📚 Formula & Theorem Quick-Reference Guide for Year 12 Cambridge Statistics | 剑桥12年级统计：公式定理速查手册
This quick-reference guide compiles every core formula and theorem from the Cambridge AS Level Probability & Statistics 1 (S1) syllabus, generally studied in Year 12. Use it for rapid revision, to check notation, and to reinforce the precise wording you need in exams.
本速查手册汇编了剑桥AS级概率与统计1（S1）课程中的所有核心公式和定理，通常在12年级学习。可用它快速复习、核对符号，并巩固考试所需的精准表述。
1. Data Representation & Graphs | 数据表示与图表
Histograms display continuous data using frequency density on the vertical axis. Frequency density = Frequency / Class width. The area of each bar is proportional to the frequency.
直方图用频率密度作纵轴来展示连续数据。频率密度 = 频率 / 组距。每个条形的面积与频率成正比。
A cumulative frequency graph plots the running total of frequencies against the upper class boundary. It is used to estimate the median, quartiles, and percentiles without assuming the shape of the distribution within classes.
累积频率图将累积频数相对于上组界描点。它可用于估算中位数、四分位数和百分位数，无需假设组内分布的形状。
Box-and-whisker plots summarise a dataset using five values: minimum, Q1, median, Q3 and maximum. An outlier is often defined as any value less than Q1 – 1.5 × IQR or greater than Q3 + 1.5 × IQR.
箱形图用五个值概括数据集：最小值、Q1、中位数、Q3和最大值。异常值通常定义为小于Q1 – 1.5×四分位距或大于Q3 + 1.5×四分位距的值。
2. Measures of Central Tendency | 集中趋势测量
The mean is the arithmetic average. For raw data: x̄ = Σx / n. For grouped data: x̄ = Σfx / Σf, where x is the class midpoint and f is the frequency.
均值是算术平均数。对于未分组数据：x̄ = Σx / n。对于分组数据：x̄ = Σfx / Σf，其中x为组中点，f为频数。
Mean (x̄) = Σx / n or Σfx / Σf
均值 x̄ = Σx / n 或 Σfx / Σf
The median is the middle value when data are ordered. For n data values, the position of the median is (n + 1)/2. In a grouped frequency table, linear interpolation is used: Median = L + ( (n/2 – cf) / f ) × w, where L is the lower boundary of the median class, cf is the cumulative frequency before the median class, f is the frequency of the median class, and w is the class width.
中位数是排序后中间的值。对于n个数据值，中位数的位置是 (n + 1)/2。在分组频数表中，使用线性插值：中位数 = L + ( (n/2 – cf) / f ) × w，其中L是中位数所在组的下界，cf是该组之前的累积频数，f是该组的频数，w是组距。
The mode is the value that occurs most frequently. In histograms, the modal class is the class with the highest frequency density.
众数是出现次数最多的值。在直方图中，众数组是频率密度最高的组。
3. Measures of Dispersion & Coding | 离散程度测量与编码
Range = Maximum – Minimum. It is sensitive to extreme values. A more robust measure is the interquartile range.
极差 = 最大值 – 最小值。它对极值敏感。更稳健的度量是四分位距。
IQR = Q3 – Q1
四分位距 IQR = 上四分位数 – 下四分位数
Variance measures the average squared deviation from the mean. For a population or set of n values: σ² = Σ(x – x̄)² / n = Σx² / n – x̄². For grouped data: σ² = Σf(x – x̄)² / Σf = Σfx² / Σf – x̄².
方差衡量与均值的平均平方偏差。对于总体或一组n个数值：σ² = Σ(x – x̄)² / n = Σx² / n – x̄²。对于分组数据：σ² = Σf(x – x̄)² / Σf = Σfx² / Σf – x̄²。
σ² = Σ(x – x̄)² / n = Σx² / n – x̄²
方差 σ² = Σ(x – x̄)² / n = Σx² / n – x̄²
Standard deviation is the square root of the variance: σ = √(variance). It has the same units as the original data.
标准差是方差的平方根：σ = √(方差)。它具有与原始数据相同的单位。
Coding simplifies arithmetic. If you transform data using y = (x – a) / b, then x̄ = a + b·ȳ and σ_x² = b²·σ_y² (standard deviation σ_x = |b|·σ_y). The coding does not change the shape of the distribution.
编码可简化计算。若用 y = (x – a) / b 转换数据，则 x̄ = a + b·ȳ，σ_x² = b²·σ_y²（标准差 σ_x = |b|·σ_y）。编码不会改变分布形态。
4. Basic Probability Rules | 概率基本法则
The probability of an event A, P(A), satisfies 0 ≤ P(A) ≤ 1. The complement rule: P(A’) = 1 – P(A).
事件A的概率满足0 ≤ P(A) ≤ 1。互补规则：P(A’) = 1 – P(A)。
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
For mutually exclusive events, A ∩ B = ∅, so P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).
若事件互斥，A ∩ B = ∅，则 P(A ∩ B) = 0，且 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). They are not independent if this equality fails.
若一个事件的发生不影响另一事件的概率，则两个事件独立：P(A ∩ B) = P(A) × P(B)。若不满足此等式，则不为独立。
Conditional probability: the probability of A given B is P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0. It is central in tree diagrams and sequential trials.
条件概率：在B发生的条件下A的概率为 P(A | B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。它在树状图和序贯试验中尤为重要。
Rearranging gives the multiplication rule: P(A ∩ B) = P(B) × P(A | B). For independent events, P(A | B) = P(A).
移项可得乘法法则：P(A ∩ B) = P(B) × P(A | B)。对于独立事件，P(A | B) = P(A)。
5. Permutations & Combinations | 排列与组合
The number of ways of arranging n distinct objects in a line is n! = n × (n-1) × … × 2 × 1.
将n个不同物体排成一排的方法数是 n! = n × (n-1) × … × 2 × 1。
The number of permutations of r objects chosen from n distinct objects, where order matters, is nPr = n! / (n – r)!.
从n个不同物体中选取r个并考虑顺序的排列数是 nPr = n! / (n – r)!。
nPr = n! / (n – r)!
排列数 ⁿPᵣ = n! / (n – r)!
The number of combinations of r objects chosen from n, where order does not matter, is nCr or C(n, r). nCr = n! / (r!(n – r)!). This is also the binomial coefficient.
从n个物体中选取r个且不考虑顺序的组合数是 nCr 或 C(n, r)。nCr = n! / (r!(n – r)!)。这也是二项式系数。
nCr = n! / (r!(n – r)!)
组合数 ⁿCᵣ = n! / (r!(n – r)!)
These are used to count outcomes in probability problems, e.g. the number of ways to obtain exactly r successes in n trials in the binomial distribution.
这些用于计算概率问题中的结果数，例如二项分布中n次试验恰有r次成功的方法数。
6. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of values x₁, x₂, … with probabilities P(X = xᵢ) given by a probability mass function. The sum of all probabilities is 1.
离散随机变量X取可数个值x₁, x₂, …，其概率由概率质量函数 P(X = xᵢ) 给出。所有概率之和为1。
Σ P(X = x) = 1
Σ P(X = x) = 1
The expected value (mean) of X
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