📚 Year 12 AQA Statistics: Quick Reference Formula & Theorem Handbook | AQA 统计：公式定理速查手册
This quick reference handbook collects all the essential formulas, theorems, and definitions needed for the Year 12 AQA Statistics course. It is designed to help you efficiently review probability rules, common distributions, hypothesis testing procedures, and correlation/regression techniques before your exams. Each section presents the core mathematical expressions with paired English and Chinese explanations.
本速查手册汇集了 Year 12 AQA 统计课程所需的所有关键公式、定理和定义。它旨在帮助你在考试前高效地复习概率规则、常见分布、假设检验程序以及相关与回归技术。每个小节都以英中对照的方式呈现核心数学表达式。
1. Measures of Central Tendency & Spread | 集中趋势与离散度量
The sample mean is the arithmetic average of the data, providing a measure of central location. The sample variance quantifies the average squared deviation from the mean, using n-1 to give an unbiased estimate of the population variance.
样本均值是数据的算术平均值，用来度量数据的中心位置。样本方差量化了各数据点与均值之差的平方的平均值，使用 n-1 是为了得到总体方差的无偏估计。
x̄ = (Σxᵢ) / n
x̄ = (Σxᵢ) / n
s² = Σ(xᵢ - x̄)² / (n - 1)
s² = Σ(xᵢ - x̄)² / (n - 1)
The standard deviation is the positive square root of the variance, s = √s include the median (middle value when ordered) and the interquartile range IQR = Q₃ - Q₁.
标准差是方差的正平方根，s = √s²。其他重要度量包括中位数（排序后中间的值）以及四分位距 IQR = Q₃ - Q₁。
| Statistic | Formula | 中文 |
|---|---|---|
| Mean | x̄ = Σxᵢ / n | 均值 |
| Variance (sample) | s² = Σ(xᵢ - x̄)² / (n-1) | 样本方差 |
| Standard deviation | s = √[Σ(xᵢ - x̄)²/(n-1)] | 标准差 |
| Range | max - min | 极差 |
| IQR | Q₃ - Q₁ | 四分位距 |
2. Basic Probability Rules | 概率基本规则
For any two events A and B, the probability that at least one of them occurs is given by the general addition rule. When events are mutually exclusive, the intersection term becomes zero.
对于任何两个事件 A 和 B，至少有一个发生的概率由一般加法规则给出。当事件互斥时，交集项为零。
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
For independent events, the joint probability is simply the product of the individual probabilities. The complement rule states that the probability of an event not occurring is 1 minus the probability of the event.
对于独立事件，联合概率就是个别的概率之积。互补规则指出，一个事件不发生的概率等于 1 减去该事件发生的概率。
P(A ∩ B) = P(A) × P(B) (if independent / 如果独立)
P(A ∩ B) = P(A) × P(B) （如果独立）
P(A') = 1 - P(A)
P(A') = 1 - P(A)
3. Conditional Probability & Independence | 条件概率与独立性
The conditional probability of B given A is the probability that B occurs under the condition that A has already occurred. It is defined only when P(A) > 0. This formula forms the basis of Bayes' Theorem when dealing with reversed conditions.
在 A 已发生的条件下 B 发生的条件概率记为 P(B|A)。它仅在 P(A) > 0 时有定义。该公式是处理条件反转问题时贝叶斯定理的基础。
P(B|A) = P(A ∩ B) / P(A)
P(B|A) = P(A ∩ B) / P(A)
Two events A and B are independent if and only if P(A ∩ B) = P(A) P(B) or equivalently P(B| A tree diagram is a powerful tool for visualising sequences of conditional probabilities.
两个事件 A 和 B 独立的充要条件是 P(A ∩ B) = P(A) P(B) 或等价地 P(B|A) = P(B)。树图是可视化一系列条件概率的强大工具。
P(A|B) = [P(B|A) P(A)] / P(B) (Bayes' Theorem / 贝叶斯定理)
P(A|B) = [P(B|A) P(A)] / P(B) （贝叶斯定理）
4. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable set of values with associated probabilities. The expected value E(X) gives the long-run average, while Var(X) measures the spread of the distribution.
离散随机变量 X 取可数个可能值，每个值都有对应的概率。期望值 E(X) 给出了长期的平均值，而方差 Var(X) 衡量分布的离散程度。
E(X) = Σ x P(X = x)
E(X) = Σ x P(X = x)
Var(X) = E(X²) - [E(X)]² = Σ x² P(X = x) - μ²
Var(X) = E(X²) - [E(X)]² = Σ x² P(X = x) - μ²
For a linear transformation of a random variable, the expectation and variance follow simple rules. Adding a constant shifts the mean but does not affect the variance; multiplying by a constant scales both the mean and the variance (squared).
对于随机变量的线性变换，期望和方差遵循简单的规则。加一个常数平移均值但不影响方差；乘一个常数同时缩放均值和方差（方差要乘以常数的平方）。
E(aX + b) = a E(X) + b
E(aX + b) = a E(X) + b
Var(aX + b) = a² Var(X)
Var(aX + b) = a² Var(X)
5. Binomial Distribution | 二项分布
The binomial distribution models the number of successes in n independent trials, each with constant probability of success p. The distribution is denoted by X ~ B(n, p). The probability of exactly r successes is given by the binomial formula.
二项分布用于描述在 n 次独立试验中成功的次数，每次成功的概率恒为 p。该分布记作 X ~ B(n, p)。恰好得到 r 次成功的概率由二项式公式给出。
P(X = r) = ⁿCᵣ pʳ (1 - p)ⁿ⁻ʳ, r = 0,1,...,n
P(X = r) = ⁿCᵣ pʳ (1 - p)ⁿ⁻ʳ, r = 0,1,...,n
The mean and variance of a binomial random variable are simple multiples of n and p. When n is large and p is close to 0.5, the binomial distribution is approximately symmetric; it can also be approximated by a normal distribution under certain conditions.
二项随机变量的均值和方差都是 n 与 p 的简单乘积。当 n 很大且 p 接近 0.5 时，二项分布近似对称；在某些条件下还可用正态分布近似。
E(X) = np
E(X) = np
Var(X) = np(1 - p)
Var(X) = np(1 - p)
6. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution defined by two parameters: the mean μ and the standard deviation σ. Its probability density function
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