📚 Pre-U Cambridge Statistics: Formula & Theorem Quick Reference | 公式定理速查手册
This quick reference guide covers essential formulas and theorems in the Cambridge Pre-U Statistics syllabus. Each section is designed for efficient revision with paired English-Chinese explanations, followed by centred key formulas where appropriate. Use it to reinforce your understanding before the exam.
本速查手册涵盖剑桥 Pre-U 统计课程中的核心公式与定理。每个主题均提供英中对照解释，重要公式以居中加粗方式突出显示，旨在帮助你在考前高效复习、巩固要点。
1. Descriptive Statistics | 描述统计
For a sample of size n, the sample mean x̄ is the sum of all observations divided by n. It measures the central location of the data.
对于容量为 n 的样本，样本均值 x̄ 等于所有观测值之和除以 n，它反映了数据的集中位置。
x̄ = ( Σ xi ) / n
The sample variance s² quantifies the spread of the data around the mean. We divide by n−1 (degrees of freedom) to obtain an unbiased estimator of the population variance.
样本方差 s² 衡量数据围绕均值的离散程度。除数为 n−1（自由度），从而得到总体方差的无偏估计量。
s² = Σ (xi − x̄)² / (n−1)
The corresponding population parameters are the population mean μ and the population variance σ² (dividing by N). The standard deviation is the positive square root of the variance.
对应的总体参数为总体均值 μ 和总体方差 σ²（除数为 N）。标准差是方差的正平方根。
σ² = Σ (xi − μ)² / N, s = √s²
most frequent value. The interquartile range IQR = Q₃ − Q₁ is a robust measure of spread.
中位数是排序后处于中间位置的数值；众数是出现频率最高的值。四分位距 IQR = Q₃ − Q₁ 是一种稳健的离散度量。
2. Probability Fundamentals | 概率基础
For any two events A and B, the addition rule gives the probability of their union: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0.
对于任意两事件 A 和 B，加法法则给出其并集的概率：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。互斥事件满足 P(A ∩ B) = 0。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
The conditional probability of A given B is defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. Two events are independent if and only if P(A ∩ B) = P(A)P(B) or equivalently P(A|B) = P(A).
在 B 发生的条件下 A 的条件概率定义为 P(A|B) = P(A ∩ B) / P(B)，要求 P(B) > 0。两事件相互独立当且仅当 P(A ∩ B) = P(A)P(B) 或等价地 P(A|B) = P(A)。
P(A|B) = P(A ∩ B) / P(B)
Bayes' theorem reverses the conditioning and is widely used in updating probabilities: P(A|B) = [ P(B|A)·P(A) ] / P(B). The denominator can be expanded using the law of total probability.
贝叶斯定理逆转了条件顺序，广泛用于概率更新：P(A|B) = [ P(B|A)·P(A) ] / P(B)。分母可利用全概率公式展开。
P(A|B) = [ P(B|A) P(A) ] / [ P(B|A)P(A) + P(B|A')P(A') ]
3. Discrete Random Variables | 离散随机变量
the expected value (mean) is E(X) = Σ x·p(x) and the variance is Var(X) = E(X²) − [E(X)]².
对于概率质量函数为 p(x) 的离散随机变量 X，其期望值（均值）为 E(X) = Σ x·p(x)，方差为 Var(X) = E(X²) − [E(X)]²。
E(X) = Σ x p(x), Var(X) = E(X²) − [E(X)]²
A binomial distribution models the number of successes in n independent trials with success probability p. Write X ~ B(n, p). The probability of exactly k successes is given by the binomial formula.
二项分布适用于 n 次独立试验中成功次数的建模，每次成功概率为 p。记作 X ~ B(n, p)。恰好 k 次成功的概率由二项式公式给出。
P(X = k) = nCk pk (1−p)n−k, E(X) = np, Var(X) = np(1−p)
The Poisson distribution is appropriate for counting rare events in a fixed interval. If X ~ Po(λ), then P(X = k) = (e−λ λk) / k!. The mean and variance are both equal to λ.
泊松分布适用于固定区间内稀有事件发生的次数。若 X ~ Po(λ)，则 P(X = k) = (e−λ λk) / k!，其均值和方差都等于 λ。
P(X = k) = e−λ λk / k!, E(X) = λ, Var(X) = λ
For both distributions, conditions must be checked: binomial – fixed n, independent trials, constant p; Poisson – events occur singly, randomly, and at a constant average rate.
两种分布都需要满足条件：二项分布——固定 n、独立试验、概率 p 恒定；泊松分布——事件独立发生、单个出现且平均发生率恒定。
4. Continuous Random Variables | 连续随机变量
lity density function (PDF) f(x). Probabilities are found by integration: P(a ≤ X ≤ b) = ∫ab f(x) dx. The cumulative distribution function (CDF) is F(x) = P(X ≤ x) = ∫−∞x f(t) dt.
连续随机变量 X 由其概率密度函数（PDF）f(x) 描述。概率通过积分求得：P(a ≤ X ≤ b) = ∫ab f(x) dx。累积分布函数（CDF）为 F(x) = P(X ≤ x) = ∫−∞x f(t) dt。
∫−∞∞ f(x) dx = 1, E(X) = ∫ x f(x) dx, Var(X) = ∫ x² f(x) dx − μ²
The normal distribution is central to statistics. X ~ N(μ, σ²) has PDF f(x) = (1/( σ√2π )) exp( −(x−μ)²/(2σ²) ). The total area under the curve equals 1.
正态分布在统计中占据核心地位。X ~ N(μ, σ²) 的 PDF 为 f(x) = (1/( σ√2π )) exp( −(x−μ)²/(2σ²) )，曲线下总面积为 1。
f(x) = 1/(σ√2π) e−(x−μ)²/(2σ²)
)/σ follows N(0,1). Probabilities are obtained from tables. A binomial distribution B(n, p) may be approximated by a normal distribution when np > 5 and n(1−p) > 5, using a continuity correction.
标准正态变量 Z = (X − μ)/σ 服从 N(0,1)，概率可查表得到。当 np > 5 和 n(1−p) > 5 时，二项分布可用正态分布近似，并需使用连续性校正。
Z = (X − μ) / σ, approx. X ~ B(n,p) ≈ N(np, np(1−p))
5. Correlation and Regression | 相关与回归
For bivariate data, the sample covariance measures the direction of linear association: sxy = Σ (xi − x̄)(yi − ȳ) / (n−1). The Pearson correlation coefficient r is a dimensionless measure of linear strength.
对于二元数据，样本协方差度量线性关联的方向：sxy = Σ (xi − x̄)(yi − ȳ) / (n−1)。皮尔逊相关系数 r 是无量纲的线性强度度量。
r = sxy / (sx sy) = Sxy / √Sxx Syy, −1 ≤ r ≤ 1
sum of squared residuals. The slope and intercept are calculated from summary statistics.
最小二乘回归线 y = a + bx 使得残差平方和最小。斜率和截距由以下汇总统计量计算得出。
b = Sxy / Sxx, a = ȳ − b x̄
The coefficient of determination r² indicates the proportion of variability in y that is explained by the regression on x. Residual plots help assess model assumptions.
决定系数 r² 表示 y 的变异性中可由 x 的回归解释的比例。残差图有助于检验模型假设。
6. Sampling Distributions & CLT | 抽样分布与中心极限定理
If X ~ N(μ, σ²) and samples of size n are drawn, the sample mean follows X̄ ~ N(μ, σ²/n). The standard deviation of the sample mean is the standard error, σ/√n.
若 X ~ N(μ, σ²) 且抽取容量为 n 的样本，则样本均值服从 X̄ ~ N(μ, σ²/n)。样本均值的标准差称为标准误，σ/√n。
X̄ ~ N(μ, σ²/n), SE = σ/√n
ntral Limit Theorem (CLT) states that for large n (commonly n ≥ 30), the distribution of X̄ ximately normal regardless of the population shape, provided observations are independent.
中心极限定理指出，对于大样本（通常 n ≥ 30），无论总体分布形状如何，只要观测值独立，样本均值的分布近似正态。
X̄ ≈ N(μ, σ²/n) for n ≥ 30 (CLT)
For a sample proportion p̂ sampling distribution is approximately normal with mean p and variance p(1−p)/n, provided np and n(1−p) are at least 10.
对于来自二项总体的样本比例 p̂ = X/n，当 np 和 n(1−p) 至少为 10 时，其抽样分布近似正态，均值为 p，方差为 p(1−p)/n。
p̂ ≈ N(p, p(1−p)/n)
7. Confidence Intervals | 置信区间
A 100(1−α)% confidence interval for the population mean (σ known) is x̄ ± zα/2 · (σ/√n). For σ unknown, we replace σ with s and use the t-distribution with n−1 degrees of freedom.
总体均值的 100(1−α)% 置信区间（σ 已知）为 x̄ ± zα/2 · (σ/√n)。σ 未知时，用 s 代替 σ 并使用自由度为 n−1 的 t 分布。
CI for μ (σ known): x̄ ± zα/2 · σ/√n
CI for μ (σ unknown): x̄ ± tn−1,α/2 · s/√n
A confidence interval for a population proportion uses p̂ and the standard error √(p̂(1−p̂)/n): p̂ ± zα/2 · √(p̂(1−p̂ comparisons, intervals for the difference of means or proportions follow analogous forms.
总体比例的置信区间使用 p̂ 和标准误 √(p̂(1−p̂)/n)：p̂ ± zα/2 · √(p̂(1−p̂)/n)。对于两样本比较，均值差或比例差的置信区间具有类似形式。
CI for p: p̂ ± zα/2 √(p̂(1−
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