X = xᵢ)，其期望值 (均值) 为 E(X) = μ = Σ xᵢ pᵢ。方差为 Var(X) = E(X²) − [E(X)]²，其中 E(X²) = Σ xᵢ² pᵢ。期望具有线性性质：E(aX + b) = aE(X) + b，Var(aX + b) = a² Var(X)，a、b 为常数。
If two discrete random variables X and Y are independent, then E(XY) = E(X)E(Y) and Var(X ± Y) = Var(X) + Var(Y). These relationships are regularly used when dealing with sums of observations in sampling theory.
若两离散随机变量 X 与 Y 独立，则 E(XY) = E(X)E(Y) 且 Var(X ± Y) = Var(X) + Var(Y)。抽样理论中处理观测值之和时经常用到这些关系。
3. Binomial Distribution | 二项分布
A binomial distribution models the number of successes in n independent trials, each with constant probability of success p. Denote X ~ B(n, p). The probability mass function is:
二项分布描述 n 次独立试验中成功的次数，每次成功概率为常数 p。记作 X ~ B(n, p)。其概率质量函数为：
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ , r = 0, 1, …, n
Mean and variance are E(X) = np and Var(X) = np(1 − p). The binomial distribution is symmetric when p = 0.5 and skewed otherwise. Tables or calculators provide cumulative probabilities, but you must be able to work with the formula directly for small n.
均值与方差为 E(X) = np、Var(X) = np(1 − p)。当 p = 0.5 时分布对称，否则偏斜。表格或计算器可提供累积概率，但对于较小的 n 必须能直接用公式计算。
4. Poisson Distribution | 泊松分布
The Poisson distribution models the number of events occurring in a fixed interval of time or space, at a constant average rate λ. Write X ~ Po(λ). The probability of observing exactly r events is:
泊松分布用于描述固定时间或空间内事件发生的次数，平均发生率为常数 λ。记作 X ~ Po(λ)。恰好观测到 r 次事件的概率为：
P(X = r) = e⁻λ λʳ / r! , r = 0, 1, 2, …
Both the mean and variance equal λ: E(X) = Var(X) = λ. The Poisson distribution arises naturally as an approximation to the binomial when n is large and p is small, with λ = np. It is also additive: if X ~ Po(λ₁) and Y ~ Po(λ₂) are independent, then X + Y ~ Po(λ₁ + λ₂).
均值与方差都等于 λ：E(X) = Var(X) = λ。当 n 很大、p 很小时，泊松分布可作为二项分布的近似（令 λ = np）。分布还具有可加性：若 X ~ Po(λ₁) 与 Y ~ Po(λ₂) 独立，则 X + Y ~ Po(λ₁ + λ₂)。
5. Normal Distribution | 正态分布
The normal distribution N(μ, σ²) is a continuous distribution with mean μ and variance σ². Its probability density is symmetric about μ, but the OCR course focuses on the standardised z‑score rather than the density function.
正态分布 N(μ, σ²) 是均值为 μ、方差为 σ² 的连续型分布，其概率密度关于 μ 对称，但 OCR 课程侧重标准化的 z 分数而非密度函数。
Z = (X − μ) / σ → Z ~ N(0, 1)
Probabilities are obtained from standard normal tables or inverse normal calculations. Key critical values to remember (two‑tailed): z₀.₀₅ = 1.96, z₀.₀₁ = 2.576, z₀.₁₀ = 1.645. When working with sample means, always state the distribution of the sample mean clearly before standardising.
使用标准正态表或逆正态计算求得概率。需记住常用双侧临界值：z₀.₀₅ = 1.96、z₀.₀₁ = 2.576、z₀.₁₀ = 1.645。处理样本均值时，先明确样本均值的分布再标准化。
6. Normal Approximations | 正态近似
When evaluating binomial or Poisson probabilities with large parameters, exact calculation becomes tedious. The normal distribution provides accurate approximations subject to continuity correction.
当二项或泊松分布的参数较大时，精确计算变得繁琐。正态分布可通过连续性校正提供准确的近似。
Binomial: X ~ B(n, p) ≈ N(np, np(1−p)) if np ≥ 5 and n(1−p) ≥ 5
Poisson: X ~ Po(λ) ≈ N(λ, λ) if λ ≥ 10 (rule of thumb)
Always apply a continuity correction: e.g. for P(X ≤ 12) use P(Z ≤ (12.5 − μ)/σ). This adjustment is essential for a discrete‑to‑continuous approximation and is examined regularly.
必须使用连续性校正，例如求 P(X ≤ 12) 时使用 P(Z ≤ (12.5 − μ)/σ)。这一调整对于离散到连续的近似至关重要，也是考试常考内容。
7. Sampling and the Central Limit Theorem | 抽样与中心极限定理
When a random sample of size n is drawn from any distribution with mean μ and variance σ², the sample mean X̄ has mean μ and variance σ²/n. If the population is normal, X̄ is exactly normally distributed.
从均值为 μ、方差为 σ² 的任意总体中抽取容量为 n 的随机样本，样本均值 X̄ 的期望为 μ，方差为 σ²/n。若总体服从正态分布，则 X̄ 也精确服从正态分布。
The Central Limit Theorem (CLT) states that for large n (usually n ≥ 30), X̄ is approximately normally distributed regardless of the population shape:
中心极限定理指出，当 n 足够大（通常 n ≥ 30）时，不论总体形状如何，X̄ 近似服从正态分布：
X̄ ~ N(μ, σ²/n) approximately
Unbiased estimators are used throughout: the sample mean estimates μ, and the sample variance s² = Σ(x − x̄)²/(n−1) estimates σ².
我们使用无偏估计量：样本均值估计 μ，样本方差 s² = Σ(x − x̄)²/(n−1) 估计 σ²。
8. Confidence Intervals for Means and Proportions | 均值和比例的置信区间
A confidence interval gives a range of plausible values for a population parameter. For a normal population with known variance σ², the 100(1 − α)% confidence interval for μ is:
置信区间给出总体参数的可能范围。已知方差 σ² 的正态总体，其均值 μ 的 100(1 − α)% 置信区间为：
x̄ ± z_{α/2} × σ / √n
When σ is unknown (the usual case), use the t‑distribution with n−1 degrees of freedom:
当 σ 未知（常见情形）时，使用自由度为 n−1 的 t 分布：
x̄ ± t_{n−1, α/2} × s / √n
For a population proportion p estimated by p̂ = x/n, a large‑sample confidence interval is:
对于由 p̂ = x/n 估计的总体比例 p，大样本置信区间为：
p̂ ± z_{α/2} × √(p̂(1−p̂) / n)
Interpreting the interval correctly – “we are 95% confident that the true mean lies between …” – is a regular examination requirement.
正确解读区间——“我们有 95% 的信心认为真实均值落在……之间”——是考试中的常见要求。
9. Hypothesis Testing – Binomial, Poisson, Normal | 假设检验 – 二项、泊松、正态
A hypothesis test decides whether sample evidence supports a stated null hypothesis H₀. Define H₁ as the alternative (one‑tailed or two‑tailed) and choose a significance level α, commonly 5% or 1%.
假设检验判断样本证据是否支持原假设 H₀。设定备择假设 H₁（单尾或双尾），选择显著性水平 α，常取 5% 或 1%。
For a binomial test on proportion p: calculate the probability of obtaining the observed result (or more extreme) assuming H₀ is true. Compare this p‑value with α, or find the critical region using P(reject | H₀ true) = α.
对比例 p 的二项检验：在 H₀ 为真时计算得到当前结果（或更极端）的概率。将此 p 值与 α 比较，或利用 P(拒绝 | H₀ 为真) = α 确定临界域。
For a Poisson mean λ the logic is identical, using the Poisson distribution under H₀: λ = λ₀. For a normal mean with known variance, the test statistic is Z = (x̄ − μ₀) / (σ/√n), referring to the standard normal distribution.
泊松均值 λ 的检验逻辑相同，基于 H₀ 下的泊松分布：λ = λ₀。已知方差时正态均值的检验统计量为 Z = (x̄ − μ₀) / (σ/√n)，参照标准正态分布。
10. Chi-Squared Tests | 卡方检验
Chi‑squared (χ²) tests are used for frequency data. There are two main types: goodness‑of‑fit test and test for independence (contingency tables). The test statistic is:
卡方 (χ²) 检验用于频数数据。主要分两种：拟合优度检验与独立性检验（列联表）。检验统计量为：
χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ
Goodness‑of‑fit: degrees of freedom ν = number of classes − 1 − m, where m is the number of parameters estimated from the data. For independence in an r×c table, ν = (r−1)(c−1). Expected frequencies Eᵢ must be at least 5 for the approximation to be valid; combine categories if necessary.
拟合优度检验自由度 ν = 组数 − 1 − m，m 为由数据估计的参数个数。r×c 列联表独立性检验的自由度 ν = (r−1)(c−1)。为保证近似有效，期望频数 Eᵢ 应不小于 5；必要时合并类别。
Reject H₀ if the calculated χ² exceeds the critical value from χ² tables at ν degrees of freedom. The test is always one‑tailed on the right because large discrepancies inflate the statistic.
若计算所得 χ² 大于自由度为 ν 的 χ² 分布临界值，则拒绝 H₀。该检验始终为右侧单尾检验，因为较大偏差会使统计量增大。
11. Correlation (PMCC and Spearman’s Rank) | 相关性 (皮尔逊积矩和斯皮尔曼秩)
Product‑moment correlation coefficient (Pearson’s r) measures linear correlation between two variables x and y. Its formula is:
积矩相关系数（皮尔逊 r）度量两个变量 x 和 y 之间的线性相关程度。公式如下：
r = S_{xy} / √(S_{xx} S_{yy})
where S_{xx} = Σx² − (Σx)²/n, S_{yy} = Σy² − (Σy)²/n, S_{xy} = Σxy − (Σx)(Σy)/n.
其中 S_{xx} = Σx² − (Σx)²/n，S_{yy} = Σy² − (Σy)²/n，S_{xy} = Σxy − (Σx)(Σy)/n。
Spearman’s rank correlation coefficient assesses monotonic association. Rank the data separately, then compute rₛ = 1 − 6 Σd² / [n(n² − 1)], where d is the difference in ranks for each pair. Spearman’s coefficient is robust to outliers and is tested using special critical values or, with large n, a normal approximation.
斯皮尔曼秩相关系数评估单调关联。分别对数据排序后计算 rₛ = 1 − 6 Σd² / [n(n² − 1)]，d 是每对数据的秩差。斯皮尔曼系数对异常值稳健，检验时使用专门的临界值表，或在大样本时用正态近似。
12. Linear Regression | 线性回归
Once a linear model y = a + bx is fitted, the least‑squares estimates for gradient and intercept are:
拟合线性模型 y = a + bx 后，斜率和截距的最小二乘估计为：
b = S_{xy} / S_{xx}
a = ȳ − b x̄
The regression line passes through the mean point (x̄, ȳ). Residuals eᵢ = yᵢ − (a + bxᵢ) should have no pattern and sum to zero. The strength of the linear association is measured by r², the coefficient of determination; it represents the proportion of variation in y explained by x.
回归直线经过均值点 (x̄, ȳ)。残差 eᵢ = yᵢ − (a + bxᵢ) 应无模式且总和为零。线性关联强度用决定系数 r² 衡量，它表示 y 的变异中由 x 解释的比例。
When using the regression equation for prediction, avoid extrapolation beyond the range of observed x‑values. Hypothesis tests for the gradient (H₀: β = 0) can be performed using the t‑distribution, linking regression to the earlier confidence interval techniques.
使用回归方程进行预测时，避免外推到观测 x 值范围之外。对斜率 (H₀: β = 0) 的假设检验可使用 t 分布，将回归与前面的置信区间方法联系起来。
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