Quick Reference Handbook: CCEA Year 12 Statistics Formulae & Theorems | 公式定理速查手册：CCEA 12年级统计

📚 Quick Reference Handbook: CCEA Year 12 Statistics Formulae & Theorems | 公式定理速查手册：CCEA 12年级统计

This quick reference guide condenses every essential formula and theorem you need for the CCEA Year 12 Statistics course. Use it as a last-minute revision tool to remind yourself of key definitions, notation, and procedures. Keep it close while working through past papers, and you will have a one-stop summary of descriptive statistics, probability rules, discrete and continuous distributions, sampling theory, confidence intervals, and hypothesis testing.

本速查手册凝练了CCEA 12年级统计课程所需的全部核心公式与定理。可作为考前快速回顾的利器，帮助你重温关键定义、符号与计算步骤。做题时放在手边，你便能一站式查阅描述性统计、概率法则、离散与连续分布、抽样理论、置信区间与假设检验的所有要点。

1. Descriptive Statistics & Notation | 描述性统计与符号

Measures of central tendency and spread form the basis of data analysis. For a sample of n observations x₁, x₂, …, xₙ, the fundamental quantities are:

数据的集中趋势与离散程度是数据分析的基础。对于包含n个观测值 x₁, x₂, …, xₙ 的样本，基本统计量如下：

The sample mean x̄ = (∑x)/n, the population mean μ = (∑x)/N. The unbiased sample variance s² = ∑(x – x̄)²/(n-1) and the population variance σ² = ∑(x – μ)²/N. Standard deviation is the square root of variance: s = √s², σ = √σ².

样本均值 x̄ = (∑x)/n，总体均值 μ = (∑x)/N。无偏样本方差 s² = ∑(x – x̄)²/(n-1)，总体方差 σ² = ∑(x – μ)²/N。标准差为方差的平方根：s = √s²，σ = √σ²。

The median is the middle value when data are ordered; the lower quartile Q₁ and upper quartile Q₃ divide the ordered list into quarters. The interquartile range IQR = Q₃ – Q₁. The range = max – min. Percentiles: the kth percentile is the value below which k% of observations fall.

中位数是排序后位于中间的值；下四分位数 Q₁ 和上四分位数 Q₃ 将有序数据分为四等份。四分位距 IQR = Q₃ – Q₁。全距 = 最大值 – 最小值。百分位数：第k百分位数是指有k%的观测值落在其下方的数值。

2. Probability Rules & Bayes’ Theorem | 概率法则与贝叶斯定理

For any event A, 0 ≤ P(A) ≤ 1. The complement rule: P(A’) = 1 – P(A). Addition rule for any two events: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0 so P(A ∪ B) = P(A) + P(B).

对于任意事件A，0 ≤ P(A) ≤ 1。互补法则：P(A’) = 1 – P(A)。任意两事件的加法法则：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若事件互斥，P(A ∩ B) = 0，则P(A ∪ B) = P(A) + P(B)。

Conditional probability: P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0. The multiplication rule: P(A ∩ B) = P(A|B)P(B) = P(B|A)P(A). Independence: A and B are independent if P(A ∩ B) = P(A) P(B), which is equivalent to P(A|B) = P(A) (and P(B|A) = P(B)) when P(B)>0.

Bayes’ theorem updates prior beliefs with new evidence: P(A|B) = [P(B|A)P(A)] / P(B). The denominator can be expanded using the law of total probability: P(B) = P(B|A)P(A) + P(B|A’)P(A’) for a partition {A, A’}.

贝叶斯定理用新证据更新先验信念：P(A|B) = [P(B|A)P(A)] / P(B)。分母可利用全概率公式展开：对分割{A, A’}，P(B) = P(B|A)P(A) + P(B|A’)P(A’)。

3. Combinatorics & Factorials | 排列组合与阶乘

The factorial of a non-negative integer n is n! = n × (n-1) × … × 1, with 0! = 1. The number of ways to arrange n distinct objects in order is n!.

非负整数n的阶乘为 n! = n × (n-1) × … × 1，且定义 0! = 1。将n个不同物体按顺序排列的方法数为 n!。

Permutations of r objects chosen from n distinct objects (order matters): ⁿPᵣ = n!/(n-r)!. Combinations (order does not matter): the binomial coefficient ⁿC_r = ⁿCᵣ = (ⁿᵣ) = n! / [r!(n-r)!].

从n个不同物体中选取r个的排列数（顺序重要）：ⁿPᵣ = n!/(n-r)!。组合数（顺序不重要）：二项式系数 ⁿC_r = ⁿCᵣ = (ⁿᵣ) = n! / [r!(n-r)!]。

Useful identities: ⁿC₀ = ⁿC_n = 1, ⁿC_r = ⁿC_n-r, and ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r (Pascal’s rule). The total number of subsets of a set of size n is 2ⁿ.

常用恒等式：ⁿC₀ = ⁿC_n = 1, ⁿC_r = ⁿC_n-r，以及 ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r（帕斯卡法则）。大小为n的集合的所有子集总数为 2ⁿ。

4. Discrete Random Variables & Expectation | 离散随机变量与期望

A discrete random variable X takes values x with probabilities P(X = x) that sum to 1. The probability mass function (p.m.f.) describes the distribution.

离散随机变量X的取值x具有概率P(X = x)，且所有概率之和为1。概率质量函数（p.m.f.）描述了该分布。

The expected value (mean) of X is E(X) = μ = ∑ x P(X = x). The expectation of a function g(X) is E[g(X)] = ∑ g(x) P(X = x).

期望值（均值）为 E(X) = μ = ∑ x P(X = x)。函数 g(X) 的期望为 E[g(X)] = ∑ g(x) P(X = x)。

The variance of X is Var(X) = E[(X – μ)²] = ∑ (x – μ)² P(X = x). A simpler computational formula is Var(X) = E(X²) – [E(X)]². The standard deviation is SD(X) = √Var(X).

方差为 Var(X) = E[(X – μ)²] = ∑ (x – μ)² P(X = x)。简化计算公式为 Var(X) = E(X²) – [E(X)]²。标准差为 SD(X) = √Var(X)。

For constants a and b, E(aX + b) = a E(X) + b, Var(aX + b) = a² Var(X). The sum of probabilities over all possible x equals 1: ∑ P(X = x) = 1.

对于常数a与b，有E(aX + b) = a E(X) + b，Var(aX + b) = a² Var(X)。所有可能x的概率之和等于1：∑ P(X = x) = 1。

5. Binomial Distribution | 二项分布

If X counts the number of successes in n independent Bernoulli trials, each with success probability p, then X ~ B(n, p). The probability mass function is P(X = r) = ⁿC_r p^r (1-p)^n-r, for r = 0, 1, …, n.

若X记录n次独立伯努利试验中成功的次数，每次成功概率为p，则 X ~ B(n, p)。概率质量函数为 P(X = r) = ⁿC_r p^r (1-p)^n-r，其中 r = 0, 1, …, n。

Mean: E(X) = np. Variance: Var(X) = np(1-p). The mode occurs around floor[(n+1)p]; if (n+1)p is integer, both (n+1)p and (n+1)p – 1 are modes.

均值：E(X) = np。方差：Var(X) = np(1-p)。众数大约在 floor[(n+1)p]；若(n+1)p为整数，则(n+1)p与(n+1)p – 1均为众数。

The binomial is symmetric when p = 0.5, and skewed otherwise. For large n, B(n, p) can be approximated by a normal distribution N(np, np(1-p)) provided np > 5 and n(1-p) > 5. When p is small and n large, it can be approximated by a Poisson distribution Po(np).

当 p = 0.5 时二项分布对称，否则呈偏态。对于大n，若 np > 5 且 n(1-p) > 5，可用正态分布 N(np, np(1-p)) 近似。当p很小而n很大时，可用泊松分布 Po(np) 近似。

Cumulative probabilities P(X ≤ r) are often found using tables or a calculator. The complement rule P(X ≥ r) = 1 – P(X ≤ r-1) is handy.

累积概率 P(X ≤ r) 通常通过查表或计算器获得。利用互补规则 P(X ≥ r) = 1 – P(X ≤ r-1) 非常方便。

6. Poisson Distribution | 泊松分布

The Poisson distribution models the number of events occurring in a fixed interval of time or space, assuming events happen independently at a constant average rate λ. Then X ~ Po(λ) with probability mass function P(X = r) = (e^-λ λ^r) / r!, for r = 0, 1, 2, ….

泊松分布描述在固定时间或空间间隔内发生的事件数，假设事件独立且以恒定平均率 λ 发生。则 X ~ Po(λ)，概率质量函数为 P(X = r) = (e^-λ λ^r) / r!，r = 0, 1, 2, …。

Mean: E(X) = λ. Variance: Var(X) = λ. The parameter λ is both the mean and the variance, making the distribution uniquely defined by its mean.

均值：E(X) = λ。方差：Var(X) = λ。参数λ既是均值也是方差，因此该分布由其均值唯一确定。

When λ is large (say λ > 10), the Poisson can be approximated by a normal distribution N(λ, λ). The sum of independent Poisson variables is also Poisson: if X ~ Po(λ₁) and Y ~ Po(λ₂) are independent, then X + Y ~ Po(λ₁ + λ₂).

当 λ 较大（如 λ > 10）时，泊松分布可用正态分布 N(λ, λ) 近似。独立泊松变量之和仍为泊松分布：若 X ~ Po(λ₁) 与 Y ~ Po(λ₂) 独立，则 X + Y ~ Po(λ₁ + λ₂)。

7. Normal Distribution & Z-Scores | 正态分布与Z分数

The normal distribution with mean μ and variance σ², denoted N(μ, σ²), is the most important continuous distribution. Its probability density function is bell-shaped and symmetric about μ.

均值为μ、方差为σ²的正态分布，记作 N(μ, σ²)，是最重要的连续分布。其概率密度函数呈钟形，关于μ对称。

To calculate probabilities, we standardise: the Z-score is Z = (X – μ) / σ. Z follows the standard normal distribution N(0, 1). The cumulative distribution function Φ(z) = P(Z ≤ z) is tabled. Probabilities for any normally distributed X are found by converting to Z: P(X ≤ x) = Φ((x – μ)/σ).

计算概率时进行标准化：Z分数为 Z = (X – μ) / σ。Z服从标准正态分布 N(0, 1)。累积分布函数 Φ(z) = P(Z ≤ z) 可查表。任何正态变量X的概率通过转换为Z求得：P(X ≤ x) = Φ((x – μ)/σ)。

Key percentiles for N(0,1): 68% of data lie within μ ± 1σ, 95% within μ ± 1.96σ, and 99.7% within μ ± 2.576σ (or roughly μ ± 3σ).

N(0,1) 的关键百分位：68%的数据落在 μ ± 1σ 内，95% 落在 μ ± 1.96σ 内，99.7% 落在 μ ± 2.576σ（或大致 μ ± 3σ）内。

If X ~ N(μ, σ²), then a linear transformation aX + b ~ N(aμ + b, a²σ²). The sum of independent normal variables is also normal: if Xᵢ ~ N(μᵢ, σᵢ²), then ΣXᵢ ~ N(Σμᵢ, Σσᵢ²).

若 X ~ N(μ, σ²)，则线性变换 aX + b ~ N(aμ + b, a²σ²)。独立正态变量之和亦为正态：若 Xᵢ ~ N(μᵢ, σᵢ²)，则 ΣXᵢ ~ N(Σμᵢ, Σσᵢ²)。

8. Sampling & Central Limit Theorem | 抽样与中心极限定理

A statistic is a function of sample data. The sampling distribution of a statistic describes its behavior over all possible samples of size n. The sample mean X̄ is a random variable with mean μ and variance σ²/n when sampling from an infinite population (or finite with replacement).

统计量是样本数据的函数。统计量的抽样分布描述了在所有容量为n的可能样本中的行为。样本均值 X̄ 是一个随机变量，当从无限总体（或从有限总体有放回）抽样时，其均值为μ，方差为σ²/n。

The standard error of the mean is SE(X̄) = σ/√n. When sampling from a normal population, X̄ is exactly N(μ, σ²/n), regardless of sample size.

均值的标准误为 SE(X̄) = σ/√n。当从正态总体抽样时，无论样本量大小，X̄ 精确服从 N(μ, σ²/n)。

The Central Limit Theorem (CLT) states that for a random sample from any population with mean μ and finite variance σ², the distribution of the sample mean X̄ approaches N(μ, σ²/n) as n becomes large (commonly n ≥ 30 is sufficient). This justifies normal-based inference even for non-normal data.

中心极限定理（CLT）指出，对于来自任何具有均值μ和有限方差σ²的总体的随机样本，随着n增大（通常 n ≥ 30 足够），样本均值 X̄ 的分布趋近于 N(μ, σ²/n)。这为即使数据非正态时使用正态推断提供了依据。

For sample totals T = ΣX, T is approximately N(nμ, nσ²) by the CLT. For proportions, the sample proportion p̂ = X/n has mean p and variance p(1-p)/n, and is approximately N(p, p(1-p)/n) when np>5 and n(1-p)>5.

对于样本总和 T = ΣX，由CLT近似服从 N(nμ, nσ²)。对于比例，样本比例 p̂ = X/n 的均值为 p，方差为 p(1-p)/n，当 np>5 且 n(1-p)>5 时近似服从 N(p, p(1-p)/n)。

9. Confidence Intervals | 置信区间

A 100(1-α)% confidence interval gives a range of plausible values for a population parameter. The critical value z* (or t*) determines the width.

100(1-α)% 置信区间给出总体参数的一个可信范围。关键值 z*（或 t*）决定了区间的宽度。

Confidence interval for a population mean μ (σ known):

x̄ ± z* × (σ/√n), where z* = Φ⁻¹(1 – α/2) for normal distribution.

总体均值 μ 的置信区间（σ已知）：

x̄ ± z* × (σ/√n)，其中 z* = Φ⁻¹(1 – α/2) 为正态分布分位点。

Common z* values: 1.645 (90% CI), 1.960 (95% CI), 2.576 (99% CI).

常用 z* 值：1.645（90%置信区间），1.960（95%置信区间），2.576（99%置信区间）。

When σ is unknown, use the t-distribution with n-1 degrees of freedom: x̄ ± t* × (s/√n). The t-distribution is wider for small n and converges to the normal as df increases.

当σ未知时，使用自由度为 n-1 的 t 分布：x̄ ± t* × (s/√n)。t 分布在小样本时比正态分布更宽，随着自由度增大收敛于正态分布。

Confidence interval for a population proportion p (large sample): p̂ ± z* × √[p̂(1-p̂)/n], where p̂ is the sample proportion. Ensure n is large enough to apply the normal approximation.

总体比例 p 的置信区间（大样本）：p̂ ± z* × √[p̂(1-p̂)/n]，其中 p̂ 为样本比例。需确保 n 足够大以使用正态近似。

Interpretation: if we repeated the sampling process many times, 100(1-α)% of the constructed intervals would capture the true parameter.

解释：若多次重复抽样过程，则构造出的区间中有 100(1-α)% 会包含真实参数。

10. Hypothesis Testing (One-Sample) | 假设检验（单样本）

A hypothesis test assesses the strength of evidence against a null hypothesis H₀ in favour of an alternative H₁. The significance level α is the probability of rejecting H₀ when it is true (Type I error, typically 0.05 or 0.01).

假设检验评估反对原假设 H₀ 而支持备择假设 H₁ 的证据强度。显著性水平 α 是当 H₀ 为真时拒绝它的概率（第一类错误），通常取 0.05 或 0.01。

Test for a mean with known σ: test statistic z = (x̄ – μ₀)/(σ/√n). Compare |z| with the critical z* or use the p-value approach. Reject H₀ if p-value < α.

已知总体标准差σ时的均值检验：检验统计量 z = (x̄ – μ₀)/(σ/√n)。将 |z| 与临界值 z* 比较，或使用 p 值法。若 p 值 < α，则拒绝 H₀。

Test for a mean with unknown σ: use t-test with statistic t = (x̄ – μ₀)/(s/√n) under H₀, with df = n-1. Refer to the t-distribution table.

未知σ时的均值检验：使用 t 检验，统计量为 t = (x̄ – μ₀)/(s/√n)，自由度为 n-1，查 t 分布表。

Test for a population proportion: z = (p̂ – p₀) / √[p₀(1-p₀)/n], using the normal approximation. Check conditions np₀ ≥ 5 and n(1-p₀) ≥ 5.

总体比例检验：z = (p̂ – p₀) / √[p₀(1-p₀)/n]，使用正态近似。需检查条件 np₀ ≥ 5 且 n(1-p₀) ≥ 5。

One-tailed test: H₁: μ > μ₀ or μ < μ₀, rejection region in one tail. Two-tailed test: H₁: μ ≠ μ₀, rejection regions in both tails, and p-value is twice the one-tailed probability. Always state a conclusion in context: "There is (insufficient / sufficient) evidence at the α level to suggest that ... "

单尾检验：H₁: μ > μ₀ 或 μ < μ₀，拒绝域在单侧。双尾检验：H₁: μ ≠ μ₀，拒绝域在两侧，p值为单尾概率的两倍。一定要结合背景陈述结论：“在 α 水平下，（不）存在充分证据表明……”。

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Quick Reference Handbook: CCEA Year 12 Statistics Formulae & Theorems | 公式定理速查手册：CCEA 12年级统计