📚 IGCSE CAIE Statistics: Formula & Theorem Quick Reference | IGCSE CAIE 统计：公式定理速查手册
This quick reference collects the formulas, definitions and theorems most commonly required for the Cambridge IGCSE Statistics syllabus. Use it alongside past papers to check your recall and to see how the same result can be applied in data, probability, correlation and distribution questions.
本速查手册汇总了剑桥 IGCSE 统计课程最常考的公式、定义和定理。配合历年真题使用，可以快速检查记忆，并理解同一结果如何应用于数据、概率、相关和分布等题型。
1. Data Types and Sampling | 数据类型与抽样
Data can be qualitative (non-numeric categories such as colour or gender) or quantitative (numeric). Quantitative data is either discrete, taking exact countable values, or continuous, taking any value in an interval. A census collects data from every member of a population, while a sample collects data from part of the population.
数据可以分为定性数据（如颜色、性别等非数值类别）和定量数据（数值）。定量数据又分为离散数据，取可数的精确值，以及连续数据，取某一区间内的任意值。普查从总体的每一个成员收集数据，而抽样只从总体的一部分收集数据。
Common sampling methods include simple random sampling, systematic sampling, stratified sampling, quota sampling and cluster sampling. In stratified sampling, the sample size for each stratum is proportional to the stratum size in the population.
常见抽样方法包括简单随机抽样、系统抽样、分层抽样、配额抽样和整群抽样。分层抽样中，每层的样本量与该层在总体中的大小成比例。
2. Frequency Distributions and Charts | 频数分布与图表
A frequency table shows how often each value or class occurs. For grouped continuous data, the class width is the difference between the upper and lower class boundaries. A histogram uses area to represent frequency, so frequency density = frequency ÷ class width.
频数表显示每个数值或组出现的次数。对于分组连续数据，组距是上组界与下组界之差。直方图用面积表示频数，因此频数密度 = 频数 ÷ 组距。
Common diagrams include bar charts for categorical data, pie charts for proportions, cumulative frequency curves for estimating medians and quartiles, and box-and-whisker plots for comparing distributions. A cumulative frequency graph is plotted at the upper class boundary.
常用图表有条形图（用于分类数据）、饼图（用于比例）、累积频数曲线（用于估计中位数和四分位数）以及箱线图（用于比较分布）。累积频数图在上组界处描点。
3. Measures of Central Tendency | 集中趋势度量
The mean of ungrouped data is x̄ = Σx / n. For grouped data, use x̄ = Σfx / Σf, where x is the class midpoint and f is the frequency. The median is the middle value when the data are ordered; for grouped data it can be estimated from the cumulative frequency curve or by interpolation.
未分组数据的均值是 x̄ = Σx / n。分组数据的均值使用 x̄ = Σfx / Σf，其中 x 为组中点，f 为频数。中位数是数据排序后的中间值；分组数据可由累积频数曲线或插值法估计。
The mode is the value or class with the highest frequency. For a symmetrical distribution, mean ≈ median ≈ mode. If the distribution is positively skewed, mean > median > mode; if negatively skewed, mean < median < mode.
众数是频数最高的数值或组。对称分布的均值约等于中位数约等于众数。如果分布为正偏态，均值 > 中位数 > 众数；若为负偏态，均值 < 中位数 < 众数。
4. Measures of Dispersion | 离散程度度量
The range is the difference between the largest and smallest values. The interquartile range (IQR) is Q₃ – Q₁, where Q₁ is the lower quartile and Q₃ is the upper quartile. Percentiles divide ordered data into 100 equal parts.
极差是最大值与最小值之差。四分位距（IQR）为 Q₃ – Q₁，其中 Q₁ 为下四分位数，Q₃ 为上四分位数。百分位数将有序数据分为 100 等份。
Variance and standard deviation measure the average squared deviation from the mean. For ungrouped data:
方差和标准差衡量数据偏离均值的平均平方距离。未分组数据的公式为：
σ² = Σ(x – x̄)² / n
σ = √[Σ(x – x̄)² / n]
For grouped data, replace x by the class midpoint and multiply each squared deviation by f:
对于分组数据，将 x 替换为组中点，并将每个平方偏差乘以频数 f：
σ² = Σf(x – x̄)² / Σf
When using a calculator, the equivalent form Σx²/n – x̄² can be quicker, but be careful to square the mean correctly.
使用计算器时，等价公式 Σx²/n – x̄² 可能更快，但要注意正确计算均值的平方。
5. Basic Probability Rules | 基本概率法则
Probability is measured on a scale from 0 (impossible) to 1 (certain). For any event A, P(A) + P(A’) = 1, where A’ is the complement of A. For equally likely outcomes, P(A) = number of favourable outcomes ÷ total number of outcomes.
概率用 0（不可能）到 1（必然）的尺度衡量。对于任何事件 A，P(A) + P(A’) = 1，其中 A’ 是 A 的补集。对于等可能结果，P(A) = 有利结果数 ÷ 总结果数。
For two events A and B, the addition rule is:
对于两个事件 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, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
若 A 与 B 互斥，则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。
6. Conditional Probability and Tree Diagrams | 条件概率与树状图
Conditional probability is the probability of A given that B has happened:
条件概率是在 B 已经发生的情况下 A 发生的概率：
P(A | B) = P(A ∩ B) / P(B)
Two events are independent if P(A ∩ B) = P(A) × P(B), or equivalently P(A | B) = P(A). Multiplication along branches of a probability tree gives the probability of the combined path; add path probabilities for the same final outcome.
若 P(A ∩ B) = P(A) × P(B)，或等价地 P(A | B) = P(A)，则两事件独立。沿概率树的分支相乘得到组合路径的概率；同一最终结果的不同路径概率相加。
Tree diagrams are especially useful for successive trials such as drawing balls without replacement, where probabilities change at each stage.
树状图特别适用于连续试验，如不放回摸球，此时每一阶段的概率都会改变。
7. Discrete Random Variables and Expectation | 离散随机变量与期望
A discrete random variable X
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