📚 Year 13 WJEC Statistics: Formula & Theorem Quick Reference | Year 13 WJEC 统计：公式定理速查手册
This quick-reference handbook brings together every essential formula and theorem examined in Year 13 WJEC Statistics. Use it to review probability rules, distributions, estimation, hypothesis tests, correlation, and regression before your exam.
本速查手册汇集了 Year 13 WJEC 统计考试中所有必备的公式与定理，涵盖概率规则、分布、估计、假设检验、相关与回归，适合考前集中复习。
1. Probability and Set Theory | 概率与集合论
For any two events A and B, the addition rule states that P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
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)。
If A and B are mutually exclusive, then 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)。
The conditional probability of A given B is P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.
在 B 发生的条件下 A 的条件概率为 P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。
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 相互独立，当且仅当 P(A ∩ B) = P(A) × P(B)，或等价地 P(A | B) = P(A)。
2. Discrete Random Variables and Expectation | 离散随机变量与期望
The expected value (mean) of a discrete random variable X is E(X) = μ = Σ x · P(X = x).
离散随机变量 X 的期望（均值）为 E(X) = μ = Σ x · P(X = x)。
The variance of X is Var(X) = σ² = Σ (x − μ)² P(X = x) = E(X²) − [E(X)]².
X 的方差为 Var(X) = σ² = Σ (x − μ)² P(X = x) = E(X²) − [E(X)]²。
For a linear transformation Y = aX + b, the mean and variance are E(Y) = a E(X) + b and Var(Y) = a² Var(X).
对于线性变换 Y = aX + b，均值和方差满足 E(Y) = a E(X) + b，Var(Y) = a² Var(X)。
The standard deviation is σ = √Var(X).
标准差为 σ = √Var(X)。
3. Binomial Distribution | 二项分布
If X ~ B(n, p), the probability of exactly r successes is P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, for r = 0, 1, 2, … , n.
若 X ~ B(n, p)，则恰好获得 r 次成功的概率为 P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，r = 0, 1, 2, … , n。
The binomial coefficient is ⁿCᵣ = n! / [r! (n − r)!].
二项式系数为 ⁿCᵣ = n! / [r! (n − r)!]。
The mean and variance of a binomial distribution are E(X) = np and Var(X) = np(1 − p).
二项分布的均值和方差为 E(X) = np，Var(X) = np(1 − p)。
Conditions: fixed number of trials, two outcomes per trial, constant probability of success, independent trials.
适用条件：试验次数固定、每次试验只有两种结果、成功概率恒定、各次试验相互独立。
4. Poisson Distribution | 泊松分布
If X ~ Po(λ), the probability of x occurrences is P(X = x) = (e⁻λ λˣ) / x! for x = 0, 1, 2, …
若 X ~ Po(λ)，则发生 x 次的概率为 P(X = x) = (e⁻λ λˣ) / x!，x = 0, 1, 2, …
The parameter λ is both the mean and the variance: E(X) = λ, Var(X) = λ.
参数 λ 既是均值也是方差：E(X) = λ，Var(X) = λ。
For large n and small p, a binomial distribution B(n, p) can be approximated by Po(λ) with λ = np. The approximation is suitable when n is large and p is small (typically n > 50 and np < 5).
当 n 很大且 p 很小时，二项分布 B(n, p) 可用泊松分布 Po(λ) 近似，其中 λ = np。通常要求 n > 50 且 np < 5。
The sum of independent Poisson variables also follows a Poisson distribution: if X ~ Po(λ₁) and Y ~ Po(λ₂) are independent, then X + Y ~ Po(λ₁ + λ₂).
独立的泊松变量之和仍服从泊松分布：若 X ~ Po(λ₁) 与 Y ~ Po(λ₂) 独立，则 X + Y ~ Po(λ₁ + λ₂)。
5. Normal Distribution | 正态分布
If X ~ N(μ, σ²), the standardised variable Z = (X − μ) / σ follows the standard normal distribution N(0, 1).
若 X ~ N(μ, σ²)，则标准化变量 Z = (X − μ) / σ 服从标准正态分布 N(0, 1)。
Probabilities are found using tables of Φ(z) = P(Z ≤ z). Remember P(Z > z) = 1 − Φ(z) and P(Z < −z) = Φ(−z) = 1 − Φ(z).
使用标准正态分布表 Φ(z) = P(Z ≤ z) 求概率。注意 P(Z > z) = 1 − Φ(z)，P(Z < −z) = Φ(−z) = 1 − Φ(z)。
When using the normal approximation to the binomial or Poisson, apply a continuity correction. For a discrete value k, use the interval (k − 0.5, k + 0.5) to improve accuracy.
用正态分布近似二项分布或泊松分布时，需应用连续性校正。对于离散值 k，使用区间 (k − 0.5, k + 0.5) 以提高精度。
For the binomial B(n, p), the normal approximation N(np, np(1 − p)) is valid when np > 5 and n(1 − p) > 5.
对于二项分布 B(n, p)，当 np > 5 且 n(1 − p) > 5 时，可用正态分布 N(np, np(1 − p)) 进行近似。
6. Sampling and the Central Limit Theorem | 抽样与中心极限定理
A statistic is a function of the sample data; the sample mean X̄ is a random variable with its own distribution called the sampling distribution.
统计量是样本数据的函数；样本均值 X̄ 是一个随机变量，其概率分布称为抽样分布。
If the population is normal, X̄ ~ N(μ, σ²/n) exactly for any sample size n.
若总体服从正态分布，则对于任意样本量 n，均有 X̄ ~ N(μ, σ²/n)。
The Central Limit Theorem (CLT) states that for a large sample size (usually n ≥ 30), the distribution of X̄ is approximately normal regardless of the population shape, with mean μ and variance σ²/n.
中心极限定理指出，当样本量足够大（通常 n ≥ 30）时，无论总体分布形状如何，样本均值的分布近似正态，均值为 μ，方差为 σ²/n。
The standard error of the sample mean is SE = σ/√n. When σ is unknown, it is estimated by s/√n using the sample standard deviation s.
样本均值的标准误为 SE = σ/√n。当 σ 未知时，用样本标准差 s 估计，即 s/√n。
7. Confidence Intervals | 置信区间
A 95% confidence interval for the population mean μ when σ is known is given by x̄ ± z₀.₀₂₅ × (σ/√n), where z₀.₀₂₅ = 1.96 from the standard normal distribution.
当总体标准差 σ 已知时，μ 的 95% 置信区间为 x̄ ± z₀.₀₂₅ × (σ/√n)，其中 z₀.₀₂₅ = 1.96 来自标准正态分布。
When σ is unknown, use the t-distribution with n − 1 degrees of freedom: x̄ ± t₀.₀₂₅, ₙ₋₁ × (s/√n).
当 σ 未知时，使用自由度为 n − 1 的 t 分布：x̄ ± t₀.₀₂₅, ₙ₋₁ × (s/√n)。
A confidence interval for a population proportion p is p̂ ± zₐ/₂ × √[p̂(1 − p̂)/n], where p̂ is the sample proportion. The margin of error must satisfy the condition that n p̂ and n(1 − p̂) are both at least 10.
总体比例 p 的置信区间为 p̂ ± zₐ/₂ × √[p̂(1 − p̂)/n]，其中 p̂ 为样本比例。要求 n p̂ 和 n(1 − p̂) 都至少为 10，以保证正态近似有效。
The width of the interval decreases as the sample size increases; for a given confidence level, larger samples give more precise estimates.
区间宽度随样本量增大而减小；在给定置信水平下，样本越大估计越精确。
8. Hypothesis Testing | 假设检验
A hypothesis test involves stating a null hypothesis H₀ and an alternative H₁. The test statistic is computed under the assumption that H₀ is true.
假设检验需要陈述原假设 H₀ 与备择假设 H₁。检验统计量是在假定 H₀ 成立的条件下计算的。
For a test of a population mean with known σ, use the z-statistic: z = (x̄ − μ₀) / (σ/√n).
在 σ 已知时对总体均值进行检验，使用 z 统计量：z = (x̄ − μ₀) / (σ/√n)。
When σ is unknown, use the t-statistic: t = (x̄ − μ₀) / (s/√n), which follows a t-distribution with n − 1 degrees of freedom under H₀.
当 σ 未知时，使用 t 统计量：t = (x̄ − μ₀) / (s/√n)，在原假设下服从自由度为 n − 1 的 t 分布。
For a proportion, the test statistic is z = (p̂ − p₀) / √[p₀(1 − p₀)/n], compared with critical values from N(0,1).
对于比例的检验，检验统计量为 z = (p̂ − p₀) / √[p₀(1 − p₀)/n]，与标准正态分布的临界值进行比较。
The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, given that H₀ is true. Reject H₀ if p-value < significance level α.
p 值是在 H₀ 成立时，获得比观测值更极端检验统计量的概率。若 p 值 < 显著性水平 α，则拒绝 H₀。
Type I error (α) is rejecting a true H₀; Type II error (β) is failing to reject a false H₀. The power of a test is 1 − β.
第一类错误（α）是拒绝了真实的 H₀；第二类错误（β）是没有拒绝错误的 H₀。检验的功效为 1 − β。
9. Chi-Squared Tests | 卡方检验
The chi-squared test statistic is X² = Σ [(O − E)² / E], where O are observed frequencies and E are expected frequencies under the null hypothesis.
卡方检验统计量为 X² = Σ [(O − E)² / E]，其中 O 为观测频数，E 为在原假设下的期望频数。
For a goodness-of-fit test, degrees of freedom = number of categories − 1 − number of estimated parameters.
对于拟合优度检验，自由度 = 类别数 − 1 − 被估计参数的个数。
In a contingency table test for independence, degrees of freedom = (r − 1)(c − 1) where r is the number of rows and c the number of columns. Expected frequency = (row total × column total) / grand total.
在列联表独立性检验中，自由度 = (r − 1)(c − 1)，其中 r 为行数，c 为列数。期望频数 = (行合计 × 列合计) / 总计。
The test is valid only when all expected frequencies are at least 5. If this condition is not met, categories should be combined.
该检验仅在所有期望频数均不小于 5 时有效。若不满足，应将类别合并。
10. Correlation and Regression | 相关与回归
The product-moment correlation coefficient (PMCC) is r = Sₓᵧ / √(Sₓₓ Sᵧᵧ), where Sₓᵧ = Σ (x − x̄)(y − ȳ), Sₓₓ = Σ (x − x̄)² and Sᵧᵧ = Σ (y − ȳ)².
积矩相关系数 (PMCC) 为 r = Sₓᵧ / √(Sₓₓ Sᵧᵧ)，其中 Sₓᵧ = Σ (x − x̄)(y − ȳ)，Sₓₓ = Σ (x − x̄)²，Sᵧᵧ = Σ (y − ȳ)²。
Spearman’s rank correlation coefficient rₛ is computed by ranking the data and applying the same formula to the ranks. It is used when the relationship is monotonic but not necessarily linear.
Spearman 等级相关系数 rₛ 是通过对数据排序并将相同公式应用于等级计算得到的。它适用于单调但不一定是线性的关系。
The least squares regression line of y on x is y = a + b x, where b = Sₓᵧ / Sₓₓ and a = ȳ − b x̄.
y 对 x 的最小二乘回归直线为 y = a + b x，其中 b = Sₓᵧ / Sₓₓ，a = ȳ − b x̄。
A residual is the difference between an observed value and the value predicted by the regression line: residual = y − ŷ. A residual plot should show no pattern if the linear model is appropriate.
残差是观测值与回归线预测值之差：残差 = y − ŷ。若线性模型合适，残差图应不呈现任何模式。
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