📚 AS AQA Statistics Quick Formula Reference | AS AQA 统计公式速查手册
This concise handbook brings together the essential formulas and theorems for the AS AQA Statistics component. Use it to review probability rules, distributions, sampling theory, inference, and regression – all in one place for quick revision and exam confidence.
本速查手册汇总了 AS AQA 统计学所需的核心公式与定理，涵盖概率法则、分布、抽样理论、推断和回归，帮助你在考前快速回顾、加深理解。
1. Probability Addition Rule | 概率加法法则
For any two events A and B, the probability that A or B occurs is given by the general addition rule:
对于任意两个事件 A 和 B，A 或 B 发生的概率由一般的加法公式给出：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B are mutually exclusive, they have no outcomes in common, so P(A ∩ B) = 0 and the formula simplifies to P(A ∪ B) = P(A) + P(B).
若 A 与 B 互斥，即没有公共结果，则 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。
2. Conditional Probability and Multiplication Rule | 条件概率与乘法法则
The probability of A given that B has occurred is the conditional probability:
在事件 B 已发生的条件下 A 发生的概率为条件概率：
P(A | B) = P(A ∩ B) / P(B), P(B) > 0
The multiplication rule follows directly from this definition:
由此直接得到乘法法则：
P(A ∩ B) = P(A | B) · P(B) = P(B | A) · P(A)
When solving problems, always identify which events are conditioned and check whether the events are independent before applying the rule.
解题时务必先明确条件事件，并判断事件是否独立，再套用乘法法则。
3. Mutually Exclusive and Independent Events | 互斥事件与独立事件
Two events are mutually exclusive if they cannot happen together: P(A ∩ B) = 0.
若两事件不能同时发生，则它们互斥：P(A ∩ B) = 0。
Two events are independent if the occurrence of one does not affect the probability of the other:
若一事件的发生不影响另一事件的概率，则两事件独立：
P(A ∩ B) = P(A) × P(B) or P(A | B) = P(A)
Mutual exclusivity and independence are distinct concepts. Do not confuse them: mutually exclusive events with non-zero probabilities are never independent.
互斥与独立是两个不同的概念，切勿混淆：非零概率的互斥事件一定不独立。
4. Permutations and Combinations | 排列与组合
The number of ways to arrange r items chosen from n distinct items (order matters) is the permutation:
从 n 个不同物品中有序选取 r 个的排列数为：
ⁿPᵣ = n! / (n − r)!
The number of ways to choose r items from n distinct items (order does not matter) is the combination:
从 n 个不同物品中无序选取 r 个的组合数为：
ⁿCᵣ = n! / [ r! (n − r)! ]
Use permutations for arrangements, passwords or races; use combinations for selecting teams or lottery numbers where the arrangement is irrelevant.
涉及排序、密码或赛跑用排列；只关心选取对象的团队成员或彩票号码则用组合。
5. Expectation and Variance of Discrete Random Variables | 离散随机变量的期望与方差
For a discrete random variable X with values xᵢ and probabilities pᵢ = P(X = xᵢ):
对取值为 xᵢ、概率 pᵢ = P(X = xᵢ) 的离散随机变量 X：
E(X) = μ = Σ xᵢ pᵢ
Var(X) = E[(X − μ)²] = Σ (xᵢ − μ)² pᵢ = E(X²) − [E(X)]²
For a linear transformation Y = aX + b:
对线性变换 Y = aX + b：
E(aX + b) = aE(X) + b, Var(aX + b) = a² Var(X)
These formulas underpin all modelling with random variables, especially the binomial distribution that follows.
这些公式是所有随机变量建模的基础，尤其是接下来要讲的二项分布。
6. Binomial Distribution | 二项分布
When a fixed number n of independent trials each has the same probability p of success, the number of successes X follows a binomial distribution:
若进行 n 次独立试验，每次成功概率均为 p，则成功次数 X 服从二项分布：
X ~ B(n, p)
P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ, for k = 0, 1, …, n
The mean and variance are:
其期望与方差分别为：
E(X) = np, Var(X) = np(1 − p)
Always check the conditions: fixed n, independent trials, two outcomes per trial, constant p. The binomial model is widely used for proportions and count data.
使用前务必验证条件：n 固定、试验独立、每次只有两个结果、p 不变。二项模型广泛用于比例和计数数据。
7. Normal Distribution and Standardisation | 正态分布与标准化
A continuous random variable X that is normally distributed with mean μ and variance σ² is written as:
若连续随机变量 X 服从均值为 μ、方差为 σ² 的正态分布，记作：
X ~ N(μ, σ²)
We standardise to the standard normal Z ~ N(0, 1) using:
通过下式转化为标准正态分布 Z ~ N(0, 1)：
Z = (X − μ) / σ
The probability P(X < x) equals P(Z < (x−μ)/σ) and is found from statistical tables. Remember the 68–95–99.7 rule: about 68% of values lie within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean.
概率 P(X < x) 等于 P(Z < (x−μ)/σ)，可查标准正态表。记住 68–95–99.7 经验法则：约 68% 的数据落在均值 ±1σ 内，95% 在 ±2σ 内，99.7% 在 ±3σ 内。
8. Sampling Distribution of the Sample Mean and the Central Limit Theorem | 样本均值的抽样分布与中心极限定理
If random samples of size n are drawn from a population with mean μ and variance σ², the sample mean X̄ has the following distribution:
从均值为 μ、方差为 σ² 的总体中抽取容量为 n 的随机样本，样本均值 X̄ 的分布为：
E(X̄) = μ, Var(X̄) = σ² / n
If the population is normal, then X̄ ~ ation is not normal, the Central Limit Theorem states that for large n (usually n ≥ 30), X̄ is approximately normal:
若总体服从正态分布，则 X̄ 精确服从 N(μ, σ²/n)。即使总体非正态，中心极限定理指出：当样本量 n 足够大（通常 n ≥ 30）时，X̄ 近似服从正态分布：
X̄ ~ N(μ, σ²/n) approximately
9. Confidence Interval for a Population Mean (σ Known) | 总体均值的置信区间（σ 已知）
When the population standard deviation σ is known, a C% confidence interval for the population mean μ is constructed from a sample mean x̄ as:
当总体标准差 σ 已知时，由样本均值 x̄ 构建的总体均值 μ 的 C% 置信区间为：
CI = x̄ ± z* × (σ / √n)
where z* is the critical value from N(0,1) corresponding to the confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%. The interval gives the range of plausible values for μ.
其中 z* 是对应置信水平的 N(0,1) 临界值：90% 用 1.645，95% 用 1.96，99% 用 2.576。该区间给出了 μ 的合理取值范围。
10. Hypothesis Testing: The p-value Approach | 假设检验：p 值法
A hypothesis test on a population mean (σ known) begins with null and alternative hypotheses:
对总体均值（σ 已知）的假设检验从原假设和备择假设开始：
H₀: μ = μ₀ , H₁: μ ≠ μ₀ (or μ > μ₀, μ < μ₀)
The test statistic is calculated from the sample:
基于样本计算检验统计量：
Z = (x̄ − μ₀) / (σ / √n)
statistic at least as extreme as the observed one. For a two-tailed test, p = 2 × P(Z > |z|). If p < α (the significance level, often 0.05), we reject H₀.
p 值是在 H₀ 成立时，得到比观测值更极端的检验统计量的概率。双侧检验 p = 2 × P(Z > |z|)。若 p < α（显著性水平，常取 0.05），则拒绝 H₀。
11. Hypothesis Testing: The Critical Value Method | 假设检验：临界值法
Instead of computing a p-value, we can compare the test statistic Z directly with the critical value(s) that mark the rejection region.
除了计算 p 值，也可以直接将检验统计量 Z 与划分拒绝域的临界值进行比较。
two-tailed test at significance level α, the critical values are ±z(α/2), e.g. ±1.96 for 5% > +z(α/2). For a one-tailed test, the rejection region depends on the direction of H₁.
对于显著性水平为 α 的双侧检验，临界值为 ±z(α/2)，例如 5% 时为 ±1.96。若 Z 落入拒绝域（Z < −z(α/2) 或 Z > +z(α/2)），则拒绝 H₀。单侧检验的拒绝域取决于备择假设的方向。
The critical value method is mathematically equivalent to the p-value method – both lead to the same conclusion.
临界值法与 p 值法在数学上等价，会得出相同的结论。
12. Linear Regression and the Correlation Coefficient | 线性回归与相关系数
For a set of paired data (x, y), the least-squares regression line of y on x is:
对成对数据 (x, y)，y 对 x 的最小二乘回归直线方程为：
y = a + bx
where the slope b and intercept a are given by:
其斜率 b 与截距 a 由下式确定：
b = Sxy / Sxx, a = ȳ − b x̄
Sxx = Σ(x − x̄)², Sxy = Σ(x − x̄)(y − ȳ)
The product-moment correlation coefficient (Pearson’s r) measures the strength of linear association:
积矩相关系数（皮尔逊 r）衡量线性关联的强度：
r = Sxy / √Sxx × Syy, −1 ≤ r ≤ 1
r ≈ 0 suggests no linear relationship. Remember that correlation does not imply causation.
r 接近 +1 或 −1 表示强线性相关；r ≈ 0 表明无线性关系。务必牢记相关不代表因果。
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