AQA A-Level Statistics: Formulae & Theorems Quick Reference | AQA A-Level 统计：公式定理速查手册

📚 AQA A-Level Statistics: Formulae & Theorems Quick Reference | AQA A-Level 统计：公式定理速查手册

This handbook summarises the essential formulae and theorems required for the Year 13 AQA Statistics specification. It covers discrete and continuous distributions, sampling theory, confidence intervals, hypothesis testing, correlation and regression, and the chi-squared test. Each section is presented in paired English–Chinese paragraphs to support bilingual learners and revision. Use this as a quick-reference guide alongside your class notes and past papers.

本手册总结了 Year 13 AQA 统计课程所需的核心公式和定理，涵盖离散与连续分布、抽样理论、置信区间、假设检验、相关与回归以及卡方检验。每个部分均采用英中对照的段落形式，以帮助双语学习者复习巩固。可配合课堂笔记和历年真题用作快速查阅指南。

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1. Key Discrete Distributions | 关键离散分布

A discrete random variable X takes a countable number of values. In AQA Statistics the main discrete distributions are the binomial and the Poisson. For any discrete distribution, the sum of all probabilities equals 1: Σ P(X = x) = 1. The expected value (mean) is E(X) = Σ x P(X = x), and the variance is Var(X) = E(X²) – [E(X)]².

离散型随机变量 X 取可数个值。在 AQA 统计中主要离散分布是二项分布和泊松分布。对任何离散分布，所有概率之和为 1：Σ P(X = x) = 1。期望值（均值）为 E(X) = Σ x P(X = x)，方差为 Var(X) = E(X²) – [E(X)]²。

When working with probability mass functions (PMFs), always check that the probabilities are valid (0 ≤ P(X = x) ≤ 1) and that the total probability sums to 1. The mode is the value with the highest probability.

使用概率质量函数（PMF）时，一定要检查概率是否有效（0 ≤ P(X = x) ≤ 1）且总概率之和为 1。众数是具有最高概率的取值。

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2. Binomial Distribution | 二项分布

If X ~ B(n, p) then X represents the number of successes in n independent trials, each with probability of success p. The probability mass function is P(X = x) = ⁿCₓ pˣ (1 – p)ⁿ⁻ˣ, for x = 0, 1, 2, …, n. The binomial coefficient ⁿCₓ = n! / [x!(n – x)!].

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 = x) = ⁿCₓ pˣ (1 – p)ⁿ⁻ˣ，其中 x = 0, 1, 2, …, n。二项式系数 ⁿCₓ = n! / [x!(n – x)!]。

The mean and variance of a binomial distribution are E(X) = np and Var(X) = np(1 – p). If p is close to 0 or 1 the distribution is skewed; when p = 0.5 it is symmetric. Binomial probabilities can be calculated using the formula, tables, or a calculator.

二项分布的均值和方差分别为 E(X) = np 和 Var(X) = np(1 – p)。当 p 接近 0 或 1 时分布呈偏态；当 p = 0.5 时为对称分布。二项概率可使用公式、表格或计算器计算。

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3. Poisson Distribution | 泊松分布

A random variable X follows a Poisson distribution with parameter λ (λ > 0), written X ~ Po(λ), if it counts the number of events occurring in a fixed interval of time or space, provided events are independent and occur at a constant average rate. The probability function is P(X = x) = e⁻λ λˣ / x!, for x = 0, 1, 2, ….

若随机变量 X 表示在固定时间或空间间隔内发生的事件数，且事件独立并以恒定的平均速率发生，则 X 服从参数为 λ 的泊松分布，记作 X ~ Po(λ)，其中 λ > 0。概率函数为 P(X = x) = e⁻λ λˣ / x!，x = 0, 1, 2, …。

The mean and variance of a Poisson distribution are both equal to λ: E(X) = Var(X) = λ. The Poisson distribution can approximate the binomial B(n, p) when n is large and p is small, using λ = np.

泊松分布的均值和方差都等于 λ：E(X) = Var(X) = λ。当 n 很大而 p 很小时，泊松分布可用来近似二项分布 B(n, p)，此时取 λ = np。

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4. Continuous Probability Distributions | 连续概率分布

For a continuous random variable, probabilities are given by areas under a probability density function (PDF), f(x). The total area under the curve is 1. Probabilities are found by integration: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. The cumulative distribution function (CDF) is F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt.

对于连续型随机变量，概率由概率密度函数 (PDF) f(x) 下的面积给出。曲线下的总面积为 1。概率通过积分求得：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。累积分布函数 (CDF) 为 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。

The mean or expected value is E(X) = ∫ x f(x) dx over the domain of X. The variance is Var(X) = ∫ (x – μ)² f(x) dx = E(X²) – μ². The median m satisfies F(m) = 0.5.

均值或期望值为 E(X) = ∫ x f(x) dx（在 X 的定义域上）。方差为 Var(X) = ∫ (x – μ)² f(x) dx = E(X²) – μ²。中位数 m 满足 F(m) = 0.5。

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5. Normal Distribution | 正态分布

The normal distribution with mean μ and variance σ² is denoted N(μ, σ²). Its PDF is f(x) = (1/(σ√(2π))) e⁻⁽ˣ⁻μ⁾²⁄⁽²σ²⁾. The standard normal distribution is N(0, 1). To find probabilities for X ~ N(μ, σ²), standardise using Z = (X – μ)/σ, then use standard normal tables.

均值为 μ、方差为 σ² 的正态分布记作 N(μ, σ²)。其概率密度函数为 f(x) = (1/(σ√(2π))) e⁻⁽ˣ⁻μ⁾²⁄⁽²σ²⁾。标准正态分布为 N(0, 1)。对于 X ~ N(μ, σ²)，先通过 Z = (X – μ)/σ 将其标准化，再查找标准正态分布表。

Key properties: symmetric about μ; mean = median = mode; about 68% of data within 1σ of μ, 95% within 2σ, and 99.7% within 3σ. For a sample mean x̄ from a normal population, x̄ ~ N(μ, σ²/n). If the population is not normal, the Central Limit Theorem states that x̄ is approximately normal for large n.

关键性质：关于 μ 对称；均值 = 中位数 = 众数；大约 68% 的数据在 μ ± 1σ 内，95% 在 μ ± 2σ 内，99.7% 在 μ ± 3σ 内。对于来自正态总体的样本均值 x̄，有 x̄ ~ N(μ, σ²/n)。若总体非正态，由中心极限定理，当 n 较大时 x̄ 近似正态。

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

When random samples of size n are drawn from a population with mean μ and variance σ², the sample mean x̄ has expected value μ and standard error σ/√n. The distribution of x̄ depends on the population distribution and the sample size.

从均值为 μ、方差为 σ² 的总体中抽取容量为 n 的随机样本，样本均值 x̄ 的期望值为 μ，标准误差为 σ/√n。x̄ 的分布取决于总体分布和样本容量。

The Central Limit Theorem (CLT) says that for sufficiently large n (usually n ≥ 30), the sample mean x̄ is approximately normally distributed regardless of the shape of the population distribution. For a proportion p̂ from a binomial distribution, p̂ ≈ N(p, p(1 – p)/n) when np ≥ 5 and n(1 – p) ≥ 5.

中心极限定理 (CLT) 指出，当 n 足够大时（通常 n ≥ 30），无论总体分布形状如何，样本均值 x̄ 都近似正态分布。对于来自二项分布的样本比例 p̂，当 np ≥ 5 且 n(1 – p) ≥ 5 时，p̂ ≈ N(p, p(1 – p)/n)。

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7. Confidence Intervals | 置信区间

A confidence interval gives a range of plausible values for a population parameter. For a normal population with known variance, a 95% confidence interval for the mean μ is x̄ ± z₀.₀₂₅ × (σ/√n), where z₀.₀₂₅ ≈ 1.96. When σ is unknown, use the t-distribution: x̄ ± tₙ₋₁, α/₂ × (s/√n).

置信区间给出了总体参数的一个合理取值范围。对于方差已知的正态总体，均值 μ 的 95% 置信区间为 x̄ ± z₀.₀₂₅ × (σ/√n)，其中 z₀.₀₂₅ ≈ 1.96。当 σ 未知时，使用 t 分布：x̄ ± tₙ₋₁, α/₂ × (s/√n)。

For a population proportion p, an approximate confidence interval is p̂ ± z × √[p̂(1 – p̂)/n]. The width of the interval decreases with larger sample size and increases with higher confidence level. Interpretation: if we repeated the sampling process many times, about 95% of such intervals would contain the true parameter.

对于总体比例 p，近似置信区间为 p̂ ± z × √[p̂(1 – p̂)/n]。区间宽度随样本量增大而缩小，随置信水平提高而增大。解释：如果多次重复抽样过程，大约 95% 的此类区间会包含真实参数。

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8. Hypothesis Testing – One Sample | 单样本假设检验

A hypothesis test evaluates evidence against a null hypothesis H₀. The alternative hypothesis H₁ can be one-tailed or two-tailed. The test statistic is computed from the sample and compared to a critical value, or a p-value is found. If p-value < significance level α, reject H₀.

假设检验用于评估反对原假设 H₀ 的证据。备择假设 H₁ 可以是单尾或双尾的。通过样本计算检验统计量，并与临界值比较，或者计算 p 值。若 p 值 < 显著性水平 α，则拒绝 H₀。

For a normal mean with known variance: z = (x̄ – μ₀)/(σ/√n). With unknown variance: t = (x̄ – μ₀)/(s/√n) with df = n – 1. For a binomial proportion: z = (p̂ – p₀)/√[p₀(1 – p₀)/n]. Always state the conclusion in context.

方差已知时正态均值的检验：z = (x̄ – μ₀)/(σ/√n)。方差未知时：t = (x̄ – μ₀)/(s/√n)，自由度为 n – 1。二项比例的检验：z = (p̂ – p₀)/√[p₀(1 – p₀)/n]。记得结合情境给出结论。

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9. Hypothesis Testing – Two Samples | 双样本假设检验

When comparing two population means, we often test H₀: μ₁ = μ₂. If both populations are normal with known variances, use z-test: z = (x̄₁ – x̄₂)/√(σ₁²/n₁ + σ₂²/n₂). For unknown but equal variances, use pooled t-test: t = (x̄₁ – x̄₂)/[sₚ √(1/n₁ + 1/n₂)], where sₚ² = [(n₁ – 1)s₁² + (n₂ – 1)s₂²]/(n₁ + n₂ – 2).

比较两个总体均值时，通常检验 H₀: μ₁ = μ₂。若两个总体均正态且方差已知，使用 z 检验：z = (x̄₁ – x̄₂)/√(σ₁²/n₁ + σ₂²/n₂)。若方差未知但相等，使用合并 t 检验：t = (x̄₁ – x̄₂)/[sₚ √(1/n₁ + 1/n₂)]，其中 sₚ² = [(n₁ – 1)s₁² + (n₂ – 1)s₂²]/(n₁ + n₂ – 2)。

For proportions, test H₀: p₁ = p₂ using a pooled estimate p̂ = (x₁ + x₂)/(n₁ + n₂) and test statistic z = (p̂₁ – p̂₂)/√[p̂(1 – p̂)(1/n₁ + 1/n₂)]. Degrees of freedom and assumptions must be checked.

对于比例，检验 H₀: p₁ = p₂ 时，使用合并估计 p̂ = (x₁ + x₂)/(n₁ + n₂) 及统计量 z = (p̂₁ – p̂₂)/√[p̂(1 – p̂)(1/n₁ + 1/n₂)]。必须检查自由度和假设条件。

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10. Chi-Squared Tests | 卡方检验

The chi-squared (χ²) test is used for categorical data. For a goodness-of-fit test, the statistic is χ² = Σ [(Oᵢ – Eᵢ)² / Eᵢ], where Oᵢ are observed frequencies and Eᵢ are expected frequencies under H₀. Degrees of freedom = number of categories – 1 – number of estimated parameters.

卡方检验 (χ²) 用于分类数据。对于拟合优度检验，统计量为 χ² = Σ [(Oᵢ – Eᵢ)² / Eᵢ]，其中 Oᵢ 为观测频数，Eᵢ 为 H₀ 下的期望频数。自由度 = 类别数 – 1 – 估计的参数个数。

In a test for independence (contingency table), the expected frequency for a cell is (row total × column total)/grand total. Degrees of freedom = (r – 1)(c – 1). Use the χ² distribution with the appropriate df. The test is valid only if all expected frequencies are at least 5.

在独立性检验（列联表）中，单元格的期望频数为 (行合计 × 列合计)/总计。自由度 = (r – 1)(c – 1)。使用相应自由度的 χ² 分布。只有当所有期望频数都至少为 5 时检验才有效。

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11. Correlation and Regression | 相关与回归

Pearson’s product–moment correlation coefficient r measures the strength and direction of a linear relationship between two variables. Its formula is r = Sₓᵧ / √(Sₓₓ Sᵧᵧ), where Sₓᵧ = Σ(x – x̄)(y – ȳ), Sₓₓ = Σ(x – x̄)², Sᵧᵧ = Σ(y – ȳ)². Values range from –1 to 1.

皮尔逊积差相关系数 r 衡量两个变量之间线性关系的强度和方向。其公式为 r = Sₓᵧ / √(Sₓₓ Sᵧᵧ)，其中 Sₓᵧ = Σ(x – x̄)(y – ȳ)，Sₓₓ = Σ(x – x̄)²，Sᵧᵧ = Σ(y – ȳ)²。取值范围从 –1 到 1。

The least-squares regression line of y on x is y = a + bx, where b = Sₓᵧ / Sₓₓ and a = ȳ – b x̄. The residual for a point is observed y – predicted y. The coefficient of determination R² = r² indicates the proportion of variation in y explained by x.

y 对 x 的最小二乘回归直线为 y = a + bx，其中 b = Sₓᵧ / Sₓₓ，a = ȳ – b x̄。一个点的残差为观测值 y – 预测值 y。决定系数 R² = r² 表示 y 的变异中由 x 解释的比例。

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12. Summary of Key Formulas | 关键公式汇总

The table below collects the most frequently used formulas in the AQA Year 13 Statistics course. Use it as a quick check before assessments.

下表汇集了 AQA Year 13 统计课程中最常用的公式，可在评估前用于快速查阅。

Concept / 概念	Formula / 公式
Binomial mean & variance	E(X) = np, Var(X) = np(1 – p)
Poisson mean & variance	E(X) = Var(X) = λ
Standardisation (normal)	Z = (X – μ) / σ
Standard error of sample mean	SE = σ / √n (or s / √n)
Confidence interval for μ (σ known)	x̄ ± z × (σ / √n)
Confidence interval for μ (σ unknown)	x̄ ± tₙ₋₁ × (s / √n)
One-sample z-test for mean	z = (x̄ – μ₀) / (σ/√n)
Two-sample z-test for means	z = (x̄₁ – x̄₂) / √(σ₁²/n₁ + σ₂²/n₂)
Pooled t-test statistic	t = (x̄₁ – x̄₂) / [sₚ √(1/n₁ + 1/n₂)]
Chi-squared statistic	χ² = Σ (O – E)² / E
Correlation coefficient	r = Sₓᵧ / √(Sₓₓ Sᵧᵧ)
Regression slope	b = Sₓᵧ / Sₓₓ

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