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

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

This comprehensive quick reference covers essential formulas and theorems for the CAIE A-Level Statistics course (Papers 1 & 2). It is designed to support revision by presenting each concept with a concise English explanation immediately followed by its Chinese equivalent. Use it as a handy companion for mastering data analysis, probability, distributions, estimation, and hypothesis testing.

这份全面的速查手册涵盖了 CAIE A-Level 统计（试卷1与2）的必备公式和定理。每个概念都先用英文简述，紧接着提供对应的中文解释，旨在辅助复习。无论您需要回顾数据分析、概率分布还是推断统计，都能在这里快速找到核心内容。

1. Measures of Central Tendency and Spread | 集中趋势与离散程度的度量

For raw data x₁, x₂, …, xₙ, the sample mean is x̄ = Σx / n. The median is the middle value when data are ordered; for an even number of observations, it is the average of the two middle values.

对于原始数据 x₁, x₂, …, xₙ，样本均值为 x̄ = Σx / n。中位数是有序数据的中间值；当观测数为偶数时，取中间两个值的平均数。

If data are presented in a frequency table with values xᵢ and frequencies fᵢ, the mean is x̄ = Σfᵢ xᵢ / Σfᵢ. The variance is given by σ² = Σfᵢ (xᵢ – x̄)² / Σfᵢ, or equivalently by the working formula σ² = Σfᵢ xᵢ² / Σfᵢ – x̄².

若数据以频数表给出，取值为 xᵢ、频数为 fᵢ，则均值为 x̄ = Σfᵢ xᵢ / Σfᵢ。方差可用 σ² = Σfᵢ (xᵢ – x̄)² / Σfᵢ 计算，也可使用等价的工作公式 σ² = Σfᵢ xᵢ² / Σfᵢ – x̄²。

For grouped continuous data, replace each class by its midpoint and treat as a frequency distribution. The interquartile range (IQR) is Q₃ – Q₁, where Q₁ and Q₃ are the first and third quartiles, respectively. Linear interpolation may be used to estimate quartiles from cumulative frequency graphs.

对于分组连续数据，将每组替换为组中值并按频数分布处理。四分位距 (IQR) 为 Q₃ – Q₁，其中 Q₁ 和 Q₃ 分别为第一和第三四分位数。可通过累积频率图进行线性插值来估计四分位数。

Linear coding: For a transformation y = a + bx, we have ȳ = a + b x̄, and the standard deviation satisfies s_y = |b| s_x. The variance transforms as Var(Y) = b² Var(X).

线性编码：对于变换 y = a + bx，有 ȳ = a + b x̄，标准差满足 s_y = |b| s_x，方差满足 Var(Y) = b² Var(X)。

x̄ = Σx / n, σ² = Σ(x – x̄)² / n = Σx² / n – x̄²

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2. Correlation and Regression | 相关与回归

The product moment correlation coefficient (PMCC) measures the strength of linear association between two variables X and Y. It is defined as r = S_{xy} / √(S_{xx} S_{yy}), where S_{xy} = Σ(x – x̄)(y – ȳ) = Σxy – n x̄ ȳ, and similarly for S_{xx}, S_{yy}.

积矩相关系数 (PMCC) 衡量两个变量 X 与 Y 之间线性关系的强度。它定义为 r = S_{xy} / √(S_{xx} S_{yy})，其中 S_{xy} = Σ(x – x̄)(y – ȳ) = Σxy – n x̄ ȳ，S_{xx} 和 S_{yy} 依此类推。

The least squares regression line of Y on X is y = a + bx, where b = S_{xy} / S_{xx} and a = ȳ – b x̄. This line minimises the sum of squared vertical deviations. The coefficient b represents the estimated change in Y for a unit increase in X.

Y 对 X 的最小二乘回归直线为 y = a + bx，其中 b = S_{xy} / S_{xx}，a = ȳ – b x̄。这条直线使垂直离差的平方和最小。系数 b 表示 X 每增加一单位时 Y 的估计变化量。

To predict a Y-value from a given X, substitute X into the regression line, but be cautious about extrapolation. The correlation coefficient r satisfies -1 ≤ r ≤ 1; values near ±1 indicate strong linear correlation.

给定 X 值预测 Y 时，将其代入回归直线即可，但需注意外推风险。相关系数 r 满足 -1 ≤ r ≤ 1；接近于 ±1 的值表明线性相关性强。

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3. Probability Basics | 概率基础

For any event A, 0 ≤ P(A) ≤ 1. The complement rule: P(A’) = 1 – P(A). The addition rule for two events: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). Events A and B are mutually exclusive if A ∩ B = ∅, so P(A ∪ B) = P(A) + P(B).

对于任意事件 A，0 ≤ P(A) ≤ 1。补集法则：P(A’) = 1 – P(A)。两个事件的加法法则：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若 A ∩ B = ∅，则称 A 与 B 互斥，此时 P(A ∪ B) = P(A) + P(B)。

Conditional probability: P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0. The multiplication rule follows: P(A ∩ B) = P(A | B) P(B) = P(B | A) P(A).

条件概率：P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。乘法法则推导出：P(A ∩ B) = P(A | B) P(B) = P(B | A) P(A)。

Events A and B are independent if P(A ∩ B) = P(A) P(B). Equivalently, P(A | B) = P(A) when P(B) > 0. Tree diagrams often help in solving multi-stage probability problems.

若 P(A ∩ B) = P(A) P(B)，则称事件 A 与 B 独立。等价地，当 P(B) > 0 时，有 P(A | B) = P(A)。树状图常用于解决多阶段概率问题。

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4. Discrete Random Variables | 离散随机变量

A discrete random variable X takes values xᵢ with probabilities pᵢ = P(X = xᵢ), where Σpᵢ = 1. The expected value (mean) is E(X) = μ = Σxᵢ pᵢ. The variance is Var(X) = σ² = Σ(xᵢ – μ)² pᵢ = Σxᵢ² pᵢ – μ².

离散随机变量 X 以概率 pᵢ = P(X = xᵢ) 取值 xᵢ，并满足 Σpᵢ = 1。期望值（均值）为 E(X) = μ = Σxᵢ pᵢ。方差为 Var(X) = σ² = Σ(xᵢ – μ)² pᵢ = Σxᵢ² pᵢ – μ²。

For a linear function g(X) = a + bX, the expectation and variance follow: E(a + bX) = a + b E(X), and Var(a + bX) = b² Var(X). The constant a does not affect the variance.

对于线性函数 g(X) = a + bX，期望与方差满足：E(a + bX) = a + b E(X)，Var(a + bX) = b² Var(X)。常数 a 不影响方差。

The cumulative distribution function is F(x) = P(X ≤ x) = Σ_{t≤x} P(X = t). The median m is the smallest value for which F(m) ≥ 0.5.

累积分布函数为 F(x) = P(X ≤ x) = Σ_{t≤x} P(X = t)。中位数 m 是满足 F(m) ≥ 0.5 的最小数值。

E(X) = Σxᵢ pᵢ, Var(X) = Σxᵢ² pᵢ – [E(X)]²

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5. Binomial Distribution | 二项分布

A binomial experiment consists of n independent trials, each with two outcomes (success/failure) and constant probability of success p. The random variable X ~ B(n, p) counts the number of successes. Its probability mass function is P(X = k) = ⁿCₖ pᵏ (1-p)ⁿ⁻ᵏ, for k = 0, 1, …, n.

二项试验由 n 次独立试验组成，每次试验有两种结果（成功/失败）且成功概率 p 不变。随机变量 X ~ B(n, p) 表示成功的次数。其概率质量函数为 P(X = k) = ⁿCₖ pᵏ (1-p)ⁿ⁻ᵏ，其中 k = 0, 1, …, n。

The mean and variance of a binomial count are E(X) = np and Var(X) = np(1-p). These formulas are derived directly from the properties of the sum of indicator variables and are essential for hypothesis tests and normal approximations.

二项计数的均值和方差分别为 E(X) = np 与 Var(X) = np(1-p)。这两个公式由指示变量总和的性质导出，对假设检验和正态近似至关重要。

The binomial assumption requires trials to be independent and the probability p to remain constant. When n is large and p is near 0.5, the binomial distribution can be approximated by a normal distribution with continuity correction.

使用二项分布的前提是试验独立且概率 p 恒定。当 n 较大且 p 接近 0.5 时，二项分布可由带连续性修正的正态分布来近似。

X ~ B(n, p): P(X = k) = ⁿCₖ pᵏ qⁿ⁻ᵏ, q = 1-p

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6. Normal Distribution | 正态分布

The normal distribution with mean μ and variance σ² is denoted by X ~ N(μ, σ²). The standard normal distribution Z ~ N(0, 1) is obtained by the transformation Z = (X – μ) / σ. Tables give Φ(z) = P(Z ≤ z) for z ≥ 0; symmetry is used for negative z.

均值为 μ、方差为 σ² 的正态分布记作 X ~ N(μ, σ²)。通过变换 Z = (X – μ) / σ，得到标准正态分布 Z ~ N(0, 1)。正态分布表给出 z ≥ 0 时的 Φ(z) = P(Z ≤ z)；负 z 值利用对称性处理。

To calculate probabilities: P(X < a) = Φ((a - μ) / σ). For a range, P(a < X < b) = Φ((b - μ)/σ) - Φ((a - μ)/σ). The inverse normal is used to find percentiles.

计算概率时：P(X < a) = Φ((a - μ) / σ)。对于区间概率，P(a < X < b) = Φ((b - μ)/σ) - Φ((a - μ)/σ)。反查正态表用于求百分位数。

When using the normal approximation to a binomial (n large), apply continuity correction: P(X = k) is approximated by P(k – 0.5 < Y < k + 0.5) where Y ~ N(np, npq). Similarly for Poisson to normal approximation (λ large), use continuity correction.

用正态分布近似二项分布（n 较大）时，需进行连续性修正：P(X = k) 近似为 P(k – 0.5 < Y < k + 0.5)，其中 Y ~ N(np, npq)。泊松分布正态近似（λ 大）也需类似修正。

Z = (X – μ) / σ ~ N(0, 1)

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7. Poisson Distribution | 泊松分布

A Poisson random variable X ~ Po(λ) models the number of events occurring in a fixed interval when events happen independently at a constant average rate λ. Its probability function is P(X = x) = e⁻ᵸ λˣ / x!, for x = 0, 1, 2, … . Both the mean and the variance equal λ.

泊松随机变量 X ~ Po(λ) 用于描述在固定区间内独立事件发生的次数，其中 λ 是平均发生率。其概率函数为 P(X = x) = e⁻ᵸ λˣ / x!，x = 0, 1, 2, …。均值和方差均为 λ。

Conditions for a Poisson model: events occur singly, independently, at a constant average rate, and randomly in a continuum. The sum of independent Poisson variables is also Poisson: if X ~ Po(λ₁) and Y ~ Po(λ₂) are independent, then X + Y ~ Po(λ₁ + λ₂).

泊松模型的条件：事件单独发生、相互独立、平均速率恒定且随机出现于连续区间内。独立泊松变量的和仍为泊松分布：若 X ~ Po(λ₁) 与 Y ~ Po(λ₂) 独立，则 X + Y ~ Po(λ₁ + λ₂)。

When np < 5 and n is large, the binomial B(n, p) can be approximated by Poisson with λ = np. Similarly, the normal approximation N(λ, λ) is suitable when λ is large (commonly λ > 15).

当 np < 5 且 n 很大时，二项分布 B(n, p) 可用泊松分布（λ = np）近似。当 λ 较大（通常 λ > 15）时，可使用正态近似 N(λ, λ)。

X ~ Po(λ): P(X = x) = e⁻ᵸ λˣ / x!

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8. Continuous Random Variables | 连续随机变量

A continuous random variable X is described by its probability density function (pdf) f(x) ≥ 0, with ∫ f(x) dx = 1 over the domain. Probabilities are found by integration: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. The cumulative distribution function is F(x) = P(X ≤ x) = ∫_{-∞}ˣ f(t) dt.

连续随机变量 X 由其概率密度函数 (pdf) f(x) ≥ 0 描述，且在整个定义域上 ∫ f(x) dx = 1。概率通过积分求得：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。累积分布函数为 F(x) = P(X ≤ x) = ∫_{-∞}ˣ f(t) dt。

The expected value of X is E(X) = ∫ x f(x) dx, and E(g(X)) = ∫ g(x) f(x) dx. Variance is computed as Var(X) = E(X²) – [E(X)]², where E(X²) = ∫ x² f(x) dx.

X 的期望值为 E(X) = ∫ x f(x) dx，且 E(g(X)) = ∫ g(x) f(x) dx。方差的计算公式为 Var(X) = E(X²) – [E(X)]²，其中 E(X²) = ∫ x² f(x) dx。

The median m is the value for which F(m) = 0.5. Quartiles and percentiles are found similarly by solving F(q) = p. The mode is the value of x that maximises f(x).

中位数 m 是满足 F(m) = 0.5 的值。四分位数和百分位数通过求解 F(q) = p 得到。众数是使 f(x) 最大的 x 值。

F(x) = ∫_{-∞}ˣ f(t) dt, E(X) = ∫ x f(x) dx

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9. Linear Combinations of Random Variables | 随机变量的线性组合

For any random variables X and Y, the expectation of a linear combination is E(aX + bY) = aE(X) + bE(Y), regardless of independence. This extends to any finite number of variables: E(a₁X₁ + … + aₙXₙ) = Σ aᵢ E(Xᵢ).

对于任意随机变量 X 和 Y，线性组合的期望为 E(aX + bY) = aE(X) + bE(Y)，且不依赖于独立性。此性质可推广至有限个变量：E(a₁X₁ + … + aₙXₙ) = Σ aᵢ E(Xᵢ)。

If X and Y are independent, then Var(aX + bY) = a² Var(X) + b² Var(Y). For the difference, Var(aX – bY) = a² Var(X) + b² Var(Y). The addition of a constant shifts the mean but does not affect the variance.

若 X 与 Y 独立，则 Var(aX + bY) = a² Var(X) + b² Var(Y)。对于差，Var(aX – bY) = a² Var(X) + b² Var(Y)。加上常数会平移均值但不影响方差。

Special cases: the sum of independent normals is normal with mean μ₁ + μ₂ and variance σ₁² + σ₂². The sum of independent Poissons is Poisson, as noted earlier. For a binomial, it can be seen as a sum of independent Bernoulli indicators, leading to the np and npq formulas.

特例：独立正态变量的和仍为正态分布，均值为 μ₁ + μ₂，方差为 σ₁² + σ₂²。如前所述，独立泊松变量之和仍为泊松分布。二项分布可视为独立伯努利指示变量的和，由此导出均值和方差公式。

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10. Sampling and Estimation | 抽样与估计

A sample of size n drawn from a population with mean μ and variance σ² yields a sample mean x̄. The sampling distribution of x̄ has mean μ and variance σ²/n. If the population is normal, x̄ is exactly N(μ, σ²/n). The central limit theorem states that for large n, x̄ is approximately normal regardless of the population distribution.

从均值为 μ、方差为 σ² 的总体中抽取容量为 n 的样本，得到样本均值 x̄。x̄ 的抽样分布均值为 μ，方差为 σ²/n。若总体为正态，则 x̄ 精确服从 N(μ, σ²/n)。中心极限定理指出，当 n 较大时，无论总体分布如何，x̄ 近似服从正态分布。

An unbiased estimator of the population variance σ² is s² = Σ(x – x̄)² / (n – 1). The standard error of the mean is σ/√n if σ is known, or s/√n when σ is estimated by s.

总体方差 σ² 的无偏估计量为 s² = Σ(x – x̄)² / (n – 1)。均值的标准误为 σ/√n（当 σ 已知时）或 s/√n（当用 s 估计 σ 时）。

A 95% confidence interval for the population mean μ (when population variance is known) is x̄ ± 1.96 × σ/√n. If σ is unknown and the sample is large, the interval is x̄ ± 1.96 × s/√n. For a proportion p, the 95% CI is p̂ ± 1.96 × √(p̂(1-p̂)/n). The critical value 1.96 comes from the standard normal distribution for a two-tailed 5% significance.

当总体方差已知时，总体均值 μ 的 95% 置信区间为 x̄ ± 1.96 × σ/√n。若 σ 未知且样本较大，则区间为 x̄ ± 1.96 × s/√n。对于比例 p，95% CI 为 p̂ ± 1.96 × √(p̂(1-p̂)/n)。临界值 1.96 来自标准正态分布的双侧 5% 显著性水平。

CI for μ (σ known): x̄ ± z × σ/√n

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11. 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 from the appropriate distribution. The significance level α is the probability of rejecting H₀ when it is true (Type I error). The p-value is the probability of obtaining a result at least as extreme as the observed one, assuming H₀ is true.

假设检验将原假设 H₀ 与备择假设 H₁ 进行比对。检验统计量与来自相应分布的临界值比较。显著性水平 α 是当 H₀ 为真时拒绝 H₀ 的概率（第一类错误）。p 值是在假定 H₀ 为真的前提下，得到至少与观测值一样极端的结果的概率。

For a binomial test with H₀: p = p₀, the test statistic X ~ B(n, p₀). Critical regions are constructed using the binomial distribution. For a test concerning a population mean with known variance, the z-test uses the statistic Z = (x̄ – μ₀) / (σ/√n). For a difference between two means, use Z = (x̄₁ – x̄₂) / √(σ₁²/n₁ + σ₂²/n₂). If variances are unknown but large samples, replace σ by s.

对原假设 H₀: p = p₀ 的二项检验，检验统计量 X ~ B(n, p₀)。临界域由二项分布确定。对已知方差的总体均值检验，z 检验使用统计量 Z = (x̄ – μ₀) / (σ/√n)。对于两均值之差的检验，使用 Z = (x̄₁ – x̄₂) / √(σ₁²/n₁ + σ₂²/n₂)；若方差未知但样本较大，用 s 替代 σ。

A Type II error occurs when H₀ is false but not rejected. The power of a test is 1 – P(Type II error). Increasing sample size reduces both error probabilities and increases power.

当 H₀ 为假但未被拒绝时，发生第二类错误。检验的功效为 1 – P(第二类错误)。增大样本量可降低两类错误概率并提升功效。

z = (x̄ – μ₀) / (σ/√n)

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Published by TutorHao | Statistics Revision Series | alevel

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