📚 Year 13 OCR Statistics: Core Concepts Revision | Year 13 OCR 统计：核心知识点梳理
In Year 13 OCR Statistics, you build upon the foundations laid in Year 12 to tackle more advanced inferential techniques. This article synthesises the essential topics: probability models such as the binomial, Poisson, normal and continuous uniform distributions; sampling theory and the Central Limit Theorem; estimation and confidence intervals; a suite of hypothesis tests including z, t, chi-squared and non‑parametric procedures; and the modelling of relationships through correlation and regression. Every concept is presented as a concise revision note, pairing an English explanation with its Chinese equivalent so that you can cross‑reference terminology and deepen your understanding.
在 Year 13 OCR 统计课程中，同学们将在 Year 12 的基础上进一步掌握高级推断方法。本文梳理了核心要点：二项分布、泊松分布、正态分布与连续均匀分布等概率模型；抽样理论与中心极限定理；估计与置信区间；从 z 检验、t 检验到卡方检验和非参数检验的整套假设检验流程；以及用相关与回归刻画变量之间的关系。每个知识点都采用英文解释与中文对照的形式，便于你熟习术语并加深理解。
1. Random Variables and Probability Distributions | 随机变量与概率分布
A random variable (r.v.) maps outcomes of an experiment to real numbers. Discrete r.v.s take countable values; continuous r.v.s take all values within an interval. The probability distribution of a discrete r.v. is described by a probability mass function (p.m.f.) P(X = x), while a continuous r.v. has a probability density function (p.d.f.) f(x) where P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. The cumulative distribution function (c.d.f.) F(x) = P(X ≤ x) works for both types.
随机变量将试验结果映射为实数。离散型随机变量取可数个值，连续型随机变量在某个区间内取任意值。离散型的概率分布由概率质量函数 P(X = x) 给出，连续型则具有概率密度函数 f(x)，且有 P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。累积分布函数 F(x) = P(X ≤ x) 对两类随机变量均适用。
Expectation E(X) = Σ x·P(X = x) (discrete) or E(X) = ∫ x f(x) dx (continuous); variance Var(X) = E[(X – μ)²] = E(X²) – [E(X)]². For a linear transformation Y = aX + b, E(Y) = a E(X) + b and Var(Y) = a² Var(X).
数学期望 E(X) = Σ x·P(X = x)（离散）或 E(X) = ∫ x f(x) dx（连续）；方差 Var(X) = E[(X – μ)²] = E(X²) – [E(X)]²。对线性变换 Y = aX + b，有 E(Y) = a E(X) + b，Var(Y) = a² Var(X)。
2. Binomial Distribution | 二项分布
A binomial distribution models the number of successes X in a fixed number n of independent Bernoulli trials, each with constant success probability p. If X ~ B(n, p), its p.m.f. is:
二项分布用于描述在固定次数 n 的独立伯努利试验中成功的次数 X，每次试验的成功概率 p 保持不变。若 X ~ B(n, p)，其概率质量函数为：
P(X = k) = ⁿCₖ pᵏ (1–p)ⁿ⁻ᵏ
Mean μ = np, variance σ² = np(1–p). The distribution is symmetric when p = 0.5 and skewed otherwise. A normal approximation N(np, np(1–p)) is appropriate when np ≥ 10 and n(1–p) ≥ 10.
均值 μ = np，方差 σ² = np(1–p)。当 p = 0.5 时分布对称，否则偏斜。当 np ≥ 10 且 n(1–p) ≥ 10 时，可用正态近似 N(np, np(1–p))。
3. Poisson Distribution | 泊松分布
The Poisson distribution models the number of events occurring randomly in a fixed interval of time or space. If X ~ Po(λ), the parameter λ is both the mean and the variance.
泊松分布用于描述在固定时间或空间区间内随机发生的事件个数。若 X ~ Po(λ)，参数 λ 既是均值也是方差。
P(X = k) = (λᵏ e⁻λ) / k!
Events must occur singly and at a constant average rate. The sum of independent Poisson variables is also Poisson: X₁~Po(λ₁), X₂~Po(λ₂) → X₁+X₂ ~ Po(λ₁+λ₂). A normal approximation N(λ, λ) is used when λ ≥ 15.
事件须独立发生且平均速率恒定。独立泊松变量之和仍服从泊松分布：X₁~Po(λ₁), X₂~Po(λ₂) → X₁+X₂ ~ Po(λ₁+λ₂)。当 λ ≥ 15 时，可用正态近似 N(λ, λ)。
4. Normal Distribution | 正态分布
The normal distribution N(μ, σ²) is fundamental to statistical inference. Its bell‑shaped curve is symmetric about the mean μ; about 68% of data lie within one standard deviation σ of the mean, 95% within 1.96σ and 99.7% within 3σ.
正态分布 N(μ, σ²) 是统计推断的基石。其钟形曲线关于均值 μ 对称；约 68% 的数据落在 μ±σ 内，95% 落在 μ±1.96σ 内，99.7% 落在 μ±3σ 内。
Standardising X ~ N(μ, σ²) gives Z = (X – μ)/σ ~ N(0, 1). Tables of Φ(z) provide cumulative probabilities. To find an unknown μ or σ, use the inverse normal function with given probabilities.
将 X ~ N(μ, σ²) 标准化得到 Z = (X – μ)/σ ~ N(0, 1)。标准正态分布表 Φ(z) 可查累积概率。若需求解未知的 μ 或 σ，可借助给定的概率和逆正态函数。
The continuous uniform distribution U(a, b) also features in OCR; its p.d.f. is f(x) = 1/(b–a) for a ≤ x ≤ b, mean (a+b)/2, variance (b–a)²/12.
OCR 也会涉及连续均匀分布 U(a, b)，其概率密度函数 f(x) = 1/(b–a)（a ≤ x ≤ b），均值 (a+b)/2，方差 (b–a)²/12。
5. Sampling and the Central Limit Theorem | 抽样与中心极限定理
A sample of size n drawn from a population yields statistics such as the sample mean X̄ = (Σxᵢ)/n and sample variance s² = Σ(xᵢ − X̄)²/(n−1). The sampling distribution of X̄ has mean μ and standard error σ/√n. When the population is normal, X̄ is exactly normal. The Central Limit Theorem (CLT) states that for large n (usually n ≥ 30), X̄ is approximately N(μ, σ²/n) regardless of the population’s shape.
从总体中抽取容量为 n 的样本，可计算样本均值 X̄ = (Σxᵢ)/n 与样本方差 s² = Σ(xᵢ − X̄)²/(n−1)。X̄ 的抽样分布均值为 μ，标准误差为 σ/√n。若总体服从正态分布，X̄ 精确服从正态分布。中心极限定理 (CLT) 指出，当 n 较大（通常 n ≥ 30）时，无论总体分布形状如何，X̄ 均近似服从 N(μ, σ²/n)。
When the population standard deviation σ is unknown, we estimate it with s and use the t‑distribution. This underpins most real‑world inference procedures.
当总体标准差 σ 未知时，我们用 s 估计，并使用 t 分布。这是大多数现实推断方法的基础。
6. Estimation and Confidence Intervals | 估计与置信区间
A point estimate, such as X̄, gives a single plausible value for a parameter. An interval estimate states a range. A 95% confidence interval for a population mean μ (σ known) is X̄ ± 1.96 × σ/√n. The confidence level C% means that if we repeatedly sampled, C% of the constructed intervals would contain μ.
点估计，如 X̄，给出参数的单一合理值。区间估计则给出一个范围。总体均值 μ 的 95% 置信区间（σ 已知）为 X̄ ± 1.96 × σ/√n。置信水平 C% 意味着，若重复抽样，有 C% 构建的区间会包含 μ。
When σ is unknown, the interval becomes X̄ ± tₙ₋₁, α/2 × s/√n, where tₙ₋₁, α/2 is the critical value from Student’s t‑distribution with n−1 degrees of freedom. For proportions, a confidence interval for p is p̂ ± z* √(p̂(1−p̂)/n), provided np̂ and n(1−p̂) are both large enough.
当 σ 未知时，区间变为 X̄ ± tₙ₋₁, α/2 × s/√n，其中 tₙ₋₁, α/2 是自由度为 n−1 的 t 分布临界值。对于比例，p 的置信区间为 p̂ ± z* √(p̂(1−p̂)/n)（需满足 np̂ 与 n(1−p̂) 足够大）。
7. Hypothesis Testing Fundamentals | 假设检验基础
A hypothesis test evaluates a claim about a population parameter. The null hypothesis H₀ assumes no effect or status quo; the alternative H₁ states the suspected effect. A test statistic, computed from the sample, measures how far the data are from H₀. The p‑value is the probability of observing a result as extreme as (or more extreme than) the test statistic, assuming H₀ is true.
假设检验用于评估关于总体参数的某个声明。原假设 H₀ 假定无效应或维持现状；备择假设 H₁ 陈述疑似效应。由样本计算检验统计量，衡量数据与 H₀ 的偏离程度。p 值是在 H₀ 为真的条件下，观察到当前及更极端结果的概率。
Decision rule: reject H₀ if the p‑value < significance level α (commonly 0.05), otherwise do not reject H₀. A critical region approach defines a rejection region of the test statistic, and H₀ is rejected if the calculated statistic falls within it. Type I error (α) rejects a true H₀; Type II error (β) fails to reject a false H₀.
决策规则：若 p 值 < 显著性水平 α（常取 0.05），则拒绝 H₀；否则不拒绝 H₀。临界区域法为检验统计量划定拒绝域，若计算值落入其中则拒绝 H₀。第 I 类错误 (α) 为拒真，第 II 类错误 (β) 为存伪。
8. Tests for the Population Mean | 总体均值的假设检验
When σ is known, use the one‑sample z‑test: Z = (X̄ – μ₀)/(σ/√n). Compare Z with the standard normal critical value ±zα/2 for two‑tailed tests, or use the p‑value derived from Φ. OCR also expects you to test a mean from paired differences using a paired t‑test or the non‑parametric alternatives.
当 σ 已知时，使用单样本 z 检验：Z = (X̄ – μ₀)/(σ/√n)。双侧检验将 Z 与临界值 ±zα/2 比较，或通过 Φ 计算 p 值。OCR 也要求对配对差值进行均值检验，可采用配对 t 检验或非参数替代方法。
When σ is estimated by s, the one‑sample t‑test applies: t = (X̄ – μ₀)/(s/√n), df = n–1. The two‑sample t‑test (unpaired) assumes equal or unequal variances; a pooled estimate of variance may be used when equality is plausible. Always check assumptions of normality and independence.
当 σ 用 s 估计时，做单样本 t 检验：t = (X̄ – μ₀)/(s/√n)，自由度 n–1。双样本 t 检验（独立样本）需根据方差齐性选择方法，若方差齐性合理可采用合并方差估计。务必检验正态性与独立性假定。
9. Chi‑Squared Tests | 卡方检验
Chi‑squared (χ²) tests assess association in contingency tables and goodness‑of‑fit for categorical data. The test statistic is:
卡方 (χ²) 检验用于列联表关联性分析和分类数据的拟合优度检验。检验统计量为：
χ² = Σ (Oⱼ – Eⱼ)² / Eⱼ
where Oⱼ are observed frequencies and Eⱼ expected frequencies under H₀. For a goodness‑of‑fit test, degrees of freedom = (number of categories – 1 – number of estimated parameters). For a test of association in an r×c table, df = (r–1)(c–1). The test is valid only when expected frequencies are ≥ 5; if not, merge categories or use Fisher’s exact test.
Oⱼ 为观测频数，Eⱼ 为 H₀ 下的期望频数。拟合优度检验的自由度 =（类别数 – 1 – 估计参数个数）。r×c 列联表独立性检验的自由度 df = (r–1)(c–1)。仅当所有期望频数 ≥ 5 时检验有效，否则需合并类别或使用 Fisher 精确检验。
A high χ² value suggests a poor fit or strong association. Compute the p‑value from χ²‑distribution with the appropriate df and compare with α.
χ² 值越大，表明拟合越差或关联越强。通过相应自由度的 χ² 分布计算 p 值，并与 α 比较。
10. Non‑Parametric Tests | 非参数检验
When data violate normality assumptions, OCR expects familiarity with three rank‑based tests. The sign test tests a population median using the count of positive or negative differences from a hypothesised median; the test statistic is the smaller number of signs, and it follows a binomial distribution with p = 0.5.
当数据违反正态性假定时，OCR 要求掌握三种基于秩的检验。符号检验用与假设中位数之差的正负号个数检验总体中位数；检验统计量为正负号中较小的数目，在原假设下服从 p = 0.5 的二项分布。
The Wilcoxon signed‑rank test (paired data) ranks the absolute differences, attaches their original signs, and compares the sum of positive ranks T⁺ to critical values. The Wilcoxon rank‑sum test (or Mann–Whitney U test) for two independent samples combines the data, ranks them, and tests whether the sums of ranks in the two groups differ significantly. All non‑parametric tests involve fewer assumptions about the underlying distribution.
威尔科克森符号秩检验（配对数据）将绝对差排序，恢复原符号，比较正秩和 T⁺ 与临界值。威尔科克森秩和检验（或曼‑惠特尼 U 检验）用于两个独立样本，将数据混合排序后检验两组秩和是否有显著差异。所有非参数检验对总体分布的要求都更宽松。
11. Correlation and Regression | 相关与回归
The product‑moment correlation coefficient (Pearson’s r) measures linear association between two variables X and Y:
积矩相关系数（皮尔逊 r）测量两变量 X 与 Y 间的线性关联强度：
r = Σ(xᵢ – x̄)(yᵢ – ȳ) / √[ Σ(xᵢ – x̄)² Σ(yᵢ – ȳ)² ]
Spearman’s rank correlation ρₛ uses ranks to measure monotonic association and is more robust to outliers. Hypothesis tests for ρ = 0 or ρₛ = 0 rely on critical values from the t‑distribution or a table of Spearman critical values.
斯皮尔曼秩相关系数 ρₛ 使用数据的秩来度量单调关联，对异常值更稳健。检验 ρ = 0 或 ρₛ = 0 可依据 t 分布临界值或斯皮尔曼专门临界值表。
Simple linear regression models the relationship as Y = a + bX + ε. The least‑squares estimates are:
简单线性回归模型写作 Y = a + bX + ε。最小二乘估计量为：
b = Sxy / Sxx, a = ȳ – b x̄
where Sxy = Σ(xᵢ – x̄)(yᵢ – ȳ) and Sxx = Σ(xᵢ – x̄)². Residuals eᵢ = yᵢ – (a + bxᵢ) should be randomly scattered. OCR may also ask you to use the regression line for prediction, comment on the coefficient of determination r², and examine residual plots for model adequacy.
其中 Sxy = Σ(xᵢ – x̄)(yᵢ – ȳ)，Sxx = Σ(xᵢ – x̄)²。残差 eᵢ = yᵢ – (a + bxᵢ) 应随机散布。OCR 还可能要求用回归直线进行预测，评论判定系数 r² 的含义，并通过残差图判断模型适宜性。
Published by TutorHao | Statistics Revision Series | aleveler.com
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