AS WJEC Statistics: Formula and Theorem Quick Reference Handbook | AS WJEC 统计：公式定理速查手册

📚 AS WJEC Statistics: Formula and Theorem Quick Reference Handbook | AS WJEC 统计：公式定理速查手册

This comprehensive reference guide brings together all the essential formulas, theorems, and key concepts required for the AS WJEC Statistics specification. Whether you are revising probability laws, calculating confidence intervals, or interpreting hypothesis tests, this handbook provides a concise and exam‑ready summary. Each section pairs an English explanation with a parallel Chinese version, ensuring clarity for learners from diverse backgrounds. Familiar symbols and standard notation are used throughout, with no reliance on LaTeX, making this guide easily readable on any device.

这份速查手册汇集了 AS WJEC 统计课程要求的所有核心公式、定理和关键概念。无论你是在复习概率法则、计算置信区间，还是解读假设检验，这本手册都能提供简洁、贴合考纲的总结。每个部分均采用英文与中文逐段对照，确保不同背景的学习者都能清晰理解。全文使用标准符号和常见记法，不依赖 LaTeX，方便在任何设备上阅读。

1. Basic Probability Formulas | 基本概率公式

For any event A, the probability P(A) satisfies 0 ≤ P(A) ≤ 1. The sum of probabilities of all mutually exclusive and exhaustive outcomes in a sample space is 1.

对于任何事件 A，概率 P(A) 满足 0 ≤ P(A) ≤ 1。样本空间中所有互斥且穷举的结果的概率之和为 1。

The complement rule: P(A’) = 1 – P(A), where A’ denotes ‘not A’. The addition law for any two events: 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).

互补法则：P(A’) = 1 – P(A)，其中 A’ 表示“非 A”。任意两个事件的加法法则：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若 A 与 B 互斥，则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。

Key formulas at a glance:

核心公式一览：

0 ≤ P(A) ≤ 1
Σ P(outcome) = 1
P(not A) = 1 – P(A)
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

2. Conditional Probability and Independence | 条件概率与独立性

Conditional probability measures the likelihood of event A given that event B has occurred. It is defined as:
P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

条件概率衡量在事件 B 已经发生的条件下事件 A 发生的可能性。其定义为：
P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。

The multiplication law follows directly: P(A ∩ B) = P(A | B) × P(B) = P(B | A) × P(A). For independent events, the occurrence of one does not affect the probability of the other. A and B are independent if and only if P(A ∩ B) = P(A) × P(B), or equivalently P(A | B) = P(A) when P(B) > 0.

乘法法则直接推出：P(A ∩ B) = P(A | B) × P(B) = P(B | A) × P(A)。对于独立事件，一个事件的发生不影响另一个事件的概率。A 与 B 独立当且仅当 P(A ∩ B) = P(A) × P(B)，或者在 P(B) > 0 时等价于 P(A | B) = P(A)。

Tree diagrams often help visualise conditional sequences. Multiply along branches for joint probabilities; add relevant branch‑end probabilities for combined events.

树状图通常有助于直观展示条件序列。沿分支相乘得到联合概率；将相关分支末端概率相加可计算组合事件的概率。

3. Expectation and Variance of Discrete Random Variables | 离散随机变量的期望与方差

For a discrete random variable X taking values xᵢ with probabilities pᵢ = P(X = xᵢ), the expected value (mean) is:
E(X) = Σ xᵢ pᵢ

对于一个取值 xᵢ、对应概率为 pᵢ = P(X = xᵢ) 的离散随机变量 X，其期望值（均值）为：
E(X) = Σ xᵢ pᵢ

The variance measures spread and is defined as Var(X) = E[(X – μ)²] where μ = E(X). A simpler computational formula is:
Var(X) = E(X²) – [E(X)]², where E(X²) = Σ xᵢ² pᵢ

方差衡量离散程度，定义为 Var(X) = E[(X – μ)²]，其中 μ = E(X)。更简便的计算公式为：
Var(X) = E(X²) – [E(X)]²，其中 E(X²) = Σ xᵢ² pᵢ

The standard deviation is the positive square root: σ = √Var(X). For any constants a and b, the following linear transformation rules hold: E(aX + b) = aE(X) + b, and Var(aX + b) = a² Var(X).

标准差是方差的正平方根：σ = √Var(X)。对于任意常数 a 和 b，下列线性变换规则成立：E(aX + b) = aE(X) + b，Var(aX + b) = a² Var(X)。

4. The Binomial Distribution | 二项分布

A binomial distribution models the number of successes in a fixed number n of independent Bernoulli trials, each with constant success probability p. If X ~ B(n, p), the probability of exactly r successes is:
P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ, where ⁿCᵣ = n! / [r! (n – r)!] and r = 0, 1, 2, …, n.

二项分布模拟在固定次数 n 的独立伯努利试验中成功的次数，每次试验的成功概率恒为 p。若 X ~ B(n, p)，则恰好有 r 次成功的概率为：
P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ，其中 ⁿCᵣ = n! / [r! (n – r)!]，r = 0, 1, 2, …, n。

Key summary values: mean μ = np, variance σ² = np(1 – p). The shape of the distribution is symmetric when p = 0.5, skewed for other p values, and becomes more symmetric as n increases.

关键汇总值：均值 μ = np，方差 σ² = np(1 – p)。当 p = 0.5 时分布对称，其他 p 值时偏斜；随着 n 增大，分布趋于对称。

Binomial probabilities can be found using statistical tables, calculators, or the formula. Cumulative probabilities P(X ≤ r) are often required for hypothesis testing.

二项概率可用统计表格、计算器或公式求得。累积概率 P(X ≤ r) 常用于假设检验。

5. The Poisson Distribution | 泊松分布

The Poisson distribution models the number of randomly occurring events in a fixed interval of time or space when events happen independently at a constant average rate λ. If X ~ Po(λ), then:
P(X = r) = e⁻λ λʳ / r!, for r = 0, 1, 2, …

泊松分布模拟在固定时间或空间间隔内随机发生的事件次数，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!，r = 0, 1, 2, …

For a Poisson random variable, mean = variance = λ, i.e. E(X) = Var(X) = λ. This equality is a hallmark property used to check if data follow a Poisson distribution.

对于泊松随机变量，均值 = 方差 = λ，即 E(X) = Var(X) = λ。这一相等性是判断数据是否服从泊松分布的关键特征。

The Poisson distribution can approximate a binomial B(n, p) when n is large and p is small, setting λ = np. Typically, the approximation works well when n > 50 and np < 5.

当 n 很大且 p 很小时，泊松分布可近似二项分布 B(n, p)，取 λ = np。通常 n > 50 且 np < 5 时近似效果良好。

6. Continuous Random Variables and Probability Density Functions | 连续随机变量与概率密度函数

A continuous random variable X has a probability density function (pdf) f(x) such that the total area under the curve equals 1: ∫ f(x) dx = 1 over the domain. Probabilities are found as areas: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.

连续随机变量 X 具有概率密度函数 f(x)，使得曲线下的总面积为 1：在定义域上 ∫ f(x) dx = 1。概率通过面积求得：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。

The cumulative distribution function (cdf) is F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt. For any continuous variable, P(X = c) = 0 for a single value c. The mean (expectation) and variance are defined as E(X) = ∫ x f(x) dx, Var(X) = ∫ (x – μ)² f(x) dx = E(X²) – μ², over the specified range.

累积分布函数为 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。对于任何连续变量，单个值 c 的概率 P(X = c) = 0。均值（期望）和方差定义为：在指定范围内 E(X) = ∫ x f(x) dx，Var(X) = ∫ (x – μ)² f(x) dx = E(X²) – μ²。

In the WJEC AS exam, you may be given a piecewise linear pdf and required to find probabilities, medians, or quartiles by integration or using areas of triangles and rectangles.

在 WJEC AS 考试中，可能会给出一分段线性概率密度函数，要求通过积分或利用三角形、矩形面积求概率、中位数或四分位数。

7. The Normal Distribution | 正态分布

The normal distribution N(μ, σ²) is a symmetric bell‑shaped continuous distribution characterised by its mean μ and variance σ². The standard normal variable Z = (X – μ) / σ follows N(0, 1).

正态分布 N(μ, σ²) 是一种对称的钟形连续分布，由均值 μ 和方差 σ² 决定。标准正态变量 Z = (X – μ) / σ 服从 N(0, 1)。

Probability calculations are performed by standardising X to Z and using the standard normal table: P(X < a) = Φ((a – μ) / σ). The symmetry property Φ(–z) = 1 – Φ(z) is frequently needed.

概率计算通过将 X 标准化为 Z 并查标准正态表完成：P(X < a) = Φ((a – μ) / σ)。对称性质 Φ(–z) = 1 – Φ(z) 需要经常使用。

Important tail percentages: about 68% within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. For WJEC, the exam provides the relevant normal tables; you must know how to interpolate or extract critical values.

重要的尾部百分比：约 68% 位于均值 ±1σ 内，95% 位于 ±2σ 内，99.7% 位于 ±3σ 内。WJEC 考试会提供相关的正态分布表；考生需掌握如何内插或提取临界值。

8. Sampling Distributions and the Central Limit Theorem | 抽样分布与中心极限定理

A statistic (e.g. sample mean x̄) varies from sample to sample. The distribution of a statistic is called its sampling distribution. For a sample of size n from a N(μ, σ²) population, the sample mean follows an exact normal distribution: x̄ ~ N(μ, σ²/n). The standard error of the mean is σ/√n.

统计量（如样本均值 x̄）随样本不同而变化。统计量的分布称为抽样分布。对来自 N(μ, σ²) 总体的容量为 n 的样本，样本均值严格服从正态分布：x̄ ~ N(μ, σ²/n)。均值的标准误为 σ/√n。

The Central Limit Theorem (CLT) states that for a large enough sample (usually n ≥ 30), the sample mean is approximately normally distributed regardless of the population distribution, provided the population has a finite variance.

中心极限定理指出，当样本容量足够大（通常 n ≥ 30）时，无论总体分布如何，只要总体方差有限，样本均值近似服从正态分布。

When the population variance is unknown, the sample variance s² is used, and the t‑distribution may be needed; however, for AS WJEC, the normal approximation is typically sufficient when n ≥ 30 and σ is replaced by s.

当总体方差未知时，使用样本方差 s²，可能需要 t 分布；但在 AS WJEC 中，通常当 n ≥ 30 且用 s 代替 σ 时，正态近似已经足够。

9. Confidence Intervals | 置信区间

A confidence interval provides a range of plausible values for an unknown population parameter. For the population mean μ with known variance σ², a 95% confidence interval based on a sample mean x̄ from a normal distribution is:
x̄ ± z × σ/√n, where z = 1.96 for 95% confidence.

置信区间给出未知总体参数的可能取值范围。对于已知方差 σ² 的总体均值 μ，基于正态分布样本均值 x̄ 的 95% 置信区间为：
x̄ ± z × σ/√n，其中对于 95% 置信水平，z = 1.96。

If the population variance is unknown and the sample is large (n ≥ 30), the normal confidence interval can still be used with σ replaced by the sample standard deviation s. Common z‑values: 90% → 1.645, 95% → 1.96, 99% → 2.576. The width of the interval decreases with larger n or lower confidence level.

若总体方差未知且样本量大（n ≥ 30），仍可使用正态置信区间，并用样本标准差 s 代替 σ。常见 z 值：90% → 1.645，95% → 1.96，99% → 2.576。区间宽度随 n 增大或置信水平降低而减小。

Interpretation: if we repeatedly take samples and construct 95% confidence intervals, about 95% of them would contain the true μ.

解读：若重复抽样并构造 95% 置信区间，其中约有 95% 的区间会包含真实的 μ。

10. Hypothesis Testing | 假设检验

A hypothesis test assesses the evidence against a null hypothesis H₀ in favour of an alternative H₁. The test statistic measures how far the observed sample result deviates from what is expected under H₀. The p‑value is the probability of obtaining a result at least as extreme as the observed one, assuming H₀ is true. For a binomial test, the critical region is determined using the binomial distribution with p = p₀ under H₀.

假设检验评估数据对原假设 H₀ 的支持程度，并考虑备择假设 H₁。检验统计量衡量观察到的样本结果偏离 H₀ 预期值的程度。p 值是在 H₀ 为真时，得到至少与观测结果同样极端的结果的概率。对于二项检验，利用 H₀ 下 p = p₀ 的二项分布确定临界区域。

For a Poisson test of rate λ, the test statistic is the observed count and the distribution is Po(λ₀). For a normal test of a population mean, the test statistic is Z = (x̄ – μ₀) / (σ/√n), compared against critical z‑values. The significance level α (usually 5% or 1%) defines the rejection region. If the p‑value < α, reject H₀.

对于比率 λ 的泊松检验，检验统计量为观测计数，分布为 Po(λ₀)。对总体均值的正态检验，检验统计量为 Z = (x̄ – μ₀) / (σ/√n)，并与临界 z 值比较。显著性水平 α（通常 5% 或 1%）定义拒绝域。若 p 值 < α，则拒绝 H₀。

Remember to state conclusions in context: ‘There is sufficient evidence at the 5% level to suggest that…’ or ‘There is insufficient evidence…’.

记得结合背景陈述结论：“在 5% 水平上有充分证据表明……”或“证据不足，无法表明……”。

11. Correlation and Regression | 相关与回归

Pearson’s product‑moment correlation coefficient r measures the strength of a linear relationship between two variables. It is given by:
r = Sxy / √(Sxx Syy), where Sxx = Σ(x – x̄)², Syy = Σ(y – ȳ)², Sxy = Σ(x – x̄)(y – ȳ).

皮尔逊积矩相关系数 r 衡量两个变量之间线性关系的强度。公式为：
r = Sxy / √(Sxx Syy)，其中 Sxx = Σ(x – x̄)²，Syy = Σ(y – ȳ)²，Sxy = Σ(x – x̄)(y – ȳ)。

The value of r lies between –1 and 1. An r close to 1 indicates strong positive correlation, close to –1 strong negative correlation, and near 0 weak or no linear correlation. Note that correlation does not imply causation.

r 的取值范围在 –1 到 1 之间。r 接近 1 表示强正相关，接近 –1 表示强负相关，接近 0 表示弱或无线性相关。注意，相关并不意味着因果。

The equation of the least‑squares regression line of y on x is y = a + bx, where slope b = Sxy / Sxx and intercept a = ȳ – b x̄. This line is used to predict y from x, but extrapolation beyond the data range is unreliable.

y 对 x 的最小二乘回归直线方程为 y = a + bx，其中斜率 b = Sxy / Sxx，截距 a = ȳ – b x̄。该直线用于从 x 预测 y，但在数据范围之外进行外推是不可靠的。

12. Quick Reference of Key Distribution Parameters | 关键分布参数速查

Here is a summary of the most important distributions in AS WJEC Statistics and their parameters, means, and variances.

以下是 AS WJEC 统计中最重要的分布及其参数、均值和方差的总结。

Distribution	Parameters	Mean	Variance
Binomial	n, p	np	np(1 – p)
Poisson	λ	λ	λ
Normal	μ, σ²	μ	σ²
Standard Normal	0, 1	0	1

Also remember the critical values for two‑tailed tests: z = 1.96 at 5% significance, z = 2.576 at 1%. For one‑tailed tests: 1.645 (5%) and 2.326 (1%). These values are essential for both confidence intervals and hypothesis tests involving the normal distribution.

同时记住双侧检验的临界值：5% 显著性水平下 z = 1.96，1% 下 z = 2.576。单侧检验：1.645（5%）和 2.326（1%）。这些值对于涉及正态分布的置信区间和假设检验都不可或缺。

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AS WJEC Statistics: Formula and Theorem Quick Reference Handbook | AS WJEC 统计：公式定理速查手册