📚 Core Knowledge Review for CIE Year 12 Statistics | CIE十二年级统计核心知识点梳理
This article summarises the essential topics in CIE AS Level Mathematics (Probability & Statistics 1), which forms the foundation for Year 12 students. Understanding these core concepts will help you tackle exam questions with confidence.
本文梳理了 CIE AS 数学（概率与统计 1）中的核心知识点，适合十二年级学生打基础。掌握这些关键概念将帮助你自信应对考试题目。
1. Types of Data and Data Representation | 数据类型与表示
Categorical or qualitative data describe qualities, while numerical or quantitative data consist of numbers. Numerical data can be discrete (countable, e.g. number of students) or continuous (measurable, e.g. height).
分类（定性）数据描述特征，而数值（定量）数据由数字组成。数值数据可以是离散型（可数，如学生人数）或连续型（可测量，如身高）。
In histograms, frequency is represented by area, not height. The frequency density = frequency / class width is used to draw bars so that unequal class widths are comparable.
在直方图中，频数由面积而非高度表示。绘制时使用频数密度 = 频数 / 组距，使不等宽的组具有可比性。
Stem-and-leaf diagrams display raw data while preserving each value; back-to-back stems can compare two datasets.
茎叶图展示原始数据并保留每个值；背靠背的茎叶图可以比较两组数据。
2. Measures of Central Tendency | 集中趋势度量
The mean for a set of n values is calculated as x̄ = Σx / n. For grouped data we use midpoints: x̄ = Σfx / Σf.
一组 n 个数据的均值计算为 x̄ = Σx / n。对于分组数据，使用组中点：x̄ = Σfx / Σf。
The median is the middle value when data are ordered; for n observations, its position is (n+1)/2 if raw data. For grouped data, linear interpolation within the median class is used.
中位数是排序后中间的值；对于原始数据，位置为 (n+1)/2。对于分组数据，在中位数所在组内使用线性插值。
The mode is the most frequent value; a data set can be multimodal.
众数是出现频率最高的值；数据集可以是多峰的。
3. Measures of Dispersion | 离散程度度量
The range is the difference between the largest and smallest values. It is easily affected by outliers.
极差是最大值与最小值之差，容易受异常值影响。
The interquartile range (IQR) = Q3 – Q1, covering the middle 50% of data and is more robust.
四分位距 IQR = Q3 – Q1，涵盖中间 50% 的数据，更稳健。
Variance measures average squared deviation from the mean: σ2 = Σ(x – μ)2 / N for a population; for a sample we often use s2 = Σ(x – x̄)2 / (n-1). The standard deviation is the square root of variance, s or σ.
方差衡量与均值的平均平方偏差：总体方差 σ2 = Σ(x – μ)2 / N；样本常用 s2 = Σ(x – x̄)2 / (n-1)。标准差是方差的平方根，记为 s 或 σ。
4. Cumulative Frequency and Percentiles | 累积频率与百分位数
A cumulative frequency graph plots upper class boundaries against cumulative totals. It can be used to estimate the median, quartiles and percentiles.
累积频率图将各组上限与累积频数描点，可用于估计中位数、四分位数和百分位数。
The pth percentile is the value below which p% of the data fall. For example, the 90th percentile is found by reading off the graph at 90% of total frequency.
第 p 百分位数是使得 p% 的数据低于此值的数。例如，第 90 百分位数可通过总频数的 90% 在图线上读取。
5. Basic Probability | 概率基础
The probability of an event A is written as P(A), and 0 ≤ P(A) ≤ 1. The sample space S includes all possible outcomes.
事件 A 的概率记为 P(A)，且 0 ≤ P(A) ≤ 1。样本空间 S 包含所有可能结果。
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, P(A ∩ B) = 0.
对于任意两个事件 A 和 B，加法公式为：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若 A 和 B 互斥，则 P(A ∩ B) = 0。
Independent events satisfy P(A ∩ B) = P(A) × P(B). Independence means the occurrence of one event does not affect the probability of the other.
独立事件满足 P(A ∩ B) = P(A) × P(B)。独立指一个事件的发生不影响另一个事件的概率。
6. Conditional Probability and Tree Diagrams | 条件概率与树状图
Conditional probability is defined as P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0. It represents the probability of A given that B has occurred.
条件概率定义为 P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。它表示在 B 发生的条件下 A 发生的概率。
Tree diagrams help solve multistage probability problems: multiply along branches for joint probabilities, and add probabilities from different branches for combined events. Always label probabilities clearly.
树状图帮助解决多阶段概率问题：沿枝相乘得到联合概率，不同枝的概率相加得到组合事件概率。务必清晰标注概率。
7. Permutations and Combinations | 排列与组合
The number of ways to arrange n distinct objects in order is n! (n factorial). For example, 5! = 120.
将 n 个不同物体排序的方法数为 n!（n 的阶乘）。例如 5! = 120。
The number of permutations of r objects chosen from n is nPr = n! / (n-r)!. Order matters in permutations.
从 n 个物体中选取 r 个的排列数为 nPr = n! / (n-r)!。排列中顺序重要。
The number of combinations (selections where order does not matter) is nCr = n! / [r!(n-r)!]. This is the number of ways to choose r items from n.
组合数（无序选取）为 nCr = n! / [r!(n-r)!]。这是从 n 个中选 r 个的方式数。
8. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of values. Its probability distribution is given by P(X = x) for each value x, and the sum of all probabilities must equal 1.
离散随机变量 X 取可数个值。其概率分布由每个值 x 的 P(X = x) 给出，且所有概率之和必须为 1。
The expected value (mean) of X is E(X) = Σ x P(X = x). It represents the long-run average.
X 的期望值（均值）为 E(X) = Σ x P(X = x)，代表长期平均结果。
The variance is Var(X) = E(X2) – [E(X)]2, where E(X2) = Σ x2 P(X = x).
方差为 Var(X) = E(X2) – [E(X)]2，其中 E(X2) = Σ x2 P(X = x)。
9. Binomial Distribution | 二项分布
A binomial experiment consists of n independent trials, each with two outcomes (success/failure), and a constant probability of success p. Then X ~ B(n, p).
二项试验由 n 次独立试验组成，每次有两种结果（成功/失败），且成功的概率恒定为 p。记为 X ~ B(n, p)。
The probability of exactly r successes is:
恰好成功 r 次的概率为：
P(X = r) = nCr pr (1-p)n-r
Mean and variance for B(n, p): E(X) = np, Var(X) = np(1-p).
二项分布的均值和方差：E(X) = np，Var(X) = np(1-p)。
10. Normal Distribution | 正态分布
The normal distribution is a continuous distribution with a symmetric bell-shaped curve. It is defined by mean μ and standard deviation σ: X ~ N(μ, σ2).
正态分布是一种连续分布，具有对称的钟形曲线。由均值 μ 和标准差 σ 定义：X ~ N(μ, σ2)。
To find probabilities, we standardise to the standard normal distribution Z ~ N(0, 1) using z = (x –
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