Year 13 OCR Statistics: Formula & Theorem Quick Reference Handbook | Year 13 OCR 统计：公式定理速查手册

📚 Year 13 OCR Statistics: Formula & Theorem Quick Reference Handbook | Year 13 OCR 统计：公式定理速查手册

This handbook distills the essential formulas, theorems, and statistical procedures required for the Year 13 OCR A-Level Mathematics Statistics component. Whether you are revising for mock exams or the final paper, this resource organises key results from probability, discrete and continuous distributions, sampling, estimation, hypothesis testing, correlation, regression, and chi-squared analysis into one easily navigable reference.

这本手册提炼了 Year 13 OCR A-Level 数学统计部分必需的核心公式、定理和统计步骤。无论你是在准备模拟考试还是最终大考，这份资源都将概率、离散与连续分布、抽样、估计、假设检验、相关与回归以及卡方分析的关键结论整合为一本便于查阅的速查资料。

1. Probability and Combinatorics | 概率与组合基础

For any events A and B, the addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, then P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B). The conditional probability of A given B is P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0. Bayes’ theorem connects inverse conditionals: P(A | B) = [P(B | A) × P(A)] / P(B). Two events are independent if and only if P(A ∩ B) = P(A) × P(B). The number of ways to choose r items from n, denoted C(n, r) or nCr, is given by nCr = n! / [r!(n − r)!], where n! = n × (n−1) × … × 1.

对于任意事件 A 和 B，加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若 A 与 B 互斥，则 P(A ∩ B) = 0，故 P(A ∪ B) = P(A) + P(B)。给定 B 发生时 A 的条件概率为 P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。贝叶斯定理连接了相反的条件概率：P(A | B) = [P(B | A) × P(A)] / P(B)。两事件独立当且仅当 P(A ∩ B) = P(A) × P(B)。从 n 个物品中选取 r 个的组合数记为 C(n, r) 或 nCr，公式为 nCr = n! / [r!(n − r)!]，其中 n! = n × (n−1) × … × 1。

2. Binomial Distribution | 二项分布

If X ~ B(n, p), then X counts the number of successes in n independent trials, each with success probability p. The probability mass function is:

若 X ~ B(n, p)，则 X 表示 n 次独立试验中成功的次数，每次成功概率为 p。概率质量函数为：

P(X = r) = nCr × pʳ × (1 − p)ⁿ⁻ʳ, for r = 0, 1, 2, …, n.

Key summaries: E(X) = np, Var(X) = np(1 − p). The distribution is symmetric when p = 0.5, positively skewed when p < 0.5, and negatively skewed when p > 0.5. Use binomial tables or cumulative probabilities to calculate P(X ≤ r) and P(X ≥ r) = 1 − P(X ≤ r − 1).

关键概括：期望 E(X) = np，方差 Var(X) = np(1 − p)。当 p = 0.5 时分布对称；p < 0.5 时呈正偏态；p > 0.5 时呈负偏态。可使用二项分布表或累积概率计算 P(X ≤ r) 以及 P(X ≥ r) = 1 − P(X ≤ r − 1)。

Parameter	Formula
P(X = r)	nCr pʳ (1−p)ⁿ⁻ʳ
Mean	np
Variance	np(1−p)

3. Poisson Distribution | 泊松分布

The Poisson distribution models the number of events occurring in a fixed interval of time or space, assuming events occur independently at a constant average rate λ. If X ~ Po(λ), then:

泊松分布用于建模在固定时间或空间间隔内事件发生的次数，假设事件以恒定的平均速率 λ 独立发生。若 X ~ Po(λ)，则：

P(X = r) = (e^(−λ) × λʳ) / r!, for r = 0, 1, 2, …

Here e is Euler’s number. Both the mean and variance equal λ: E(X) = Var(X) = λ. The Poisson distribution is positively skewed but becomes more symmetric as λ increases. When λ is large (typically λ > 15), a normal approximation N(λ, λ) can be used. Cumulative probabilities are found from tables.

其中 e 为欧拉数。期望和方差均等于 λ：E(X) = Var(X) = λ。泊松分布呈正偏态，但随着 λ 增大趋于对称。当 λ 较大（通常 λ > 15）时，可采用正态近似 N(λ, λ)。累积概率可通过查表获得。

Parameter	Formula
P(X = r)	e^(-λ) λʳ / r!
Mean	λ
Variance	λ

4. Normal Distribution | 正态分布

The normal distribution N(μ, σ²) is a continuous, bell-shaped curve symmetrical about its mean μ. The probability density function is not used directly in OCR calculations; instead, we standardise the variable:

正态分布 N(μ, σ²) 是一条关于均值 μ 对称的钟形连续曲线。在 OCR 计算中不直接使用概率密度函数，而是将变量标准化：

Z = (X − μ) / σ

Z follows the standard normal distribution N(0, 1). Important empirical values: approximately 68% of data lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. To find probabilities, transform the boundary x to z = (x − μ) / σ and use the standard normal table. Remember that P(Z > z) = 1 − Φ(z) and by symmetry Φ(−z) = 1 − Φ(z).

Z 服从标准正态分布 N(0, 1)。重要的经验法则：约 68% 的数据落在均值 ±1σ 内，95% 落在 ±2σ 内，99.7% 落在 ±3σ 内。计算概率时，将边界 x 转换为 z = (x − μ) / σ，再查标准正态表。注意 P(Z > z) = 1 − Φ(z)，且由对称性 Φ(−z) = 1 − Φ(z)。

Inverse normal calculations find the value x satisfying P(X ≤ x) = p. First locate the z-value such that Φ(z) = p, then back-transform: x = μ + zσ.

逆向正态计算求满足 P(X ≤ x) = p 的 x 值。先找出使 Φ(z) = p 的 z 值，再回代：x = μ + zσ。

5. Continuous Uniform Distribution | 连续均匀分布

If X ~ U(a, b), the probability density function is constant over [a, b]: f(x) = 1/(b − a). The cumulative distribution function is F(x) = (x − a)/(b − a) for a ≤ x ≤ b. For any subinterval [c, d] within [a, b], P(c ≤ X ≤ d) = (d − c)/(b − a).

若 X ~ U(a, b)，概率密度函数在 [a, b] 上为常数：f(x) = 1/(b − a)。累积分布函数为 F(x) = (x − a)/(b − a)，a ≤ x ≤ b。对于 [a, b] 内的任意子区间 [c, d]，P(c ≤ X ≤ d) = (d − c)/(b − a)。

E(X) = (a + b) / 2, Var(X) = (b − a)² / 12

The median equals the mean, and the distribution is symmetric with no mode. It is frequently used to model measurement rounding errors or waiting times with fixed bounds.

中位数等于均值，分布对称且无极值。该分布常用于对给定界限内的测量舍入误差或等待时间建模。

6. Sampling and the Central Limit Theorem | 抽样与中心极限定理

A sample statistic is a random variable; its probability distribution is called a sampling distribution. For a random sample of size n from a normal population N(μ, σ²), the sample mean X̄ is distributed as N(μ, σ²/n). The standard error of the mean is σ/√n. The Central Limit Theorem (CLT) states that for any population with mean μ and variance σ², the sample mean X̄ for a large sample (often n ≥ 30) is approximately N(μ, σ²/n), regardless of the population shape.

样本统计量是一个随机变量，其概率分布称为抽样分布。对于来自正态总体 N(μ, σ²)、容量为 n 的随机样本，样本均值 X̄ 服从分布 N(μ, σ²/n)。均值的标准误为 σ/√n。中心极限定理指出：对于均值为 μ、方差为 σ² 的任意总体，当样本容量较大（通常 n ≥ 30）时，样本均值 X̄ 近似服从 N(μ, σ²/n)，无论总体分布形状如何。

When σ is unknown and n is large, the sample standard deviation s can be used to estimate the standard error as s/√n, and the normal approximation still holds. This underpins many confidence intervals and test procedures.

当 σ 未知且 n 较大时，可用样本标准差 s 估计标准误 s/√n，正态近似仍然成立。这为大量置信区间和检验方法提供了基础。

7. Confidence Intervals | 置信区间

A confidence interval gives a range of plausible values for a population parameter. For a population mean μ with known variance σ², the 95% (or 90%, 99%) confidence interval is:

置信区间给出了总体参数可能的取值范围。对于已知方差 σ² 的总体均值 μ，95%（或 90%, 99%）置信区间为：

x̄ ± z* × (σ / √n)

where z* is the critical value from N(0,1). Common values: 95% → z* = 1.960, 90% → z* = 1.645, 99% → z* = 2.576. Interpretation: If we take many samples and compute a 95% CI each time, 95% of those intervals will contain the true μ.

其中 z* 是 N(0,1) 的临界值。常用值：95% → z* = 1.960, 90% → z* = 1.645, 99% → z* = 2.576。解读：若多次抽样并每次计算 95% 置信区间，则 95% 的区间将包含真实的 μ。

When σ is unknown but n is large, replace σ with the sample standard deviation s. For a population proportion p, the CI for the binomial parameter uses p̂ = X/n and the approximate interval:

当 σ 未知但 n 很大时，可用样本标准差 s 代替 σ。对于总体比例 p，二项参数的置信区间用 p̂ = X/n 和近似区间：

p̂ ± z* × √[p̂(1 − p̂) / n]

This requires the success/failure condition np̂ ≥ 10 and n(1−p̂) ≥ 10 for the normal approximation to be valid.

这要求满足成功/失败条件 np̂ ≥ 10 且 n(1−p̂) ≥ 10，正态近似方可使用。

8. Hypothesis Testing: Binomial & Poisson | 假设检验：二项与泊松

A hypothesis test assesses evidence against a null hypothesis H₀. For a binomial test with H₀: p = p₀, the p-value is the probability of obtaining a result at least as extreme as the observed test statistic, assuming H₀ is true. For a one-tailed test, p-value = P(X ≥ x_obs) or P(X ≤ x_obs) depending on the direction; for two-tailed, double the smaller tail probability. Compare the p-value with the significance level α (commonly 0.05) – reject H₀ if p-value ≤ α.

假设检验评估反对零假设 H₀ 的证据。对于二项检验，H₀: p = p₀，p 值是在 H₀ 成立的条件下，获得至少与实际观测统计量一样极端结果的概率。单尾检验的 p 值 = P(X ≥ x_obs) 或 P(X ≤ x_obs)，视方向而定；双尾检验则将较小一尾的概率加倍。将 p 值与显著性水平 α（通常 0.05）比较，若 p 值 ≤ α 则拒绝 H₀。

The critical region is the set of values of X that lead to rejection. For example, in a two-tailed test at 5% significance, the critical regions are the lower 2.5% and upper 2.5% tails of the null distribution. Similar principles apply to Poisson tests with H₀: λ = λ₀; substitute the Poisson distribution for calculations.

临界域是导致拒绝 H₀ 的 X 值的集合。例如，在 5% 显著性水平的双尾检验中，临界域是零假设分布的下侧 2.5% 和上侧 2.5% 尾部。泊松检验 H₀: λ = λ₀ 采用相同原理，计算时替换为泊松分布即可。

Always state the conclusion in context: ‘there is (or is not) sufficient evidence at the α level to suggest that …’

结论务必放入背景中表述：‘在 α 水平上，有（或没有）足够证据表明……’

9. Hypothesis Testing: Normal Mean | 假设检验：正态均值

For a normal population with known σ, test H₀: μ = μ₀ using the test statistic:

对于已知 σ 的正态总体，用以下检验统计量检验 H₀: μ = μ₀：

Z = (x̄ − μ₀) / (σ / √n)

Compare Z with critical z-values or compute the p-value from N(0,1). If σ is unknown but n ≥ 30, replace σ with sample standard deviation s: Z ≈ (x̄ − μ₀) / (s / √n). The test remains approximately normal. Conditions: random sample, normality (or n large for CLT), and independence.

将 Z 值与临界 z 值比较，或基于 N(0,1) 计算 p 值。若 σ 未知但 n ≥ 30，以样本标准差 s 代替 σ：Z ≈ (x̄ − μ₀) / (s / √n)，检验仍近似正态。条件：随机样本、正态性（或大样本借助 CLT）以及独立性。

Be careful to identify one-tailed or two-tailed alternatives. For H₁: μ > μ₀

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Year 13 OCR Statistics: Formula & Theorem Quick Reference Handbook | Year 13 OCR 统计：公式定理速查手册