Core Knowledge Review of Year 13 CIE Statistics | Year 13 CIE 统计核心知识点梳理

📚 Core Knowledge Review of Year 13 CIE Statistics | Year 13 CIE 统计核心知识点梳理

Welcome to this comprehensive revision guide covering the most important topics in Year 13 CIE Statistics. Whether you are preparing for A2 exams or revisiting fundamentals, mastering these core concepts will help you build a solid foundation, from probability and distributions to estimation and hypothesis testing. Let's dive into the key knowledge points one by one.

欢迎阅读本篇全面复习指南，涵盖 Year 13 CIE 统计中最重要的主题。无论你是在为 A2 考试做准备，还是回顾基础知识，掌握这些核心概念——从概率和分布到估计和假设检验——都将为你打下坚实基础。让我们逐一梳理关键知识点。

1. Data Representation and Summary Statistics | 数据表示与汇总统计

Effective data analysis begins with clear representation. A histogram uses area to represent frequency, with frequency density calculated as frequency divided by class width. Cumulative frequency graphs are used to estimate median, quartiles, and percentiles directly from grouped data.

有效的数据分析始于清晰的表示。直方图用面积表示频率，频率密度由频率除以组距得出。累积频率图用于直接从分组数据中估计中位数、四分位数和百分位数。

Box-and-whisker plots visually summarise the five‑ , upper quartile (Q₃), and maximum, along with any outliers defined by the 1.5 × IQR rule.

箱线图直观地概括了五数摘要：最小值、下四分位数（Q₁）、中位数（Q₂）、上四分位数（Q₃）和最大值，以及由 1.5×IQR 规则确定的异常值。

Key numerical measures include the sample mean x̄ and sample variance s², defined as:

s² = Σ (xᵢ – x̄)² / (n – 1)

关键数值度量包括样本均值 x̄ 和样本方差 s²，其定义为：

s² = Σ (xᵢ – x̄)² / (n – 1)

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2. Probability and Counting Techniques | 概率与计数技巧

nty. For any event A, 0 ≤ P(A) ≤ 1. The addition rule for mutually exclusive events is P(A ∪ B) = P(A) + P(B), while the multiplication rule for independent events is P(A ∩ B) = P(A) × P(B). Conditional probability is given by P(A|B) = P(A ∩ B) / P(B).

概率模拟不确定性。对于任何事件 A，有 0 ≤ P(A) ≤ 1。互斥事件的加法公式为 P(A ∪ B) = P(A) + P(B)，而独立事件的乘法公式为 P(A ∩ B) = P(A) × P(B)。条件概率由 P(A|B) = P(A ∩ B) / P(B) 给出。

Permutations count ordered arrangements, e.g. ⁿPᵣ = n!/(n – r)!, while combinations count selections where order does not matter: ⁿCᵣ = n!/(r!(n – r)!). Tree diagrams help break down multi‑stage experiments and apply these rules systematically.

排列计数有序安排，例如 ⁿPᵣ = n!/(n – r)!，而组合计数不计顺序的选择：ⁿCᵣ = n!/(r!(n – r)!)。树状图有助于分解多阶段实验并系统地应用这些规则。

Always check whether events are independent or mutually exclusive when solving probability problems. Venn diagrams are powerful tools for visualising intersections and unions.

在解决概率问题时，务必检查事件是独立还是互斥。文氏图是可视化交集与并集的强大工具。

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3. Discrete Random Variables | 离散型随机变量

ution satisfies Σ P(X = x) = 1. The expected value E(X) = Σ x P(X = x) represents the long‑run average.

离散型随机变量 X 取有限或可数集合的值。其概率分布满足 Σ P(X = x) = 1。期望值 E(X) = Σ x P(X = x) 代表长期的平均值。

Variance measures spread: Var(X) = E[(X – μ)²] = E(X²) – [E(X)]². For a linear transformation Y = aX + b, the properties are:

E(aX + b) = a E(X) + b,    Var(aX + b) = a² Var(X)

方差衡量分散程度：Var(X) = E[(X – μ)²] = E(X²) – [E(X)]²。对于线性变换 Y = aX + b，则有：

E(aX + b) = a E(X) + b，   Var(aX + b) = a² Var(X)

These rules are essential when coding data or working with composite variables. Always ensure you sum probabilities and use the correct intervals for expectation.

在处理数据编码或复合变量时，这些规则至关重要。务必确保概率求和，并使用正确的区间计算期望。

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4. Binomial and Geometric Distributions | 二项分布与几何分布

The binomial distribution models the number of successes in n independent trials with constant probability p. Notation: X ~ B(n, p). Its probability mass function is:

P(X = k) = ⁿCₖ pᵏ (1 – p)ⁿ⁻ᵏ

Mean and variance are E(X) = np, Var(X) = np(1 – p). Conditions: fixed number of trials, two outcomes, independence, constant p.

二项分布模拟 n 次独立试验且每次成功概率 p 不变的成功次数。记作 X ~ B(n, p)。其概率质量函数为：

P(X = k) = ⁿCₖ pᵏ (1 – p)ⁿ⁻ᵏ

均值与方差为 E(X) = np，Var(X) = np(1 – p)。条件：固定试验次数、两种结果、独立性、常数 p。

The geometric distribution counts the number of trials until the first success: X ~ Geo(p). Its pmf is P(X = k) = (1 – p)ᵏ⁻¹ p, with E(X) = 1/p and Var(X) = (1 – p)/p². It is memoryless.

几何分布统计直到首次成功所需的试验次数：X ~ Geo(p)。其概率质量函数为 P(X = k) = (1 – p)ᵏ⁻¹ p，期望 1/p，方差 (1 – p)/p²。该分布具有无记忆性。

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5. The Poisson Distribution | 泊松分布

The Poisson distribution models the number of randomly occurring events in a fixed interval of time or space. If X ~ Po(λ), then:

P(X = k) = e⁻λ λᵏ / k!

Both the mean and variance equal λ. Conditions: events occur independently at a constant average rate, and two events cannot occur exactly simultaneously.

泊松分布模拟在固定时间或空间间隔内随机发生的事件数量。如果 X ~ Po(λ)，则：

P(X = k) = e⁻λ λᵏ / k!

均值与方差都等于 λ。条件：事件以恒定平均速率独立发生，且两个事件不能完全同时发生。

When λ is large (typically λ > 15), a normal approximation N(λ, λ) can be used, applying a continuity correction. The sum of independent Poisson variables is also Poisson: X₁ ~ Po(λ₁), X₂ ~ Po(λ₂) ⇒ X₁+X₂ ~ Po(λ₁+λ₂).

当 λ 很大时（通常 λ > 15），可使用正态近似 N(λ, λ)，并应用连续性校正。独立泊松变量之和也服从泊松分布：X₁ ~ Po(λ₁)，X₂ ~ Po(λ₂) ⇒ X₁+X₂ ~ Po(λ₁+λ₂)。

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6. Normal Distribution and Approximations | 正态分布与近似

The normal distribution is the most important continuous distribution. Notation X ~ N(μ, σ²), where μ is the mean and σ² is the variance. The standard normal variable is Z ~ N(0, 1), obtained by standardisation:

Z = (X – μ) / σ

Probabilities are found using standard normal tables, remembering that P(Z > z) = 1 – Φ(z) and P(Z < –z) = 1 – Φ(z).

正态分布是最重要的连续分布。记作 X ~ N(μ, σ²)，其中 μ 为均值，σ² 为方差。标准化得到标准正态变量 Z ~ N(0, 1)：

Z = (X – μ) / σ

通过标准正态表查找概率，记住 P(Z > z) = 1 – Φ(z) 且 P(Z < –z) = 1 – Φ(z)。

For binomial B(n, p) when np > 5 and n(1–p) > 5, we use N(np, np(1–p)). For Poisson Po(λ) with λ > 15, use N(λ, λ). In both cases a continuity correction is applied by adding or subtracting 0.5 to the discrete boundary.

对于二项分布 B(n, p)，若 np > 5 且 n(1–p) > 5，我们用 N(np, np(1–p)) 近似；对于泊松分布 Po(λ)，若 λ > 15，用 N(λ, λ) 近似。两种情形都必须通过将离散边界加或减 0.5 来实施连续性校正。

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7. Continuous Random Variables | 连续型随机变量

otal area under the curve equal to 1. Probabilities are areas: P(a < X < b) = ∫ₐᵇ f(x) dx.

连续型随机变量由概率密度函数 (pdf) f(x) ≥ 0 描述，曲线下的总面积等于 1。概率即为面积：P(a < X < b) = ∫ₐᵇ f(x) dx。

he lower bound to x. The median m satisfies F(m) = 0.5, and quartiles are found similarly.

累积分布函数 (cdf) 为 F(x) = P(X ≤ x) = ∫ f(t) dt（从下限积到 x）。中位数 m 满足 F(m) = 0.5，四分位数同理求得。

Expectation and variance are calculated through integration:

E(X) = ∫ x f(x) dx,    Var(X) = ∫ x² f(x) dx – [E(X)]²

期望与方差通过积分计算：

E(X) = ∫ x f(x) dx,    Var(X) = ∫ x² f(x) dx – [E(X)]²

A simple continuous distribution is the uniform distribution on [a, b], with f(x) = 1/(b–a) and E(X) = (a+b)/2. For piecewise pdfs, break the integration at each knot.

一个简单的连续分布是 [a, b] 上的均匀分布，f(x) = 1/(b–a)，E(X) = (a+b)/2。对于分段 pdf，须在每个结点处分割积分。

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8. Linear Combinations of Random Variables | 随机变量的线性组合

For any random variables X and Y, the expectation is always linear:

E(aX + bY) = aE(X) + bE(Y)

对于任意随机变量 X 和 Y，期望始终是线性的：

E(aX + bY) = aE(X) + bE(Y)

When X and Y are independent, the variance of the sum is the sum of variances:

Var(aX + bY) = a²Var(X) + b²Var(Y)

同样，当 X 和 Y 独立时，和的方差等于方差之和：

Var(aX + bY) = a²Var(X) + b²Var(Y)

If X ~ N(μₓ, σₓ²) and Y ~ N(μᵧ, σᵧ²) are independent, then aX + bY ~ N(aμₓ + bμᵧ, a²σₓ² + b²σᵧ²). This property is crucial when dealing with sample means and differences between normal variables.

如果 X ~ N(μₓ, σₓ²) 与 Y ~ N(μᵧ, σᵧ²) 独立，则 aX + bY ~ N(aμₓ + bμᵧ, a²σₓ² + b²σᵧ²)。这一性质在处理样本均值和正态变量之差时至关重要。

Note that Var(X – , not subtraction. These results extend naturally to linear combinations of more than two variables.

注意，当独立时 Var(X – Y) = Var(X) + Var(Y)，而不是相减。这些结果自然推广到两个以上变量的线性组合。

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9. Sampling and Estimation | 抽样与估计

A statistic is a function of sample data used to estimate a population parameter. The sample mean X̄ is an unbiased estimator of the population mean μ. For variance, the unbiased estimator is:

s² = Σ (Xᵢ – X̄)² / (n – 1)

统计量是用于估计总体参数的样本数据函数。样本均值 X̄ 是总体均值 μ 的无偏估计量。对于方差，无偏估计量为：

s² = Σ (Xᵢ – X̄)² / (n – 1)

According to the Central Limit Theorem, for large n the sample mean is approximately normally distributed regardless of the original population, with mean μ and variance σ²/n.

根据中心极限定理，对于大样本量 n，无论原始总体如何，样本均值近似服从正态分布，均值为 μ，方差为 σ²/n。

Confidence intervals provide a range of plausible values. When σ is known: X̄ ± z × σ/√n. When σ is unknown, use the t‑distribution with n–1 degrees of freedom: X̄ ± t × s/√n. For a population proportion p, the large‑sample interval is p̂ ± z √(p̂(1 – p̂)/n).

置信区间提供一个合理值的范围。当 σ 已知时：X̄ ± z × σ/√n。当 σ 未知时，使用自由度为 n–1 的 t 分布：X̄ ± t × s/√n。对于总体比例 p，大样本的置信区间为 p̂ ± z √(p̂(1 – p̂)/n)。

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10. Hypothesis Testing | 假设检验

A hypothesis test assesses evidence against a null hypothesis H₀ in favour of an alternative H₁. The significance level α is the probability of a Type I error (rejecting H₀ when it is true). The p‑value is the probability of obtaining a result at least as extreme as the observed value, assuming H₀ is true.

假设检验评估反对原假设 H₀、支持备择假设 H₁ 的证据。显著性水平 α 是 I 类错误（H₀ 为真时拒绝 H₀）的概率。p 值是在 H₀ 为真的条件下，获得至少与观测值一样极端结果的概率。

For a binomial test of p, calculate exact cumulative probabilities under H₀ and compare with α. For a Poisson test of λ, use the Poisson distribution. For a normal mean test with known σ, use the Z‑statistic:

Z = (X̄ – μ₀) / (σ / √n)

对于二项分布参数 p 的检验，计算 H₀ 下的精确累积概率并与 α 比较。对于泊松分布参数 λ 的检验，使用泊松分布。对于已知 σ 的正态均值检验，使用 Z 统计量：

Z = (X̄ – μ₀) / (σ / √n)

When σ is unknown, use the t‑test: t = (X̄ – μ₀) / (s / √n) with n – 1 degrees of freedom. Chi‑squared tests are used for categorical data. The goodness‑of‑fit test statistic is:

χ² = Σ (Oᵢ – Eᵢ)² / Eᵢ

当 σ 未知时使用 t 检验：t = (X̄ – μ₀) / (s / √n)，自由度为 n – 1。卡方检验用于分类数据。拟合优度检验统计量为：

χ² = Σ (Oᵢ – Eᵢ)² / Eᵢ

For contingency tables, degrees of freedom are (r – 1)(c – 1). Always check that expected frequencies are at least 5 to ensure validity of the χ² approximation.

对于列联表，自由度为 (r – 1)(c – 1)。务必检查期望频数至少为 5 以确保 χ² 近似的有效性。

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Published by TutorHao | Statistics Revision Series | aleveler.com

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