📚 AS CAIE Statistics: Formula & Theorem Quick Reference Guide | AS CAIE 统计：公式定理速查手册
This quick-reference guide summarises the essential formulas and theorems required for the CAIE AS Level Probability & Statistics 1 (S1). All key definitions, notation, and computational methods are collected here to support exam revision and problem-solving. Use it alongside past papers and your formula booklet to strengthen recall and accuracy.
本速查手册总结了 CAIE AS 阶段概率与统计 1（S1）所需的核心公式和定理。所有关键定义、符号和计算方法集中于此，帮助复习和解题。请结合历年真题和公式手册使用，强化记忆与准确性。
1. Measures of Central Tendency | 集中趋势度量
x̄ = Σx / n
The arithmetic mean for ungrouped data is the sum of all observations divided by the number of observations.
未分组数据的算术平均数等于所有观测值之和除以观测值的个数。
Estimated mean for grouped data: x̄ = Σfx / Σf
For grouped data, multiply each class midpoint (x) by its frequency (f), sum these products and divide by total frequency. The midpoint is (upper boundary + lower boundary) / 2.
对于分组数据，将每个组中值 (x) 乘以频数 (f)，求和后除以总频数。组中值为（上限+下限）/ 2。
Median = value at position (n+1)/2 for odd n; ½(xn/2 + xn/2+1) for even n
s lower boundary of the median class, F is cumulative frequency before the class, f is class frequency, and w is class width.
中位数为排序后中间的值。分组数据用线性插值：中位数 = L + ((n/2 - F) / f) × w，其中 L 为中位数组下限，F 为该组之前累积频数，f 为该组频数，w 为组距。
The mode is the most frequent value. In a histogram, the modal class is the class with the highest frequency density, not necessarily the highest frequency.
众数是出现次数最多的值。在直方图中，众数组是频数密度最高的组，而不一定是频数最高的组。
2. Measures of Spread & Box Plots | 离散程度与箱线图
Sample variance: s² = Σ(x - x̄)² / (n - 1)
Often computed faster via: s² = (Σx² - (Σx)²/n) / (n - 1). The standard deviation is s = √s².
常用简便公式：s² = (Σx² - (Σx)²/n) / (n - 1)。标准差 s = √s²。
Interquartile range: IQR = Q₃ - Q₁
Q₃ at ¾. Outliers are usually defined as points outside [Q₁ - 1.5 × IQR, Q₃ + 1.5 × IQR].
四分位数的求法与中位数类似。Q₁ 位于排序数据的 ¼ 处，Q₃ 位于 ¾ 处。异常值通常定义为落在 [Q₁ - 1.5 × IQR, Q₃ + 1.5 × IQR] 之外的点。
A box-and-whisker plot shows the minimum, Q₁, median, Q₃, and maximum, with outliers plotted as individual crosses. It reveals skewness: if median closer to Q₁, data might be positively skewed.
箱线图显示最小值、Q₁、中位数、Q₃ 和最大值，异常值用叉号单独标出。它可以反映偏态：若中位数靠近 Q₁，则数据可能右偏。
3. Basic Probability | 概率基础
P(A) = number of outcomes in A / total number of outcomes
asures how likely an event is, always between 0 and 1. The complement rule: P(not A) = P(A') = 1 - P(A).
概率衡量事件发生的可能性，取值在 0 到 1 之间。互补法则：P(A') = 1 - P(A)。
Addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
If A and B are mutually exclusive (cannot occur together), 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，此时 P(A ∪ B) = P(A) + P(B)。
Multiplication rule: P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)
For independent events, knowing B does not change the probability of A, thus P(A|B) = P(A) and P(A ∩ B) = P(A) × P(B).
对于独立事件，已知 B 不改变 A 的概率，因此 P(A|B) = P(A)，且 P(A ∩ B) = P(A) × P(B)。
4. Conditional Probability & Independence | 条件概率与独立性
P(A|B) = P(A ∩ B) / P(B), P(B) > 0
This reads as 'probability of A given B'. The sample space is reduced to only those outcomes where B has occurred.
读作“在 B 发生的条件下 A 发生的概率”。此时样本空间缩小为仅含 B 已发生的结果。
Two events are independent if and only if P(A ∩ B) = P(A) × P(B), or equivalently P(A| nct concepts; mutually exclusive events with non-zero probabilities cannot be independent.
两个事件独立当且仅当 P(A ∩ B) = P(A) × P(B) 或等价地 P(A|B) = P(A)。独立与互斥是不同的概念；非零概率的互斥事件不可能独立。
Always draw a Venn diagram or a tree diagram when dealing with conditional probabilities. A tree diagram multiplies along branches and adds probabilities of different paths.
处理条件概率时务必画维恩图或树状图。树状图沿分支相乘，不同路径的概率相加。
5. Permutations & Combinations | 排列与组合
n! = n × (n-1) × ... × 2 × 1, 0! = 1
Factorial counts the number of ways to arrange n distinct objects in a line.
阶乘计算将 n 个不同物体排成一排的方法数。
Permutations: ⁿPᵣ = n! / (n - r)!
The number of ways to choose and arrange r objects from n distinct objects when order matters.
从 n 个不同物体中选出 r 个并按顺序排列的方法数，顺序重要。
Combinations: ⁿCᵣ = n! / [r!(n - r)!] = (n
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