📚 Year 13 CIE Statistics: Quick Reference Handbook of Formulas and Theorems | Year 13 CIE 统计：公式定理速查手册
This handbook consolidates the essential formulas and theorems required for the Year 13 CIE Statistics syllabus. It covers continuous distributions, sampling theory, hypothesis tests, regression and chi-squared tests, presented in a clear dual-language format to support both revision and quick reference during exam preparation.
本手册汇总了 Year 13 CIE 统计学所需的必备公式与定理，涵盖连续分布、抽样理论、假设检验、回归分析和卡方检验等内容，以清晰的双语形式呈现，方便复习与考前速查。
1. Continuous Probability Distributions | 连续型概率分布
A continuous random variable X has a probability density function (PDF) f(x) defined for a range of x, such that the total area under the curve is 1: ∫ f(x) dx = 1 over its domain. The probability P(a ≤ X ≤ b) is given by ∫ab f(x) dx.
连续型随机变量 X 具有概率密度函数 (PDF) f(x)，定义域内曲线下总面积为 1：∫ f(x) dx = 1。概率 P(a ≤ X ≤ b) 由 ∫ab f(x) dx 给出。
The cumulative distribution function (CDF) is F(x) = P(X ≤ x) = ∫-∞x f(t) dt. For a valid PDF: f(x) ≥ 0 for all x, and ∫-∞∞ f(x) dx = 1.
累积分布函数 (CDF) 为 F(x) = P(X ≤ x) = ∫-∞x f(t) dt。有效的 PDF 必须满足：对所有 x，f(x) ≥ 0，以及 ∫-∞∞ f(x) dx = 1。
Key distributions in Year 13:
Year 13 中的关键分布：
- Uniform distribution: X ~ U[a, b], f(x) = 1/(b-a) for a ≤ x ≤ b; E(X) = (a+b)/2; Var(X) = (b-a)²/12.
- 均匀分布: X ~ U[a, b], f(x) = 1/(b-a)（a ≤ x ≤ b）；E(X) = (a+b)/2; Var(X) = (b-a)²/12.
- Exponential distribution: X ~ Exp(λ), f(x) = λe⁻λx for x ≥ 0; E(X) = 1/λ; Var(X) = 1/λ². The CDF is F(x) = 1 – e⁻λx.
- 指数分布: X ~ Exp(λ), f(x) = λe⁻λx（x ≥ 0）；E(X) = 1/λ; Var(X) = 1/λ²。CDF 为 F(x) = 1 – e⁻λx。
- Normal distribution: X ~ N(μ, σ²), f(x) = (1/(σ√(2π))) e^{-(x-μ)²/(2σ²)}. Standardised variable Z = (X-μ)/σ ~ N(0,1).
- 正态分布: X ~ N(μ, σ²), f(x) = (1/(σ√(2π))) e^{-(x-μ)²/(2σ²)}。标准化变量 Z = (X-μ)/σ ~ N(0,1)。
Remember: For a continuous distribution, P(X = c) = 0 for any single value c.
注意：对于连续分布，任何单点值的概率为 0，即 P(X = c) = 0。
2. Expectation and Variance of Continuous Variables | 连续变量的期望与方差
For a continuous random variable X with PDF f(x), the expectation (mean) is μ = E(X) = ∫ x f(x) dx over the domain. The variance is Var(X) = E[(X – μ)²] = E(X²) – [E(X)]², where E(X²) = ∫ x² f(x) dx.
对于具有 PDF f(x) 的连续随机变量 X，期望（均值）为 μ = E(X) = ∫ x f(x) dx 在其定义域上积分。方差为 Var(X) = E[(X – μ)²] = E(X²) – [E(X)]²，其中 E(X²) = ∫ x² f(x) dx。
For a function g(X), E[g(X)] = ∫ g(x) f(x) dx. Useful transformations include E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X).
对于函数 g(X)，E[g(X)] = ∫ g(x) f(x) dx。常用变换如 E(aX + b) = aE(X) + b，Var(aX + b) = a² Var(X)。
When working with the normal distribution, remember that the median, mode and mean all coincide at μ. The points of inflection are at μ ± σ.
在处理正态分布时，记住中位数、众数和均值都位于 μ 处，拐点位于 μ ± σ。
3. Sampling and the Central Limit Theorem | 抽样与中心极限定理
If X₁, X₂, …, Xₙ form a random sample of size n from any distribution with mean μ and variance σ², the sample mean X̄ has E(X̄) = μ and Var(X̄) = σ²/n. The standard error of the sample mean is σ/√n.
若 X₁, X₂, …, Xₙ 是来自任意分布（均值 μ、方差 σ²）的大小为 n 的随机样本，则样本均值 X̄ 满足 E(X̄) = μ，Var(X̄) = σ²/n。样本均值的标准误为 σ/√n。
The Central Limit Theorem (CLT) states that for sufficiently large n (typically n ≥ 30), the distribution of X̄ is approximately normal: X̄ ~ N(μ, σ²/n), regardless of the shape of the original population distribution. This allows the use of normal-based inference even when the data are not normally distributed.
中心极限定理 (CLT) 指出，当 n 足够大（通常 n ≥ 30）时，无论原始总体分布形状如何，X̄ 的分布近似正态：X̄ ~ N(μ, σ²/n)。这使得即使数据非正态，也可使用基于正态的推断方法。
When the population variance σ² is unknown and estimated by the sample variance s², and the sample size is small (n < 30) from a normal population, we use the t-distribution with (n-1) degrees of freedom for inference about the mean.
当总体方差 σ² 未知且用样本方差 s² 估计，且样本容量较小（n < 30）并来自正态总体时，我们使用自由度为 (n-1) 的 t 分布对均值进行推断。
4. Point Estimation and Confidence Intervals | 点估计与置信区间
A point estimate is a single value computed from a sample to estimate a population parameter. Common unbiased estimators are: sample mean X̄ for μ, sample proportion p̂ for p, sample variance s² = Σ(xᵢ – X̄)²/(n-1) for σ².
点估计是根据样本计算出的单一值，用于估计总体参数。常见的无偏估计量有：样本均值 X̄ 估计 μ，样本比例 p̂ 估计 p，样本方差 s² = Σ(xᵢ – X̄)²/(n-1) 估计 σ²。
A confidence interval (CI) gives an estimated range of values which is likely to include the parameter. For a normal mean with known variance σ²:
X̄ ± zα/2 × (σ/√n)
置信区间 (CI) 给出一个可能包含参数的值范围。对于已知方差 σ² 的正态均值：
X̄ ± zα/2 × (σ/√n)
If σ² is unknown and replaced by s², and the data are normal, use the t-interval:
X̄ ± tn-1, α/2 × (s/√n)
若 σ² 未知并用 s² 代替，且数据正态，则用 t-区间：
X̄ ± tn-1, α/2 × (s/√n)
For a population proportion p, an approximate CI is:
p̂ ± zα/2 × √[p̂(1-p̂)/n]
对于总体比例 p，近似置信区间为：
p̂ ± zα/2 × √[p̂(1-p̂)/n]
5. Hypothesis Testing for a Normal Mean | 正态均值的假设检验
A hypothesis test evaluates two competing statements about a population parameter. The null hypothesis H₀ typically represents the status quo, while the alternative H₁ is what we suspect. The test statistic and rejection region are chosen to control the significance level α, the probability of a Type I error.
假设检验评估关于总体参数的两个对立陈述。原假设 H₀ 通常代表现状，备择假设 H₁ 是我们怀疑的情况。检验统计量和拒绝域的选择要控制显著性水平 α，即第一类错误的概率。
For a normal mean with known variance, the test statistic under H₀: μ = μ₀ is:
Z = (X̄ – μ₀) / (σ/√n) ~ N(0,1)
对于已知方差的正态均值，H₀: μ = μ₀ 下的检验统计量为：
Z = (X̄ – μ₀) / (σ/√n) ~ N(0,1)
When σ is unknown, use the t-test:
t = (X̄ – μ₀) / (s/√n) ~ tn-1
当 σ 未知时，使用 t 检验：
t = (X̄ – μ₀) / (s/√n) ~ tn-1
Decision rules: For a two-tailed test at level α, reject H₀ if |test statistic| > critical value. For one-tailed tests, compare the sign and magnitude accordingly. The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one observed, given H₀ is true.
决策规则：对于双侧检验，显著性水平 α，若 |检验统计量| > 临界值，则拒绝 H₀。对于单侧检验，则根据方向和大小比较。p 值是在 H₀ 为真时，观察到当前及更极端检验统计量的概率。
6. Hypothesis Tests for Difference of Means | 两均值差的假设检验
When comparing two independent normal populations with means μ₁ and μ₂, we often test H₀: μ₁ – μ₂ = d₀ (commonly d₀ = 0). If both population variances are known, the test statistic is:
Z = [(X̄₁ – X̄₂) – d₀] / √(σ₁²/n₁ + σ₂²/n₂) ~ N(0,1)
比较两个独立正态总体的均值 μ₁ 和 μ₂ 时，通常检验 H₀: μ₁ – μ₂ = d₀（常用 d₀ = 0）。若两个总体方差已知，检验统计量为：
Z = [(X̄₁ – X̄₂) – d₀] / √(σ₁²/n₁ + σ₂²/n₂) ~ N(0,1)
If variances are unknown but assumed equal, we pool the sample variances to estimate the common variance:
sp² = [(n₁-1)s₁² + (n₂-1)s₂²] / (n₁ + n₂ – 2)
若方差未知但假设相等，则合并样本方差来估计共同方差：
sp² = [(n₁-1)s₁² + (n₂-1)s₂²] / (n₁ + n₂ – 2)
Then the t-test statistic is:
t = [(X̄₁ – X̄₂) – d₀] / [sp√(1/n₁ + 1/n₂)] ~ tn₁+n₂-2
然后 t 检验统计量为：
t = [(X̄₁ – X̄₂) – d₀] / [sp√(1/n₁ + 1/n₂)] ~ tn₁+n₂-2
When variances are unknown and unequal, use Welch’s t-test with adjusted degrees of freedom (not explicitly derived in CIE but may be referenced).
当方差未知且不等时，使用 Welch t 检验并调整自由度（CIE 不要求推导，但可能提及）。
7. Chi-Squared Tests | 卡方检验
The chi-squared (χ²) distribution is used for goodness-of-fit tests and tests of independence in contingency tables. The test statistic is:
χ² = Σ [(O – E)² / E]
卡方 (χ²) 分布用于拟合优度检验和列联表独立性检验。检验统计量为：
χ² = Σ [(O – E)² / E]
where O are observed frequencies and E are expected frequencies under the null hypothesis. For a goodness-of-fit test, the expected frequencies are calculated from the assumed distribution; degrees of freedom (ν) = number of classes – 1 – number of estimated parameters. For a test of independence in an r × c table, ν = (r-1)(c-1).
其中 O 为观测频数，E 为在原假设下的期望频数。拟合优度检验中，期望频数由假设分布计算；自由度 ν = 类别数 – 1 – 估计的参数个数。对于 r × c 列联表独立性检验，ν = (r-1)(c-1)。
Conditions: All expected frequencies should be at least 5; if not, combine adjacent categories. The test is always one-tailed (upper tail) for the χ² statistic.
条件：所有期望频数应至少为 5；否则合并相邻类别。χ² 检验总是单侧（上尾）检验。
8. Linear Regression and Correlation | 线性回归与相关
The simple linear regression model assumes a relationship Yᵢ = α + βxᵢ + εᵢ, where εᵢ are independent N(0, σ²) errors. The least squares estimates of α and β are obtained by minimising Σ(yᵢ – α – βxᵢ)². Formulas for the estimates on a sample of n pairs (xᵢ, yᵢ):
β = Sxy / Sxx (slope), α = Ȳ – β X̄ (intercept)
简单线性回归模型假设关系为 Yᵢ = α + βxᵢ + εᵢ，其中 εᵢ 是独立 N(0, σ²) 误差。α 和 β 的最小二乘估计通过最小化 Σ(yᵢ – α – βxᵢ)² 获得。对于 n 对数据 (xᵢ, yᵢ) 的计算公式：
β = Sxy / Sxx （斜率）， α = Ȳ – β X̄ （截距）
where Sxy = Σ(xᵢ – X̄)(yᵢ – Ȳ) and Sxx = Σ(xᵢ – X̄)². The product-moment correlation coefficient is r = Sxy / √(Sxx Syy), with -1 ≤ r ≤ 1.
其中 Sxy = Σ(xᵢ – X̄)(yᵢ – Ȳ)，Sxx = Σ(xᵢ – X̄)²。积矩相关系数为 r = Sxy / √(Sxx Syy)，取值范围是 -1 ≤ r ≤ 1。
Testing the significance of the slope H₀: β = 0 uses the t-statistic:
t = β / SE(β) ~ tn-2, where SE(β) = s / √(Sxx) and s² = Σ(yᵢ – ŷᵢ)²/(n-2).
检验斜率显著性 H₀: β = 0 使用 t 统计量：
t = β / SE(β) ~ tn-2, 其中 SE(β) = s / √(Sxx)，s² = Σ(yᵢ – ŷᵢ)²/(n-2)。
Confidence intervals for β and for the mean response at a given x can be constructed using the t-distribution.
β 和给定 x 下均值响应的置信区间可利用 t 分布构建。
9. Type I and Type II Errors, Power of a Test | 第一类与第二类错误，检验功效
A Type I error occurs when H₀ is true but rejected: P(Type I error) = α. A Type II error occurs when H₀ is false but not rejected: P(Type II error) = β. The power of a test is 1 – β, the probability of correctly rejecting a false H₀.
第一类错误发生在 H₀ 为真却被拒绝时：P(第一类错误) = α。第二类错误发生在 H₀ 为假却未被拒绝时：P(第二类错误) = β。检验的功效是 1 – β，即正确拒绝一个错误的 H₀ 的概率。
Power depends on the true value of the parameter under H₁, the sample size, and the significance level. Increasing sample size or α generally increases power. In planning studies, we often work backwards to determine the necessary sample size to achieve a desired power.
功效取决于 H₁ 下参数的真值、样本容量和显著性水平。增大样本容量或 α 通常能提高功效。在研究设计中，常反过来求所需样本量以达到期望的功效。
10. Distributions of Sample Variance and Confidence Intervals for Variance | 样本方差的分布与方差置信区间
If X₁, …, Xₙ are a random sample from N(μ, σ²), then the statistic (n-1)s²/σ² follows a chi-squared distribution with (n-1) degrees of freedom. This result underlies inference for the population variance.
若 X₁, …, Xₙ 是来自 N(μ, σ²) 的随机样本，则统计量 (n-1)s²/σ² 服从自由度为 (n-1) 的卡方分布。这一结果是总体方差推断的基础。
A 100(1-α)% confidence interval for σ² is given by:
[ (n-1)s²/χ²n-1, α/2 , (n-1)s²/χ²n-1, 1-α/2 ]
σ² 的 100(1-α)% 置信区间为：
[ (n-1)s²/χ²n-1, α/2 , (n-1)s²/χ²n-1, 1-α/2 ]
Taking the square root gives a CI for σ. These intervals assume normality of the original data.
取平方根可得 σ 的置信区间。这些区间假定原始数据服从正态分布。
11. Nonparametric Tests: Wilcoxon Signed-Rank and Rank-Sum | 非参数检验：Wilcoxon 符号秩与秩和检验
When normality cannot be assumed, nonparametric alternatives are used. For paired samples, the Wilcoxon signed‑rank test tests whether the median difference is zero. The test statistic T is the smaller of the sum of positive ranks and the sum of negative ranks; small T values lead to rejection.
当不能假定正态性时，使用非参数替代方法。对于配对样本，Wilcoxon 符号秩检验检验中位差是否为零。检验统计量 T 是正秩和与负秩和的较小者；较小的 T 值导致拒绝原假设。
For two independent samples, the Mann–Whitney U test (Wilcoxon rank‑sum) can be used. It assesses whether one population tends to have larger values than the other. The test statistic U is calculated from the ranks; tables give critical values for small sample sizes, and a normal approximation applies for larger samples.
对于两个独立样本，可使用 Mann–Whitney U 检验（Wilcoxon 秩和检验）。它检验一个总体是否倾向于比另一个总体具有更大的值。统计量 U 由秩次计算得出；小样本使用临界值表，大样本则采用正态近似。
12. Using Statistical Tables and Calculator Techniques | 统计表使用与计算器技巧
You must be proficient in reading standard normal, t, chi-squared and Wilcoxon tables. Always note whether the table gives upper-tail, lower-tail, or two-tail probabilities. For the normal distribution, use the symmetry: Φ(-z) = 1 – Φ(z). For t-tables, the degrees of freedom and significance level determine the critical value.
你必须熟练阅读标准正态、t、卡方和 Wilcoxon 表。始终注意表给出的是上尾、下尾还是双侧概率。对于正态分布，利用对称性：Φ(-z) = 1 – Φ(z)。对于 t 表，自由度和显著性水平决定临界值。
Modern calculators can compute p-values, confidence intervals, and distribution probabilities directly. However, always write down the formulas and substitution steps in exam answers to earn method marks.
现代计算器可直接计算 p 值、置信区间和分布概率。但在考试作答时，务必写出公式和代入步骤以获取方法分。
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