📚 A-Level CCEA Statistics: Key Terminology Revision Guide | A-Level CCEA 统计：关键术语速记指南
Mastering statistical vocabulary is the first step towards confidently tackling A-Level CCEA Statistics. Without a clear grasp of the precise language used in exam questions, even the strongest mathematical skills can be derailed. This revision guide pairs every key term with its Chinese equivalent and a concise explanation, creating a bilingual memory bridge that speeds up recall. Work through each section consistently, and the terminology will become second nature.
掌握统计词汇是自信应对A-Level CCEA统计学考试的第一步。如果不能在考卷上准确理解那些精确的语言，再强的数学能力也可能跑偏。这份复习指南为每一个关键术语配上中文对照和简洁的解释，搭建起一座双语记忆桥梁，能让你更快回想起定义。坚持一个一个小节学完，术语就会成为你的第二天性。
1. Populations, Samples & Data Types | 总体、样本与数据类型
Population: The entire set of individuals, items or measurements that we are interested in studying. For example, all A-Level students in Northern Ireland taking CCEA Statistics.
总体：我们所希望研究的全体个体、项目或测量值的完整集合。例如，所有参加CCEA统计考试的北爱尔兰A-Level学生。
Sample: A subset of the population selected to represent it. A random sample is one where every member of the population has an equal chance of being chosen.
样本：从总体中选出的、用以代表总体的一个子集。随机样本指总体中每一个成员都有均等机会被选中。
Parameter: A numerical measure that describes a characteristic of a population, such as the population mean μ.
参数：描述总体某个特征的数值度量，例如总体均值 μ。
Statistic: A numerical measure calculated from a sample, used to estimate a population parameter. For instance, the sample mean x̄ estimates μ.
统计量：由样本计算得到的数值度量，用来估计总体参数。例如，样本均值 x̄ 用来估计 μ。
Qualitative data: Non-numerical information, often called categorical data, such as eye colour or favourite subject.
定性数据：非数值的信息，常称为分类数据，如眼睛颜色或最喜欢的科目。
Quantitative data: Numerical data that can be discrete (countable, e.g. number of siblings) or continuous (measurable, e.g. height).
定量数据：数值型数据，可以是离散的（可数，如兄弟姐妹人数）或连续的（可量测，如身高）。
2. Probability Essentials | 概率基础
Experiment: A repeatable process that gives rise to a set of outcomes, such as rolling a die.
试验：一个可重复的过程，会产生一组结果，例如掷一枚骰子。
Outcome: A single possible result of an experiment. For the roll of a die, the outcomes are 1, 2, 3, 4, 5 or 6.
结果：试验的一个单一可能结果。掷骰子的结果就是1,2,3,4,5或6。
Event: A set of one or more outcomes, usually denoted by a capital letter. ‘Rolling an even number’ is an event containing outcomes {2, 4, 6}.
事件：由一个或多个结果组成的集合，通常用大写字母表示。“掷出偶数”就是一个包含{2,4,6}的事件。
Probability: A measure of how likely an event is, on a scale from 0 (impossible) to 1 (certain). P(A) = number of favourable outcomes / total number of equally likely outcomes.
概率：衡量事件发生可能性的度量，范围从0（不可能）到1（必然）。P(A) = 有利结果数 / 所有等可能结果总数。
Conditional probability: The probability of event A occurring given that event B has already occurred, written as P(A|B). P(A|B) = P(A ∩ B) / P(B).
条件概率：已知事件B已发生的情况下事件A发生的概率，写作P(A|B)。P(A|B) = P(A ∩ B) / P(B)。
Independent events: Two events are independent if the occurrence of one does not affect the probability of the other; mathematically P(A ∩ B) = P(A) × P(B).
独立事件：如果一个事件的发生不影响另一个事件的概率，则两者独立；数学表达为P(A ∩ B) = P(A) × P(B)。
3. Random Variables & Distributions | 随机变量与分布
Random variable: A variable whose value is a numerical outcome of a random phenomenon. They are usually denoted by capital letters such as X.
随机变量：其数值为随机现象的结果的变量，通常用大写字母如X表示。
Discrete random variable: A random variable that can take only a countable number of distinct values, like the score on a die.
离散随机变量：只能取可数个不同值的随机变量，例如骰子的点数。
Continuous random variable: A random variable that can take any value within an interval, such as the exact mass of an apple.
连续随机变量：在一个区间内可取任意值的随机变量，例如一个苹果的精确质量。
Probability mass function (PMF): For a discrete X, the function P(X = x) lists the probability for each possible value x; ΣP(X=x) = 1.
概率质量函数：对于离散型X，函数P(X = x)列出了每个可能取值x的概率；∑P(X=x) = 1。
Probability density function (PDF): For a continuous X, the function f(x) is used so that the area under the curve between a and b equals P(a < X < b).
概率密度函数：对于连续型X，函数f(x)用来表示，曲线下a与b之间的面积等于P(a < X < b)。
4. Expectation & Variance | 期望与方差
Expected value E(X): The long-run average value of a random variable. For a discrete X, E(X) = Σ x·P(X=x). For continuous X, it is found by integration.
期望值E(X)：随机变量的长期平均取值。离散型：E(X) = ∑ x·P(X=x)。连续型则通过积分求得。
Variance Var(X): A measure of how spread out the values are around the mean. Var(X) = E[(X − μ)²] = E(X²) − [E(X)]².
方差Var(X)：衡量数据在均值附近离散程度的量。Var(X) = E[(X − μ)²] = E(X²) − [E(X)]²。
Standard deviation σ: The positive square root of the variance, σ = √Var(X). It has the same units as the original data.
标准差σ：方差的正平方根，σ = √Var(X)。它的单位与原始数据一致。
Linear transformation: If Y = aX + b, then E(Y) = aE(X) + b and Var(Y) = a² Var(X). This is vital for standardising variables.
线性变换：若Y = aX + b，则E(Y) = aE(X) + b，Var(Y) = a² Var(X)。这条性质对变量标准化至关重要。
5. Discrete Distributions: Binomial & Poisson | 离散分布：二项与泊松
Binomial distribution B(n, p): Models the number of successes in n independent trials, each with success probability p. X ~ B(n, p).
二项分布B(n, p)：对n次独立试验中成功的次数建模，每次成功概率为p。记为X ~ B(n, p)。
P(X = r) = nCr pr (1 − p)n−r
P(X = r) = C(n, r)·pr(1 − p)n−r
Poisson distribution Po(λ): Models the number of events occurring in a fixed interval when events happen at a constant average rate λ and are independent. X ~ Po(λ).
泊松分布Po(λ)：在固定区间内，事件以恒定平均速率λ发生且各事件相互独立时，对事件发生次数的建模。X ~ Po(λ)。
P(X = r) = e−λ λr / r!
P(X = r) = e−λ·λr / r!
Conditions for binomial: Fixed n; trials independent; two outcomes per trial; constant p. ‘BINS’: Binary, Independent, Number fixed, Same probability.
二项分布条件：固定n；各次试验独立；每次试验两种结果；p恒定。记忆口诀“BINS”：二元结果(Binary)、独立(Independent)、固定次数(Number)、相同概率(Same probability)。
Conditions for Poisson: Events occur singly, at a constant rate, independently, in a given interval. ‘RISC’: Random, Independent, Single (no simultaneous events), Constant rate.
泊松分布条件：事件在给定区间内单个发生、速率恒定且相互独立。记忆“RISC”：随机(Random)、独立(Independent)、单个(Single)、恒定速率(Constant)。
6. The Normal Distribution | 正态分布
Normal distribution N(μ, σ²): A continuous distribution with a bell-shaped curve, symmetric about the mean μ. The spread is described by variance σ².
正态分布N(μ, σ²)：具有钟形曲线的连续分布，关于均值μ对称。离散程度由方差σ²描述。
Z = (X − μ) / σ ~ N(0, 1)
Z = (X − μ) / σ ~ N(0, 1)
Standard normal distribution: The distribution with mean 0 and variance 1. Any normal variable can be standardised using the Z‑score. Tables give Φ(z) = P(Z ≤ z).
标准正态分布：均值为0、方差为1的分布。任何正态变量都可以通过Z分数标准化。标准正态表给出Φ(z) = P(Z ≤ z)。
Empirical rule (68‑95‑99.7%): In a normal distribution, about 68% of data lie within ±1σ of μ, 95% within ±2σ, and 99.7% within ±3σ.
经验法则（68‑95‑99.7%）：在正态分布中，约68%的数据落在μ ± 1σ内，95%落在μ ± 2σ内，99.7%落在μ ± 3σ内。
InvNorm / backward normal: Given a probability p, find the value k such that P(X < k) = p. This requires the inverse normal function or reading standard normal tables in reverse.
逆正态：给定概率p，找到值k使得P(X < k) = p。需要用逆正态函数或反向查标准正态表。
7. Sampling & the Central Limit Theorem | 抽样与中心极限定理
Sampling distribution: The probability distribution of a statistic, such as the sample mean. It describes how the statistic varies from sample to sample.
抽样分布：某个统计量（如样本均值）的概率分布。它描述了该统计量在反复抽样中的变动规律。
Central Limit Theorem (CLT): For a large sample size n (usually n ≥ 30), the sampling distribution of the sample mean is approximately normal, regardless of the population’s shape. The mean of X̄ is μ and its variance is σ²/n.
中心极限定理：当样本量n足够大（通常n ≥ 30）时，不论总体形状如何，样本均值的抽样分布都近似正态。X̄的均值为μ，方差为σ²/n。
X̄ ~ N(μ, σ²/n) approximately
X̄ ~ N(μ, σ²/n) 近似成立
Standard error: The standard deviation of a sampling distribution. For the sample mean, SE = σ/√n. It measures the precision of the estimate.
标准误差：抽样分布的标准差。对于样本均值，SE = σ/√n。它衡量估计量的精准程度。
8. Confidence Intervals | 置信区间
Confidence interval: An interval of values calculated from sample data that is believed to contain the unknown population parameter with a certain level of confidence, e.g. 95%.
置信区间：根据样本数据计算出的一个数值区间，相信该区间以一定置信水平（如95%）包含未知的总体参数。
95% confidence interval for μ (σ known): x̄ ± 1.96 × σ/√n. The multiplier 1.96 comes from the standard normal distribution (z0.025).
σ已知时μ的95%置信区间：x̄ ± 1.96 × σ/√n。乘数1.96来自标准正态分布（z0.025）。
x̄ ± z* × σ/√n
x̄ ± z* × σ/√n
Interpretation: If we were to take many samples and construct a 95% CI each time, 95% of those intervals would capture μ. It is not saying there is a 95% chance that a single interval contains μ.
解释：如果我们重复抽取许多样本并每次都构造一个95%置信区间，那么其中95%的区间会捕获μ。这并不是说某个特定区间有95%的概率包含μ。
Margin of error: The quantity z* × σ/√n (or t* × s/√n). It shows how far the estimate and the true parameter may differ due to sampling variability.
边际误差：z* × σ/√n（或t* × s/√n）的值。它反映出由于抽样波动，估计值与真实参数可能相差多远。
9. Hypothesis Testing & Errors | 假设检验与错误类型
Null hypothesis H₀: A statement that a population parameter equals a specified value, often the status quo. H₀: μ = μ₀.
原假设H₀：陈述总体参数等于某一指定值，通常是“现状”假设。H₀: μ = μ₀。
Alternative hypothesis H₁: The statement we are looking for evidence to support, e.g. H₁: μ > μ₀ (one-tailed) or μ ≠ μ₀ (two-tailed).
备择假设H₁：我们寻找证据去支持的陈述，例如H₁: μ > μ₀（单尾）或μ ≠ μ₀（双尾）。
Significance level α: The probability of rejecting H₀ when it is actually true. Commonly α = 0.05. This is a Type I error rate.
显著性水平α：当H₀实际为真时拒绝H₀的概率。常用α = 0.05。这就是第一类错误率。
Test statistic: A standardised value computed from the sample, used to decide whether to reject H₀. For a mean, z = (x̄ − μ₀)/(σ/√n).
检验统计量：由样本计算出的标准化值，用于决定是否拒绝H₀。对于均值，z = (x̄ − μ₀)/(σ/√n)。
p-value: The probability of obtaining a test statistic at least as extreme as the observed one, assuming H₀ is true. If p-value < α, reject H₀.
p值：在H₀为真的前提下，得到至少和观测值一样极端的检验统计量的概率。若p值 < α，则拒绝H₀。
Type I error: Rejecting a true H₀. The probability is α. Type II error: Failing to reject a false H₀. The probability is β.
第一类错误：拒绝了真实的H₀，概率为α。第二类错误：未能拒绝错误的H₀，概率为β。
Power of a test: 1 − β, the probability of correctly rejecting a false H₀.
检验功效：1 − β，即正确拒绝错误H₀的概率。
10. Chi-Squared Tests | 卡方检验
Chi-squared statistic χ²: A measure of the discrepancy between observed and expected frequencies. χ² = Σ (O − E)² / E.
卡方统计量χ²：衡量观测频数与期望频数
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