📚 Year 12 CIE Statistics: Glossary & Terminology Quick-Memorization Guide | CIE 统计：词汇术语速记指南
Mastering the language of statistics is half the battle in CIE AS-Level Mathematics. This guide pairs every key term with its precise definition, a memory hook, and a contextual example, so you can stop confusing ‘variance’ with ‘variation’ and start using the correct notation with confidence.
掌握统计语言，是 CIE AS 数学成功的一半。本指南为每个关键术语配上了准确定义、记忆口诀和情境示例，帮助你不再混淆“方差”与“变异”，自信地使用正确记号。
1. Population and Sample | 总体与样本
Population: The entire set of individuals or items of interest. A parameter is a numerical summary describing the population (e.g. population mean μ, population variance σ²). Think P for Parameter and Population — both start with P.
总体：研究对象的全部个体或项目。参数是描述总体的数值概括（如总体均值 μ、总体方差 σ²）。记忆诀窍：Parameter 和 Population 都以 P 开头——P 对 P。
Sample: A subset of the population selected for study. A statistic is a numerical summary calculated from a sample (e.g. sample mean x̄, sample variance s²). The word statistic contains sample — think ‘s’ for ‘sample’.
样本：从总体中选出来用于研究的一个子集。统计量是从样本算出的数值概括（如样本均值 x̄、样本方差 s²）。统计量 statistic 里藏着 sample — 记忆：s 代表样本。
Census: An attempt to collect data from every member of the population. A sample is usually preferred because a census is often time‑consuming, expensive, or impossible.
普查：试图从总体的每一个成员那里收集数据。实际中多采用样本，因为普查通常耗时、昂贵或无法实现。
2. Types of Data | 数据类型
Quantitative data: Numerical data that can be measured or counted. Subdivided into discrete (countable, e.g. number of students, shoe size 38, 39, …) and continuous (measurable, e.g. height 172.5 cm, time 3.21 s).
定量数据：可测量或计数的数值数据。又分为离散型（可数，如学生人数、鞋码 38、39……）和连续型（可测，如身高 172.5 cm、时间 3.21 s）。
Qualitative (categorical) data: Non‑numerical data describing qualities or categories (e.g. eye colour, brand of car). Often further split into nominal (no natural order, e.g. hair colour) and ordinal (natural order exists, e.g. satisfaction rating: poor, fair, good).
定性（分类）数据：描述性质或类别的非数值数据（如眼睛颜色、汽车品牌）。常再分为名义数据（无自然顺序，如发色）和有序数据（存在自然顺序，如满意度评级：差、一般、好）。
Bivariate data: Pairs of values from two related variables, often analysed for correlation or regression (e.g. height and weight of students).
二元数据：来自两个相关变量的成对数值，常用于相关或回归分析（如学生的身高与体重）。
3. Measures of Central Tendency | 集中趋势的度量
Mean (x̄ or μ): The arithmetic average, sum of values divided by number of values. Sensitive to outliers. For grouped data, use midpoints: x̄ = Σf x / Σf.
均值（x̄ 或 μ）：算术平均数，总和除以个数。对异常值敏感。分组数据用组中值：x̄ = Σf x / Σf。
Median: The middle value when data are ordered. Not affected by extreme values, making it the preferred measure for skewed distributions. For n data values, median position is (n+1)/2.
中位数：排序后位于中间的值。不受极端值影响，因此是偏态分布下首选的集中趋势度量。n 个数据时中位数位置为 (n+1)/2。
Mode: The most frequently occurring value. A data set can be unimodal, bimodal, or multimodal. The only measure of central tendency suitable for qualitative data.
众数：出现频率最高的值。数据集可以是单峰、双峰或多峰的。是唯一适用于定性数据的集中趋势度量。
4. Measures of Dispersion | 离散程度的度量
Range: Maximum minus minimum. Quick to compute but heavily influenced by outliers. Useful as a rough guide to spread.
极差：最大值减最小值。计算快，但受异常值影响大。可作为离散程度的粗略参考。
Interquartile range (IQR): Q₃ − Q₁, the range of the middle 50% of data. Robust to outliers. The lower quartile Q₁ is the median of the lower half, the upper quartile Q₃ is the median of the upper half.
四分位距 (IQR)：Q₃ − Q₁，中间 50% 数据的区间。对异常值稳健。下四分位数 Q₁ 是下半部分数据的中位数，上四分位数 Q₃ 是上半部分数据的中位数。
Variance (σ² or s²): Average squared deviation from the mean. For a population: σ² = Σ(x − μ)² / N. For a sample: s² = Σ(x − x̄)² / (n−1). The n−1 denominator corrects bias.
方差（σ² 或 s²）：偏离均值差方的平均值。总体方差：σ² = Σ(x−μ)² / N。样本方差：s² = Σ(x−x̄)² / (n−1)。除以 n−1 是为了校正偏差。
Standard deviation (σ or s): Square root of variance. Has the same units as the original data, making it more interpretable. s = √[Σ(x−x̄)² / (n−1)].
标准差（σ 或 s）：方差的平方根。量纲与原始数据相同，更易解释。s = √[Σ(x−x̄)² / (n−1)]。
5. Graphical Representations | 图表表示
Histogram: For continuous or grouped discrete data; area of each bar is proportional to frequency. Frequency density = frequency / class width. No gaps between bars.
直方图：用于连续或分组离散数据；每个条形面积与频数成正比。频数密度 = 频数 / 组距。条形之间无间隙。
Cumulative frequency curve (ogive): Plots cumulative frequency against upper class boundary. Used to estimate medians, quartiles, and percentiles by reading off the graph.
累积频数曲线（肩形图）：绘制累积频数对上组界。用于通过图线读取中位数、四分位数和百分位数。
Box‑and‑whisker plot: Displays minimum, Q₁, median, Q₃, maximum. Outliers are often plotted as separate points beyond whiskers. Excellent for comparing distributions.
箱线图：展示最小值、Q₁、中位数、Q₃ 和最大值。异常值通常绘制在须线之外作为单独点。非常适合分布比较。
6. Probability Vocabulary | 概率词汇
Experiment: A repeatable process with observable outcomes (e.g. tossing a coin). The sample space (S) is the set of all possible outcomes.
试验：一个可重复的、能观察到结果的过程（如抛硬币）。样本空间 (S) 是所有可能结果的集合。
Event (A): A subset of the sample space. P(A) = number of outcomes in A / total number of outcomes, provided outcomes are equally likely.
事件 (A)：样本空间的一个子集。若结果等可能，则 P(A) = A 中的结果数 / 总结果数。
Complement (A’): All outcomes not in A. P(A’) = 1 − P(A). The little ‘ mark reminds you it’s the ‘opposite’ of A.
补集 (A’)：不在 A 中的所有结果。P(A’) = 1 − P(A)。右上角的小撇暗示它是 A 的反面。
Mutually exclusive events: Events that cannot occur together. P(A ∩ B) = 0. Addition rule simplifies: P(A ∪ B) = P(A) + P(B).
互斥事件：不能同时发生的事件。P(A ∩ B) = 0。加法法则简化为：P(A ∪ B) = P(A) + P(B)。
Independent events: Occurrence of one does not affect the probability of the other. P(A ∩ B) = P(A) × P(B). Do not confuse ‘independent’ with ‘mutually exclusive’.
独立事件：一个事件的发生不影响另一个事件发生的概率。P(A ∩ B) = P(A) × P(B)。切勿把“独立”与“互斥”混淆。
7. Permutations and Combinations | 排列与组合
Factorial (n!): n! = n × (n−1) × … × 2 × 1. By definition, 0! = 1. Used to count arrangements of distinct objects.
阶乘 (n!)：n! = n × (n−1) × … × 2 × 1。定义 0! = 1。用于计算不同物体的排列方式数。
Permutation (ⁿPᵣ): Number of ways to arrange r objects from n distinct objects when order matters. ⁿPᵣ = n! / (n−r)!. P for position — order is important.
排列 (ⁿPᵣ)：从 n 个不同物体中取出 r 个并按顺序排列的方式数。ⁿPᵣ = n! / (n−r)!。P 代表位置 —— 顺序重要。
Combination (ⁿCᵣ): Number of ways to choose r objects from n when order does not matter. ⁿCᵣ = n! / [r!(n−r)!]. C for choose — just a selection.
组合 (ⁿCᵣ)：从 n 个不同物体中选出 r 个且不计顺序的方式数。ⁿCᵣ = n! / [r!(n−r)!]。C 代表选择 —— 仅仅是选出。
Distinguishable arrangements: For n objects with n₁ identical of type 1, n₂ of type 2, etc., number of arrangements = n! / (n₁! n₂! …).
可区分排列：若 n 个物体中有 n₁ 个第 1 类完全相同，n₂ 个第 2 类完全相同……，则排列数 = n! / (n₁! n₂! …)。
8. Discrete Random Variables | 离散随机变量
Random variable (X): A variable whose value is determined by the outcome of a random experiment. Denoted by capital letters; specific values by lowercase, e.g. P(X = x).
随机变量 (X)：取值由随机试验结果决定的变量。用大写字母表示；具体取值小写，如 P(X = x)。
Probability distribution: A table, graph, or formula giving the probability for each possible value of a discrete random variable. Two conditions: 0 ≤ P(X = x) ≤ 1 and Σ P(X = x) = 1.
概率分布：给出离散随机变量每个可能取值概率的表、图或公式。两个条件：0 ≤ P(X = x) ≤ 1 且 Σ P(X = x) = 1。
Expected value E(X): Mean of the random variable. E(X) = Σ x · P(X = x). If the distribution describes a game, E(X) is the fair price to pay.
期望值 E(X)：随机变量的均值。E(X) = Σ x·P(X = x)。如果分布描述一个游戏，E(X) 就是应付的公平价格。
Variance Var(X): Var(X) = E(X²) − [E(X)]², where E(X²) = Σ x² · P(X = x). The shortcut formula is usually quicker than Σ (x−μ)² p.
方差 Var(X)：Var(X) = E(X²) − [E(X)]²，其中 E(X²) = Σ x²·P(X = x)。使用这个快捷公式通常比 Σ (x−μ)² p 更快。
9. Binomial Distribution | 二项分布
Conditions for B(n, p): Fixed number of trials n; each trial independent; only two outcomes (success/failure); probability of success p constant. X ~ B(n, p) means X follows a binomial distribution.
二项分布 B(n, p) 的条件：试验次数 n 固定；各次试验独立；每次只有两种结果（成功/失败）；成功概率 p 不变。X ~ B(n, p) 表示 X 服从二项分布。
Probability formula: P(X = r) = ⁿCᵣ p^r (1−p)^(n−r). The ⁿCᵣ counts the number of ways to choose which r trials are successes.
概率公式：P(X = r) = ⁿCᵣ p^r (1−p)^(n−r)。ⁿCᵣ 计算选哪 r 次为成功的方式数。
Mean and variance: E(X) = np, Var(X) = np(1−p). Learn the derivation: E(X) = Σ x·P(X=x), but these results are in the formula booklet.
均值与方差：E(X) = np，Var(X) = np(1−p)。理解推导过程有益，但公式可直接使用。
10. Geometric Distribution | 几何分布
Definition: The number of trials up to and including the first success in a series of independent trials with constant p. X ~ Geo(p). Trials continue until success occurs.
定义：在一系列独立试验中，直到首次成功为止（含成功的那一次）的试验次数，每次成功概率 p 不变。X ~ Geo(p)。试验一直进行到成功出现。
Probability: P(X = r) = p (1−p)^(r−1) for r = 1, 2, 3, … The (1−p)^(r−1) represents r−1 failures followed by 1 success.
概率：P(X = r) = p (1−p)^(r−1)，r = 1, 2, 3, …。(1−p)^(r−1) 对应前 r−1 次失败、最后 1 次成功。
Mean and variance: E(X) = 1/p, Var(X) = (1−p) / p². Both are given in the formula booklet. If p is small, you will wait a long time (large E(X)).
均值与方差：E(X) = 1/p，Var(X) = (1−p) / p²。公式手册中直接提供。如果 p 很小，平均等待时间就很长。
11. Normal Distribution | 正态分布
Characteristics: Symmetric, bell‑shaped continuous distribution defined by mean μ and variance σ². Notation: X ~ N(μ, σ²). Total area under the curve equals 1.
特征：对称的钟形连续分布，由均值 μ 和方差 σ² 定义。记作 X ~ N(μ, σ²)。曲线下总面积为 1。
Standard normal Z: Z = (X − μ) / σ. Z ~ N(0, 1). The z‑value tells how many standard deviations X is from the mean. Use the standard normal table to find probabilities.
标准正态 Z：Z = (X − μ) / σ。Z ~ N(0, 1)。z 值表示 X 距离均值多少倍标准差。用标准正态分布表查概率。
Inverse normal: Given a probability (area to the left), find the corresponding z‑value, then X = μ + zσ. Essential for finding percentiles or thresholds.
逆正态：已知左侧概率（面积），求对应的 z 值，再用 X = μ + zσ。对于查找百分位数或阈值必不可少。
Continuity correction: When approximating a binomial or other discrete distribution with a normal, adjust the boundary by ±½. For example, P(X ≥ 10) becomes P(X > 9.5).
连续性校正：用正态近似二项等离散分布时，边界要 ±½ 调整。例如 P(X ≥ 10) 变为 P(X > 9.5)。
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