📚 Year 12 CAIE Statistics: Quick Reference Handbook of Formulas and Theorems | CAIE 12年级统计：公式定理速查手册
This revision handbook covers the essential formulas, definitions and key theorems for the Year 12 CAIE Statistics syllabus (Paper 5 Probability & Statistics 1). Use it to consolidate your understanding and for last‑minute revision. Each section pairs an English explanation with a Chinese translation to ensure clarity.
本速查手册涵盖CAIE 12年级统计（试卷5 概率与统计1）的核心公式、定义与关键定理。可用以巩固理解并进行考前突击。每节均以英中对照形式呈现，确保清晰易懂。
1. Measures of Central Tendency | 集中趋势的度量
For a dataset {x₁, x₂, …, xₙ}, the mean (μ for a population, x̄ for a sample) is given by the sum of all values divided by the number of observations.
对于数据集{x₁, x₂, …, xₙ}，均值（总体用μ，样本用x̄）等于所有数值之和除以观测个数。
Mean: x̄ = Σx / n
The median is the middle value when the data are ordered; for an even number of observations it is the mean of the two middle values. The mode is the most frequent value.
中位数是数据按大小排序后位于中间的值；当观测个数为偶数时，中位数等于中间两个数的均值。众数是出现次数最多的值。
For grouped data, the mean is estimated using class midpoints m: x̄ = Σ(fm) / Σf.
对于分组数据，用组中值m估算均值：x̄ = Σ(fm) / Σf。
The mode for grouped data is the class with the highest frequency, usually refined by linear interpolation.
分组数据的众数为频率最高的组，通常借助线性插值精确定位。
In a frequency table the cumulative frequency helps locate the median class, and interpolation gives the median value.
在频数表中，累积频数可用于确定中位数所在组，再通过插值求出中位数值。
2. Measures of Spread | 离散程度的度量
The range is the difference between the largest and smallest observation: range = x_max − x_min.
极差是最大观测值与最小观测值之差：range = x_max − x_min。
The interquartile range (IQR) is Q₃ − Q₁, where Q₁ and Q₃ are the lower and upper quartiles.
四分位距(IQR)等于上四分位数Q₃减去下四分位数Q₁。
Variance measures the average squared deviation from the mean. For a population: σ² = Σ(x − μ)² / N. For a sample the unbiased estimator uses denominator n − 1.
方差衡量各数据与均值之差的平方的平均值。总体方差：σ² = Σ(x − μ)² / N。样本方差的无偏估计使用分母n − 1。
Sample variance: s² = Σ(x − x̄)² / (n − 1)
The standard deviation is the positive square root of the variance: s = √[Σ(x − x̄)² / (n − 1)].
标准差是方差的正平方根：s = √[Σ(x − x̄)² / (n − 1)]。
An alternative computational formula for variance is: s² = [Σx² − (Σx)²/n] / (n − 1). This is quicker for hand calculations.
方差的计算公式可写为：s² = [Σx² − (Σx)²/n] / (n − 1)，便于手工计算。
When data are given in a frequency table with values x and frequencies f, the formulas become Σfx, Σfx², and the denominator remains (Σf − 1) for sample variance.
当数据以频数表形式给出（值为x，频数为f），公式中的Σx用Σfx、Σx²用Σfx²代替，样本方差的分母保持为(Σf − 1)。
3. Quartiles and Percentiles | 四分位数与百分位数
Quartiles divide an ordered dataset into four equal parts. Q₁ is the median of the lower half, Q₂ the overall median, and Q₃ the median of the upper half.
四分位数将有序数据等分为四份。Q₁为下半部分的中位数，Q₂为全体的中位数，Q₃为上半部分的中位数。
For discrete data, the position of the lower quartile is often taken as (n + 1)/4, and the upper quartile as 3(n + 1)/4; linear interpolation may be used when the position is not an integer.
对于离散数据，下四分位数位置常用(n + 1)/4，上四分位数位置用3(n + 1)/4；当位置不是整数时可采用线性插值。
The p‑th percentile is the value below which p% of the data fall. In a cumulative frequency graph, reading across from p% on the vertical axis gives the corresponding percentile.
第p百分位数是指有p%的数据低于该值。在累积频数图上，从纵轴上p%处水平引线即可读取对应的百分位数。
Box‑and‑whisker plots use the five‑number summary: minimum, Q₁, median, Q₃, maximum, and identify outliers using fences 1.5 × IQR beyond the quartiles.
箱线图利用五数概括：最小值、Q₁、中位数、Q₃、最大值，并以1.5×IQR为界标记异常值。
4. Probability Basics | 概率基础
The probability of an event A, denoted P(A), satisfies 0 ≤ P(A) ≤ 1. The sum of probabilities of all possible outcomes in a sample space is 1.
事件A的概率P(A)满足0 ≤ P(A) ≤ 1。样本空间中所有可能结果的概率之和为1。
The complement of A is A′ or A̅, and P(A′) = 1 − P(A).
事件A的补集记为A′或A̅，P(A′) = 1 − P(A)。
For any two events A and B, the addition rule states: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
对于任意两事件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（不能同时发生），则P(A ∩ B) = 0，因此P(A ∪ B) = P(A) + P(B)。
Empirical probability is based on observed frequencies: P(A) = (number of times A occurs) / (total number of trials).
经验概率基于观测频率：P(A) = (A出现的次数) / (总试验次数)。
5. Conditional Probability | 条件概率
The probability of event A given that B has occurred is defined as P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.
在事件B已发生的条件下事件A发生的概率定义为：P(A | B) = P(A ∩ B) / P(B)，其中P(B) > 0。
This leads to the multiplication rule: P(A ∩ B) = P(A | B) × P(B) = P(B | A) × P(A).
由此得到乘法法则：P(A ∩ B) = P(A | B) × P(B) = P(B | A) × P(A)。
Two‑way tables and tree diagrams are powerful tools for visualizing conditional probabilities. On a tree diagram, multiply along branches and add probabilities of the relevant final nodes.
双向表和树图是可视化条件概率的强大工具。在树图中，沿分支相乘概率，再将相关末端节点的概率相加。
Bayes’ theorem, though not explicitly required in AS, is implicitly used when updating probabilities using P(A | B) after new information is given.
贝叶斯定理虽非AS阶段明确要求，但在给定新信息后使用P(A | B)更新概率时实质上已被应用。
6. Mutually Exclusive and Independent Events | 互斥与独立事件
Two events are mutually exclusive if they cannot happen at the same time: P(A ∩ B) = 0.
两事件互斥意味着它们不能同时发生：P(A ∩ B) = 0。
Two events are independent if the occurrence of one does not affect the probability of the other. Mathematically, P(A | B) = P(A) and P(B | A) = P(B).
两事件独立意味着一个事件的发生不影响另一事件的概率。数学上，P(A | B) = P(A)且P(B | A) = P(B)。
The test for independence is: P(A ∩ B) = P(A) × P(B). This is equivalent to the definition above and is the most commonly checked condition in exam problems.
独立性的检验是：P(A ∩ B) = P(A) × P(B)。这与上述定义等价，是试题中最常检查的条件。
Mutually exclusive events cannot be independent unless at least one of them has probability zero. Students must be careful not to confuse the two concepts.
互斥事件不可能是独立的，除非其中至少一个事件概率为零。学生必须注意不要混淆这两个概念。
7. Permutations and Combinations | 排列与组合
The factorial of a positive integer n is n! = n × (n − 1) × … × 1, with 0! = 1.
正整数n的阶乘定义为n! = n × (n − 1) × … × 1，并规定0! = 1。
The number of ways to arrange r distinct objects chosen from n distinct objects, where order matters, is the permutation: ⁿPᵣ = n! / (n − r)!.
从n个不同物体中选出r个进行排列（顺序重要）的方法数为：ⁿPᵣ = n! / (n − r)!。
The number of ways to choose r objects from n distinct objects, where order does not matter, is the combination: ⁿCᵣ = n! / [r!(n − r)!].
从n个不同物体中选出r个进行组合（顺序不重要）的方法数为：ⁿCᵣ = n! / [r!(n − r)!]。
ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿC₀ = ⁿCₙ = 1
Combinations are used for binomial probabilities, selections, and draws without replacement where order is ignored.
组合用于二项概率、选择问题以及忽略顺序的无放回抽取。
When objects are rearranged and some are identical, the number of distinct arrangements of n objects with n₁ alike of type 1, n₂ alike of type 2, … is n! / (n₁! n₂! …).
当排列中存在相同的物体时，n个对象中有n₁个类型1相同、n₂个类型2相同……的不同排列数为n! / (n₁! n₂! …)。
8. Discrete Random Variables | 离散随机变量
A discrete random variable X takes values x₁, x₂, … with corresponding probabilities P(X = xᵢ). The probability distribution must satisfy ΣP(X = xᵢ) = 1 and each P(X = xᵢ) ≥ 0.
离散随机变量X取值x₁, x₂, …，对应概率为P(X = xᵢ)。概率分布须满足ΣP(X = xᵢ) = 1且每个P(X = xᵢ) ≥ 0。
The expected value (mean) of X is E(X) = μ = Σ[xᵢ · P(X = xᵢ)].
X的期望值（均值）为E(X) = μ = Σ[xᵢ · P(X = xᵢ)]。
The variance of X is Var(X) = E[(X − μ)²] = Σ[(xᵢ − μ)² · P(X = xᵢ)]. The computational form is Var(X) = E(X²) − [E(X)]².
X的方差为Var(X) = E[(X − μ)²] = Σ[(xᵢ − μ)² · P(X = xᵢ)]。计算公式为Var(X) = E(X²) − [E(X)]²。
Var(X) = Σ xᵢ² P(X = xᵢ) − (Σ xᵢ P(X = xᵢ))²
For a linear function aX + b, the following rules hold: E(aX + b) = a·E(X) + b, and Var(aX + b) = a²·Var(X).
对于线性函数aX + b，满足：E(aX + b) = a·E(X) + b，Var(aX + b) = a²·Var(X)。
The standard deviation is SD(X) = √Var(X). These rules extend to sums of independent random variables, where E(X + Y) = E(X) + E(Y) and Var(X + Y) = Var(X) + Var(Y).
标准差为SD(X) = √Var(X)。这些规则可推广至独立随机变量之和，此时E(X + Y) = E(X) + E(Y)且Var(X + Y) = Var(X) + Var(Y)。
9. Binomial Distribution | 二项分布
A binomial distribution models the number of successes X in n independent trials, each with the same probability of success p. This is written as X ~ B(n, p).
二项分布描述在n次独立试验中成功的次数X，每次试验成功的概率均为p。记为X ~ B(n, p)。
The probability of exactly r successes is: P(X = r) = ⁿCᵣ × pʳ × (1 − p)ⁿ⁻ʳ, for r = 0, 1, …, n.
恰好成功r次的概率为：P(X = r) = ⁿCᵣ × pʳ × (1 − p)ⁿ⁻ʳ，r = 0, 1, …, n。
The mean and variance of a binomial distribution are: μ = E(X) = np, and σ² = Var(X) = np(1 − p).
二项分布的均值和方差为：μ = E(X) = np，σ² = Var(X) = np(1 − p)。
When n is large, the binomial distribution can be approximated by a normal distribution with the same mean and variance, provided np > 5 and n(1 − p) > 5 (a continuity correction is often applied).
当n较大时，若np > 5且n(1 − p) > 5，二项分布可用相同均值和方差的正态分布近似（通常需使用连续性校正）。
Cumulative binomial probabilities P(X ≤ r) can be found using statistical tables or calculators. Remember: P(X ≥ r) = 1 − P(X ≤ r − 1) and P(X < r) = P(X ≤ r − 1).
累积二项概率P(X ≤ r)可通过统计表或计算器获得。注意：P(X ≥ r) = 1 − P(X ≤ r − 1)，P(X < r) = P(X ≤ r − 1)。
10. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution defined by its mean μ and variance σ². The random variable is written as X ~ N(μ, σ²). Its graph is a symmetric bell‑shaped curve.
正态分布是由均值μ和方差σ²定义的连续概率分布。随机变量记为X ~ N(μ, σ²)，其图像是一条对称的钟形曲线。
The total area under the probability density curve is 1. The probability P(a < X < b) equals the area under the curve between a and b.
概率密度曲线下的总面积为1。P(a < X < b)等于曲线下方介于a和b之间的面积。
Standardisation converts X to the standard normal variable Z = (X − μ) / σ, where Z ~ N(0, 1). Tables provide values of Φ(z) = P(Z < z).
标准化将X转化为标准正态变量Z = (X − μ) / σ，其中Z ~ N(0, 1)。标准正态分布表提供Φ(z) = P(Z < z)的值。
To find P(X < a), compute z = (a − μ) / σ and use Φ(z). For P(X > a), use 1 − Φ(z). For an interval, P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ).
求P(X < a)时，计算z = (a − μ) / σ并使用Φ(z)。求P(X > a)时用1 − Φ(z)。求区间概率时，P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ)。
Sometimes you are given a probability and need to find the corresponding x value. Use the inverse normal table: find z such that Φ(z) = given probability, then x = μ + zσ.
有时题目给出概率要求反求x值。使用逆正态分布表：查找满足Φ(z) = 给定概率的z，然后得x = μ + zσ。
The normal distribution can model many natural phenomena, and also approximates the distribution of sample means (Central Limit Theorem), though the latter is beyond AS Level.
正态分布可模拟许多自然现象，也可近似样本均值的分布（中心极限定理），尽管后者超出AS阶段要求。
The empirical rule (68–95–99.7%) states that approximately 68% of data fall within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
经验法则（68–95–99.7%）指出，约68%的数据落在μ ± σ内，95%落在μ ± 2σ内，99.7%落在μ ± 3σ内。
11. Data Representation with Cumulative Frequency | 累积频数图与数据展示
In a cumulative frequency diagram, points are plotted at the upper class boundary for grouped data and joined with a smooth curve. The graph can be used to estimate the median, quartiles, and percentiles.
在累积频数图中，分组数据在每组的上组界处描点并连成光滑曲线。此图可用于估计中位数、四分位数和百分位数。
To estimate the median, draw a horizontal line from the 50% mark on the cumulative percentage axis (or half of total frequency) to the curve, then drop down to the x‑axis. The same method gives Q₁ and Q₃ from 25% and 75%.
估计中位数时，从累积百分率轴上50%处（或总频数的一半）画水平线交曲线，再垂直向下读取x轴。类似地从25%和75%可得Q₁和Q₃。
Histograms are used for continuous grouped data. The frequency of each class is proportional to the area of the bar; the height is given by frequency density = frequency ÷ class width.
直方图用于连续分组数据。每组的频率与条形面积成正比；条形高度由频率密度 = 频数 ÷ 组距给出。
These representations are essential for understanding the shape of a distribution (symmetric, skewed, bimodal) and for spotting outliers before formal analysis.
这些图示对于理解分布形态（对称、偏态、双峰）以及在正式分析前识别异常值至关重要。
12. Key Skills for Exam Success | 考试成功的关键技巧
Always identify the type of variable and distribution first: discrete or continuous, binomial or normal. This decides which table or formula to apply.
首先务必识别变量和分布类型：离散还是连续，二项还是正态。这决定了应使用哪个表格或公式。
Write down all given information using correct notation, and convert probabilities into mathematical statements before calculating.
用正确符号写下所有已知信息，并在计算前将概率表述转化为数学表达式。
For normal probability problems, always draw a bell curve, shade the required area, and standardise to Z. Keep track of signs when using symmetry, e.g. P(Z < −k) = 1 − Φ(k).
处理正态概率题时，始终画出钟形曲线，标出所求区域，再标准化为Z。利用对称性时注意符号，例如P(Z < −k) = 1 − Φ(k)。
In binomial questions, check for independence and constant probability. Use the complementary rule to shorten calculations, especially for cumulative tails.
在二项问题中，检查独立性和固定概率。善用补集规则简化计算，尤其在处理累积尾概率时。
For grouped data, remember to use midpoints for mean and variance estimates, and understand that interpolation with cumulative frequency is approximate. Always state which formulas you are using.
对于分组数据，记得用组中值估算均值和方差，并理解通过累积频数插值得到的是近似值。始终注明所引用的公式。
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