A-Level AQA Statistics: Formula & Theorem Quick Reference Handbook | A-Level AQA 统计：公式定理速查手册

📚 A-Level AQA Statistics: Formula & Theorem Quick Reference Handbook | A-Level AQA 统计：公式定理速查手册

This quick reference handbook compiles the essential formulae, theorems and critical concepts required for the AQA A-Level Statistics syllabus. From descriptive measures to hypothesis testing, each section presents formulas in a clear bilingual format, making revision efficient and structured.

本速查手册汇集了AQA A-Level统计课程所需的核心公式、定理与关键概念。从描述性统计量到假设检验，每个小节以清晰的双语形式呈现公式，帮助高效、有条理地复习。

1. Descriptive Statistics | 描述性统计

For a sample of n observations x₁, x₂, …, xₙ, the key location and dispersion measures are defined as follows.

对于包含n个观测值x₁, x₂, …, xₙ的样本，位置与离散程度的主要度量定义如下。

Sample mean x̄ = Σxᵢ / n

样本均值 x̄ = Σxᵢ / n

Sample variance s² = Σ(xᵢ − x̄)² / (n − 1)

样本方差 s² = Σ(xᵢ − x̄)² / (n − 1)

Standard deviation s = √[ Σ(xᵢ − x̄)² / (n − 1) ]

标准差 s = √[ Σ(xᵢ − x̄)² / (n − 1) ]

The quartiles divide ordered data into four equal parts: Q₁ is the median of the lower half, Q₂ is the median, Q₃ is the median of the upper half. The interquartile range IQR = Q₃ − Q₁.

四分位数将有序数据分成四等份：Q₁是下半部分的中位数，Q₂是中位数，Q₃是上半部分的中位数。四分位距 IQR = Q₃ − Q₁。

For grouped data with frequencies fᵢ and midpoints mᵢ, the estimated mean is x̄ = Σfᵢmᵢ / Σfᵢ, and variance s² = Σfᵢ(mᵢ − x̄)² / (Σfᵢ − 1) when treating as a sample.

对于分组数据，设频数为fᵢ、组中点为mᵢ，估计均值 x̄ = Σfᵢmᵢ / Σfᵢ，样本方差 s² = Σfᵢ(mᵢ − x̄)² / (Σfᵢ − 1)。

2. Probability Fundamentals | 概率基础

The probability of an event A is P(A). For any event A, 0 ≤ P(A) ≤ 1. The complement rule states P(A’) = 1 − P(A).

事件A的概率为P(A)。对任一事件A，0 ≤ P(A) ≤ 1。互补规则为 P(A’) = 1 − P(A)。

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

加法法则: 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).

若A与B互斥，则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。

Conditional probability: P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0

条件概率: P(A | B) = P(A ∩ B) / P(B)，前提 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独立当且仅当 P(A ∩ B) = P(A) × P(B)，或等价地 P(A | B) = P(A)。

The law of total probability for a partition {B₁, B₂, …, Bₖ} gives P(A) = Σ P(A | Bᵢ) × P(Bᵢ). Bayes’ theorem then states P(Bⱼ | A) = [P(A | Bⱼ) × P(Bⱼ)] / Σ P(A | Bᵢ) P(Bᵢ).

全概率公式对划分{B₁, B₂, …, Bₖ}给出 P(A) = Σ P(A | Bᵢ) × P(Bᵢ)。贝叶斯定理进而给出 P(Bⱼ | A) = [P(A | Bⱼ) × P(Bⱼ)] / Σ P(A | Bᵢ) P(Bᵢ)。

3. Discrete Random Variables | 离散随机变量

A discrete random variable X has probability mass function P(X = x) = p(x). The sum of all probabilities is Σ p(x) = 1.

离散随机变量X的概率质量函数为 P(X = x) = p(x)。所有概率之和为 Σ p(x) = 1。

Expected value: E(X) = Σ x p(x)

期望值: E(X) = Σ x p(x)

Variance: Var(X) = E[(X − μ)²] = Σ x² p(x) − μ², where μ = E(X)

方差: Var(X) = E[(X − μ)²] = Σ x² p(x) − μ²，其中 μ = E(X)

For constants a and b, E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X).

对常数a与b，E(aX + b) = aE(X) + b，Var(aX + b) = a² Var(X)。

If X and Y are independent, E(XY) = E(X)E(Y) and Var(X ± Y) = Var(X) + Var(Y).

若X与Y独立，则 E(XY) = E(X)E(Y)，且 Var(X ± Y) = Var(X) + Var(Y)。

Standard discrete distributions: the binomial distribution X ~ B(n, p) has P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ, with E(X) = np, Var(X) = np(1 − p). The Poisson distribution X ~ Poisson(λ) has P(X = k) = (e⁻λ λᵏ) / k!, with E(X) = Var(X) = λ.

标准离散分布：二项分布 X ~ B(n, p) 有 P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ，E(X) = np，Var(X) = np(1 − p)。泊松分布 X ~ Poisson(λ) 有 P(X = k) = (e⁻λ λᵏ) / k!，E(X) = Var(X) = λ。

4. Continuous Random Variables | 连续随机变量

A continuous random variable X is described by its probability density function f(x) ≥ 0, with ∫₋∞⁺∞ f(x) dx = 1. The probability P(a < X < b) = ∫ₐᵇ f(x) dx.

连续随机变量X由其概率密度函数 f(x) ≥ 0 描述，且 ∫₋∞⁺∞ f(x) dx = 1。概率 P(a < X < b) = ∫ₐᵇ f(x) dx。

E(X) = ∫₋∞⁺∞ x f(x) dx, Var(X) = ∫₋∞⁺∞ (x − μ)² f(x) dx = ∫₋∞⁺∞ x² f(x) dx − μ²

E(X) = ∫₋∞⁺∞ x f(x) dx，Var(X) = ∫₋∞⁺∞ (x − μ)² f(x) dx = ∫₋∞⁺∞ x² f(x) dx − μ²

The cumulative distribution function F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt. The median m satisfies F(m) = 0.5. For the uniform distribution on [a, b], f(x) = 1/(b − a), E(X) = (a + b)/2, Var(X) = (b − a)²/12.

累积分布函数 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。中位数m满足 F(m) = 0.5。对于[a, b]上的均匀分布，f(x) = 1/(b − a)，E(X) = (a + b)/2，Var(X) = (b − a)²/12。

5. Normal Distribution | 正态分布

If X ~ N(μ, σ²), then the standardised variable Z = (X − μ) / σ follows N(0, 1). The density of Z is φ(z) = (1/√(2π)) e^(−z²/2).

若 X ~ N(μ, σ²)，则标准化变量 Z = (X − μ) / σ 服从 N(0, 1)。Z的密度函数为 φ(z) = (1/√(2π)) e^(−z²/2)。

Probabilities are found using the standard normal table: P(Z ≤ z) = Φ(z). For X ~ N(μ, σ²), P(X ≤ x) = Φ((x − μ)/σ).

使用标准正态表计算概率：P(Z ≤ z) = Φ(z)。对于 X ~ N(μ, σ²)，P(X ≤ x) = Φ((x − μ)/σ)。

Critical values: for a right-tail probability α, find zₐ such that P(Z > zₐ) = α

临界值：对右侧尾部概率α，求zₐ使 P(Z > zₐ) = α

The sum of independent normal variables is normal: if Xᵢ ~ N(μᵢ, σᵢ²) and independent, then Σ aᵢXᵢ + b ~ N(Σ aᵢμᵢ + b, Σ aᵢ² σᵢ²).

独立正态变量的和仍为正态：若 Xᵢ ~ N(μᵢ, σᵢ²) 且独立，则 Σ aᵢXᵢ + b ~ N(Σ aᵢμᵢ + b, Σ aᵢ² σᵢ²)。

The normal approximation to the binomial (with continuity correction): if X ~ B(n, p) with np > 5 and n(1 − p) > 5, then X ≈ N(np, np(1 − p)), and P(X ≤ k) ≈ Φ((k + 0.5 − np)/√(np(1 − p))).

二项分布的正态近似（含连续性校正）：若 X ~ B(n, p) 且 np > 5, n(1 − p) > 5，则 X ≈ N(np, np(1 − p))，P(X ≤ k) ≈ Φ((k + 0.5 − np)/√(np(1 − p)))。

6. Sampling Distributions & Central Limit Theorem | 抽样分布与中心极限定理

For a random sample of size n from a population with mean μ and variance σ², the sample mean x̄ has E(x̄) = μ and Var(x̄) = σ²/n.

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满足 E(x̄) = μ，Var(x̄) = σ²/n。

Central Limit Theorem: regardless of the population distribution, for large n, x̄ ≈ N(μ, σ²/n)

中心极限定理：不论总体分布如何，当n较大时，x̄ ≈ N(μ, σ²/n)

If the population is normal, x̄ ~ N(μ, σ²/n) exactly for any n. When σ² is unknown, the statistic T = (x̄ − μ) / (s/√n) follows a t-distribution with n − 1 degrees of freedom.

若总体为正态，则对任意n有 x̄ ~ N(μ, σ²/n) 精确成立。当σ²未知时，统计量 T = (x̄ − μ) / (s/√n) 服从自由度为n − 1的t分布。

The sample proportion p̂ = X/n has E(p̂) = p, Var(p̂) = p(1 − p)/n, and for large n, p̂ ≈ N(p, p(1 − p)/n) by the CLT.

样本比例 p̂ = X/n 满足 E(p̂) = p, Var(p̂) = p(1 − p)/n，当n较大时由中心极限定理 p̂ ≈ N(p, p(1 − p)/n)。

7. Point Estimation & Confidence Intervals | 点估计与置信区间

An estimator is unbiased if E(θ̂) = θ. The sample mean x̄ is an unbiased estimator of μ; s² = Σ(xᵢ − x̄)²/(n − 1) is unbiased for σ².

若 E(θ̂) = θ，则估计量是无偏的。样本均值x̄是μ的无偏估计量；s² = Σ(xᵢ − x̄)²/(n − 1) 是σ²的无偏估计量。

Confidence interval for μ (known σ): x̄ ± zₐ/₂ × (σ/√n)

μ的置信区间（σ已知）：x̄ ± zₐ/₂ × (σ/√n)

Confidence interval for μ (unknown σ): x̄ ± tₙ₋₁,ₐ/₂ × (s/√n)

μ的置信区间（σ未知）：x̄ ± tₙ₋₁,ₐ/₂ × (s/√n)

Confidence interval for population proportion p: p̂ ± zₐ/₂ × √[p̂(1 − p̂)/n], provided np̂ > 5, n(1 − p̂) > 5

总体比例p的置信区间：p̂ ± zₐ/₂ × √[p̂(1 − p̂)/n]，前提 np̂ > 5, n(1 − p̂) > 5

The width of a confidence interval decreases with larger sample size n and increases with higher confidence level (1 − α).

置信区间的宽度随样本量n增大而减小，随置信水平(1 − α)提高而增大。

8. Hypothesis Testing | 假设检验

A hypothesis test evaluates a null hypothesis H₀ against an alternative H₁. The test statistic is compared with a critical value, and the null is rejected if the p-value < α.

假设检验衡量原假设H₀与备择假设H₁。将检验统计量与临界值比较，若p值 < α则拒绝原假设。

One-sample z-test for μ (σ known): Z = (x̄ − μ₀) / (σ/√n) ~ N(0,1) under H₀

单样本z检验（σ已知）：Z = (x̄ − μ₀) / (σ/√n) 在H₀下~N(0,1)

One-sample t-test for μ (σ unknown): T = (x̄ − μ₀) / (s/√n) ~ tₙ₋₁ under H₀

单样本t检验（σ未知）：T = (x̄ − μ₀) / (s/√n) 在H₀下~tₙ₋₁

Two-sample t-test (equal variances): t = (x̄₁ − x̄₂) / (sₚ √(1/n₁ + 1/n₂)), where sₚ² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2)

双样本t检验（等方差）：t = (x̄₁ − x̄₂) / (sₚ √(1/n₁ + 1/n₂))，其中 sₚ² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2)

For proportion tests, use Z = (p̂ − p₀) / √[p₀(1 − p₀)/n] for H₀: p = p₀. Type I error (α) is rejecting H₀ when true; Type II error (β) is failing to reject H₀ when false. Power = 1 − β.

比例检验使用 Z = (p̂ − p₀) / √[p₀(1 − p₀)/n] 检验 H₀: p = p₀。第一类错误(α)是H₀真时拒绝；第二类错误(β)是H₀假时未拒绝。功效 = 1 − β。

9. Chi-Squared Tests | 卡方检验

The chi-squared statistic measures the goodness of fit or association in contingency tables. For observed frequencies Oᵢ and expected frequencies Eᵢ,

卡方统计量用于拟合优度检验或列联表关联性检验。对于观测频数Oᵢ和期望频数Eᵢ，

χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ

For a goodness-of-fit test, Eᵢ are derived from the hypothesised model, and degrees of freedom ν = number of categories − 1 (or minus extra estimated parameters).

对拟合优度检验，Eᵢ由假设模型给出，自由度ν = 类别数 − 1（或减去额外估计的参数个数）。

For an r × c contingency table, the expected frequency for cell (i, j) is Eᵢⱼ = (row total × column total) / grand total, with ν = (r − 1)(c − 1).

对r × c列联表，第(i,j)格期望频数 Eᵢⱼ = (行合计 × 列合计) / 总和，自由度 ν = (r − 1)(c − 1)。

The test rejects H₀ if χ² > critical value from the χ²ₗ table at α significance level. All expected frequencies should be at least 5 for the approximation to be valid.

若 χ² > χ²ₗ 表在显著性水平α下的临界值，则拒绝H₀。为保证近似有效，所有期望频数应至少为5。

10. Correlation & Regression | 相关与回归

The product-moment correlation coefficient r measures linear association between two variables x and y:

r = Sₓᵧ / √(Sₓₓ Sᵧᵧ), where Sₓₓ = Σ(x − x̄)², Sᵧᵧ = Σ(y − ȳ)², Sₓᵧ = Σ(x − x̄)(y − ȳ)

积矩相关系数r度量两变量x与y之间的线性关联：

r = Sₓᵧ / √(Sₓₓ Sᵧᵧ)，其中 Sₓₓ = Σ(x − x̄)²，Sᵧᵧ = Σ(y − ȳ)²，Sₓᵧ = Σ(x − x̄)(y − ȳ)

The value of r lies between −1 and 1. A positive r indicates a positive linear trend; negative r indicates a negative trend.

r的值介于−1和1之间。正r表示正线性趋势；负r表示负线性趋势。

The equation of the least squares regression line of y on x is y = a + bx, where

b = Sₓᵧ / Sₓₓ, a = ȳ − b x̄

y关于x的最小二乘回归线方程为 y = a + bx，其中

b = Sₓᵧ / Sₓₓ，a = ȳ − b x̄

The residual for observation i is eᵢ = yᵢ − (a + bxᵢ). Residual plots help assess model assumptions. The coefficient of determination R² = (Sₓᵧ)² / (Sₓₓ Sᵧᵧ) = r² indicates the proportion of variation in y explained by x.

观测值i的残差为 eᵢ = yᵢ − (a + bxᵢ)。残差图有助于评估模型假设。决定系数 R² = (Sₓᵧ)² / (Sₓₓ Sᵧᵧ) = r² 表示y的变异中由x解释的比例。

A hypothesis test for the population correlation coefficient ρ uses the test statistic t = r√(n − 2) / √(1 − r²), which follows a t-distribution with n − 2 df under H₀: ρ = 0.

对总体相关系数ρ的假设检验使用统计量 t = r√(n − 2) / √(1 − r²)，在 H₀: ρ = 0 下服从自由度为n − 2的t分布。

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A-Level AQA Statistics: Formula & Theorem Quick Reference Handbook | A-Level AQA 统计：公式定理速查手册