📚 AS CIE Statistics: Vocabulary & Terminology Quick-Reference Guide | AS CIE 统计：词汇术语速记指南
Mastering statistical vocabulary is the first decisive step towards excelling in AS CIE Mathematics (9709). This bilingual guide arranges the core terms by topic, with every English definition immediately followed by its Chinese equivalent, to help you internalise concepts rather than just memorise disjointed translations.
掌握统计词汇是在 AS CIE 数学 (9709) 中取得优异成绩的关键第一步。这份双语指南按主题梳理核心术语，每个英文定义立刻配以对应的中文解释，帮助你内化概念，而不仅仅是死记硬背脱节的翻译。
1. Measures of Central Tendency | 集中趋势的度量
The mean is the arithmetic average found by summing all observations and dividing by the number of observations, denoted by x̄ for a sample and μ for a population.
均值是算术平均值，通过将所有观测值求和再除以观测值个数得到，样本均值记作 x̄，总体均值记作 μ。
The median is the middle value when data are ordered from smallest to largest; for an even number of observations it is the average of the two central values.
中位数是将数据从小到大排序后中间位置的值；当观测值个数为偶数时，中位数是中间两个数值的平均数。
The mode is the most frequently occurring value in a data set; a data set may have one mode (unimodal), two modes (bimodal), or no mode at all in the case of a uniform frequency distribution.
众数是数据集中出现频率最高的值；一个数据集可以有一个众数（单峰）、两个众数（双峰），或者在均匀频率分布的情况下没有众数。
For grouped data, the modal class is the class interval with the highest frequency, and the mean is estimated using class midpoints multiplied by their frequencies.
对于分组数据，众数组是频率最高的组距，而均值则用组中值乘以对应的频率来估计。
2. Measures of Dispersion | 离散程度的度量
Range is the simplest measure of spread, calculated as the difference between the largest and the smallest observations. It is heavily affected by outliers.
极差是最简单的离散度量，计算为最大观测值与最小观测值之差。它极易受到异常值的影响。
The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁); it contains the middle 50% of the data and is resistant to outliers.
四分位距 (IQR) 是上四分位数 (Q₃) 与下四分位数 (Q₁) 的差；它包含中间 50% 的数据，并且能抵抗异常值的影响。
Variance measures the average squared deviation from the mean. The sample variance s² uses n − 1 in the denominator (Bessel’s correction), while the population variance σ² divides by N.
方差衡量各数据与均值之差的平方的平均值。样本方差 s² 分母用 n − 1（贝塞尔校正），而总体方差 σ² 分母用 N。
Standard deviation is the square root of the variance and is expressed in the same units as the original data, making it more interpretable than variance.
标准差是方差的平方根，以原始数据的相同单位表示，因此比方差更容易解读。
The standard deviation for a data set x is given by σ = √[Σ(x − μ)² / N] or s = √[Σ(x − x̄)² / (n − 1)].
数据集 x 的标准差计算公式为 σ = √[Σ(x − μ)² / N] 或 s = √[Σ(x − x̄)² / (n − 1)]。
3. Statistical Diagrams and Data Presentation | 统计图与数据呈现
A histogram uses the area of bars to represent frequency; for unequal class widths, frequency density = frequency ÷ class width must be plotted on the vertical axis.
直方图用条形的面积来表示频率；对于不等组距，必须将频率密度（频率 ÷ 组距）绘制在纵轴上。
A cumulative frequency curve (ogive) is used to estimate medians, quartiles, and percentiles by reading off the value corresponding to a given cumulative frequency.
累积频率曲线（形如 S 的 ogive）用于通过在曲线上读取给定累积频率对应的数值来估计中位数、四分位数和百分位数。
A box‑and‑whisker plot displays the minimum, Q₁, median, Q₃, and maximum, offering a visual five‑number summary of the data. Outliers are often shown as separate points.
箱线图显示最小值、Q₁、中位数、Q₃ 和最大值，提供了数据的五数概括。异常值通常以独立点显示。
Stem‑and‑leaf diagrams preserve the original data values while showing the shape of the distribution; a back‑to‑back stem‑and‑leaf diagram compares two related data sets efficiently.
茎叶图在保留原始数据值的同时展示分布形状；背靠背茎叶图能高效地比较两个相关的数据集。
4. Probability Fundamentals | 概率基础
An experiment is any repeatable process that yields a single outcome from a well‑defined set of possible outcomes, called the sample space.
试验是任何可重复的过程，它从一组明确定义的可能结果（称为样本空间）中产生一个单一的结果。
An event is a subset of the sample space. Mutually exclusive events cannot happen simultaneously; for mutually exclusive A and B, P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).
事件是样本空间的子集。互斥事件不可能同时发生；对于互斥事件 A 和 B，P(A ∩ B) = 0，且 P(A ∪ B) = P(A) + P(B)。
Independent events satisfy P(A ∩ B) = P(A) × P(B). Independence relates to whether one event influences the probability of another, not to mutual exclusivity.
独立事件满足 P(A ∩ B) = P(A) × P(B)。独立性指的是一个事件是否影响另一事件的概率，与互斥性无关。
Conditional probability is given by P(A|B) = P(A ∩ B) / P(B), representing the probability of A given that B has already occurred.
条件概率由 P(A|B) = P(A ∩ B) / P(B) 给出，表示在事件 B 已经发生的条件下，事件 A 发生的概率。
5. Permutations and Combinations | 排列与组合
The factorial of a non‑negative integer n, written n!, is the product of all positive integers ≤ n. By definition, 0! = 1.
非负整数 n 的阶乘，记作 n!，是所有 ≤ n 的正整数的乘积。定义 0! = 1。
A permutation is an ordered arrangement of r distinct items chosen from n distinct items. The number of permutations is ⁿPᵣ = n! / (n − r)!.
排列是从 n 个不同的项目中选出 r 个不同项目的有序安排。排列数为 ⁿPᵣ = n! / (n − r)!。
A combination is a selection of r items from n distinct items where order does not matter. The number of combinations is ⁿCᵣ = n! / [r!(n − r)!], also written as binomial coefficient (n r).
组合是从 n 个不同项目中选出 r 个项目，且顺序无关紧要。组合数为 ⁿCᵣ = n! / [r!(n − r)!]，也写作二项式系数 (n r)。
When items are not all distinct, the number of distinct permutations of n items with repetitions n₁, n₂, …, nₖ is n! / (n₁! n₂! … nₖ!).
当项目不全相同时，n 个物品中分别有 n₁, n₂, …, nₖ 个相同物品的不同排列数为 n! / (n₁! n₂! … nₖ!)。
6. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of distinct values. Its probability distribution lists each value x with the corresponding probability P(X = x), where 0 ≤ P(X = x) ≤ 1 and Σ P(X = x) = 1.
离散随机变量 X 取可数个不同的值。其概率分布列出每个值 x 以及对应的概率 P(X = x)，满足 0 ≤ P(X = x) ≤ 1 且 Σ P(X = x) = 1。
The expectation (or expected value) of X, denoted E(X) or μ, is the long‑run average, calculated as E(X) = Σ x·P(X = x).
X 的期望（或期望值），记作 E(X) 或 μ，是长期平均值，计算为 E(X) = Σ x·P(X = x)。
The variance of X is Var(X) = E(X²) − [E(X)]², where E(X²) = Σ x²·P(X = x). The standard deviation σ is the square root of Var(X).
X 的方差为 Var(X) = E(X²) − [E(X)]²，其中 E(X²) = Σ x²·P(X = x)。标准差 σ 是 Var(X) 的平方根。
For a linear transformation Y = aX + b, we have E(Y) = a E(X) + b and Var(Y) = a² Var(X). Adding a constant shifts the mean but does not affect variance.
对于线性变换 Y = aX + b，有 E(Y) = a E(X) + b 以及 Var(Y) = a² Var(X)。加上一个常数会平移均值，但不影响方差。
7. The Binomial Distribution | 二项分布
A binomial distribution arises from a fixed number n of independent trials, each with two possible outcomes (success or failure) and a constant probability of success p.
二项分布产生自固定次数 n 的独立试验，每次试验只有两种可能的结果（成功或失败），且成功的概率 p 恒定。
If X ~ B(n, p), then P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ for k = 0, 1, 2, …, n.
如果 X ~ B(n, p)，则 P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ，其中 k = 0, 1, 2, …, n。
For a binomial random variable, the mean is E(X) = np and the variance is Var(X) = np(1 − p), often written as npq where q = 1 − p.
对于二项随机变量，均值为 E(X) = np，方差为 Var(X) = np(1 − p)，时常写作 npq，其中 q = 1 − p。
The conditions for a binomial model: fixed number of trials, trials are independent, two outcomes per trial, probability of success remains constant.
二项模型的条件：试验次数固定，试验之间相互独立，每次试验只有两种结果，成功的概率保持不变。
8. The Geometric Distribution | 几何分布
The geometric distribution models the number of trials up to and including the first success in a series of independent Bernoulli trials, each with success probability p.
几何分布模拟在重复独立伯努利试验中，直到第一次成功为止所进行的试验次数，每次试验的成功概率为 p。
If Y ~ Geo(p), then P(Y = k) = (1 − p)ᵏ⁻¹ p for k = 1, 2, 3, …. Note that the lowest possible value is 1, unlike the binomial which can give 0 successes.
若 Y ~ Geo(p)，则 P(Y = k) = (1 − p)ᵏ⁻¹ p，其中 k = 1, 2, 3, …。注意最小值是 1，这与二项分布可获得 0 次成功不同。
The expected value of a geometric distribution is E(Y) = 1 / p, and the variance is Var(Y) = (1 − p) / p².
几何分布的期望值为 E(Y) = 1 / p，方差为 Var(Y) = (1 − p) / p²。
A key feature is the memoryless property: P(Y > m + n | Y > m) = P(Y > n). Past failures do not change the distribution of remaining trials to the first success.
一个关键特征是无记忆性：P(Y > m + n | Y > m) = P(Y > n)。过去的失败不会改变为达到第一次成功所需的剩余试验次数的分布。
9. The Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution with a symmetric, bell‑shaped curve defined by two parameters: the mean μ and the variance σ². We write X ~ N(μ, σ²).
正态分布是一种连续概率分布，具有由均值 μ 和方差 σ² 两个参数定义的对称钟形曲线。我们记作 X ~ N(μ, σ²)。
The total area under the normal probability density curve equals 1. The probability that X lies in a given interval is found by integrating the density function over that interval; tables or calculators give the cumulative probability Φ(z).
正态概率密度曲线下的总面积等于 1。X 落在给定区间的概率通过在该区间上对密度函数积分求得；表格或计算器可提供累积概率 Φ(z)。
The standard normal variable Z is defined as Z = (X − μ) / σ and follows N(0, 1). Standardising allows the use of a single set of probability tables.
标准正态变量 Z 定义为 Z = (X − μ) / σ，并服从 N(0, 1)。标准化使我们能够使用单一的一套概率表。
For any normal distribution, approximately 68% of the data lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ, often called the empirical rule.
对于任何正态分布，大约 68% 的数据落在 μ ± σ 范围内，95% 落在 μ ± 2σ 范围内，99.7% 落在 μ ± 3σ 范围内，这常被称为经验法则。
To find an unknown mean or standard deviation, set up an equation using Z = (x − μ) / σ and the known cumulative probability, solving for the missing parameter.
要找出未知的均值或标准差，可利用 Z = (x − μ) / σ 和已知的累积概率建立方程，进而求解缺失的参数。
10. Sampling and Estimation | 抽样与估计
A population is the entire set of individuals or items of interest, while a sample is a subset selected from the population. A simple random sample gives every member an equal chance of being chosen.
总体是所有感兴趣的个体或项目的完整集合，而样本是从总体中选出的一个子集。简单随机抽样使每个成员被抽中的机会相等。
A statistic is a numerical summary calculated from sample data, such as the sample mean x̄. A parameter is a numerical summary of the population, such as μ.
统计量是从样本数据计算出的数值概括，例如样本均值 x̄。参数是总体的数值概括，例如 μ。
The sampling distribution of the mean describes how x̄ varies across repeated samples. If X ~ N(μ, σ²), then x̄ ~ N(μ, σ² / n). The standard deviation of the sampling distribution, σ/√n, is called the standard error.
均值的抽样分布描述了 x̄ 在不同重复样本中如何变化。如果 X ~ N(μ, σ²)，那么 x̄ ~ N(μ, σ² / n)。抽样分布的标准差 σ/√n 被称为标准误差。
The Central Limit Theorem states that for a large sample (usually n ≥ 30), x̄ is approximately normally distributed regardless of the shape of the population, with mean μ and variance σ² / n.
中心极限定理指出，对于大样本（通常 n ≥ 30），无论总体的分布形状如何，x̄ 都近似服从均值为 μ、方差为 σ² / n 的正态分布。
11. Confidence Intervals | 置信区间
A confidence interval provides a range of plausible values for an unknown population parameter. A 95% confidence interval for the mean is x̄ ± z × σ/√n when σ is known, where z = 1.96 for the 95% level.
置信区间为未知总体参数提供了一个合理的取值范围。当 σ 已知时，均值的 95% 置信区间为 x̄ ± z × σ/√n，其中 95% 水平对应的 z = 1.96。
The width of a confidence interval depends on the confidence level, the sample size, and the variability in the data. A larger sample narrows the interval, increasing precision.
置信区间的宽度取决于置信水平、样本容量和数据的变异性。较大的样本会缩小区间宽度，提高估计的精确度。
When σ is unknown and the sample size is small, the t‑distribution is used instead of the normal distribution; however, for AS level, the normal‑based interval with known or large‑sample σ dominates questions.
当 σ 未知且样本较小时，应使用 t 分布替代正态分布；不过，在 AS 阶段，已知 σ 或大样本条件下的正态置信区间是主要的考查形式。
Interpreting a confidence interval correctly: “We are 95% confident that the true population mean lies between the lower and upper bounds.” It does not mean there is a 95% chance the mean is in that specific interval.
正确解读置信区间：“我们有 95% 的信心认为真正的总体均值位于下限与上限之间。”这并不意味着该具体区间包含均值的概率为 95%。
12. Key Connectors and Exam Terminology | 关键连接词与考试术语
“State” means write down the answer with no working needed; “Calculate” or “Find” requires a numerical answer with supporting steps; “Show that” means derive a given result, with all reasoning fully displayed.
“State” 表示直接写出答案，无需步骤；”Calculate” 或 “Find” 要求给出带计算步骤的数值答案；”Show that” 意为推导出一个给定结果，并完整展示推理过程。
Common phrases: “mutually exclusive” (互斥的), “independent” (独立的), “distribution” (分布), “frequency density” (频率密度), “cumulative” (累积的), “permutation” (排列), “combination” (组合), “standardised” (标准化的).
常见短语：”mutually exclusive”（互斥的），”independent”（独立的），”distribution”（分布），”frequency density”（频率密度），”cumulative”（累积的），”permutation”（排列），”combination”（组合），”standardised”（标准化的）。
Always check whether the given variance is for the population or for the sample; the distinction between σ² and s² determines whether to divide by n or by n − 1 in calculations.
务必检查所给方差是针对总体的还是样本的；σ² 与 s² 的区别决定了计算时是除以 n 还是除以 n − 1。
When a question asks for an assumption, mention key conditions like independence of trials for binomial, constant p for geometric, or normality of the population when using the normal model.
当题目要求写出假设时，要提及关键条件，如二项分布试验的独立性、几何分布中 p 不变，或使用正态模型时总体服从正态分布。
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