📚 AS CAIE Statistics: Core Knowledge Review | AS CAIE 统计：核心知识点梳理
Welcome to this comprehensive walkthrough of the essential topics in AS CAIE Statistics. Whether you are revising stem‑and‑leaf plots or getting to grips with the geometric distribution, this article brings every core idea into one structured, bilingual guide. Each concept is explained in clear English and matched immediately with Chinese translation so you can reinforce understanding in both languages.
欢迎来到这篇 AS CAIE 统计核心知识的梳理指南。无论你是在复习茎叶图，还是想掌握几何分布，这篇文章都将所有重点整合到一份结构清晰的双语资料中。每个概念先用简洁的英文解释，立即配以对应的中文翻译，帮助你在两种语言中同步巩固理解。
1. Representing and Summarising Data | 数据的表示与概括
A stem‑and‑leaf diagram orders raw data while preserving every original value. The stem consists of all digits except the last, and the leaf is the final digit. A key, such as ‘3 | 5 means 35’, must always be included. It is easy to spot the median and quartiles from an ordered stem‑and‑leaf display.
茎叶图在对原始数据进行排序的同时保留了每一个原始数值。茎由除最后一位以外的所有数位构成，叶是最后一位数字。必须附带图例，例如 ‘3 | 5 表示 35’。从有序的茎叶图上可以很方便地找出中位数和四分位数。
For grouped continuous data we use histograms, where the area of each bar is proportional to frequency. The vertical axis is frequency density, calculated as frequency ÷ class width. Unlike a bar chart, there are no gaps between bars unless a class has zero frequency.
对于分组连续数据，我们使用直方图，其中每个矩形的面积与频数成正比。纵轴为频率密度，由频率 ÷ 组距算出。与条形图不同，除非某组频数为零，否则相邻矩形之间不留空隙。
A cumulative frequency curve (or ogive) plots running totals against the upper class boundary. We use it to estimate the median, quartiles and percentiles by drawing horizontal lines from the cumulative frequency axis. The graph must have a smooth, S‑shaped curve through the plotted points.
累积频率曲线将累积频数相对于组上限描点作图。通过从累积频率轴画水平线，我们可以估计中位数、四分位数和百分位数。图形必须经过描点画出平滑的 S 形曲线。
2. Measures of Central Tendency | 集中趋势的度量
The mean of a set of numbers is the sum divided by the count. For raw data it is x̄ = Σx / n, while for a frequency distribution it becomes x̄ = Σfx / Σf. The mean is suitable for symmetric data but can be pulled by extreme values.
一组数据的均值等于总和除以数据个数。对于原始数据，x̄ = Σx / n；对于频数分布，x̄ = Σfx / Σf。均值适用于对称数据，但可能被极端值拉偏。
The median is the middle value when data are ordered. If n is odd, it is the (n+1)/2 th value; if n is even, it is the average of the n/2 th and (n/2 +1)th values. From a cumulative frequency graph, the median is read at the 50% cumulative frequency mark, often using linear interpolation for grouped data.
中位数是数据排序后位于中间的数值。当 n 为奇数时，中位数为第 (n+1)/2 个值；当 n 为偶数时，则为第 n/2 与第 (n/2+1) 个值的平均数。在累积频率图上，中位数在 50% 累积频率处读出，对于分组数据常用线性插值法求中位数。
The mode is the value that occurs most frequently. For discrete data it is simply the value with the highest frequency. For grouped continuous data we identify the modal class – the class with the highest frequency density – and may estimate a specific mode within that class.
众数是出现次数最多的数值。对于离散数据，它就是频数最高的那一个值。对于分组连续数据，我们找出众数组——频率密度最高的组——并可以在该组内估计出一个更具体的众数值。
3. Measures of Dispersion | 离散程度的度量
Range is the simplest measure, defined as largest value minus smallest value. However, it only uses two observations and is therefore sensitive to outliers. A more robust alternative is the interquartile range: IQR = Q3 − Q1, which covers the middle 50% of the data.
全距是最简单的离散程度指标，定义为最大值减最小值。但它只用到了两个观测值，因此对异常值非常敏感。更稳健的替代指标是四分位距：IQR = Q3 − Q1，涵盖了中间 50% 的数据。
Variance measures the average squared deviation from the mean. For a list of values, s² = Σ(x − x̄)² / (n − 1) when we treat the data as a sample. A shortcut formula for calculation is s² = (Σx² / n) − (x̄)², which can be adjusted with frequencies Σfx² / Σf − (x̄)² for a frequency distribution.
方差衡量的是各个数值与均值之差的平方的平均值。当数据视为样本时，s² = Σ(x − x̄)² / (n − 1)。用于计算的简便公式为 s² = (Σx² / n) − (x̄)²，对于频数分布可调整为 Σfx² / Σf − (x̄)²。
The standard deviation is the positive square root of the variance: s = √s². It is expressed in the same units as the original data, which makes interpretation more straightforward than variance.
标准差是方差的正平方根：s = √s²。它与原始数据具有相同的单位，因此比方差更易于解释。
4. Basic Probability Rules | 概率基本法则
Probability is a measure of how likely an event is to occur, always lying between 0 and 1. For an experiment with equally likely outcomes, P(A) = number of favourable outcomes / total number of outcomes. The complement rule states P(A′) = 1 − P(A).
概率衡量某个事件发生的可能性大小，其值永远介于 0 和 1 之间。对于等可能结果的试验，P(A) = 有利结果个数 / 总结果个数。互补事件法则为 P(A′) = 1 − P(A)。
The addition rule for any two events is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, they cannot happen together, so P(A ∩ B) = 0 and the formula simplifies to P(A ∪ B) = P(A) + P(B).
两个事件的加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若 A 与 B 互斥，它们不可能同时发生，因此 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。
Two events are independent if the occurrence of one does not affect the probability of the other: P(A ∩ B) = P(A) × P(B). The multiplication rule for dependent events is P(A ∩ B) = P(A) × P(B | A), where the conditional probability is introduced in the next section.
如果两个事件中一个的发生不影响另一个发生的概率，则它们相互独立：P(A ∩ B) = P(A) × P(B)。对于不独立事件，乘法法则为 P(A ∩ B) = P(A) × P(B | A)，其中条件概率将在下一节介绍。
5. Conditional Probability and Tree Diagrams | 条件概率与树状图
Conditional probability is the probability of event A given that B has occurred, written P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0. It helps us update probabilities when partial information is available.
条件概率是指在事件 B 已经发生的条件下事件 A 发生的概率，记作 P(A | B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。当掌握部分信息时，条件概率帮我们更新事件的可能性。
Tree diagrams display all possible outcomes of a sequence of trials. The probability is written on each branch, and the final probability of a path is found by multiplying along the branches. For two-stage experiments involving conditional probabilities, the second‑stage probabilities depend on the first stage outcome.
树状图展示一系列试验的所有可能结果。每个分枝上都标有概率，一条路径的最终概率通过将沿线各分枝概率相乘得到。对于涉及条件概率的两阶段实验，第二阶段的概率取决于第一阶段的结果。
When we sample without replacement, the probabilities change from one pick to the next. Tree diagrams make it easy to track these changes and to apply the law of total probability when an event can happen through several mutually exclusive paths.
在不放回抽样中，每次抽取的概率都会改变。树状图便于追踪这些变化，并在一个事件可通过多条互斥路径发生时应用全概率公式。
6. Permutations and Combinations | 排列与组合
A permutation is an arrangement of items in a specific order. The number of ways to arrange n distinct objects is n! (n factorial). The number of permutations of r objects chosen from n is given by ⁿPᵣ = n! / (n − r)!.
排列是指将物品按一定顺序排成一列。将 n 个不同物品全排列的方式数为 n!（n 的阶乘）。从 n 个中选取 r 个进行排列的方式数为 ⁿPᵣ = n! / (n − r)!。
A combination is a selection of items where the order does not matter. The number of combinations of r objects from n is ⁿCᵣ = n! / [r!(n − r)!]. This is the same as the binomial coefficient and appears in expansions and probability distributions.
组合是指不考虑顺序的选取方式。从 n 个中选取 r 个的组合数为 ⁿCᵣ = n! / [r!(n − r)!]。这与二项式系数相同，并出现在展开式与概率分布中。
When items are repeated, the number of distinguishable permutations of n objects where there are groups of identical items is n! / (p! q! …) where p, q, … are the frequencies of each repeated type.
当物品有重复时，n 个物品中含有若干组相同物品的可区分排列数为 n! / (p! q! …)，其中 p、q 等为每种重复物品的个数。
7. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of values, each with a probability P(X = x). All probabilities must sum to 1. The probability distribution is often presented as a table showing each x and the corresponding P(X = x).
离散随机变量 X 取有限个或可数个值，每个值有对应的概率 P(X = x)。所有概率之和必须等于 1。概率分布常用表格来列出每个 x 及其相应的 P(X = x)。
| x | 1 | 2 | 3 |
| P(X=x) | 0.2 | 0.5 | 0.3 |
The expected value E(X) = μ = Σ x P(X = x) is the long‑run average. It is a measure of central tendency for the distribution. A linear transformation follows the rule E(aX + b) = a E(X) + b.
期望值 E(X) = μ = Σ x P(X = x) 表示长期平均，是分布集中趋势的度量。线性变换遵循规则 E(aX + b) = a E(X) + b。
Variance measures the spread of the distribution: Var(X) = E[(X − μ)²] = Σ (x − μ)² P(X = x). The calculation formula Var(X) = E(X²) − [E(X)]² is often quicker to apply. For a linear transformation, Var(aX + b) = a² Var(X), and the standard deviation scales by |a|.
方差衡量分布的离散程度：Var(X) = E[(X − μ)²] = Σ (x − μ)² P(X = x)。计算公式 Var(X) = E(X²) − [E(X)]² 通常更快捷。线性变换下方差变为 Var(aX + b) = a² Var(X)，标准差则缩放 |a| 倍。
8. The Binomial Distribution | 二项分布
A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. If X ~ B(n, p), then P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ for r = 0, 1, 2, …, n.
二项分布模型描述在固定次数的独立试验中成功的次数，每次试验成功概率相同，记为 p。若 X ~ B(n, p)，则 P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，r = 0, 1, 2, …, n。
The conditions for a binomial distribution are: a fixed number of trials n, each trial has exactly two outcomes (success/failure), trials are independent, and the probability of success p remains constant. If these conditions are satisfied, X is binomially distributed.
应用二项分布的条件为：试验次数 n 固定，每次试验只有两种结果（成功/失败），各次试验相互独立，且每次成功的概率 p 保持不变。当满足这些条件时，X 服从二项分布。
The mean and variance of a binomial variable are E(X) = np and Var(X) = np(1 − p). These formulas allow quick calculation without needing the entire probability distribution table. You can also use cumulative binomial tables to find P(X ≤ r) or P(X ≥ r).
二项变量的期望与方差分别为 E(X) = np 和 Var(X) = np(1 − p)。利用这些公式无需列出整个概率分布表即可快速计算。你还可以借助二项累积分布表求 P(X ≤ r) 或 P(X ≥ r)。
9. The Geometric Distribution | 几何分布
The geometric distribution models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials. If X ~ Geo(p), then P(X = r) = p (1 − p)ʳ⁻¹ for r = 1, 2, 3, …, where p is the constant probability of success at each trial.
几何分布模型描述在一连串独立伯努利试验中首次成功所需要的试验次数。若 X ~ Geo(p)，则 P(X = r) = p (1 − p)ʳ⁻¹，r = 1, 2, 3, …，其中 p 为每次试验成功的恒定概率。
The key features of a geometric distribution are: each trial is independent, trials continue until the first success, and the probability of success p is the same for every trial. The distribution has an infinite right tail, so it has no fixed upper limit for r.
几何分布的主要特征有：每次试验相互独立，试验持续到第一次成功为止，每次成功的概率 p 相同。该分布具有无穷长的右尾，因此 r 没有固定的上限。
For X ~ Geo(p), the expected value is E(X) = 1 / p, reflecting the fact that a smaller success probability leads to a longer wait for the first success. The variance is Var(X) = (1 − p) / p². These formulas are provided in the CAIE formula booklet.
对于 X ~ Geo(p)，期望值为 E(X) = 1 / p，反映出成功概率越小，等待首次成功的时间越长。方差为 Var(X) = (1 − p) / p²。这些公式在 CAIE 公式手册中都有提供。
10. The Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution with a bell‑shaped curve, denoted X ~ N(μ, σ²). The parameters μ (mean) and σ² (variance) determine the centre and spread. The total area under the curve is 1, and the distribution is symmetric about μ.
正态分布是一种连续型概率分布，钟形曲线记作 X ~ N(μ, σ²)。参数 μ（均值）和 σ²（方差）决定了分布的中心和散布程度。曲线下总面积为 1，分布关于 μ 对称。
To calculate probabilities for any normal variable we standardise it, turning X into the standard normal variable Z = (X − μ) / σ, which follows Z ~ N(0, 1). The tables give values of Φ(z
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