📚 A-Level Maths Statistics: Core Concepts, Models and Hypothesis Testing | A-Level 数学统计：核心概念、模型与假设检验
This revision guide covers the essential statistical topics in A-level Mathematics, including sampling, data summaries, probability models, the binomial and normal distributions, hypothesis testing, correlation and regression. Each section pairs a concise English explanation with a Chinese equivalent to support bilingual learners preparing for exams.
本复习指南涵盖 A-level 数学统计模块的核心主题，包括抽样、数据概括、概率模型、二项分布与正态分布、假设检验、相关性与回归分析。每一节都提供简洁的英文讲解并配以中文对照，帮助双语学习者备考。
1. Statistical Sampling and Data Types | 统计抽样与数据类型
In A-level statistics, you must distinguish between a population and a sample. The population is the entire set of individuals or items of interest, while a sample is a subset selected to represent the population.
在 A-level 统计中，你必须区分总体和样本。总体是研究对象全部个体或项目的集合，而样本是从中选出用于代表总体的子集。
A simple random sample gives every member of the population an equal chance of selection, usually using random numbers or a sampling frame. Stratified sampling divides the population into groups and samples proportionally, which can improve representativeness.
简单随机抽样让总体中每个成员都有相等的被选取机会，通常借助随机数或抽样框。分层抽样先将总体分组，再按比例抽样，可以提高代表性。
Other methods include systematic sampling, where every kth item is chosen, and quota sampling, where interviewers select a fixed number of people from each category. Remember that only random methods allow unbiased inference from sample to population.
其他方法包括系统抽样，即每隔 k 个对象选取一个；以及配额抽样，即调查员从每个类别中选取固定数量的人。要记住，只有随机抽样方法才能从样本对总体进行无偏推断。
Data can be quantitative, meaning numerical measurements, or qualitative, meaning non-numerical categories. Quantitative data are either discrete, taking fixed values such as counts, or continuous, taking any value in an interval such as time or length.
数据可以是定量的，即数值型测量值；也可以是定性的，即非数值类别。定量数据分为离散型，如计数只能取固定值；或连续型，如时间、长度可以在区间内取任意值。
2. Measures of Location and Spread | 位置与离散程度的度量
The three main measures of location are the mean, median and mode. The mean is sensitive to outliers, while the median is resistant and often preferred for skewed data.
三种主要的位置度量是平均数、中位数和众数。平均数对异常值敏感，而中位数具有较强的抗干扰性，偏态数据通常优先使用中位数。
For grouped data, you estimate the mean by using class midpoints: mean ≈ Σfx / Σf, where f is the frequency and x is the midpoint. To estimate the median from a grouped frequency table, use linear interpolation:
对于分组数据，通常用组中值估计平均数：平均数 ≈ Σfx / Σf，其中 f 是频数，x 是组中值。若要从分组频数表估计中位数，可使用线性插值：
Median = L + [(n/2 − F) / f] × w
Here L is the lower class boundary of the median class, F is the cumulative frequency before the median class, f is the frequency of the median class, and w is the class width.
其中 L 是中位数所在组的下边界，F 是中位数所在组之前的累计频数，f 是中位数所在组的频数，w 是组距。
Spread can be measured by the range, interquartile range, variance and standard deviation. For raw data, variance is the mean of squared deviations from the mean:
离散程度可以用极差、四分位距、方差和标准差来度量。对于原始数据，方差是各数据与平均数离差平方的平均值：
Variance = Σx² / n − (Σx / n)²
The standard deviation is the positive square root of the variance. Outliers are often defined as values below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR.
标准差是方差的正平方根。异常值通常定义为小于 Q1 − 1.5 × IQR 或大于 Q3 + 1.5 × IQR 的数据。
Coding changes the original data to make calculations easier. If y = (x − a)/b, then the mean of x is a + b × mean of y, and the standard deviation of x is b × standard deviation of y.
对数据做编码可以简化计算。如果 y = (x − a)/b，则 x 的平均数等于 a + b × y 的平均数；x 的标准差等于 b × y 的标准差。
3. Probability Rules and Venn Diagrams | 概率法则与韦恩图
The probability of an event A is written P(A), with 0 ≤ P(A) ≤ 1. The sample space is the set of all possible outcomes.
事件 A 的概率记作 P(A)，满足 0 ≤ P(A) ≤ 1。样本空间是所有可能结果组成的集合。
For any two events A and B, the addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, then P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
对任意两个事件 A 与 B，加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。如果 A 与 B 互斥，则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。
Conditional probability is given by P(A | B) = P(A ∩ B) / P(B). Two events are independent if P(A ∩ B) = P(A) × P(B), or equivalently if P(A | B) = P(A).
条件概率为 P(A | B) = P(A ∩ B) / P(B)。如果 P(A ∩ B) = P(A) × P(B)，或等价地 P(A | B) = P(A)，则两个事件相互独立。
Tree diagrams are useful for multi-stage experiments. Multiply probabilities along branches and add between branches. Venn diagrams are ideal for visualising intersections, unions and complements.
树状图适合多阶段试验，沿着分支相乘概率，不同分支之间相加。韦恩图适合直观表示交、并和补事件。
4. Discrete Random Variables and Expectation | 离散随机变量与期望
A discrete random variable X takes a countable set of values with given probabilities. A valid probability distribution must satisfy Σ P(X = x) = 1 for all possible values x.
离散随机变量 X 取有限或可数个值，每个取值对应相应概率。一个有效的概率分布必须满足所有可能值 x 上 Σ P(X = x) = 1。
The expected value E(X) is the long-run average of X, calculated as E(X) = Σ x P(X = x). It is also called the mean of X, denoted μ.
期望 E(X) 是 X 的长期平均值，计算方式为 E(X) = Σ x P(X = x)。它也叫做 X 的均值，记作 μ。
Variance measures the spread of X around its mean: Var(X) = Σ x² P(X = x) − [E(X)]², or equivalently Var(X) = E(X²) − [E(X)]².
方差衡量 X 在其均值附近的离散程度：Var(X) = Σ x² P(X = x) − [E(X)]²，等价于 Var(X) = E(X²) − [E(X)]²。
For linear transformations, E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X). Adding b shifts the mean but does not change the variance.
对于线性变换，E(aX + b) = aE(X) + b，Var(aX + b) = a² Var(X)。加 b 会平移均值，但不会改变方差。
If X has a discrete uniform distribution over n values, each value has probability 1/n. Then E(X) is the average of the smallest and largest values, and Var(X) = (n² − 1)/12.
如果 X 在 n 个值上服从离散均匀分布，每个值的概率均为 1/n。此时 E(X) 等于最小值和最大值的平均数，Var(X) = (n² − 1)/12。
5. The Binomial Distribution | 二项分布
The binomial distribution applies when there are a fixed number n of independent trials, each trial has only two outcomes called success and failure, and the probability of success p is constant for every trial.
二项分布适用于固定次数 n 的独立试验，每次试验只有成功和失败两种结果，且每次成功概率 p 保持不变。
We write X ~ B(n, p). The probability of obtaining exactly x successes is:
我们记作 X ~ B(n, p)。恰好得到 x 次成功的概率为：
P(X = x) = C(n, x) pˣ (1 − p)ⁿ⁻ˣ
Here C(n, x) is the binomial coefficient, equal to n! / [x!(n − x)!]. The term pˣ is the probability of x successes, and (1 − p)ⁿ⁻ˣ is the probability of n − x failures.
其中 C(n, x) 是二项式系数，等于 n! / [x!(n − x)!]。pˣ 表示 x 次成功的概率，(1 − p)ⁿ⁻ˣ 表示 n − x 次失败的概率。
The mean and variance of X ~ B(n, p) are E(X) = np and Var(X) = np(1 − p). These results are often used in hypothesis testing and normal approximations.
X ~ B(n, p) 的均值与方差分别为 E(X) = np 和 Var(X) = np(1 − p)。这些结果常用于假设检验和正态近似。
To find P(X ≤ x) or P(X ≥ x), use cumulative binomial tables or a calculator. Remember that P(X ≥ x) = 1 − P(X ≤ x − 1) for integer x.
求 P(X ≤ x) 或 P(X ≥ x) 时，可使用二项分布累积表或计算器。记住对于整数 x，P(X ≥ x) = 1 − P(X ≤ x − 1)。
6. The Normal Distribution and Standardisation | 正态分布与标准化
The normal distribution is a continuous probability distribution with a symmetric bell-shaped curve. It is defined by its mean μ and variance σ², written X ~ N(μ, σ²).
正态分布是一种连续型概率分布，具有对称的钟形曲线。它由均值 μ 和方差 σ² 决定，记作 X ~ N(μ, σ²)。
The total area under the normal curve is 1, and the probability P(a < X < b) is the area between a and b. There is no simple formula for these areas, so we use standardisation.
正态曲线下方总面积为 1，概率 P(a < X < b) 就是 a 与 b 之间的面积。这些面积没有简单的计算公式，因此通常使用标准化。
Z = (X − μ) / σ
The standardised variable Z ~ N(0, 1). Convert x-values to z-values, then use the standard normal table to find Φ(z) = P(Z < z).
标准化变量 Z ~ N(0, 1)。将 x 值转换为 z 值，再用标准正态表查 Φ(z) = P(Z < z)。
The normal distribution is symmetric, so Φ(−z) = 1 − Φ(z). Also use P(Z > z) = 1 − Φ(z) and P(a < Z < b) = Φ(b) − Φ(a).
正态分布具有对称性，因此 Φ(−z) = 1 − Φ(z)。同时 P(Z > z) = 1 − Φ(z)，P(a < Z < b) = Φ(b) − Φ(a)。
Inverse normal problems ask for the value x given a probability. First find z from the table, then convert back using x = μ + zσ.
反查正态分布问题要求根据给定概率求 x 值。先由表格查出 z 值，再用 x = μ + zσ 转换回去。
7. Hypothesis Testing: Binomial and Normal | 假设检验：二项与正态
A hypothesis test begins with a null hypothesis H₀ and an alternative hypothesis H₁. The significance level α is the maximum probability of wrongly rejecting H₀.
假设检验首先提出原假设 H₀ 和备择假设 H₁。显著性水平 α 是错误拒绝 H₀ 的最大允许概率。
For a binomial test, assume X ~ B(n, p) under H₀. Find P(X ≤ x) for a lower-tail test or P(X ≥ x) for an upper-tail test, and compare this p-value with α.
对于二项检验，在原假设下假设 X ~ B(n, p)。下尾检验求 P(X ≤ x)，上尾检验求 P(X ≥ x)，再将 p 值与 α 比较。
The critical region is the set of outcomes that lead to rejecting H₀. For a one-tailed test at the 5% level, you may reject H₀ if the observed result falls in the lowest or highest 5% of the distribution.
拒绝域是导致拒绝 H₀ 的结果集合。对于 5% 显著性水平的单尾检验，如果观测结果落在分布最低或最高的 5% 区域，就拒绝 H₀。
For a normal test, standardise the sample mean using Z = (x̄ − μ) / (σ / √n). Then find the p-value from the standard normal distribution or compare z with the critical value.
对于正态检验，将样本均值标准化：Z = (x̄ − μ) / (σ / √n)。然后从标准正态分布中求 p 值，或将 z 值与临界值比较。
Always write a conclusion in context: if p-value < α, reject H₀ and state there is sufficient evidence for H₁; otherwise do not reject H₀.
必须在具体语境中给出结论：若 p 值小于 α，拒绝 H₀，说明有充分证据支持 H₁；否则不拒绝 H₀。
8. Correlation and Linear Regression | 相关性与线性回归
Correlation measures the strength and direction of a linear relationship between two variables. The product moment correlation coefficient r, also called PMCC, always lies between −1 and 1.
相关性衡量两个变量之间线性关系的强度和方向。积矩相关系数 r，也称 PMCC，始终在 −1 到 1 之间。
If r is close to 1, there is strong positive correlation; if r is close to −1, there is strong negative correlation. A value near 0 suggests weak or no linear correlation.
若 r 接近 1，表示强正相关；若 r 接近 −1，表示强负相关；接近 0 则表示线性相关很弱或不存在。
A scatter diagram should always be drawn first, as it reveals outliers, non-linear patterns and whether a linear model is appropriate. Correlation does not imply causation.
应首先绘制散点图，因为它能显示异常值、非线性模式以及线性模型是否合适。相关性并不意味着因果关系。
The least squares regression line has equation y = a + bx, where b = Sxy / Sxx and a = ȳ − b x̄. Here Sxy = Σxy − Σx Σy / n and Sxx = Σx² − (Σx)² / n.
最小二乘回归直线方程为 y = a +
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