📚 Year 12 WJEC Statistics: Core Knowledge Review | Year 12 WJEC 统计：核心知识点梳理
This article provides a comprehensive review of the core topics covered in the Year 12 WJEC Statistics syllabus. Mastering these foundations is essential for success in the AS exam and beyond.
本文全面梳理了 Year 12 WJEC 统计课程中的核心知识点。掌握这些基础对于在 AS 考试及后续学习中取得成功至关重要。
1. Statistical Sampling | 统计抽样
A population is the entire set of individuals or items of interest. A sample is a subset of the population selected to represent it. The sampling frame is a list of all members of the population from which the sample is drawn.
总体是感兴趣的全体个体或项目。样本是从总体中选出的一个子集，用以代表总体。抽样框是列出所有总体成员的名单，样本即从中抽取。
Simple random sampling gives every member an equal chance of being chosen, avoiding bias. Stratified sampling divides the population into strata and selects a proportional random sample from each. Systematic sampling selects every kth member from the sampling frame. Quota sampling fills specific quotas and is non‑random, while convenience sampling selects easily available members.
简单随机抽样让每个成员被选中的机会均等，避免了偏差。分层抽样将总体划分为层，从每层中按比例随机抽样。系统抽样从抽样框中每隔 k 名抽取一个。配额抽样按指定配额选取，是非随机方法；便利抽样则选择最容易获得的成员。
Advantages and disadvantages must be understood: random methods eliminate selection bias but require a full sampling frame; non‑random methods are quicker but may be unrepresentative.
需要理解各自的优缺点：随机方法消除了选择偏差但需要完整的抽样框；非随机方法较快但可能不具有代表性。
2. Data Presentation | 数据展示
Data can be displayed using frequency tables, bar charts, histograms, stem‑and‑leaf diagrams, box plots and cumulative frequency curves. Histograms show frequency density on the vertical axis with area proportional to frequency; frequency density = frequency / class width.
数据可以通过频数表、条形图、直方图、茎叶图、箱线图和累积频率曲线来展示。直方图用频率密度作为纵轴，面积与频数成正比；频率密度 = 频数 / 组距。
Box plots display minimum, lower quartile Q₁, median Q₂, upper quartile Q₃, and maximum. Outliers are often defined as values below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR. Stem‑and‑leaf diagrams retain original values while showing shape.
箱线图显示最小值、下四分位数 Q₁、中位数 Q₂、上四分位数 Q₃ 和最大值。异常值通常定义为 Q₁ − 1.5 × IQR 以下或 Q₃ + 1.5 × IQR 以上的数值。茎叶图在展示分布形态的同时保留了原始数据。
3. Measures of Central Tendency and Dispersion | 集中趋势与离散量数
Central tendency is measured by mean, median and mode. The mean x̄ = Σx / n is sensitive to extreme values, while the median is robust and the mode indicates the most frequent value.
集中趋势通过均值、中位数和众数来衡量。均值 x̄ = Σx / n 受极端值影响较大，而中位数具有稳健性，众数表示出现最多的值。
Dispersion is described by range, interquartile range (IQR = Q₃ − Q₁), variance and standard deviation. The sample variance s² = Σ(x − x̄)² / (n−1) uses n−1 for unbiased estimation. Standard deviation s = √s² has the same units as the original data.
离散程度可用极差、四分位距（IQR = Q₃ − Q₁）、方差和标准差来描述。样本方差 s² = Σ(x − x̄)² / (n−1) 以 n−1 作为除数进行无偏估计。标准差 s = √s² 的单位与原始数据相同。
4. Probability | 概率
Probability is a measure of the likelihood of an event, ranging from 0 to 1. For a sample space S, P(S) = 1. The complement rule states P(A’) = 1 − P(A).
概率衡量事件发生的可能性，范围从 0 到 1。对于样本空间 S，P(S) = 1。互补规则为 P(A’) = 1 − P(A)。
For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B). For independent events, P(A ∩ B) = P(A) × P(B). Tree diagrams help multiply probabilities along branches for compound events.
对于互斥事件 A 与 B，P(A ∪ B) = P(A) + P(B)。对于独立事件，P(A ∩ B) = P(A) × P(B)。树状图有助于沿分支相乘概率以处理复合事件。
5. Conditional Probability and Independence | 条件概率与独立性
Conditional probability P(B|A) = P(A ∩ B) / P(A), provided P(A) > 0. It represents the probability of B occurring given that A has occurred. This concept is essential in reversing probabilities and updating beliefs.
条件概率 P(B|A) = P(A ∩ B) / P(A)，前提是 P(A) > 0。它表示在已知事件 A 发生的情况下事件 B 发生的概率。这一概念对于反转概率和更新判断至关重要。
Two events are independent if P(A ∩ B) = P(A)P(B) or equivalently P(B|A) = P(B). In practice, you can check if the product of individual probabilities equals the joint probability.
若 P(A ∩ B) = P(A)P(B) 或等价地 P(B|A) = P(B)，则两个事件独立。实际应用中可以检验个体概率之积是否等于联合概率。
6. Discrete Random Variables | 离散随机变量
A discrete random variable X has a set of possible values with associated probabilities. The probability distribution table lists x and P(X = x), and must satisfy ΣP(X = x) = 1.
离散随机变量 X 拥有一组可能的取值及对应的概率。概率分布表列出 x 与 P(X = x)，且必须满足 ΣP(X = x) = 1。
The expectation E(X) = Σ x·P(X = x) represents the long‑term average. Variance Var(X) = E(X²) − [E(X)]² measures spread. Both linear transformations E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X) are examinable.
期望值 E(X) = Σ x·P(X = x) 代表长期的平均值。方差 Var(X) = E(X²) − [E(X)]² 衡量离散程度。线性变换公式 E(aX + b) = aE(X) + b 和 Var(aX + b) = a²Var(X) 都是考查内容。
7. Binomial Distribution | 二项分布
The binomial model applies when there is a fixed number of trials n, each trial is independent, each has two outcomes (success/failure), and the probability of success p is constant. We write X ~ B(n, p).
当满足固定试验次数 n、每次试验独立、每次只有成功或失败两种结果且成功概率 p 恒定不变时，适用二项分布模型。记作 X ~ B(n, p)。
The binomial probability formula is P(X = r) = C(n, r) pʳ (1−p)ⁿ⁻ʳ, where C(n, r) = n! / [r!(n−r)!]. Cumulative probabilities can be found using tables or calculators. The mean is E(X) = np and variance Var(X) = np(1−p).
二项概率公式为 P(X = r) = C(n, r) pʳ (1−p)ⁿ⁻ʳ，其中 C(n, r) = n! / [r!(n−r)!]。累积概率可通过查表或计算器获得。均值为 E(X) = np，方差为 Var(X) = np(1−p)。
8. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution with a symmetric bell‑shaped curve. It is defined by its mean μ and variance σ²; X ~ N(μ, σ²). The total area under the curve equals 1.
正态分布是一种连续型概率分布，呈对称的钟形曲线。它由均值 μ 和方差 σ² 定义，记作 X ~ N(μ, σ²)。曲线下的总面积为 1。
To find probabilities, standardise using Z = (X − μ) / σ, so Z ~ N(0, 1). Probability tables give Φ(z) = P(Z < z). For reverse calculations, you look up the z‑value corresponding to a given probability and then transform back: X = μ + zσ.
求概率时，先标准化 Z = (X − μ) / σ，使得 Z ~ N(0, 1)。概率表给出 Φ(z) = P(Z < z)。进行反向计算时，先查找对应于给定概率的 z 值，再变换回 X = μ + zσ。
9. Estimation | 估计
A point estimate gives a single value for a population parameter, such as using the sample mean x̄ to estimate the population mean μ, or the sample proportion p̂ to estimate the population proportion p.
点估计用单一数值来估计总体参数，例如用样本均值 x̄ 估计总体均值 μ，或用样本比例 p̂ 估计总体比例 p。
The standard error measures the variability of the estimator. For the sample mean, standard error = σ / √n when σ is known, or s / √n when estimated from the sample. The Central Limit Theorem states that for large samples, the distribution of x̄ is approximately normal regardless of the population distribution.
标准误衡量估计量的变异性。对于样本均值，当 σ 已知时标准误 = σ / √n，若由样本估计则为 s / √n。中心极限定理指出，在大样本下，无论总体分布如何，x̄ 的分布都近似正态。
10. Hypothesis Testing | 假设检验
A hypothesis test assesses evidence against a null hypothesis H₀. The alternative hypothesis H₁ may be one‑tailed or two‑tailed. The significance level α is the probability of rejecting H₀ when it is true.
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。显著性水平 α 是当 H₀ 为真时拒绝它的概率。
For a binomial test, the test statistic is the observed number of successes. The p‑value is P(observed or more extreme | H₀ true). If p‑value ≤ α, reject H₀. Alternatively, find the critical region where the test statistic leads to rejection. For normal population mean testing with known variance, the test statistic z = (x̄ − μ₀) / (σ/√n) is compared with critical z‑values.
对于二项检验，检验统计量是观测到的成功次数。p 值 = P(观测值或更极端结果 | H₀ 为真)。若 p 值 ≤ α，则拒绝 H₀。另一种方法是找出导致拒绝的临界区域。对于已知方差的正态总体均值检验，检验统计量 z = (x̄ − μ₀) / (σ/√n) 与临界 z 值进行比较。
11. Correlation and Regression | 相关与回归
Scatter graphs show the relationship between two variables. Pearson’s product‑moment correlation coefficient r measures the strength and direction of linear association, with −1 ≤ r ≤ 1. Values close to 1 or −1 indicate strong linear correlation.
散点图显示两个变量之间的关系。皮尔逊积矩相关系数 r 衡量线性关联的强度与方向，−1 ≤ r ≤ 1。接近 1 或 −1 的数值表示强线性相关。
The least squares regression line has equation y = a + bx, where b = Sxy / Sxx and a = ȳ − b x̄. The slope b indicates the change in y per unit increase in x. Interpolation is prediction within the range of observed x‑values; extrapolation beyond that range is unreliable.
最小二乘回归直线的方程为 y = a + bx，其中 b = Sxy / Sxx，a = ȳ − b x̄。斜率 b 表示 x 每增加一个单位时 y 的变化量。内插是在观测 x 值范围内进行预测；外推超出该范围则不可靠。
Published by TutorHao | Statistics Revision Series | aleveler.com
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