📚 Year 12 WJEC Statistics: Core Concepts Review | WJEC 12年级统计：核心知识点梳理
Mastering Year 12 Statistics is all about building a solid foundation in data handling, probability, distributions and inference. This article distills the essential topics you need to know for the WJEC AS specification, from sampling methods to hypothesis testing, with clear explanations and practical examples.
掌握12年级统计学的关键在于为数据处理、概率、分布与推断打下坚实基础。本文提炼了WJEC AS阶段你必需掌握的核心主题，从抽样方法到假设检验，并提供清晰解释与实用示例。
1. Data Types and Sampling | 数据类型与抽样
Data can be categorical (qualitative), such as eye colour, or numerical (quantitative), like height. Numerical data is further split into discrete, where values are countable (e.g. number of siblings), and continuous, where values can take any measurement on a scale (e.g. time).
数据可以是分类的（定性的），如眼睛颜色，也可以是数值的（定量的），如身高。数值数据又细分为离散型（可数的，例如兄弟姐妹数量）和连续型（可在量尺上取任意测量值的，例如时间）。
In WJEC Statistics you must be confident with sampling methods. A simple random sample gives every member of the population an equal chance of being selected. Stratified sampling divides the population into distinct groups (strata) and takes a random sample from each in proportion to size. Systematic sampling selects every kth item from a list, and quota sampling is a non-random method where interviewers fill fixed quotas of different groups.
在WJEC统计中，你需要熟练运用抽样方法。简单随机抽样使总体中每个成员都有相等被选中的机会。分层抽样将总体划分为明确的分组（层），并按比例从每层中进行随机抽样。系统抽样按名单每隔k个选取一项，而配额抽样是一种非随机方法，调查员按照不同组别的固定配额进行填充。
2. Data Presentation | 数据呈现
Appropriate diagrams make patterns visible. For discrete data, bar charts and vertical line charts are common. For continuous data, histograms are essential; note that area is proportional to frequency, so frequency density = frequency ÷ class width. Cumulative frequency curves (ogives) help locate medians and quartiles, while box plots display the five-number summary: minimum, Q1, median, Q3 and maximum, and can reveal outliers using the 1.5×IQR rule.
恰当的图示能让数据模式变得直观。对于离散数据，常用柱状图和折线图。对于连续数据，直方图是核心工具；注意面积与频数成正比，因此频数密度 = 频数 ÷ 组距。累积频率曲线（肩形图）有助于确定中位数和四分位数，而箱线图则呈现五数汇总：最小值、Q1、中位数、Q3和最大值，并能通过1.5×IQR法则揭示异常值。
Scatter diagrams show the relationship between two variables, giving a first visual impression of correlation before any formal calculation.
散点图展示两个变量之间的关系，能让你在进行正式计算前先形成对相关性的初步视觉印象。
3. Measures of Central Tendency | 集中趋势度量
The three principal measures are the mean, median and mode. For a set of n values x₁, x₂, …, xₙ, the mean is x̄ = Σxᵢ / n. The median is the middle value when data are ordered, and the mode is the most frequent value. In grouped frequency tables, the modal class is the interval with the highest frequency, and the mean is estimated using midpoints: x̄ ≈ Σfₘ / Σf, where m is the midpoint of each class.
三个主要度量指标是平均数、中位数和众数。对于 n 个数值 x₁, x₂, …, xₙ，平均数为 x̄ = Σxᵢ / n。中位数是数据排序后位于中间的值，众数是出现频率最高的值。在分组频数表中，众数组是频数最高的区间，平均数则利用组中值来估算：x̄ ≈ Σfₘ / Σf，其中 m 为每组中的组中值。
Each measure has its strengths: the mean uses all data but is sensitive to outliers; the median is resistant to extreme values; the mode is the only measure suitable for categorical data.
每种度量各有优点：平均数利用了所有数据，但对异常值敏感；中位数不受极端值影响；众数是唯一适用于分类数据的度量。
4. Measures of Dispersion | 离散度度量
Dispersion tells you how spread out the data are. The range is simply max – min, but is easily distorted by outliers. The interquartile range (IQR = Q3 – Q1) focuses on the middle 50 percent of the data and is robust.
离散度告诉你数据的分散程度。极差就是最大值减最小值，但很容易被异常值扭曲。四分位距（IQR = Q3 – Q1）关注中间50%的数据，具有稳健性。
Variance and standard deviation are the most widely used measures. For a population, variance σ² = Σ(x – μ)² / N; for a sample, variance s² = Σ(x – x̄)² / (n – 1). Standard deviation is the square root of variance. You must be able to calculate these using both raw data and grouped frequency tables, often with the formula s² = (Σfx² / Σf) – x̄².
方差和标准差是使用最广泛的度量。对于总体，方差 σ² = Σ(x – μ)² / N；对于样本，方差 s² = Σ(x – x̄)² / (n – 1)。标准差是方差的平方根。你必须能够利用原始数据和分组频数表来计算这些指标，常用公式为 s² = (Σfx² / Σf) – x̄²。
5. Probability Basics | 概率基础
Probability quantifies the chance of an event, ranging from 0 (impossible) to 1 (certain). For equally likely outcomes, P(A) = number of outcomes in A ÷ total number of outcomes. The complement rule states P(not A) = 1 – P(A).
概率量化事件发生的可能性，范围从0（不可能）到1（一定）。对于等可能结果，P(A) = 事件A中的结果数 ÷ 总结果数。互补规则为 P(非A) = 1 – P(A)。
Two events A and B are mutually exclusive if they cannot occur together; then P(A or B) = P(A) + P(B). Otherwise, the general addition rule applies: P(A or B) = P(A) + P(B) – P(A and B).
如果两个事件 A 和 B 互斥（不能同时发生），则 P(A 或 B) = P(A) + P(B)。否则，要用一般加法法则：P(A 或 B) = P(A) + P(B) – P(A 且 B)。
6. Conditional Probability and Independence | 条件概率与独立性
Conditional probability P(A|B) reads “the probability of A given B” and is defined as P(A|B) = P(A and B) / P(B), provided P(B) > 0. Events are independent if knowing one does not change the probability of the other: P(A|B) = P(A) or equivalently P(A and B) = P(A)×P(B).
条件概率 P(A|B) 读作“已知B时A的概率”，定义为 P(A|B) = P(A 且 B) / P(B)，前提是 P(B) > 0。如果知道一个事件的发生不会改变另一个事件的概率，则称两事件独立：P(A|B) = P(A) 或等价地 P(A 且 B) = P(A)×P(B)。
WJEC questions often require you to analyse two-way tables and tree diagrams. Multiply along branches for “and” probabilities; add appropriate final probabilities for “or”. Always check that branch probabilities sum to 1.
WJEC考题经常要求你分析双向表格和树形图。沿分支相乘得到“且”的概率；将合适的最终概率相加得到“或”的概率。务必确保分支概率之和为1。
7. Discrete Random Variables | 离散随机变量
A random variable X takes numerical values determined by chance. For a discrete variable, you list its probability distribution P(X = x). The sum of all probabilities must equal 1.
随机变量 X 是由机会决定的数值型变量。对于一个离散型变量，你需要列出它的概率分布 P(X = x)。所有概率之和必须等于1。
The expected value E(X) = Σ x·P(X = x) represents the long‑run average. Variance Var(X) = E(X²) – [E(X)]², where E(X²) = Σ x²·P(X = x). Standard deviation is √Var(X).
期望值 E(X) = Σ x·P(X = x) 代表长期平均值。方差 Var(X) = E(X²) – [E(X)]²，其中 E(X²) = Σ x²·P(X = x)。标准差为 √Var(X)。
Linear transformations follow simple rules: E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).
线性变换遵循简单规则：E(aX + b) = aE(X) + b 且 Var(aX + b) = a²Var(X)。
8. The Binomial Distribution | 二项分布
A binomial distribution arises when you have a fixed number of independent trials, n, each with the same probability of success, p. If X ~ B(n, p), then the probability of exactly r successes is given by the binomial formula:
当你有固定次数的独立试验 n，每次成功的概率 p 相同时，就形成二项分布。若 X ~ B(n, p)，则恰好 r 次成功的概率由以下二项式公式给出：
P(X = r) = ⁿCᵨ × pʳ × (1 – p)ⁿ⁻ʳ
You can calculate ⁿCᵨ using the formula n! / [r!(n – r)!] or your calculator. The mean is E(X) = np and the variance is Var(X) = np(1 – p). WJEC exams will expect you to use statistical tables or calculators to find cumulative binomial probabilities, and to handle phrases like “at most”, “fewer than”, “at least” and “more than”.
你可以用公式 n! / [r!(n – r)!] 或计算器计算 ⁿCᵨ。均值为 E(X) = np，方差为 Var(X) = np(1 – p)。WJEC 考试希望你能够使用统计表或计算器查找累积二项概率，并能处理如“至多”、“少于”、“至少”和“多于”等措辞。
9. The Normal Distribution | 正态分布
The normal distribution models continuous data that cluster symmetrically around a mean μ with standard deviation σ. The notation is X ~ N(μ, σ²). Because there are infinitely many normal curves, we standardise using the Z‑score: Z = (X – μ) / σ, which gives Z ~ N(0, 1²).
正态分布可用于对围绕均值 μ 对称聚集、标准差为 σ 的连续数据进行建模。记作 X ~ N(μ, σ²)。由于正态曲线有无穷多条，我们通过 Z 分数进行标准化：Z = (X – μ) / σ，从而得到 Z ~ N(0, 1²)。
To find P(X < a), standardise to P(Z < (a – μ)/σ) and use standard normal tables. For P(X > b), use 1 – P(Z < (b – μ)/σ). You must also be able to find unknown means or standard deviations given a probability.
要计算 P(X < a)，先标准化为 P(Z < (a – μ)/σ) 再查标准正态表。对于 P(X > b)，使用 1 – P(Z < (b – μ)/σ)。你还必须掌握在已知概率下反推未知均值或标准差。
Remember that the normal approximation to the binomial (with continuity correction) may appear in some Year 12 specifications, so ensure you know the conditions: np > 5 and n(1 – p) > 5.
请记住，某些12年级大纲会涉及二项分布的正态近似（带连续性校正），为此你需要明确适用条件：np > 5 且 n(1 – p) > 5。
10. Hypothesis Testing for a Binomial Proportion | 二项分布假设检验
Hypothesis testing is a formal method for deciding whether sample evidence supports a claim about a population parameter. For Year 12 WJEC, the focus is on testing a probability p in a binomial setting.
假设检验是一种规范方法，用于判断样本证据是否支持关于总体参数的某个论述。对12年级WJEC而言，重点是检验二项背景下的概率 p。
You set up a null hypothesis H₀: p = some value, against an alternative H₁: p < , p > or p ≠ that value, depending on the wording. The significance level α, often 5%, is the probability of rejecting H₀ when it is true.
你需要设定原假设 H₀: p = 某值，并根据题意选择备择假设 H₁: p < 某值、p > 某值 或 p ≠ 某值。显著性水平 α（常为5%）是在 H₀ 为真时拒绝 H₀ 的概率。
For a binomial test, you calculate the probability of obtaining the observed result, or more extreme, assuming H₀ is true. If this p‑value is less than or equal to α, you reject H₀. Otherwise, you do not reject H₀. Always state your conclusion in the context of the problem, referencing “sufficient evidence” or “insufficient evidence”.
对二项检验而言，你在假定 H₀ 为真的前提下，计算出获得当前观察结果或更极端结果的概率。如果此 p 值 ≤ α，则拒绝 H₀；否则不拒绝 H₀。必须结合问题背景陈述结论，并提及“有充分证据”或“证据不足”。
Critical regions are another approach: find the set of outcomes that would lead to rejecting H₀ at the given significance level, often by using cumulative binomial tables, and check whether the observed result lies inside that region.
拒绝域是另一种方法：利用累积二项表找出在给定显著性水平下会拒绝 H₀ 的结果集合，再检查观察结果是否落入该区域内。
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导