📚 Year 12 Edexcel Statistics: Formula & Theorem Quick Reference Handbook | Year 12 Edexcel 统计：公式定理速查手册
This quick reference handbook covers all essential formulas and theorems for Year 12 Edexcel Statistics. It is structured to follow the AS-level specification, including measures of location and spread, probability, discrete random variables, binomial distribution, normal distribution, hypothesis testing, and key data-handling techniques. Each section presents the core relationship, notation, and interpretation needed to solve exam-style problems efficiently.
本速查手册涵盖了 Year 12 Edexcel 统计课程中所有核心公式和定理。编排紧扣 AS 考试大纲，包括位置与离差度量、概率、离散随机变量、二项分布、正态分布、假设检验以及关键数据处理方法。每一节都给出核心关系、符号定义与解释，帮助读者高效应对考试题型。
1. Sampling and Data Collection | 抽样与数据收集
A population is the whole set of items of interest. A sample is a subset of the population selected to represent it. A sampling frame is a list of all members of the population. Good sampling methods minimise bias and allow valid inferences.
总体是研究对象的全体。样本是从总体中选出的代表性子集。抽样框是列出总体全部成员的名单。好的抽样方法能减少偏差，并可作出有效推断。
Simple random sampling: every member has an equal chance of selection, usually achieved with random number generators.
简单随机抽样：每个成员被选中的概率相等，通常借助随机数生成器实现。
Stratified sampling: the population is divided into mutually exclusive strata, and a random sample is taken from each stratum in proportion to its size.
分层抽样：将总体划分为互斥的层，按各层比例从中随机抽取样本。
Systematic sampling: members are chosen at regular intervals from an ordered list, with a random starting point.
系统抽样：从有序名单中以固定间隔抽取成员，起点随机确定。
Quota sampling: the interviewer selects a predetermined number of individuals from different categories. It is non-random and can introduce bias.
配额抽样：访问员从不同类别中选择预定数量的个体。此方法非随机，可能引入偏差。
Opportunity sampling: the sample is drawn from people who are readily available. It is convenient but often unrepresentative.
方便抽样：从易于接触的人群中选取样本。简便但常缺乏代表性。
2. Measures of Location and Spread | 位置与离差度量
The sample mean (x̄) measures central tendency. The median (Q₂) and quartiles (Q₁, Q₃) describe the distribution's shape together with the range and interquartile range (IQR). Variance and standard deviation quantify spread.
样本均值 (x̄) 度量集中趋势。中位数 (Q₂) 和四分位数 (Q₁, Q₃) 与极差和四分位距 (IQR) 一起描述分布形状。方差和标准差量化离散程度。
Sample mean: x̄ = Σx / n
样本均值：x̄ = Σx / n
For grouped data, use the midpoint (m) of each class: x̄ ≈ Σfm / Σf. The variance of a sample uses the divisor (n − 1):
对于分组数据，使用组中值 m：x̄ ≈ Σfm / Σf。样本方差使用除数 (n − 1)：
Sxx = Σ(x − x̄)² = Σx² − (Σx)²/n
样本方差：s² = Sxx / (n − 1)
标准差：s = √[Sxx / (n − 1)]
The interquartile range is IQR = Q₃ − Q₁. It is resistant to outliers.
四分位距：IQR = Q₃ − Q₁，它对异常值不敏感。
3. Data Presentation and Interpretation | 数据呈现与解读
In a histogram, the area of each bar is proportional to frequency. To find the frequency from a bar, multiply the class width by the frequency density.
在直方图中，每个条形的面积与频数成正比。求某条形代表的频数时，用组距乘以频率密度。
Frequency density = Frequency / Class width
频率密度 = 频数 / 组距
A box plot displays the five-number summary: minimum, Q₁, median, Q₃, maximum. Outliers are points that fall outside the fences:
箱线图展示五数概括：最小值、Q₁、中位数、Q₃、最大值。异常值指落在以下界限之外的点：
Lower fence = Q₁ − 1.5 × IQR
Upper fence = Q₃ + 1.5 × IQR
A cumulative frequency curve can be used to estimate medians, quartiles and percentiles. The curve should start at the lower boundary of the first class and reach the total frequency at the top boundary of the last class.
累积频率曲线可用于估计中位数、四分位数和百分位数。曲线应从第一组的下限开始，在最后一组的上限处达到总频数。
4. Probability Basics | 概率基础
easures the chance that an event occurs, with 0 ≤ P(A) ≤ 1. The complement rule states P(A') = 1 − P(A).
概率衡量事件发生的可能性，取值范围 0 ≤ P(A) ≤ 1。补集法则为 P(A') = 1 − P(A)。
For two events A and B, the addition rule gives the probability of A or B occurring:
对于两个事件 A 和 B，加法法则给出 A 或 B 发生的概率：
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)。
Conditional probability concerns the chance of A given that B has occurred:
条件概率考虑已知 B 发生的条件下 A 发生的概率：
P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0
From this, the multiplication rule follows:
由此得出乘法法则：
P(A ∩ B) = P(A) P(B|A) = P(B) P(A|B)
Events A and B are independent if P(A ∩ B) = P(A) P(B). Under independence, P(A|B) = P(A).
若 P(A ∩ B) = P(A) P(B)，则事件 A 与 B 独立。此时 P(A|B) = P(A)。
Bayes' theorem connects reversed conditional probabilities:
贝叶斯定理联系逆向条件概率：
P(A|B) = [P(B|A) P(A)] / P(B)
5. Correlation and Regression | 相关与回归
Pearson's product-moment correlation coefficient r measures the strength of a linear relationship. Its formula uses the sums of squares:
皮尔逊积矩相关系数 r 衡量线性关系的强度。公式使用平方和：
Sxx = Σx² − (Σx)²/n, Syy = Σy² − (Σy)²/n, Sxy = Σxy − (Σx)(Σy)/n
r = Sxy / √Sxx × Syy
r lies between –1 and 1. A value near 1 or –1 indicates strong linear correlation. r close to 0 suggests little or no linear correlation.
r 取值在 –1 到 1 之间。接近 1 或 –1 表示强线性相关；接近 0 表示几乎没有线性相关。
regression line of y on x is used to predict y from x. It has the form y = a + bx, where:
最小二乘回归线 y 对 x 的回归用于由 x 预测 y，其形式为 y = a + bx，其中：
