📚 CAIE AS-Level Statistics: Formula & Theorem Quick Reference Handbook | CAIE AS 统计：公式定理速查手册
This quick-reference handbook covers the essential formulas and theorems required for the CAIE AS-Level Statistics (9709/Paper 5). It is designed to help you review key results rapidly, with each point presented in both English and Chinese. Use it to reinforce your understanding and boost confidence before the exam.
本速查手册涵盖 CAIE AS 层统计学（9709/Paper 5）所必备的公式与定理。每个要点均以中英双语对照呈现，旨在帮助你在考前快速回顾关键结论，巩固理解并提升信心。
1. Measures of Central Tendency | 集中趋势度量
The mean (x̄) of a data set is the sum of all values divided by the number of observations. It is the most widely used measure of location.
均值（x̄）为所有数据值之和除以观测个数，是最常用的位置度量。
x̄ = Σxᵢ / n
The median is the middle value when the data are arranged in ascending order. For an odd number of observations, it is the centre value; for an even number, it is the average of the two centre values. In grouped data, linear interpolation is applied.
中位数是将数据升序排列后居于中间位置的值。奇数个观测时取正中间的数值，偶数个时取中间两数的平均值。对于分组数据，需使用线性插值法求中位数。
The mode is the most frequently occurring value in a data set. It is especially useful for categorical data.
众数是一组数据中出现频率最高的值，特别适用于类别数据。
2. Measures of Dispersion | 离散度量
Dispersion measures describe the spread or variability of a data set. The range is the difference between the maximum and minimum values.
离散度量用于描述数据的分散程度。极差为最大值与最小值之差。
The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁). It captures the middle 50% of the data.
四分位距（IQR）为上四分位数（Q₃）与下四分位数（Q₁）之差，反映了中间 50% 数据的分布范围。
IQR = Q₃ − Q₁
Variance measures the average of the squared deviations from the mean. The population variance (σ²) and the sample variance (s²) are given by:
方差衡量各数据与均值之差的平方的平均值。总体方差（σ²）和样本方差（s²）的计算公式如下：
σ² = Σ(xᵢ − μ)² / N
s² = Σ(xᵢ − x̄)² / (n − 1)
The standard deviation (σ or s) is the positive square root of the variance, expressing dispersion in the original units of measurement.
标准差（σ 或 s）是方差的正平方根，用与原数据相同的单位表示离散程度。
σ = √[Σ(xᵢ − μ)² / N]
3. Probability Rules | 概率法则
Probability theory provides the foundation for statistical inference. The probability of an event A is denoted by P(A) and satisfies 0 ≤ P(A) ≤ 1.
概率论是统计推断的基础。事件 A 的概率记作 P(A)，且满足 0 ≤ P(A) ≤ 1。
The addition rule for any two events A and B is:
对于任意两个事件 A 和 B，加法法则为：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B are mutually exclusive (cannot occur together), then P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
若 A 与 B 互斥（不可能同时发生），则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。
The conditional probability of A given B is defined as:
在事件 B 发生的条件下 A 的条件概率定义为：
P(A | B) = P(A ∩ B) / P(B), for P(B) > 0
This leads to the multiplication rule: P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B). Events A and B are independent if and only if P(A ∩ B) = P(A) × P(B), or equivalently P(A | B) = P(A).
由此可得乘法法则：P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B)。若且仅若 P(A ∩ B) = P(A) × P(B)，或等价地 P(A | B) = P(A)，则事件 A 与 B 相互独立。
4. Permutations & Combinations | 排列与组合
Counting techniques are essential for calculating probabilities in finite sample spaces. The number of ways to arrange n distinct objects in order is n! (n factorial).
计数技巧是计算有限样本空间概率的基础。将 n 个不同物体进行有序排列的方法数为 n!（n 阶乘）。
n! = n × (n−1) × (n−2) × … × 2 × 1
The number of ordered arrangements of r objects chosen from n distinct objects (permutations) is:
从 n 个不同物体中选取 r 个进行有序排列（排列数）为：
ⁿPᵣ = n! / (n − r)!
When the order does not matter, the number of ways to select r objects from n distinct objects (combinations) is:
若不考虑顺序，从 n 个不同物体中选取 r 个的方法数（组合数）为：
ⁿCᵣ = n! / (r!(n − r)!)
5. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable set of values, each with a certain probability. Its probability distribution satisfies Σ P(X = x) = 1.
离散随机变量 X 取一系列可数值，每个取值对应一定的概率。其概率分布满足 Σ P(X = x) = 1。
The expected value (mean) of X is the long-run average:
X 的期望值（均值）表示长期的平均结果：
E(X) = μ = Σ xᵢ P(X = xᵢ)
The variance of X measures the spread of the distribution, and can be computed using the alternative formula:
X 的方差衡量分布的离散程度，可通过简捷公式进行计算：
Var(X) = E(X²) − [E(X)]²
where E(X²) = Σ xᵢ² P(X = xᵢ). The standard deviation is the square root of the variance.
其中 E(X²) = Σ xᵢ² P(X = xᵢ)。标准差为方差的平方根。
6. Binomial Distribution | 二项分布
The 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:
二项分布描述在固定次数的独立试验中成功次数的分布，每次试验成功的概率同为 p。若 X ~ B(n, p)，则：
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, r = 0, 1, 2, …, n
The mean and variance of a binomial random variable are:
二项随机变量的均值与方差分别为：
E(X) = np
Var(X) = np(1 − p)
The conditions for a binomial distribution are: fixed number of trials, independence, two outcomes per trial, and constant probability of success.
二项分布的条件为：试验次数固定、各试验相互独立、每次试验只有两个结果、且成功概率不变。
7. Geometric Distribution | 几何分布
The geometric distribution counts the number of trials required to achieve the first success in a sequence of independent Bernoulli trials, each with success probability p. If X ~ Geo(p), then:
几何分布统计在一系列独立的伯努利试验中首次成功所需的试验次数，每次试验成功的概率为 p。若 X ~ Geo(p)，则：
P(X = r) = p (1 − p)ʳ⁻¹, r = 1, 2, 3, …
The expected value and variance are:
期望值与方差为：
E(X) = 1 / p
Var(X) = (1 − p) / p²
Key assumption: trials are independent and the probability of success p stays constant. The distribution has no upper bound on the number of trials.
关键假设：各试验独立且成功概率 p 恒定。该分布对试验次数没有上限。
8. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution characterised by its bell-shaped curve. It is defined by two parameters: the mean μ and the variance σ². Notation: X ~ N(μ, σ²).
正态分布是一种以钟形曲线为特征的连续概率分布，由均值 μ 和方差 σ² 两个参数决定。记作 X ~ N(μ, σ²)。
To find probabilities for any normal variable, we standardise it to the standard normal variable Z ~ N(0, 1):
若要求任意正态变量的概率，需将其标准化为标准正态变量 Z ~ N(0, 1)：
Z = (X − μ) / σ
The empirical rule (68–95–99.7 rule) states that approximately 68% of observations fall within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ. The normal distribution is symmetric about the mean.
经验法则（68–95–99.7 法则）指出，约 68% 的观测值落在 μ ± σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。正态分布关于均值对称。
Probabilities such as P(X < a) or P(X > b) are found using the standard normal table or technology, working with the cumulative distribution function Φ(z).
类似 P(X < a) 或 P(X > b) 的概率可通过标准正态表或计算工具，借助累积分布函数 Φ(z) 得出。
9. Data Representation & Outliers | 数据表示与离群值
Data can be visualised through stem-and-leaf diagrams and box-and-whisker plots. A box plot displays the minimum, Q₁, median, Q₃, and maximum, revealing skewness and spread.
数据可通过茎叶图和箱线图进行可视化。箱线图展示了最小值、Q₁、中位数、Q₃ 和最大值，用以揭示偏态和离散程度。
Outliers are observations that lie far from the central body of the data. The standard rule for identifying outliers in a box plot is:
离群值是指远离数据中心主体的观测值。箱线图中识别离群值的标准规则为：
Lower boundary = Q₁ − 1.5 × IQR
Upper boundary = Q₃ + 1.5 × IQR
Any data point below the lower boundary or above the upper boundary is typically flagged as an outlier. The five-number summary (minimum, Q₁, median, Q₃, maximum) provides a quick overview of the distribution.
任何低于下边界或高于上边界的数据点通常被标记为离群值。五数概括（最小值、Q₁、中位数、Q₃、最大值）可快速把握分布概况。
10. Linear Coding of Data | 数据的线性编码
When a data set is transformed by a linear coding Y = aX + b, the measures of central tendency and dispersion change in a predictable way. This is extremely useful in simplifying calculations and comparing distributions.
当数据经过线性变换 Y = aX + b 时，集中趋势和离散度量会按可预测的规律变化，这在简化计算和比较分布时极为有用。
Expected value (mean) of the coded variable:
编码后变量的期望值（均值）：
E(Y) = a E(X) + b
Variance of the coded variable (adding a constant does not affect variance):
编码后变量的方差（加上常数不影响方差）：
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