📚 Year 13 Cambridge Statistics: Core Knowledge Review | 13年级剑桥统计：核心知识点梳理
This article provides a structured review of the essential topics in Year 13 Cambridge Statistics, covering discrete and continuous random variables, the normal distribution, sampling theory, confidence intervals, hypothesis testing, chi‑squared tests, correlation and regression. Each section pairs English and Chinese explanations to help you consolidate key definitions, formulae, and exam skills.
本文系统梳理13年级剑桥统计课程的核心知识点，涵盖离散与连续随机变量、正态分布、抽样理论、置信区间、假设检验、卡方检验以及相关与回归。每节以中英文对照的形式讲解关键定义、公式和应试技巧，帮助你高效复习。
1. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable set of values. Its probability distribution is defined by P(X = x), with Σ P(X = x) = 1. The expected value, or mean, is E(X) = Σ x·P(X = x). The variance is Var(X) = E(X²) – [E(X)]², and the standard deviation is σ = √Var(X).
离散随机变量 X 取可数个值，概率分布由 P(X = x) 给出，满足 Σ P(X = x) = 1。期望（均值）是 E(X) = Σ x·P(X = x)，方差为 Var(X) = E(X²) – [E(X)]²，标准差为 σ = √Var(X)。
For a linear transformation Y = aX + b, the rules are E(Y) = aE(X) + b and Var(Y) = a²Var(X). The expectation of a sum is always E(X + Y) = E(X) + E(Y), and if X and Y are independent, Var(X + Y) = Var(X) + Var(Y).
对于线性变换 Y = aX + b，有 E(Y) = aE(X) + b 以及 Var(Y) = a²Var(X)。和的期望满足 E(X + Y) = E(X) + E(Y)；当 X 与 Y 独立时，Var(X + Y) = Var(X) + Var(Y)。
When calculating probabilities, the cumulative distribution function F(x) = P(X ≤ x) is often useful. For integer‑valued variables, P(X = r) = F(r) – F(r – 1).
累积分布函数 F(x) = P(X ≤ x) 常用于计算概率。对于取整数值的变量，P(X = r) = F(r) – F(r – 1)。
2. Binomial & Poisson Distributions | 二项分布与泊松分布
A binomial distribution, X ~ B(n, p), models the number of successes in n independent trials, each with success probability p. The probability mass function is P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ, for r = 0, 1, …, n. The mean is E(X) = np and the variance is Var(X) = np(1 – p).
二项分布 X ~ B(n, p) 描述在 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 = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ，r = 0, 1, …, n。均值 E(X) = np，方差 Var(X) = np(1 – p)。
The Poisson distribution, X ~ Po(λ), models the number of events occurring in a fixed interval of time or space when events happen independently at a constant average rate λ. Its probability function is P(X = r) = e−λ λʳ / r!. Both the mean and the variance equal λ.
泊松分布 X ~ Po(λ) 描述在固定时间或空间区间内，事件以恒定平均速率 λ 独立发生时的次数。概率函数为 P(X = r) = e−λ λʳ / r!。均值与方差都等于 λ。
Under certain conditions a binomial distribution can be approximated by a Poisson distribution: if n is large and p is small, use λ = np. Similarly, a binomial can be approximated by a normal distribution when np > 5 and n(1 – p) > 5; a Poisson can be approximated by a normal when λ > 10.
在一定条件下，二项分布可由泊松分布近似：当 n 大、p 小时，取 λ = np。当 np > 5 且 n(1 – p) > 5 时，二项分布可用正态分布近似；当 λ > 10 时，泊松分布可用正态分布近似。
3. Continuous Random Variables | 连续随机变量
A continuous random variable X has a probability density function (pdf) f(x) such that P(a < X < b) = ∫ₐᵇ f(x) dx. The total area under the pdf equals 1, i.e. ∫ f(x) dx = 1 over the support.
连续随机变量 X 具有概率密度函数 f(x)，满足 P(a < X < b) = ∫ₐᵇ f(x) dx。pdf 下方面积总和为 1，即在其支撑集上 ∫ f(x) dx = 1。
The cumulative distribution function is F(x) = P(X ≤ x) = ∫^x_{−∞} f(t) dt. The mean is E(X) = ∫ x·f(x) dx and the variance is Var(X) = ∫ (x – μ)² f(x) dx = E(X²) – [E(X)]². The median m satisfies F(m) = 0.5.
累积分布函数 F(x) = P(X ≤ x) = ∫^x_{−∞} f(t) dt。均值 E(X) = ∫ x·f(x) dx，方差 Var(X) = ∫ (x – μ)² f(x) dx = E(X²) – [E(X)]²。中位数 m 满足 F(m) = 0.5。
Common continuous distributions include the continuous uniform distribution on [a, b] with f(x) = 1/(b – a) and E(X) = (a + b)/2, as well as the normal distribution discussed next.
常见连续分布包括在 [a, b] 上的均匀分布，f(x) = 1/(b – a)，E(X) = (a + b)/2，以及接下来讨论的正态分布。
4. The Normal Distribution | 正态分布
The normal distribution, X ~ N(μ, σ²), has pdf f(x) = (1/(σ√(2π))) e−(x−μ)²/(2σ²). The curve is bell‑shaped and symmetric about the mean μ. The standard normal distribution has mean 0 and variance 1, denoted Z ~ N(0, 1).
正态分布 X ~ N(μ, σ²) 的概率密度函数为 f(x) = (1/(σ√(2π))) e−(x−μ)²/(2σ²)。曲线呈钟形，关于均值 μ 对称。标准正态分布均值为 0、方差为 1，记作 Z ~ N(0, 1)。
To standardise a normal variable, use Z = (X – μ) / σ. Probabilities are then found using the standard normal table or calculator. For a given probability, inverse normal calculations give the corresponding z‑value, and then X = μ + zσ.
对正态变量进行标准化，使用 Z = (X – μ) / σ。然后通过标准正态表或计算器求概率。对于给定概率，反向查找得到 z 值，再用 X = μ + zσ 求出观测值。
Many real‑world quantities follow a normal distribution, and the normal approximation to binomial and Poisson distributions is widely used in hypothesis testing and confidence intervals.
许多现实数据服从正态分布，正态近似二项分布和泊松分布被广泛用于假设检验和置信区间中。
5. Sampling & Central Limit Theorem | 抽样与中心极限定理
A statistic calculated from a sample, such as the sample mean X̄, is a random variable whose distribution is called the sampling distribution. For a random sample of size n from a population with mean μ and variance σ², E(X̄) = μ and Var(X̄) = σ²/n.
从样本计算出的统计量，例如样本均值 X̄，是一个随机变量，其分布称为抽样分布。对于来自均值为 μ、方差为 σ² 的总体的容量为 n 的随机样本，有 E(X̄) = μ、Var(X̄) = σ²/n。
The Central Limit Theorem (CLT) states that, regardless of the shape of the population distribution, the sampling distribution of X̄ becomes approximately normal as n increases (usually n ≥ 30 is sufficient). If the population itself is normal, X̄ is exactly normal for any n.
中心极限定理指出，无论总体分布形状如何，当样本容量 n 增大时（通常 n ≥ 30 即可），样本均值 X̄ 的抽样分布近似正态。若总体本身为正态，则 X̄ 对任意 n 都精确服从正态分布。
This result is crucial because it allows us to perform hypothesis tests and construct confidence intervals for means even when the population is not normal, provided n is large enough.
这一结论至关重要，因为它允许我们在总体非正态但样本足够大时，仍能对均值进行假设检验和构建置信区间。
6. Point Estimation & Confidence Intervals | 点估计与置信区间
A point estimate is a single value used to estimate a population parameter, e.g. the sample mean x̄ estimates μ, and the sample proportion p̂ estimates p. The sample variance s² estimates σ², calculated with divisor n – 1 for an unbiased estimator.
点估计是用单一数值估计总体参数，例如样本均值 x̄ 估计 μ，样本比例 p̂ 估计 p。样本方差 s² 估计 σ²，用除数 n – 1 得到无偏估计量。
A confidence interval gives a range of plausible values for the parameter. For a normal population with known σ, a 95% confidence interval for μ is x̄ ± z × (σ/√n). When σ is unknown, use the t‑distribution: x̄ ± t × (s/√n) with n – 1 degrees of freedom.
置信区间给出参数的一个合理范围。对于已知 σ 的正态总体，μ 的 95% 置信区间为 x̄ ± z × (σ/√n)。当 σ 未知时，使用 t 分布：x̄ ± t × (s/√n)，自由度为 n – 1。
For a population proportion p, a large‑sample confidence interval is p̂ ± z × √(p̂(1 – p̂)/n), provided the sample is large enough that np̂ > 5 and n(1 – p̂) > 5. The width of the interval decreases as n increases.
对于总体比例 p，大样本置信区间为 p̂ ± z × √(p̂(1 – p̂)/n)，前提是样本足够大，满足 np̂ > 5 且 n(1 – p̂) > 5。区间宽度随 n 增大而减小。
7. Hypothesis Testing – One‑Sample Tests | 假设检验——单样本检验
A hypothesis test evaluates evidence against a null hypothesis H₀ in favour of an alternative hypothesis H₁. The test statistic is calculated from the sample, and its value is compared with critical values or converted to a p‑value.
假设检验评估反对原假设 H₀ 而支持备择假设 H₁ 的证据。检验统计量由样本计算，其值与临界值比较或转化为 p 值。
For a normal mean with known σ, use the z‑test: z = (x̄ – μ₀) / (σ/√n). For an unknown σ, use the t‑test: t = (x̄ – μ₀) / (s/√n) with n – 1 degrees of freedom. The significance level α is the probability of rejecting H₀ when it is true.
对于已知 σ 的正态均值，使用 z 检验：z = (x̄ – μ₀) / (σ/√n)。σ 未知时用 t 检验：t = (x̄ – μ₀) / (s/√n)，自由度为 n – 1。显著性水平 α 为当 H₀ 为真时拒绝 H₀ 的概率。
For a binomial proportion, the test statistic can be based on the normal approximation: z = (p̂ – p₀) / √(p₀(1 – p₀)/n). For a Poisson rate, the test statistic is z = (x − λ₀) / √λ₀ under the normal approximation, or exact Poisson probabilities are used.
对于二项比例，检验统计量可基于正态近似：z = (p̂ – p₀) / √(p₀(1 – p₀)/n)。对于泊松速率，正态近似下使用 z = (x − λ₀) / √λ₀，或直接使用精确泊松概率。
8. Hypothesis Testing – Two‑Sample Tests & t‑tests | 双样本检验与 t 检验
Two‑sample tests compare means or proportions from two independent populations. For independent samples with known variances, use a two‑sample z‑test. When variances are unknown but assumed equal, use a pooled two‑sample t‑test with test statistic t = (x̄₁ – x̄₂) / (sₚ × √(1/n₁ + 1/n₂)), where sₚ² is the pooled variance.
双样本检验比较两个独立总体的均值或比例。方差已知时用双样本 z 检验。方差未知但假设相等时，使用合并双样本 t 检验，检验统计量为 t = (x̄₁ – x̄₂) / (sₚ × √(1/n₁ + 1/n₂))，其中 sₚ² 为合并方差。
When the two population variances are not assumed equal, the Welch–Satterthwaite t‑test is applied, which adjusts the degrees of freedom. For paired data (e.g. before‑and‑after studies), use a paired t‑test by analysing the differences dᵢ = xᵢ – yᵢ as a single sample.
当不假定两个总体方差相等时，使用 Welch–Satterthwaite t 检验，并对自由度进行修正。对于配对数据（如前后对照研究），将差值 dᵢ = xᵢ – yᵢ 视为单样本进行配对 t 检验。
Hypothesis tests for the difference between two proportions use the pooled estimate p̂ = (x₁ + x₂)/(n₁ + n₂) under H₀: p₁ = p₂, giving the test statistic z = (p̂₁ – p̂₂) / √(p̂(1 – p̂)(1/n₁ + 1/n₂)).
对于两个比例之差的假设检验，在 H₀: p₁ = p₂ 下使用合并估计 p̂ = (x₁ + x₂)/(n₁ + n₂)，检验统计量为 z = (p̂₁ – p̂₂) / √(p̂(1 – p̂)(1/n₁ + 1/n₂))。
9. Chi‑Squared Tests | 卡方检验
The chi‑squared test is used for categorical data. The test statistic is χ² = Σ (O − E)² / E, where O and E are observed and expected frequencies. The distribution depends on the degrees of freedom ν.
卡方检验用于分类数据。检验统计量为 χ² = Σ (O − E)² / E，其中 O 与 E 分别为观测频数和期望频数。其分布取决于自由度 ν。
In a goodness‑of‑fit test, we compare an observed frequency distribution with a theoretical distribution (e.g. binomial, Poisson). Degrees of freedom ν = number of classes – 1 – number of estimated parameters. In a contingency table test for independence, ν = (r – 1)(c – 1).
在拟合优度检验中，我们将观测频数分布与理论分布（如二项、泊松）进行比较。自由度 ν = 组数 – 1 – 估计参数个数。在列联表独立性检验中，ν = (r – 1)(c – 1)。
The test requires that expected frequencies are generally not less than 5. If some expected counts are too small, adjacent categories may be combined. The chi‑squared statistic is always compared against right‑tail critical values.
该检验要求期望频数通常不小于 5。若某些期望频数过小，可将相邻类别合并。卡方统计量总是与右侧临界值比较。
10. Correlation & Linear Regression | 相关与线性回归
The Pearson product‑moment correlation coefficient r measures the strength and direction of a linear relationship between two variables: r = Σ (xᵢ − x̄)(yᵢ − ȳ) / √[Σ (xᵢ − x̄)² Σ (yᵢ − ȳ)²]. Values of r range from –1 to 1.
Pearson 积矩相关系数 r 衡量两个变量之间线性关系的强度和方向：r = Σ (xᵢ − x̄)(yᵢ − ȳ) / √[Σ (xᵢ − x̄)² Σ (yᵢ − ȳ)²]。r 的取值范围为 –1 到 1。
The least‑squares regression line of y on x is y = a + bx, where b = Σ (xᵢ − x̄)(yᵢ − ȳ) / Σ (xᵢ − x̄)² and a = ȳ − b x̄. The line passes through the means (x̄, ȳ). The residual sum of squares measures the variation about the line.
y 对 x 的最小二乘回归直线为 y = a + bx，其中 b = Σ (xᵢ − x̄)(yᵢ − ȳ) / Σ (xᵢ − x̄)²，a = ȳ − b x̄。该直线通过均值点 (x̄, ȳ)。残差平方和衡量数据点围绕直线的变异性。
To test whether the population correlation ρ is zero, use the t‑statistic t = r √(n – 2) / √(1 – r²) with n – 2 degrees of freedom. Confidence intervals for the slope and intercept can also be constructed using standard errors from the regression output.
检验总体相关系数 ρ 是否为零，使用 t 统计量 t = r √(n – 2) / √(1 – r²)，自由度为 n – 2。利用回归输出的标准误也可构造斜率和截距的置信区间
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