📚 Year 13 WJEC Statistics: Core Knowledge Review | WJEC 统计 Year 13：核心知识点梳理
In Year 13 WJEC Statistics, you build on the probability and data-handling skills developed in the first year to master a range of advanced inference techniques. This article systematically reviews the most important topics, including probability distributions, sampling theory, estimation, hypothesis testing, correlation and regression, and non‑parametric methods. Clear definitions, key formulas, and common exam pitfalls are highlighted to support your revision.
在 WJEC 统计 Year 13 课程中，你将在第一年学习的概率和数据处理技能基础上，进一步掌握一系列高级推断方法。本文系统地梳理了最重要的考点，涵盖概率分布、抽样理论、估计、假设检验、相关与回归以及非参数方法。文中重点阐明了核心定义、关键公式和常见应试陷阱，以辅助你的复习。
1. Discrete and Continuous Random Variables | 离散与连续随机变量
A random variable (r.v.) is a function that assigns a numerical value to each outcome of a random experiment. For a discrete r.v., the probability mass function (pmf) P(X = x) specifies the probability that X takes the value x. The cumulative distribution function (cdf) F(x) = P(X ≤ x) gives the running total of probabilities.
随机变量是将随机实验的每个结果赋予一个数值的函数。对于离散随机变量，概率质量函数 P(X = x) 规定了 X 取值为 x 的概率。累积分布函数 F(x) = P(X ≤ x) 给出了概率的累计值。
For a continuous r.v., the probability density function (pdf) f(x) satisfies f(x) ≥ 0 and the area under the curve between a and b gives P(a < X < b). The cdf is F(x) = ∫₋∞ˣ f(t) dt, and the total area under the pdf is 1. The median m satisfies F(m) = 0.5, and the mode is the value of x that maximises f(x).
对于连续随机变量，概率密度函数 f(x) 满足 f(x) ≥ 0，并且在 a 和 b 之间的曲线下面积给出了 P(a < X < b)。累积分布函数为 F(x) = ∫₋∞ˣ f(t) dt，pdf 下的总面积为 1。中位数 m 满足 F(m) = 0.5，众数是使 f(x) 达到最大的 x 值。
Always check whether the random variable is discrete or continuous – this determines whether you sum probabilities or integrate the pdf. In WJEC exams, you may be required to find unknown constants in a pdf by setting the total area to 1, or to obtain the cdf by integration.
务必判断随机变量是离散还是连续——这决定了你是对概率求和还是对 pdf 积分。在 WJEC 考试中，你可能需要通过令总面积等于 1 来确定 pdf 中的未知常数，或者通过积分求出 cdf。
2. Expectation, Variance and Functions of Random Variables | 期望、方差与随机变量的函数
The expected value E(X) is the long‑run mean of the random variable. For discrete X, E(X) = Σ x P(X = x); for continuous X, E(X) = ∫ x f(x) dx over the domain. The variance Var(X) = E[(X − μ)²] = E(X²) − [E(X)]². The standard deviation is σ = √Var(X).
期望值 E(X) 是随机变量的长期平均值。对于离散 X，E(X) = Σ x P(X = x)；对于连续 X，E(X) = ∫ x f(x) dx 在定义域上积分。方差 Var(X) = E[(X − μ)²] = E(X²) − [E(X)]²。标准差为 σ = √Var(X)。
Linear transformations follow simple rules: E(aX + b) = aE(X) + b, and Var(aX + b) = a²Var(X). For two random variables X and Y, E(X + Y) = E(X) + E(Y) always, but Var(X + Y) = Var(X) + Var(Y) only if X and Y are independent. If they are dependent, Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y), where covariance is Cov(X,Y) = E[(X − μₓ)(Y − μᵧ)].
线性变换遵循简单规则：E(aX + b) = aE(X) + b，且 Var(aX + b) = a²Var(X)。对于两个随机变量 X 和 Y，总有 E(X + Y) = E(X) + E(Y)，但只有当 X 和 Y 独立时才有 Var(X + Y) = Var(X) + Var(Y)。若相关，则 Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y)，其中协方差 Cov(X,Y) = E[(X − μₓ)(Y − μᵧ)]。
You must be able to calculate E(X) and Var(X) for standard discrete distributions (binomial, Poisson) and for simple continuous pdfs. The expectation of a function of X, g(X), is E[g(X)] = Σ g(x)P(X=x) or ∫ g(x)f(x)dx. This is particularly useful when finding E(X²).
你必须能够为标准离散分布（二项分布、泊松分布）和简单的连续 pdf 计算 E(X) 和 Var(X)。随机变量函数 g(X) 的期望为 E[g(X)] = Σ g(x)P(X=x) 或 ∫ g(x)f(x)dx，这在计算 E(X²) 时特别有用。
3. The Normal Distribution and Approximations | 正态分布及其近似
The normal distribution X ~ N(μ, σ²) has the bell‑shaped pdf f(x) = (1/(σ√(2π))) exp( −½((x − μ)/σ)² ). Use the standardisation formula Z = (X − μ)/σ to convert to Z ~ N(0, 1), and then use statistical tables for probabilities. The distribution is symmetric, so P(Z < −k) = P(Z > k).
正态分布 X ~ N(μ, σ²) 具有钟形概率密度函数 f(x) = (1/(σ√(2π))) exp( −½((x − μ)/σ)² )。使用标准化公式 Z = (X − μ)/σ 转换为 Z ~ N(0, 1)，然后利用统计表求概率。该分布对称，因此 P(Z < −k) = P(Z > k)。
When a binomial distribution X ~ B(n, p) has np ≥ 5 and n(1−p) ≥ 5, it can be approximated by a normal distribution with μ = np and σ² = np(1−p). Apply a continuity correction: replace X ≤ k with X ≤ k + 0.5, and X ≥ k with X ≥ k − 0.5. Similarly, a Poisson distribution Y ~ Po(λ) with λ > 10 can be approximated by N(λ, λ) using a continuity correction.
当二项分布 X ~ B(n, p) 满足 np ≥ 5 且 n(1−p) ≥ 5 时，可用正态分布近似，其中 μ = np，σ² = np(1−p)。需应用连续性校正：将 X ≤ k 替换为 X ≤ k + 0.5，将 X ≥ k 替换为 X ≥ k − 0.5。类似地，当 λ > 10 时，泊松分布 Y ~ Po(λ) 可以用 N(λ, λ) 近似，并使用连续性校正。
In WJEC questions, precise use of the continuity correction and clear statements about the approximation are credited. Remember that the normal distribution also provides a model for sample means via the Central Limit Theorem, a separate but related topic.
在 WJEC 试题中，连续性校正的精确使用和对近似的清晰说明会得到分数。请记住，正态分布还通过中心极限定理为样本均值提供了模型，这是一个独立但相关的主题。
4. Sampling Distributions and the Central Limit Theorem | 抽样分布与中心极限定理
A statistic is a function of sample data, such as the sample mean x̄ or sample proportion p̂. Its probability distribution is called the sampling distribution. For a random sample of size n from a population with mean μ and variance σ², the sampling distribution of x̄ has mean μ and variance σ²/n. If the population is normal, x̄ is exactly normal.
统计量是样本数据的函数，如样本均值 x̄ 或样本比例 p̂。其概率分布称为抽样分布。对于从均值为 μ、方差为 σ² 的总体中抽取的容量为 n 的随机样本，x̄ 的抽样分布的均值为 μ，方差为 σ²/n。若总体服从正态分布，则 x̄ 精确服从正态分布。
The Central Limit Theorem (CLT) states that for large n (typically n ≥ 30), the sampling distribution of x̄ is approximately normal, regardless of the shape of the population distribution. This is the foundation for many inference procedures. For proportions, the sampling distribution of p̂ is approximately N(p, p(1−p)/n) provided np and n(1−p) are both at least 5.
中心极限定理指出，对于较大的 n（通常 n ≥ 30），无论总体分布的形状如何，x̄ 的抽样分布都近似正态分布。这是许多推断方法的基础。对于比例，只要 np 和 n(1−p) 均至少为 5，p̂ 的抽样分布近似为 N(p, p(1−p)/n)。
The standard error (SE) of a statistic is the standard deviation of its sampling distribution. For x̄, SE = σ/√n; for p̂, SE = √[p(1−p)/n]. When σ is unknown, we estimate it by the sample standard deviation s, leading to the t‑distribution for small samples.
统计量的标准误是其抽样分布的标准差。对于 x̄，SE = σ/√n；对于 p̂，SE = √[p(1−p)/n]。当 σ 未知时，我们用样本标准差 s 来估计，对于小样本这将引向 t 分布。
5. Point Estimation and 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. x̄ estimates μ, and s² estimates σ². Unbiased estimators have expected values equal to the true parameter; for a normal population E(s²) = σ², but x̄ is always unbiased for μ. Confidence intervals provide a range of plausible values for the parameter.
点估计是用单一数值来估计总体参数，例如 x̄ 估计 μ，s² 估计 σ²。无偏估计量的期望值等于真实参数；对于正态总体，E(s²) = σ²，而 x̄ 对 μ 始终是无偏的。置信区间为参数提供了一个可信的范围。
For the population mean with known σ: 95% CI is x̄ ± 1.96(σ/√n). When σ is unknown and the population is normal, use the t‑distribution with n−1 degrees of freedom: x̄ ± t_(n−1, α/2) × s/√n. For a population proportion p, the approximate CI is p̂ ± z_(α/2) × √[p̂(1−p̂)/n]. The width of the interval decreases as n increases.
对于已知 σ 的总体均值：95% 置信区间为 x̄ ± 1.96(σ/√n)。当 σ 未知且总体正态时，使用自由度为 n−1 的 t 分布：x̄ ± t_(n−1, α/2) × s/√n。对于总体比例 p，近似置信区间为 p̂ ± z_(α/2) × √[p̂(1−p̂)/n]。区间宽度随 n 增大而减小。
Confidence intervals for the difference of two means (independent or paired) and for the ratio of two variances may also be required. Understanding the interpretation – “we are 95% confident that the interval captures the true parameter” – is essential for writing conclusions in context.
可能还需要两个均值之差（独立或配对）的置信区间以及两个方差之比的置信区间。理解其解释——“我们有 95% 的把握该区间包含了真实参数”——对于在具体情境中写出结论至关重要。
6. Hypothesis Testing: Concepts and One‑Sample Tests | 假设检验：概念与单样本检验
A hypothesis test assesses the evidence against a null hypothesis H₀ in favour of an alternative H₁. The significance level α (often 0.05) is the probability of a Type I error: rejecting H₀ when it is true. The p‑value is the probability of obtaining a test statistic at least as extreme as the observed one, assuming H₀ is true. If p < α, reject H₀.
假设检验评估反对原假设 H₀、支持备择假设 H₁ 的证据。显著性水平 α（通常为 0.05）是发生第一类错误的概率：当 H₀ 为真时拒绝 H₀。p 值是假定 H₀ 为真时，获得一个至少与实际观测一样极端的检验统计量的概率。若 p < α，则拒绝 H₀。
One‑sample tests: For a normal mean with known σ, use the z‑test: z = (x̄ − μ₀) / (σ/√n). If σ is unknown, use the one‑sample t‑test: t = (x̄ − μ₀) / (s/√n) with df = n−1. For a binomial proportion, the test statistic z = (p̂ − p₀) / √[p₀(1−p₀)/n] is approximately standard normal under H₀. For a Poisson mean λ, you may either use the exact Poisson distribution or the normal approximation if λ > 10.
单样本检验：对于已知 σ 的正态均值，使用 z 检验：z = (x̄ − μ₀) / (σ/√n)。若 σ 未知，使用单样本 t 检验：t = (x̄ − μ₀) / (s/√n)，自由度为 n−1。对于二项比例，检验统计量 z = (p̂ − p₀) / √[p₀(1−p₀)/n] 在 H₀ 下近似标准正态。对于泊松均值 λ，可使用精确泊松分布，或当 λ > 10 时使用正态近似。
Always state hypotheses clearly, define the test statistic, compare the calculated value with the critical value or use the p‑value approach, and write a conclusion in the context of the problem. Remember to choose the correct one‑tailed or two‑tailed test based on H₁.
务必清晰陈述假设，定义检验统计量，将计算值与临界值比较或采用 p 值法，并在问题情境下写出结论。切记根据 H₁ 选择正确的单尾或双尾检验。
7. Two‑Sample Tests and Non‑Parametric Tests | 双样本检验与非参数检验
When comparing two independent samples from normal populations with unknown but equal variances, use the two‑sample t‑test with pooled variance: t = (x̄₁ − x̄₂) / (s_p √(1/n₁ + 1/n₂)), where s_p² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁+n₂−2). The degrees of freedom are n₁+n₂−2. If variances are not assumed equal, use Welch’s t‑test.
当比较来自未知但等方差的正态总体的两个独立样本时，使用合并方差的双样本 t 检验：t = (x̄₁ − x̄₂) / (s_p √(1/n₁ + 1/n₂))，其中 s_p² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁+n₂−2)。自由度为 n₁+n₂−2。若不假设方差相等，则使用 Welch t 检验。
For paired data, compute the differences dᵢ = xᵢ − yᵢ and perform a one‑sample t‑test on the dᵢ values. Non‑parametric tests do not require normality. The Wilcoxon signed‑rank test is used for paired differences: rank the absolute differences, sum the ranks for positive and negative differences, and take the smaller sum as the test statistic T. The sign test simply counts the number of positive signs and uses a binomial test.
对于配对数据，计算差值 dᵢ = xᵢ − yᵢ，并对 dᵢ 值进行单样本 t 检验。非参数检验不要求正态性。Wilcoxon 符号秩检验用于配对差异：对绝对差值排序，分别求正差值的秩和和负差值的秩和，取其中较小的秩和作为检验统计量 T。符号检验仅计算正号的个数，并采用二项检验。
The Mann‑Whitney U test compares two independent samples by ranking all observations together and computing U₁ and U₂, where the smaller U is the test statistic. The null hypothesis is that the two populations have the same distribution. These tests are particularly valuable when data are ordinal or when normality cannot be assumed.
Mann‑Whitney U 检验通过将所有观测值一起排序并计算 U₁ 和 U₂ 来比较两个独立样本，其中较小的 U 为检验统计量。原假设是两个总体具有相同分布。当数据为顺序变量或无法假定正态性时，这些检验尤其有价值。
8. Chi‑Squared Tests | 卡方检验
The chi‑squared goodness‑of‑fit test checks whether observed frequencies follow an expected distribution. The test statistic is χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ, with degrees of freedom ν = number of categories − 1 − d, where d is the number of parameters estimated from the data. If χ² > χ²_crit, reject the null hypothesis of a good fit.
卡方拟合优度检验用于检查观测频数是否服从预期分布。检验统计量为 χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ，自由度 ν = 类别数 − 1 − d，其中 d 是从数据中估计的参数个数。若 χ² > χ²_crit，则拒绝拟合良好的原假设。
The chi‑squared test for independence uses a contingency table. Expected frequencies are calculated as (row total × column total) / grand total. The test statistic has the same form, with ν = (r−1)(c−1). Yates’ correction for continuity (subtracting 0.5 from each |O − E|) may be applied for 2×2 tables, though its use is debated; follow the specification’s guidance.
独立性卡方检验使用列联表。期望频数计算公式为 (行合计 × 列合计) / 总计。检验统计量形式相同，自由度为 ν = (r−1)(c−1)。对于 2×2 表，可应用 Yates 连续性校正（将每个 |O − E| 减去 0.5），尽管其使用存在争议；请遵循考试大纲的指导。
Ensure all expected frequencies are at least 5; if not, combine adjacent categories. In WJEC exams, you are expected to state hypotheses in words, compute the test statistic, determine the critical value from tables, and interpret the result in context. The test often features in questions about distributions, genetics, and opinion surveys.
确保所有期望频数不少于 5；若不足，则合并相邻类别。在 WJEC 考试中，你需要用文字陈述假设，计算检验统计量，从表中查出临界值，并在具体情境中解释结果。该检验经常出现在关于分布、遗传学和意见调查的问题中。
9. Correlation and Linear Regression | 相关与线性回归
The product‑moment correlation coefficient (PMCC) r measures the strength and direction of a linear relationship between two variables. It is calculated as r = Sxy / √(Sxx Sxy) where Sxx = Σ(xᵢ − x̄)², Syy = Σ(yᵢ − ȳ)², Sxy = Σ(xᵢ − x̄)(yᵢ − ȳ). r always lies between −1 and 1. Spearman’s rank correlation coefficient rₛ measures monotonic association using the ranked data.
积矩相关系数 r 测量两个变量之间线性关系的强度和方向。其计算公式为 r = Sxy / √(Sxx Syy)，其中 Sxx = Σ(xᵢ − x̄)², Syy = Σ(yᵢ − ȳ)², Sxy = Σ(xᵢ − x̄)(yᵢ − ȳ)。r 始终介于 −1 和 1 之间。Spearman 秩相关系数 rₛ 使用排序数据来度量单调关联。
In simple linear regression, we model the response y as y = a + bx + ε. The least squares estimates are b = Sxy / Sxx and a = ȳ − b x̄. The fitted line can be used for prediction, but careful: extrapolation beyond the data range is unreliable. The coefficient of determination r² indicates the proportion of variation in y explained by x.
在简单线性回归中，我们将响应变量 y 建模为 y = a + bx + ε。最小二乘估计为 b = Sxy / Sxx 且 a = ȳ − b x̄。拟合直线可用于预测，但需注意：超出数据范围的外推并不可靠。决定系数 r² 表示 y 的变异中由 x 解释的比例。
Confidence intervals for the slope b and prediction intervals for a new observation are assessed using the t‑distribution. Hypothesis tests for ρ = 0 (population correlation) use a t‑test: t = r √(n−2) / √(1−r²) with n−2 degrees of freedom. Spearman’s test uses either special tables or a t‑approximation.
对斜率 b 的置信区间以及对新观测值的预测区间，均利用 t 分布进行评估。关于 ρ = 0 的假设检验使用 t 检验：t = r √(n−2) / √(1−r²)，自由度为 n−2。Spearman 检验使用专用表或 t 近似。
10. Key Formulae and Exam Tips | 关键公式与应试技巧
Below is a summary of essential formulas often required in Year 13 WJEC Statistics examinations. Familiarity with these will boost your confidence.
以下是 Year 13 WJEC 统计考试中经常需要的关键公式总结。熟练掌握这些公式将增强你的信心。
| Topic | Key Formula | Conditions |
|---|---|---|
| Mean and variance | E(X) = Σ xP(X=x) or ∫ x f(x)dx; Var(X) = E(X²) − [E(X)]² | Discrete / continuous |
| Normal distribution | Z = (X − μ)/σ | 更多咨询请联系16621398022（同微信）
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