📚 Pre-U CCEA Statistics: Formula & Theorem Quick Reference Guide | Pre-U CCEA 统计：公式定理速查手册
This quick reference guide brings together the essential formulas, distributions, theorems and inferential procedures required for the Pre-U CCEA Statistics syllabus. It is designed as a portable companion for revision, classroom work and examination preparation, with every topic presented in bilingual form so that you can reinforce both your statistical knowledge and your subject-specific English.
本速查手册汇集了 Pre-U CCEA 统计考纲所要求的核心公式、分布、定理与推断方法。它旨在成为你的便携复习伴侣，适用于备考、课堂学习和考前梳理。每个主题均以中英双语呈现，帮助你在巩固统计知识的同时，强化学科英语能力。
1. Summary Statistics and Measures of Central Tendency | 汇总统计量与集中趋势度量
Summary statistics condense a raw dataset into a few key numbers that describe its centre. The three most commonly used measures are the arithmetic mean, the median and the mode. For frequency distributions, weighted forms of the mean are used.
汇总统计量将原始数据提炼成少数几个描述中心位置的关键数字。最常用的三种度量是算术平均值、中位数和众数。对于频数分布，需要使用加权形式的均值公式。
The sample mean for a set of n observations x₁, x₂, …, xₙ is calculated by summing all values and dividing by the sample size.
对于包含 n 个观测值 x₁, x₂, …, xₙ 的样本，样本均值等于所有数值之和除以样本容量。
x̄ = (∑ xᵢ) / n
When data are organised in a frequency table with values xᵢ and corresponding frequencies fᵢ, the mean becomes the sum of frequency-weighted observations divided by total frequency.
当数据整理为频数表，带有数值 xᵢ 及对应频数 fᵢ 时，均值等于频数加权之和除以总频数。
x̄ = (∑ fᵢ xᵢ) / (∑ fᵢ)
The median is the middle value when the data are ordered from smallest to largest. If n is odd, it is the (n+1)/2 th observation; if n is even, it is the average of the n/2 th and (n/2 + 1)th observations. For grouped data, cumulative frequency graphs or linear interpolation are used.
中位数是按大小排序后居于中间位置的数值。若 n 为奇数，则为第 (n+1)/2 个观测值；若 n 为偶数，则为第 n/2 与第 (n/2 +1) 个观测值的平均。对于分组数据，需使用累积频数图或线性插值法。
The mode is the value that occurs with the highest frequency. In a grouped frequency distribution the modal class is the class with the highest frequency density.
众数是出现频数最高的数值。在分组频数分布中，众数所在的组是频数密度最高的组。
2. Measures of Dispersion and Skewness | 离散度与偏度度量
Measures of dispersion describe the spread or variability of a dataset. The most fundamental measure is the range, but the variance and standard deviation are more informative because they involve every observation.
离散度度量描述数据集的散布或变异程度。最基本的度量是极差，但方差和标准差因涉及每一个观测值而提供更丰富的信息。
The sample variance s² uses the sum of squared deviations from the sample mean, divided by (n – 1) as an unbiased estimator of the population variance.
样本方差 s² 使用各观测值与样本均值的离差平方和除以 (n–1)，作为总体方差的无偏估计量。
s² = Σ(xᵢ – x̄)² / (n – 1)
The sample standard deviation s is the positive square root of the variance and has the same units as the raw data.
样本标准差 s 是方差的正平方根，其单位与原始数据相同。
s = √[ Σ(xᵢ – x̄)² / (n – 1) ]
The interquartile range (IQR = Q₃ – Q₁) is a robust measure of spread that is unaffected by extreme values. It measures the range of the middle 50% of the data.
四分位距 (IQR = Q₃ – Q₁) 是一种不受极端值影响的稳健离散度量，它衡量中间 50% 数据的极差。
Skewness indicates the asymmetry of a distribution. A commonly used coefficient is Pearson’s first coefficient of skewness, which involves the mean, median and standard deviation. A positive value indicates a right‑skewed distribution; a negative value indicates a left‑skewed distribution.
偏度反映分布的不对称性。常用的系数是皮尔逊第一偏度系数，它涉及均值、中位数与标准差。正值表示右偏分布，负值表示左偏分布。
Pearson’s skewness = 3(x̄ – median) / s
3. Elementary Probability Rules | 基本概率规则
Probability rules form the foundation for inference. The addition rule combines probabilities of unions, while the multiplication rule handles intersections for independent and dependent events.
概率规则是统计推断的基础。加法规则用于并集的概率，乘法规则处理独立事件和相依事件的交集的概率。
For any two events A and B, the probability of A or B occurring is given by the general addition rule.
对于任意两个事件 A 与 B，A 或 B 发生的概率由一般加法规则给出。
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
If A and B are mutually exclusive (disjoint), their intersection is empty, so the addition rule simplifies.
若 A 与 B 互斥，则它们的交集为空，加法规则得以简化。
P(A ∪ B) = P(A) + P(B)
Conditional probability measures the chance of event A given that event B has occurred. It is computed as the ratio of the intersection probability to the probability of the conditioning event.
条件概率度量在事件 B 已发生的条件下事件 A 发生的可能性，它等于交集概率与条件事件概率的比值。
P(A | B) = P(A ∩ B) / P(B), P(B) > 0
Two events are independent if the occurrence of one does not influence the probability of the other. In that case the multiplication rule uses the product of individual probabilities.
如果两个事件的发生互不影响，则它们独立。此时乘法规则采用各自概率的乘积。
P(A ∩ B) = P(A) × P(B)
The law of total probability expresses the probability of an event A as a weighted sum of conditional probabilities across a partition B₁, B₂, …, Bₖ.
全概率公式将事件 A 的概率表示为在划分 B₁, B₂, …, Bₖ 下条件概率的加权和。
P(A) = Σ P(Bᵢ) P(A | Bᵢ)
4. Discrete Probability Distributions (Binomial & Poisson) | 离散概率分布（二项与泊松）
Discrete probability distributions assign probabilities to each possible integer outcome of a random variable. Two workhorse distributions in Pre-U Statistics are the binomial and the Poisson.
离散概率分布为随机变量的每一个可能整数结果分配概率。Pre-U 统计中两个核心分布是二项分布和泊松分布。
The binomial distribution models the number of successes in a fixed number n of independent Bernoulli trials, each with success probability p. The probability mass function uses the binomial coefficient C(n, k).
二项分布刻画在固定次数 n 的独立伯努利试验中成功的次数，每次成功概率为 p。概率质量函数使用二项式系数 C(n, k)。
X ~ B(n, p) ⇒ P(X = k) = C(n, k) pᵏ (1–p)ⁿ⁻ᵏ
The expected value and variance of a binomial random variable are straightforward products.
二项随机变量的期望值和方差是简单的乘积形式。
E(X) = np, Var(X) = np(1–p)
The Poisson distribution models rare events occurring randomly in a fixed interval of time or space, with λ representing the average rate of occurrence. Its probability mass function involves the exponential constant e.
泊松分布模拟在固定时间或空间区间内随机发生的稀有事件，λ 表示平均发生率。其概率质量函数含有常数 e。
X ~ Po(λ) ⇒ P(X = x) = (λˣ e⁻λ) / x!
For a Poisson variable, the mean and variance are both equal to λ. The Poisson distribution can also be used to approximate a binomial distribution when n is large and p is small (typically n > 50 and np < 5), by setting λ = np.
泊松变量的均值与方差都等于 λ。当 n 很大且 p 很小（通常 n > 50 且 np < 5）时，泊松分布可用来近似二项分布，此时令 λ = np。
5. Continuous Distributions: Normal & Related | 连续分布：正态及相关分布
The normal distribution plays a central role in statistics because of its frequent appearance in nature and its importance through the Central Limit Theorem. The notation N(μ, σ²) indicates a normal distribution with mean μ and variance σ².
正态分布在统计中居于核心地位，既因为它在自然界频繁出现，也因中心极限定理而至关重要。记号 N(μ, σ²) 表示均值为 μ、方差为 σ² 的正态分布。
Many statistical procedures rely on the standard normal distribution N(0, 1). Any normal variable X can be standardised by subtracting its mean and dividing by its standard deviation.
许多统计方法依赖标准正态分布 N(0, 1)。任意正态变量 X 可通过减去均值再除以标准差进行标准化。
Z = (X – μ) / σ ~ N(0, 1)
The empirical rule (68–95–99.7 rule) states that for a normal distribution approximately 68% of observations fall within μ ± σ, 95% within μ ± 2σ and 99.7% within μ ± 3σ.
经验法则（68–95–99.7 规则）指出，对于正态分布，约 68% 的观测值落在 μ ± σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。
When a binomial distribution has sufficiently large np and n(1–p), it can be approximated by a normal distribution with continuity correction. The usual criteria are np > 5 and n(1–p) > 5.
当二项分布满足 np 和 n(1–p) 均足够大时（通常要求 > 5），它可以用正态分布近似，并需进行连续性校正。
X ~ B(n, p) ≈ N( np, np(1–p) ), apply ±0.5 correction
The t-distribution is used for inference on a mean when the population standard deviation is unknown. It is symmetric but has heavier tails than the normal distribution. As degrees of freedom increase, it approaches the standard normal distribution.
当总体标准差未知时，对均值进行推断须使用 t 分布。它对称但尾部厚于正态分布，随着自由度增大，趋于标准正态分布。
6. Sampling Distributions and Central Limit Theorem | 抽样分布与中心极限定理
The sampling distribution of a statistic describes how the statistic varies from sample to sample. The distribution of the sample mean is particularly important.
统计量的抽样分布描述了该统计量在不同样本之间的变异方式。样本均值的抽样分布尤为重要。
If the underlying population has mean μ and variance σ², the sample mean X̄ has mean μ and variance σ²/n, regardless of the population shape. If the population is normally distributed, X̄ is exactly normal for any sample size.
若总体具有均值 μ 与方差 σ²，则样本均值 X̄ 的期望为 μ，方差为 σ²/n，这与总体形状无关。如果总体呈正态分布，无论样本容量多大，X̄ 都服从正态分布。
E(X̄) = μ, Var(X̄) = σ²/n
The Central Limit Theorem (CLT) states that for a sufficiently large sample size (usually n ≥ 30), the sampling distribution of the sample mean is approximately normal even when the population is not normal.
中心极限定理 (CLT) 指出，当样本容量足够大（通常 n ≥ 30）时，即使总体不呈正态分布，样本均值的抽样分布也近似于正态。
X̄ ~ N(μ, σ²/n) approximately, by CLT
For the sample proportion p̂, based on a binomial experiment, the sampling distribution can be approximated by a normal distribution when the success/failure condition is met. Its mean is p and its variance is p(1–p)/n.
对于基于二项试验的样本比例 p̂，当成功/失败条件满足时，其抽样分布可用正态分布近似。均值为 p，方差为 p(1–p)/n。
p̂ ~ N( p, p(1–p)/n ) approximately
7. Confidence Intervals | 置信区间
A confidence interval provides a range of plausible values for an unknown population parameter. The interval is constructed around a point estimate and its width depends on the confidence level and the standard error.
置信区间给出未知总体参数的一个合理取值范围。它围绕点估计构建，其宽度取决于置信水平和标准误。
For a population mean μ when the population standard deviation σ is known, a confidence interval uses the standard normal critical value z*.
当总体标准差 σ 已知时，总体均值 μ 的置信区间使用标准正态临界值 z*。
x̄ ± z* × (σ / √n)
When σ is unknown (the typical case), we substitute the sample standard deviation s and use the t critical value with n–1 degrees of freedom.
当 σ 未知（典型情况）时，用样本标准差 s 替代，并使用自由度为 n–1 的 t 临界值。
x̄ ± t* × (s / √n)
For a population proportion p, a large-sample confidence interval uses the sample proportion p̂ and the normal approximation.
对于总体比例 p，大样本置信区间使用样本比例 p̂ 和正态近似。
p̂ ± z* × √[ p̂(1–p̂) / n ]
For the difference between two independent population means (σ₁ and σ₂ unknown but assumed equal), a pooled two-sample t interval is constructed. The pooled standard deviation sₚ combines variability from both samples.
对于两个独立总体均值之差（σ₁ 与 σ₂ 未知但假设相等），使用合并双样本 t 区间。合并标准差 sₚ 结合了两个样本的变异性。
(x̄₁ – x̄₂) ± t* × sₚ √(1/n₁ + 1/n₂)
8. Hypothesis Testing: One-Sample Procedures | 假设检验：单样本方法
Hypothesis testing is a formal decision-making framework. The null hypothesis H₀ represents a default position, while the alternative hypothesis H₁ corresponds to the claim we seek evidence for.
假设检验是一个正式的决策框架。原假设 H₀ 代表默认立场，备择假设 H₁ 则对应我们试图寻找证据支持的主张。
For a test concerning a population mean with known σ, the test statistic follows a standard normal distribution under H₀.
对于已知
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