📚 Year 13 CIE Statistics: Quick Guide to Key Vocabulary | Year 13 CIE 统计：词汇术语速记指南
Mastering the precise terminology in Year 13 CIE Statistics is essential for interpreting examination questions, presenting clear arguments, and avoiding common misunderstandings. This guide pairs every key term with its concise definition in English and Chinese, covering probability models, sampling theory, and hypothesis testing. Use it as a rapid reference throughout your revision to sharpen both your statistical reasoning and exam technique.
掌握 Year 13 CIE 统计学中精确的术语对于理解考题、表达清晰论据并避免常见误解至关重要。本指南将每一个关键术语配以英汉简明释义，涵盖概率模型、抽样理论与假设检验。在复习期间将它作为速查手册，助你同时提升统计推理能力和应试技巧。
1. Discrete Random Variables | 离散随机变量
A random variable is a numerical outcome of a random phenomenon. A discrete random variable can take only a finite or countably infinite number of distinct values, usually integers. For example, the number of heads when tossing a coin three times is a discrete random variable that can take values 0, 1, 2, or 3.
随机变量是随机现象的数值结果。离散随机变量只能取有限个或可数无限个互异的值，通常为整数。例如，抛三次硬币正面朝上的次数就是一个离散随机变量，可取 0、1、2 或 3。
bability that X takes the exact value x, and the sum over all possible x equals 1. A distribution table lists each value alongside its probability, making the model easy to read.
离散随机变量 X 的概率分布由其概率质量函数 (PMF) 描述，记作 P(X = x)。它给出了 X 恰好取值 x 的概率，且所有可能取值的概率之和为 1。分布表将每个取值及其概率列在一起，使模型一目了然。
2. Probability Functions: PMF, PDF and CDF | 概率函数：质量函数、密度函数与累积分布函数
function (PDF) f(x), where probabilities are obtained as areas under the curve. The cumulative distribution function (CDF), denoted F(x), gives P(X ≤ x) for any real x, whether the variable is discrete or continuous.
对于离散变量，函数 f(x) = P(X = x) 是概率质量函数 (PMF)。对于连续变量，我们使用概率密度函数 (PDF) f(x)，概率通过曲线下的面积求得。累积分布函数 (CDF) 记作 F(x)，表示对任意实数 x 的概率 P(X ≤ x)，无论变量是离散还是连续。
ty; instead, the total area under f(x) equals 1. The CDF is always non-decreasing, with F(‑∞) = 0 and F(∞) = 1.
PMF 满足 0 ≤ P(X = x) ≤ 1。PDF 也满足 f(x) ≥ 0，但 f(x) 本身不是概率；取而代之的是 f(x) 下的总面积等于 1。CDF 是非减函数，且 F(‑∞) = 0、F(∞) = 1。
3. Expectation and Variance | 期望与方差
The expected value E(X), or mean μ, is the long‑ able with PDF f(x), E(X) = ∫ x f(x) dx. Expectation measures the centre of a distribution.
期望值 E(X) 或均值 μ 是随机变量的长期平均值。对于离散变量，E(X) = Σ x P(X = x)；对于具有密度 f(x) 的连续变量，E(X) = ∫ x f(x) dx。期望衡量分布的中心。
[E(X)]². The standard deviation is the positive square root of the variance, σ = √Var(X).
方差 Var(X) 衡量分布围绕均值的分散程度。它定义为 E[(X − μ)²]，并可用 Var(X) = E(X²) − [E(X)]² 计算。标准差是方差的正平方根，σ = √Var(X)。
4. The Binomial Distribution | 二项分布
A binomial distribution models the number of successes in a fixed number n of independent trials, each with the same probability p of success. We write X ~ B(n, p). The conditions are: fixed number of trials, independence, exactly two possible outcomes per trial (success/failure), and constant probability p.
二项分布模型描述在固定的 n 次独立试验中成功的次数，每次试验成功的概率 p 相同。记作 X ~ B(n, p)。其条件为：固定试验次数、每次试验独立、每次试验仅有两种可能结果（成功/失败）、且概率 p 恒定。
P(X = x) = ⁿCₓ pˣ (1 – p)ⁿ⁻ˣ
variance is σ² = np(1−p). The distribution is symmetric when p = 0.5 and skewed otherwise.
二项分布的均值为 μ = np，方差为 σ² = np(1−p)。当 p = 0.5 时分布对称，否则呈偏态。
5. The 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. We denote this as X ~ Po(λ).
泊松分布模型描述在固定时间或空间间隔内事件的发生次数，假设事件独立发生且平均发生率 λ 恒定。记作 X ~ Po(λ)。
P(X = x) = e−λ λx / x! , x = 0, 1, 2, …
n also approximate a binomial distribution when n is large and p is small, setting λ = np.
泊松变量的均值和方差都等于 λ。这一特性使它成为稀有事件的实用模型。当 n 很大而 p 很小时，泊松分布可作为二项分布的近似，取 λ = np。
6. Continuous Distributions and the Normal Distribution | 连续分布与正态分布
A continuous random variable can take any value within an interval. Its probabilities are found by integrating the PDF over that interval. The most important continuous distribution is the normal distribution, with a symmetric bell‑shaped curve defined by two parameters: the mean μ and variance σ². We write X ~ N(μ, σ²).
连续随机变量可以在一个区间内取任意值。其概率由 PDF 在该区间上的积分给出。最重要的连续分布是正态分布，其对称钟形曲线由两个参数定义：均值 μ 与方差 σ²。记作 X ~ N(μ, σ²)。
The PDF of a normal distribution is
f(x) = 1/(σ√2π) e−(x−μ)²/(2σ²). About 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
正态分布的概率密度函数为 f(x) = 1/(σ√2π) e−(x−μ)²/(2σ²)。大约 68% 的值落在均值的一个标准差范围内，95% 落在两个标准差内，99.7% 落在三个标准差内。
Many real‑world measurements, such as heights or examination scores, are approximately normally distributed. The normal distribution is also central to inference because of the Central Limit Theorem.
许多现实世界的测量值（如身高或考试成绩）都近似服从正态分布。由于中心极限定理，正态分布在推断统计中也处于核心地位。
7. Standardisation and the Standard Normal | 标准化与标准正态分布
Any normal variable X ~ y the formula Z = (X − μ)/σ. The standard normal distribution has mean 0 and variance 1: Z ~ N(0, 1).
任何正态变量 X ~ N(μ, σ²) 都可通过公式 Z = (X − μ)/σ 转换为标准正态变量 Z。标准正态分布的均值为 0、方差为 1：Z ~ N(0, 1)。
Standardisation allows us to use a single set of probability tables, the z‑tables, to find probabilities for any normal distribution. When solving problems, always draw a sketch, shade the required region, standardise the boundaries, and then use the table.
标准化使我们能借助同一套概率表（z 表）求出任意正态分布的概率。解题时务必先画草图，标出所需区域，对边界进行标准化，再查表。
For calculations involving a discrete distribution approximated by the normal (such as binomial or Poisson with continuity correction), we add or subtract 0.5 before standardising.
在正态近似离散分布（如二项或泊松分布，需用连续性校正）的计算中，标准化前要加或减 0.5。
8. Sampling and the Central Limit Theorem | 抽样与中心极限定理
A population is the whole set of items of interest, while a sample is a subset drawn to infer properties of the population. The sample mean x̅ is a statistic used to estimate the population mean μ. Different samples yield different values of x̅, so x̅ is itself a random variable with its own distribution, called the sampling distribution of the mean.
总体是所关注的全部对象的集合，而样本是从中抽取的一个子集，用于推断总体的性质。样本均值 x̅ 是用来估计总体均值 μ 的统计量。不同样本会产生不同的 x̅ 值，因此 x̅ 自身也是一个随机变量，其分布称为均值的抽样分布。
tion of the sample mean is approximately normal, regardless of the shape of the population distribution: x̅ ~ N(μ, σ²/n). The standard deviation of the sampling distribution, σ/√n, is called the standard error.
中心极限定理 (CLT) 指出，对于足够大的样本量（通常 n ≥ 30），无论总体分布的形状如何，样本均值的分布都近似正态：x̅ ~ N(μ, σ²/n)。抽样分布的标准差 σ/√n 称为标准误。
9. Confidence Intervals | 置信区间
A confidence interval (CI) provides a range of plausible values for an unknown population parameter, constructed from sample data. For a population mean with known variance σ², a 95% confidence interval is calculated as x̅ ± z0.025 × σ/√n, where z0.025 = 1.96 is the critical value from the standard normal distribution that leaves 2.5% in each tail.
置信区间 (CI) 为未知总体参数提供了一个由样本数据构造的可能取值范围。对于方差 σ² 已知的总体均值，95% 置信区间的计算公式为 x̅ ± z0.025 × σ/√n，其中 z0.025 = 1.96 是标准正态分布双侧尾部各留出 2.5% 的临界值。
The confidence level refers to the long‑run success rate of the method: if we repeated the sampling many times, 95% of the intervals so constructed would contain the true mean μ. A higher confidence level produces a wider interval, while a larger sample size narrows it.
置信水平指的是该方法在长期中的成功率：如果我们多次重复抽样，95% 这样构造的区间将包含真实均值 μ。置信水平越高，区间越宽；样本量越大，区间越窄。
10. Hypothesis Testing: Concepts | 假设检验：基本概念
A hypothesis test evaluates evidence from a sample to decide whether a statement about a population parameter should be rejected. The null hypothesis H₀ is the assumption being tested, typically a statement of no effect or no difference. The alternative hypothesis H₁ represents what we suspect might be true instead.
假设检验根据样本证据判断关于总体参数的某个陈述是否应被拒绝。原假设 H₀ 是被检验的假定，通常是“无效应”或“无差异”的陈述。备择假设 H₁ 代表了我们所怀疑可能成立的相反情形。
The test statistic, such as Z = (x̅ − μ₀) / (σ/√n), measures how far the sample result deviates from what the null predicts. The p‑ obtaining a result at least as extreme as the observed one, assuming H₀ is true. A small p‑value (typically ≤ significance level α) provides evidence against H₀.
检验统计量（如 Z = (x̅ − μ₀)/(σ/√n)）衡量样本结果与原假设预测偏离多远。p 值是假定 H₀ 为真时，得到至少与观测值同样极端结果的概率。极小的 p 值（通常 ≤ 显著性水平 α）构成了反对 H₀ 的证据。
The critical region (or rejection region) is the set of values of the test statistic for which we reject H₀. It is determined by the significance level α, often chosen as 0.05 or 0.01.
临界域（又称拒绝域）是使我们拒绝 H₀ 的检验统计量的取值集合，由显著性水平 α 决定，通常取 0.05 或 0.01。
11. Errors in Testing: Type I and Type II | 检验中的错误：第一类和第二类错误
A Type I error occurs when H₀ is true but we incorrectly reject it. The probability of a Type I error is the significance level α. A Type II error occurs when H₀ is false but we fail to reject it. Its probability is denoted β. The power of a test is 1 – β, the chance of correctly rejecting a false H₀.
第一类错误发生在 H₀ 为真却被错误拒绝时。第一类错误的概率就是显著性水平 α。第二类错误发生在 H₀ 为假却未能拒绝时，其概率记为 β。检验的功效是 1 – β，即正确拒绝错误 H₀ 的机会。
There is a trade‑off: decreasing α (making it harder to reject H₀) increases β, unless the sample size is increased. In CIE questions, you may be asked to interpret these errors in context, e.g. concluding a new drug is effective when it is not (Type I) or failing to detect a genuine effect (Type II).
这两类错误存在权衡：降低 α（使拒绝 H₀ 更难）会增大 β，除非增大样本量。在 CIE 考题中，你可能需要在情境中解释这些错误，例如，将无效的新药误判为有效（第一类错误）或未能发现真实疗效（第二类错误）。
12. Key Terminology Recap | 关键术语速查表
| English Term | 中文术语 | Brief Meaning |
|---|---|---|
| Discrete random variable | 离散随机变量 | Variable with countable values |
| Probability mass function (PMF) | 概率质量函数 | P(X = x) for discrete X |
| Probability density function (PDF) | 概率密度函数 | f(x) for continuous X; area gives probability |
| Cumulative distribution function (CDF) | 累积分布函数 | F(x) = P(X ≤ x) |
| Expectation E(X) | 期望值 | Mean (long‑run average) |
| Variance Var(X) | 方差 | Measure of spread, σ² |
| Binomial distribution B(n, p) | 二项分布 | Number of successes in n independent trials |
| Poisson distribution Po(λ) | 泊松分布 | Counts of events at constant average rate |
| Normal distribution N(μ, σ²) | 正态分布 | 广告 / Sponsored 【现货】剑桥国际高中教材 Cambridge IGCSE & O Level Additional Mathemati… 数学 备考教辅 · 正版现货 · 与本文内容相关 查看详情 / View on Taobao参考价 ¥198.0
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