A-Level CAIE Statistics: Core Knowledge Review | A-Level CAIE 统计：核心知识点梳理

📚 A-Level CAIE Statistics: Core Knowledge Review | A-Level CAIE 统计：核心知识点梳理

This article provides a structured revision guide to the core topics in the A-Level CAIE Statistics syllabus. It covers summary statistics, probability models, key distributions, sampling, estimation and hypothesis testing. Use it as a checklist before your exam.

本文为 A-Level CAIE 统计课程提供结构化复习指南，涵盖汇总统计、概率模型、重要分布、抽样、估计和假设检验。请在考试前将其作为检查清单使用。

1. Data Representation and Summary Statistics | 数据的表示与汇总统计

In CAIE Statistics, data can be qualitative or quantitative. Quantitative data may be discrete or continuous. The first step in any analysis is to summarise data using measures of centre and spread.

在 CAIE 统计中，数据可以是定性的或定量的。定量数据可以是离散的或连续的。任何分析的第一步都是使用集中趋势和离散程度的度量来汇总数据。

For a sample or population, the mean is x̄ = Σx / n. The variance is the average squared deviation from the mean, and the standard deviation is its square root.

对于样本或总体，均值是 x̄ = Σx / n。方差是偏离均值平方的平均数，标准差是方差的平方根。

Variance = Σ(x − x̄)² / n = Σx² / n − x̄²

When data are grouped, use class midpoints for calculations and state clearly whether you are using n or n−1 for sample variance. Common measures are listed below.

当数据分组时，使用组中值进行计算，并清楚说明计算样本方差时使用的是 n 还是 n−1。常用度量如下所列。

• Measures of centre: mean, median, mode | 集中趋势度量：均值、中位数、众数
• Measures of spread: range, interquartile range, variance, standard deviation | 离散程度度量：极差、四分位距、方差、标准差
• Five-number summary: minimum, Q₁, median, Q₃, maximum | 五数概括：最小值、下四分位数、中位数、上四分位数、最大值

2. Probability Laws and Counting Methods | 概率法则与计数方法

Probability is the measure of the likelihood that an event occurs, with values between 0 and 1. For events A and B, the addition law is P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

概率是事件发生可能性的度量，取值在 0 到 1 之间。对于事件 A 和 B，加法法则是 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。

Conditional probability is P(A | B) = P(A ∩ B) / P(B). Two events are independent if P(A ∩ B) = P(A) × P(B). Mutually exclusive events cannot occur together, so P(A ∩ B) = 0.

条件概率是 P(A | B) = P(A ∩ B) / P(B)。如果 P(A ∩ B) = P(A) × P(B)，则两个事件相互独立。互斥事件不能同时发生，因此 P(A ∩ B) = 0。

Counting techniques are often needed when outcomes are equally likely. The number of arrangements of n distinct objects is n!, the number of ordered selections is nPr = n! / (n − r)!, and the number of unordered selections is nCr = n! / (r!(n − r)!).

当结果等可能时，常常需要计数技巧。n 个不同对象的排列数是 n!，有序选择数是 nPr = n! / (n − r)!，无序选择数是 nCr = n! / (r!(n − r)!)。

Tree diagrams and Venn diagrams are useful tools for organising combined events and finding probabilities in multi-stage experiments.

树形图和韦恩图是组织组合事件和求多阶段试验概率的有用工具。

3. Discrete Random Variables and Expectation | 离散随机变量与期望

A discrete random variable X takes a finite or countable set of values. Its probability distribution must satisfy Σ P(X = x) = 1.

离散随机变量 X 取有限或可列个值。其概率分布必须满足 Σ P(X = x) = 1

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