📚 Year 13 Edexcel Statistics: Formula and Theorem Quick Reference Handbook | Year 13 Edexcel 统计：公式定理速查手册
This quick reference handbook compiles the essential formulas, theorems, and key concepts for the Year 13 Edexcel Statistics course. It serves as a concise revision aid, covering probability distributions, sampling theory, hypothesis testing, and regression analysis. Every effort has been made to present the material in a clear, bilingual format to support both conceptual understanding and exam preparation.
本速查手册汇集了 Year 13 Edexcel 统计课程的核心公式、定理和重要概念，是一份精炼的复习工具，覆盖概率分布、抽样理论、假设检验和回归分析。内容以中英双语清晰呈现，助力同学在理解概念的同时高效备考。
1. Probability Foundations & Expectation | 概率基础与期望
For a discrete random variable X, the expectation is the probability‑weighted average of its possible values.
对于离散随机变量 X，期望是其所有可能取值的概率加权平均。
E(X) = Σ x·P(X = x)
The variance measures the spread of the distribution and can be calculated using the alternative formula.
方差衡量分布的离散程度，可用简便公式计算。
Var(X) = E(X2) − [E(X)]2
Two events A and B are independent if and only if P(A ∩ B) = P(A)·P(B). Independence is crucial when combining random variables.
当且仅当 P(A ∩ B) = P(A)·P(B) 时，事件 A 与 B 独立。独立性的概念在随机变量的组合中至关重要。
The expectation is linear, so for any constants a and b,
期望具有线性性，对任意常数 a 和 b，
E(aX + b) = aE(X) + b
and the variance transforms as
方差变换规律为
Var(aX + b) = a2Var(X).
For independent random variables X and Y, Var(X ± Y) = Var(X) + Var(Y), noting that the variance of a difference is still the sum of variances.
若随机变量 X 与 Y 独立，Var(X ± Y) = Var(X) + Var(Y)，注意差的方差仍然是方差相加。
2. Binomial Distribution | 二项分布
A binomial distribution models the number of successes in a fixed number n of independent Bernoulli trials, each with the same success probability p.
二项分布描述在 n 次独立伯努利试验中成功的次数，每次试验成功的概率均为 p。
X ~ B(n, p)
The probability of obtaining exactly k successes is
恰好获得 k 次成功的概率为
P(X = k) = nCk pk (1 − p)n−k
where nCk = n!/(k!(n−k)!). The mean and variance are simple to compute:
其中 nCk = n!/(k!(n−k)!)。其均值与方差便于计算：
E(X) = np Var(X) = np(1 − p)
The binomial distribution is symmetric when p = 0.5 and becomes skewed for extreme p. It forms the foundation for many hypothesis tests involving proportions.
当 p = 0.5 时二项分布是对称的，p 趋近极端值时则呈现偏态。它为许多关于比例的假设检验奠定基础。
3. Poisson Distribution | 泊松分布
The Poisson distribution models the number of events occurring in a fixed interval of time or space when events happen independently at a constant average rate λ.
泊松分布用于模拟在固定时间或空间区间内事件发生的次数，事件独立且以恒定平均速率 λ 发生。
X ~ Po(λ)
The probability mass function is
概率质量函数为
P(X = k) = (e−λ λk) / k!
The mean and variance are both equal to λ, a distinctive feature of the Poisson distribution.
均值和方差均等于 λ，这是泊松分布的一个显著特征。
E(X) = λ Var(X) = λ
When n is large and p is small (typically n > 20 and p < 0.1), the binomial distribution B(n, p) can be approximated by Poisson(λ = np).
当 n 较大且 p 较小（一般 n > 20 且 p < 0.1）时，二项分布 B(n, p) 可用泊松分布 Poisson(λ = np) 近似。
Similarly, for large λ (typically λ > 15), the Poisson distribution can be approximated by a normal distribution N(λ, λ).
类似地，当 λ 较大（一般 λ > 15）时，泊松分布可用正态分布 N(λ, λ) 近似。
4. Normal Distribution | 正态分布
The normal distribution is the most important continuous distribution in statistics, characterized by its bell‑shaped curve.
正态分布是统计学中最重要的连续分布，以其钟形曲线为特征。
X ~ N(μ, σ2)
The probability density function is not required in the formula booklet, but standardisation is essential:
概率密度函数不需要记忆，但标准化是关键：
Z = (X − μ) / σ ~ N(0, 1)
Using standard normal tables, we can find probabilities such as
使用标准正态分布表可查得以下常用概率
- P(−1.96 < Z < 1.96) ≈ 0.95
- P(−2.576 < Z < 2.576) ≈ 0.99
- P(Z > 1.6449) ≈ 0.05
For any normal variable, 68% of values lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ (the empirical rule).
对于任何正态变量，约有 68% 的值落在 μ ± σ 范围内，95% 落在 μ ± 2σ，99.7% 落在 μ ± 3σ（经验法则）。
The sum or linear combination of independent normal variables is also normally distributed. This property underlies the central limit theorem.
独立正态随机变量的和或线性组合仍服从正态分布，这一性质是中心极限定理的基础。
5. Continuous Uniform Distribution | 连续均匀分布
A continuous uniform distribution describes a variable that is equally likely to take any value within a specified interval [a, b].
连续均匀分布描述在特定区间 [a, b] 内取值可能性均等的变量。
X ~ U(a, b)
The probability density function is constant over the interval:
概率密度函数在区间内为常数：
f(x) = 1 / (b − a) for a ≤ x ≤ b
The cumulative distribution function is
累积分布函数为
F(x) = (x − a) / (b − a)
The mean and variance are
均值与方差为
E(X) = (a + b) / 2 Var(X) = (b − a)2 / 12
This distribution is often used as a simple model for random number generation and as a testbed for theoretical results.
该分布常被用作随机数生成的简单模型，也用于验证理论结果。
6. Sampling Distributions & Central Limit Theorem | 抽样分布与中心极限定理
When we take a random sample of size n from a population with mean μ and variance σ2, the sample mean X̄ is a random variable with
从均值为 μ、方差为 σ2 的总体中抽取大小为 n 的随机样本，样本均值 X̄ 是一个随机变量，满足
E(X̄) = μ Var(X̄) = σ2 / n
If the population is normally distributed, then X̄ is exactly normal:
若总体正态，则样本均值精确服从正态分布：
X̄ ~ N(μ, σ2 / n)
The Central Limit Theorem (CLT) states that, even when the population is not normal, the distribution of X̄ becomes approximately normal as n increases (typically n ≥ 30 is sufficient).
中心极限定理 (CLT) 指出，即使总体不服从正态分布，当样本量 n 足够大时（通常 n ≥ 30），样本均值的分布也近似正态。
For a sample proportion p̂ = X / n from a binomial setting, the sampling distribution satisfies
对于来自二项分布总体的样本比例 p̂ = X / n，其抽样分布满足
E(p̂) = p Var(p̂) = p(1 − p) / n
and by the CLT, p̂ is approximately normal for large n.
根据中心极限定理，当样本量较大时 p̂ 近似服从正态分布。
7. Confidence Intervals | 置信区间
A confidence interval provides a range of plausible values for a population parameter, based on a sample statistic.
置信区间基于样本统计量给出总体参数的一个可能取值范围。
When the population variance is known (or for large samples), a 100(1 − α)% confidence interval for the mean μ is
当总体方差已知（或大样本）时，均值 μ 的 100(1 − α)% 置信区间为
x̄ ± zα/2 × σ / √n
If σ is unknown and the sample size is small, we use the t‑distribution with n − 1 degrees of freedom:
若 σ 未知且样本较小，则采用自由度为 n − 1 的 t 分布：
x̄ ± tn−1, α/2 × s / √n
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