📚 Core Knowledge Summary for Cambridge Pre-U Statistics | Pre-U Cambridge 统计：核心知识点梳理
This article provides a structured overview of the essential topics in the Cambridge Pre-U Statistics syllabus. It covers data description, probability theory, random variables, key distributions, sampling, estimation, hypothesis testing, correlation and regression, as well as chi-squared tests. Each concept is presented in a clear, exam-focused manner to support revision and deeper understanding.
本文系统梳理了剑桥 Pre-U 统计学课程的核心知识点，包括数据描述、概率论、随机变量、重要分布、抽样、估计、假设检验、相关与回归以及卡方检验等内容。每个概念均以清晰且贴合考试要求的方式呈现，旨在帮助复习并加深理解。
1. Types of Data and Summary Statistics | 数据类型与汇总统计
Data can be categorical (nominal or ordinal) or numerical (discrete or continuous). Graphical representations such as bar charts, histograms, box plots and cumulative frequency curves help to visualise distributions and detect outliers.
数据可分为分类数据（名义或有序）和数值数据（离散或连续）。条形图、直方图、箱线图和累积频率曲线等图表有助于直观展示分布特征并识别异常值。
Measures of central tendency include the mean, median and mode. The sample mean x̄ is given by x̄ = (Σxᵢ)/n, where n is the sample size. The median is the middle value when data are ordered, and the mode is the most frequent value.
集中趋势的度量包括均值、中位数和众数。样本均值 x̄ 定义为 x̄ = (Σxᵢ)/n，其中 n 为样本容量；中位数是数据排序后居中的数值，众数则是出现频率最高的值。
Measures of dispersion describe the spread of the data. The range, interquartile range (IQR = Q₃ − Q₁), variance (s² = Σ(xᵢ − x̄)²/(n−1)) and standard deviation (s = √s²) are commonly used.
离散程度的度量描述了数据的分散情况，常用指标包括极差、四分位距（IQR = Q₃ − Q₁）、方差（s² = Σ(xᵢ − x̄)²/(n−1)）和标准差（s = √s²）。
s² = Σ(xᵢ − x̄)²/(n − 1)
2. Probability and Conditional Probability | 概率与条件概率
Probability P(A) measures the likelihood of an event A, with 0 ≤ P(A) ≤ 1. For equally likely outcomes, P(A) = (number of favourable outcomes)/(total number of outcomes). The addition rule for mutually exclusive events states P(A ∪ B) = P(A) + P(B); for any two events, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
概率 P(A) 度量事件 A 发生的可能性，满足 0 ≤ P(A) ≤ 1。对于等可能结果，P(A) = 有利结果数 / 总结果数。互斥事件的加法法则为 P(A ∪ B) = P(A) + P(B)；对任意两事件，P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
Conditional probability P(A|B) = P(A ∩ B)/P(B) represents the probability of A given that B has occurred. Two events are independent if P(A ∩ B) = P(A)P(B), or equivalently P(A|B) = P(A).
条件概率 P(A|B) = P(A ∩ B)/P(B) 表示在 B 已发生的前提下 A 发生的概率。若 P(A ∩ B) = P(A)P(B)，或等价地 P(A|B) = P(A)，则两事件独立。
Bayes’ theorem allows reversal of conditional probabilities: P(A|B) = P(B|A)P(A)/P(B). This is particularly useful in diagnostic testing and decision analysis.
贝叶斯定理可实现条件概率的逆转：P(A|B) = P(B|A)P(A)/P(B)，在诊断检验和决策分析中尤为有用。
3. Discrete Random Variables and Distributions | 离散随机变量与分布
A discrete random variable X takes a countable number of values. Its probability mass function (PMF) is p(x) = P(X = x). The sum of all probabilities is 1: Σ p(x) = 1.
离散随机变量 X 取可数个值，其概率质量函数（PMF）为 p(x) = P(X = x)，且所有概率之和为 1：Σ p(x) = 1。
The expected value E(X) = Σ x p(x) gives the long-run average; variance Var(X) = E(X²) − [E(X)]² measures spread. For a linear transformation Y = aX + b, E(Y) = aE(X) + b and Var(Y) = a² Var(X).
期望 E(X) = Σ x p(x) 表示长期平均值，方差 Var(X) = E(X²) − [E(X)]² 衡量离散程度。对于线性变换 Y = aX + b，有 E(Y) = aE(X) + b，Var(Y) = a² Var(X)。
The binomial distribution B(n, p) models the number of successes in n independent trials with success probability p. Its PMF is P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ, with E(X) = np, Var(X) = np(1−p).
二项分布 B(n, p) 描述 n 次独立试验中成功次数，每次成功概率为 p。PMF 为 P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ，且 E(X) = np，Var(X) = np(1−p)。
The Poisson distribution Po(λ) counts events in a fixed interval with mean λ. P(X = r) = e⁻ᵅ λʳ / r!, with E(X) = Var(X) = λ. It approximates the binomial when n is large and p is small (λ ≈ np).
泊松分布 Po(λ) 用于固定区间内的事件计数，均值为 λ。PMF 为 P(X = r) = e⁻ᵅ λʳ / r!，E(X) = Var(X) = λ。当 n 大而 p 小（λ ≈ np）时，泊松分布可用于近似二项分布。
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ
4. Continuous Random Variables | 连续随机变量
A continuous random variable can take any value in an interval. Its probability density function (PDF) f(x) satisfies f(x) ≥ 0 and ∫ f(x) dx = 1 over the support. Probability is given by area under the curve: P(a < X < b) = ∫ₐᵇ f(x) dx.
连续随机变量可取某区间内的任意值。其概率密度函数（PDF）f(x) 满足 f(x) ≥ 0 且在支撑集上的积分 ∫ f(x) dx = 1。概率由曲线下面积给出：P(a < X < b) = ∫ₐᵇ f(x) dx。
The cumulative distribution function (CDF) is F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt. The median m satisfies F(m) = 0.5. For any continuous variable, P(X = c) = 0.
累积分布函数（CDF）为 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。中位数 m 满足 F(m) = 0.5。对于任意连续变量，P(X = c) = 0。
Expectation and variance are defined via integrals: E(X) = ∫ x f(x) dx, Var(X) = ∫ (x − μ)² f(x) dx = E(X²) − μ². The standard deviation is σ = √Var(X).
期望与方差通过积分定义：E(X) = ∫ x f(x) dx，Var(X) = ∫ (x − μ)² f(x) dx = E(X²) − μ²。标准差为 σ = √Var(X)。
5. The Normal Distribution | 正态分布
The normal distribution N(μ, σ²) has the PDF f(x) = 1/(σ√(2π)) exp(−(x − μ)²/(2σ²)). It is symmetric about the mean μ, and approximately 68% of data lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
正态分布 N(μ, σ²) 的 PDF 为 f(x) = 1/(σ√(2π)) exp(−(x − μ)²/(2σ²))。其关于均值 μ 对称，约 68% 的数据落在 μ ± σ 内，95% 在 μ ± 2σ 内，99.7% 在 μ ± 3σ 内。
The standard normal distribution Z ~ N(0, 1) is obtained by standardising: Z = (X − μ)/σ. Probability calculations use statistical tables or technology to find Φ(z) = P(Z ≤ z).
通过标准化 Z = (X − μ)/σ 可得到标准正态分布 Z ~ N(0, 1)。概率计算借助统计表格或技术工具求得 Φ(z) = P(Z ≤ z)。
Linear combinations of independent normal variables are also normal. If Xᵢ ~ N(μᵢ, σᵢ²) independently, then Σ aᵢ Xᵢ ~ N(Σ aᵢ μᵢ, Σ aᵢ² σᵢ²).
独立正态变量的线性组合仍服从正态分布。若 Xᵢ ~ N(μᵢ, σᵢ²) 且相互独立，则 Σ aᵢ Xᵢ ~ N(Σ aᵢ μᵢ, Σ aᵢ² σᵢ²)。
Z = (X − μ) / σ
6. Sampling Methods and the Central Limit Theorem | 抽样方法与中心极限定理
Sampling methods include simple random sampling, stratified sampling, systematic sampling and cluster sampling. Each aims to produce a representative sample while managing cost and bias. The sampling distribution of a statistic describes its variation across repeated samples.
抽样方法包括简单随机抽样、分层抽样、系统抽样和整群抽样，旨在获得代表性样本并控制成本和偏差。统计量的抽样分布描述了其在重复抽样中的变异。
The Central Limit Theorem (CLT) states that for a random sample of size n from any population with mean μ and variance σ², the sample mean X̄ is approximately N(μ, σ²/n) when n is large (usually n ≥ 30). This underpins many inferential procedures even when the population is not normal.
中心极限定理（CLT）指出，从均值为 μ、方差为 σ² 的任意总体中抽取容量为 n 的随机样本，当 n 较大（通常 n ≥ 30）时，样本均值 X̄ 近似服从 N(μ, σ²/n)。这使得许多推断方法即使总体非正态也依然有效。
The standard error of the sample mean is σ/√n; it is estimated by s/√n when σ is unknown. The larger the sample, the smaller the standard error, improving precision.
样本均值的标准误为 σ/√n；当 σ 未知时用 s/√n 估计。样本越大，标准误越小，估计精度越高。
7. Estimation and Confidence Intervals | 估计与置信区间
A point estimate is a single value used to estimate a population parameter, e.g., x̄ estimates μ. An estimate is unbiased if its expected value equals the true parameter.
点估计是用单个数值估计总体参数，例如用 x̄ 估计 μ。若估计量的期望等于真实参数，则称其为无偏估计。
A confidence interval (CI) gives a range of plausible values for the parameter. For a population mean with known variance, a 95% CI is x̄ ± 1.96 × σ/√n. When σ is unknown, use the t-distribution: x̄ ± t* × s/√n, where t* is the critical value with n−1 degrees of freedom.
置信区间（CI）给出了参数的一个合理取值范围。已知方差时，总体均值的 95% 置信区间为 x̄ ± 1.96 × σ/√n。当 σ 未知时，使用 t 分布：x̄ ± t* × s/√n，其中 t* 是自由度为 n−1 的临界值。
The width of the CI decreases with larger n and lower confidence level. For a proportion p, the 95% CI uses p̂ ± 1.96 × √[p̂(1−p̂)/n], provided np̂ and n(1−p̂) are both ≥ 5.
置信区间的宽度随 n 增大和置信水平降低而变窄。对于比例 p，95% 置信区间为 p̂ ± 1.96 × √[p̂(1−p̂)/n]，但要求 np̂ 和 n(1−p̂) 均 ≥ 5。
x̄ ± t* × s/√n
8. Hypothesis Testing – Core Principles | 假设检验 – 核心原理
A hypothesis test assesses evidence against a null hypothesis H₀. The alternative hypothesis H₁ defines the direction of the test (one-tailed or two-tailed). A test statistic is calculated from the sample; its value is compared to a critical region or used to find a p-value.
假设检验评估反对原假设 H₀ 的证据。备择假设 H₁ 定义了检验的方向（单尾或双尾）。由样本计算检验统计量，将其值与临界域比较或通过其计算 p 值。
The p-value is the probability of obtaining a test statistic at least as extreme as the observed one, assuming H₀ is true. If p ≤ α (the significance level, often 0.05), H₀ is rejected in favour of H₁.
p 值是在 H₀ 为真的条件下，观测到至少与现有结果同样极端的检验统计量的概率。若 p ≤ α（显著性水平，通常取 0.05），则拒绝 H₀ 而接受 H₁。
Type I error (α) is rejecting a true H₀; Type II error (β) is failing to reject a false H₀. Power = 1 − β is the probability of correctly rejecting H₀.
第一类错误（α）是弃真错误，第二类错误（β）是取伪错误。功效 = 1 − β 是正确拒绝假原假设的概率。
9. One-sample and Two-sample Tests for Means | 均值的单样本与双样本检验
For a single population mean with σ known, use the z-test: z = (x̄ − μ₀)/(σ/√n). With σ unknown, the one-sample t-test applies: t = (x̄ − μ₀)/(s/√n), with n−1 df. Conditions include random sampling and normality (or n ≥ 30 by CLT).
已知 σ 时，用 z 检验：z = (x̄ − μ₀)/(σ/√n)。σ 未知时，使用单样本 t 检验：t = (x̄ − μ₀)/(s/√n)，自由度为 n−1。条件包括随机抽样和正态性（或 n ≥ 30 由 CLT 支持）。
For comparing two independent means, the two-sample t-test assumes equal variances: t = (x̄₁ − x̄₂)/(sₚ √(1/n₁ + 1/n₂)), where sₚ² = [(n₁−1)s₁² + (n₂−1)s₂²]/(n₁+n₂−2). An unequal variance version (Welch’s t-test) adjusts degrees of freedom.
比较两个独立均值时，若方差齐性，可使用双样本 t 检验：t = (x̄₁ − x̄₂)/(sₚ √(1/n₁ + 1/n₂))，其中 sₚ² = [(n₁−1)s₁² + (n₂−1)s₂²]/(n₁+n₂−2)。方差不齐时可使用 Welch t 检验并调整自由度。
Paired samples arise when observations are matched; the differences dᵢ = xᵢ − yᵢ are tested using a one-sample t-test on d̄, with H₀: μ_d = 0.
配对样本发生在观测值匹配时，对差值 dᵢ = xᵢ − yᵢ 进行单样本 t 检验，H₀: μ_d = 0。
10. Correlation and Linear Regression | 相关与线性回归
The Pearson product-moment correlation coefficient r measures the strength and direction of a linear relationship between two variables. r = S_xy / √(S_xx S_yy), where S_xy = Σ(xᵢ − x̄)(yᵢ − ȳ). It ranges from −1 to +1.
皮尔逊积矩相关系数 r 度量两变量间线性关系的强度和方向。r = S_xy / √(S_xx S_yy)，其中 S_xy = Σ(xᵢ − x̄)(yᵢ − ȳ)，取值范围为 −1 到 +1。
A hypothesis test for ρ = 0 uses t = r √(n−2) / √(1−r²) with n−2 df. Correlation does not imply causation.
检验 ρ = 0 使用 t = r √(n−2) / √(1−r²)，自由度为 n−2。相关关系并不代表因果关系。
The simple linear regression model is y = α + βx + ε, where ε ~ N(0, σ²). The least squares estimates are b = S_xy / S_xx and a = ȳ − b x̄, giving fitted line ŷ = a + bx. Residuals eᵢ = yᵢ − ŷᵢ should show random scatter.
简单线性回归模型为 y = α + βx + ε，其中 ε ~ N(0, σ²)。最小二乘估计为 b = S_xy / S_xx，a = ȳ − b x̄，得拟合直线 ŷ = a + bx。残差 eᵢ = yᵢ − ŷᵢ 应呈现随机散布。
The coefficient of determination r² indicates the proportion of variance in y explained by x. Confidence intervals and hypothesis tests for β can be constructed using b and its standard error.
判定系数 r² 表示 x 解释的 y 变异的比例。可通过 b 及其标准误构建 β 的置信区间和假设检验。
r = S_xy / √(S_xx S_yy)
11. The Chi-squared Test | 卡方检验
The chi-squared (χ²) test assesses goodness-of-fit and independence in categorical data. The test statistic is χ² = Σ (O − E)² / E, where O are observed frequencies and E are expected frequencies under H₀.
卡方（χ²）检验用于分类数据的拟合优度检验和独立性检验。检验统计量为 χ² = Σ (O − E)² / E，其中 O 为观测频数，E 为 H₀ 下的期望频数。
For a goodness-of-fit test with k categories and m estimated parameters, df = k − 1 − m. For a contingency table with r rows and c columns testing independence, df = (r−1)(c−1). The null hypothesis is rejected if χ² exceeds a critical value from χ² distribution with appropriate degrees of freedom.
拟合优度检验有 k 个类别、m 个估计参数时，自由度 df = k − 1 − m。对于 r 行 c 列的列联表独立性检验，df = (r−1)(c−1)。当 χ² 大于相应自由度下 χ² 分布的临界值时拒绝原假设。
All expected frequencies should be at least 5 to ensure the approximation is valid. Yates’ correction may be applied for 2×2 tables with small counts to improve accuracy.
所有期望频数至少为 5 以保证近似有效。对于计数较小的 2×2 表，可应用 Yates 校正以提高准确性。
χ² = Σ (O − E)² / E
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply