📚 A-Level WJEC Mathematics: Statistics Key Points Guide | A-Level WJEC 数学：统计 考点精讲
This guide offers a thorough revision of the essential Statistics topics in the WJEC A-Level Mathematics specification. It covers data presentation, probability, key distributions, hypothesis testing, correlation and regression, and is designed to reinforce your understanding and exam technique.
本文是针对 WJEC A-Level 数学大纲中统计部分的核心考点精讲，系统梳理了数据表示、概率、重要分布、假设检验以及相关与回归等内容，帮助你巩固知识，提升应试能力。
1. Data Presentation | 数据表示
WJEC questions frequently ask you to draw and interpret histograms. For continuous data, the vertical scale represents frequency density, where frequency density = frequency ÷ class width. The area of each bar is proportional to the frequency.
WJEC 试题常要求绘制并解读直方图。对连续数据，纵轴表示频率密度，频率密度 = 频数 ÷ 组距。每个矩形的面积与频数成正比。
Cumulative frequency curves are used to estimate medians, quartiles and percentiles. Plot the upper class boundary against cumulative frequency, join the points with a smooth curve, and read off values at 25%, 50% and 75% of the total frequency.
累积频率曲线用于估算中位数、四分位数和百分位数。将组上限对累积频数描点，用平滑曲线连接，再按总频数的 25%、50% 和 75% 读取对应值。
Box plots display the five‑number summary: minimum, Q1, median, Q3 and maximum. Outliers can be defined as any observation lying more than 1.5 × IQR below Q1 or above Q3, where IQR = Q3 − Q1. In the exam, you may need to identify outliers and draw box plots with or without them.
箱线图展示五数概括：最小值、下四分位数 Q1、中位数 Q2、上四分位数 Q3 和最大值。异常值通常定义为低于 Q1−1.5×IQR 或高于 Q3+1.5×IQR 的观测值，其中 IQR = Q3−Q1。考试中可能需要识别异常值并绘制包含或不包含异常值的箱线图。
2. Measures of Central Tendency and Spread | 集中趋势与离散程度
For a dataset, the mean (average) is x̄ = Σx / n (sample) or μ for a population. The median is the middle value when data are ordered, and the mode is the most frequent value. For grouped data, linear interpolation is used to estimate the median from a cumulative frequency table.
集中趋势用均值、中位数和众数描述。样本均值 x̄ = Σx / n，总体均值记作 μ。中位数是排序后中间的值，众数是出现次数最多的值。对于分组数据，常用线性插值从累积频率表估算中位数。
Measures of spread include range, interquartile range (IQR = Q3 − Q1) and variance/standard deviation. The sample standard deviation is s = √[ Σ(x − x̄)² / (n−1) ]. The population standard deviation σ divides by n rather than n−1. In WJEC, both forms are used depending on context – learn when to apply each.
离散程度用极差、四分位距和标准差衡量。样本标准差公式为 s = √[ Σ(x − x̄)² / (n−1) ]；总体标准差 σ 分母为 n。WJEC 考试中需要根据语境选择正确的公式，务必清楚两者的区别。
s = √[ Σ(x − x̄)² / (n − 1) ]
3. Probability | 概率
Basic rules: P(A′) = 1 − P(A). For any two events A and B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Mutually exclusive events have P(A ∩ B) = 0. Conditional probability is given by P(A|B) = P(A ∩ B) / P(B), and A and B are independent if P(A ∩ B) = P(A) × P(B).
基本概率法则：互补事件 P(A′) = 1−P(A)；加法法则 P(A∪B) = P(A) + P(B) − P(A∩B)；互斥事件 P(A∩B) = 0。条件概率 P(A|B) = P(A∩B) / P(B)。独立性检验：若 P(A∩B) = P(A)×P(B)，则 A 与 B 独立。
Venn diagrams and tree diagrams are essential tools. Tree diagrams help with sequential events and conditional probability: multiply along branches and add the probabilities of relevant outcomes. Always check that probabilities on branches from the same point sum to 1.
文氏图和树状图是核心工具。树状图特别适合处理多步试验和条件概率问题：沿分支相乘，不同结果的概率相加。务必确保同一点出发的各分支概率之和为 1。
4. Discrete Random Variables | 离散随机变量
A discrete random variable X takes countable values with probabilities P(X = x). The sum of all probabilities must equal 1. The probability function can be given as a table or formula.
离散随机变量 X 取可数个值，每个值对应概率 P(X=x)。所有概率之和必须等于 1。概率函数可用表格或公式给出。
The expectation (mean) is E(X) = Σ x·P(X=x). The variance is Var(X) = E(X²) − [E(X)]². For a linear transformation Y = aX + b, the rules are E(Y) = aE(X) + b and Var(Y) = a² Var(X). These relations are heavily tested in WJEC.
期望（均值）为 E(X) = Σ x·P(X=x)。方差 Var(X) = E(X²) − [E(X)]²。若 Y = aX + b，则 E(Y) = aE(X) + b，Var(Y) = a² Var(X)。线性变换公式是 WJEC 的高频考点。
5. Binomial Distribution | 二项分布
If a trial has two outcomes (success/failure), with P(success) = p and the trials are independent, the number of successes X in n fixed trials follows a binomial distribution: X ~ B(n, p).
若每次试验只有“成功”和“失败”两种结果，P(成功) = p，且各次试验独立，则在固定的 n 次试验中成功的次数 X 服从二项分布：X ~ B(n, p)。
The probability of exactly r successes is P(X = r) = nCr pr (1−p)n−r, where nCr = n! / (r!(n−r)!). The mean is E(X) = np and the variance is Var(X) = np(1−p). Use binomial cumulative probability tables or your calculator to answer questions.
恰好获得 r 次成功的概率为 P(X=r) = nCr pr (1−p)n−r，其中 nCr = n! / (r!(n−r)!)。期望 E(X) = np，方差 Var(X) = np(1−p)。做题时利用二项分布累积概率表或计算器。
6. Poisson Distribution | 泊松分布
The Poisson distribution models the number of randomly occurring events in a fixed interval of time or space, with a known constant mean rate λ. We write X ~ Po(λ). The probability of exactly r events is P(X = r) = e−λ λr / r!.
泊松分布用于描述固定时间或空间内随机事件发生的次数，平均发生率为常数 λ，记作 X ~ Po(λ)。恰好发生 r 次的概率为 P(X=r) = e−λ λr / r!。
A key property is that both the mean and the variance equal λ. The Poisson distribution can approximate a binomial distribution when n is large and p is small; set λ = np. Conditions for this approximation are often tested: n > 50 and np < 5 is a typical rule of thumb.
泊松分布的重要性质是其均值与方差相等，均为 λ。当 n 很大且 p 很小时，可用泊松分布近似二项分布，取 λ = np。常见近似条件为 n > 50 且 np < 5，这是 WJEC 常见考点。
7. Normal Distribution | 正态分布
The normal distribution X ~ N(μ, σ²) is a continuous distribution with a bell‑shaped curve, symmetric about the mean μ. To find probabilities, standardise using Z = (X − μ) / σ, giving Z ~ N(0, 1). Use the standard normal table to find Φ(z).
正态分布 X ~ N(μ, σ²) 是一种钟形对称连续分布。求概率时需将其标准化：Z = (X − μ) / σ，则 Z 服从标准正态分布 N(0,1)，再查标准正态表获取 Φ(z)。
You must be able to work backwards: given a probability, find the corresponding z‑value and then solve for X, μ or σ. When the sample size is large, the sample mean X̄ is approximately normal, which underpins hypothesis testing for a population mean.
反向求解同样重要：已知概率求 z 值，再反解 X、μ 或 σ。在大样本条件下，样本均值 X̄ 近似服从正态分布，这是总体均值假设检验的理论基础。
Z = (X − μ) / σ
8. Sampling and Data Collection | 抽样与数据收集
Common sampling methods include simple random sampling, stratified sampling and systematic sampling. A stratified sample divides the population into distinct strata and samples proportionally from each. This can give a more representative sample than a simple random sample.
常见抽样方法有简单随机抽样、分层抽样和系统抽样。分层抽样先将总体分为互不重叠的层，然后按比例从各层抽取样本，通常能获得比简单随机抽样更具代表性的样本。
Bias in sampling occurs when certain groups are over‑represented or under‑represented. WJEC may ask you to identify potential sources of bias and suggest improvements, such as using a larger sample size or a random selection mechanism.
当某些群体被过度代表或代表不足时，就会产生抽样偏差。WJEC 可能要求识别偏差来源并提出改进建议，例如增大样本量或引入随机选择机制。
9. Hypothesis Testing | 假设检验
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