📚 AS CAIE Statistics: Vocabulary & Terminology Quick Memorisation Guide | AS CAIE 统计：词汇术语速记指南
Mastering the core vocabulary of AS-level Statistics is essential for interpreting questions and writing clear, accurate answers. This guide provides paired English–Chinese explanations and practical memory hooks for every key term you will meet in the CAIE syllabus – from data types and summary measures to probability distributions.
掌握 AS 阶段统计学的核心词汇，对于读懂题意并写出清晰、准确的答案至关重要。本指南针对 CAIE 考纲中你会遇到的每一个关键术语，提供英文-中文对照讲解与实用的记忆窍门，涵盖数据类型、统计量以及概率分布等内容。
1. Basic Terms: Population, Sample, and Variable Types | 基本术语：总体、样本与变量类型
Population: the entire collection of individuals or items we wish to investigate. Sample: a subset of the population selected to represent it, allowing us to draw conclusions without measuring everyone.
总体：我们希望研究的全体个体或项目的集合。样本：从总体中选出的一个子集，用来代表总体，使我们无需调查每一个体就能得出结论。
A variable is any characteristic that can take different values. Discrete data can only take certain fixed values – usually whole numbers (e.g. shoe size, number of cars). Continuous data can take any value within a given range (e.g. height, time).
变量是任何可以取不同值的特征。离散数据只能取某些固定的值——通常是整数（如鞋码、汽车数量）。连续数据在给定的范围内可以取任意值（如身高、时间）。
2. Measures of Central Tendency: Mean, Median, Mode | 集中趋势度量：均值、中位数、众数
Mean (x̄): the arithmetic average, calculated as Σx / n for raw data or Σfx / Σf for grouped data. It uses every value, so it is sensitive to outliers.
均值 (x̄)：算术平均数，对于原始数据用 Σx / n 计算，对于分组数据用 Σfx / Σf 计算。它使用了每一个数值，因此对极端值敏感。
Median: the middle value when the data are ordered. If n is even, take the average of the two middle numbers. The median is resilient to extreme values.
中位数：将数据排序后位于中间的值。如果 n 为偶数，则取中间两个数的平均值。中位数对极端值有较强的抵抗能力。
Mode: the value that appears most often. A data set can have no mode, one mode (unimodal), or more than one (bimodal, multimodal).
众数：出现次数最多的值。一组数据可能没有众数，有一个众数（单峰），或者有多个众数（双峰、多峰）。
Memory hook: Median = Middle; Mode = Most.
记忆窍门：Median（中位数）与 Middle（中间）相似；Mode（众数）与 Most（最多）首字母相同。
3. Measures of Spread: Range, Quartiles, IQR, and Percentiles | 离散度量：极差、四分位数、四分位距与百分位数
Range = maximum value – minimum value. It is the simplest measure of dispersion but is easily affected by outliers.
极差 = 最大值 – 最小值。它是最简单的离散度量，但极易受极端值影响。
Quartiles split the ordered data into four equal parts. Lower quartile Q₁ is the 25th percentile; median Q₂ is the 50th percentile; upper quartile Q₃ is the 75th percentile.
四分位数将排序后的数据分成四等份。下四分位数 Q₁ 是第 25 百分位数；中位数 Q₂ 是第 50 百分位数；上四分位数 Q₃ 是第 75 百分位数。
Interquartile range (IQR) = Q₃ – Q₁. It measures the spread of the middle 50% of values and is not influenced by extreme scores.
四分位距 (IQR) = Q₃ – Q₁。它衡量中间 50% 数据的分散程度，且不受极端值的影响。
A percentile is a score below which a given percentage of data falls. The kth percentile is found using linear interpolation in grouped data.
百分位数是指某一百分比的数据落在其下的某个分数。在分组数据中，我们通过线性插值来寻找第 k 百分位数。
4. Graphical Representations: Histograms, Cumulative Frequency, Box Plots | 图形表示：直方图、累积频数图、箱线图
Histogram: a bar chart for continuous data where the area of each bar is proportional to frequency. For unequal class widths, plot frequency density = frequency / class width on the vertical axis.
直方图：适用于连续数据的条形图，其中每个长方形的面积与频数成正比。若组距不等，纵轴应绘制频率密度 = 频数 / 组距。
Cumulative frequency curve (ogive): a graph of running total of frequencies. It is used to estimate medians, quartiles and percentiles by reading off values at the corresponding cumulative frequencies.
累积频数曲线（折线图）：显示累加频数的图像，用于通过读取相应的累积频数来估计中位数、四分位数及百分位数。
Box plot (box-and-whisker): displays minimum, Q₁, median, Q₃ and maximum. Outliers are often marked as individual points beyond 1.5×IQR from the quartiles.
箱线图（盒须图）：显示最小值、Q₁、中位数、Q₃ 和最大值。异常值通常被标记为超过四分位数 1.5 倍 IQR 范围以外的个体点。
Tip: ‘CF curve goes up, never down.’
提示：“累积频数曲线只升不降。”
5. Variance and Standard Deviation | 方差与标准差
Variance (σ² or s²) measures the 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).
方差 (σ² 或 s²) 衡量各数据点与均值之差的平方的平均值。总体方差：σ² = Σ(x – μ)² / N。样本方差：s² = Σ(x – x̄)² / (n – 1)。
Standard deviation (σ or s) is the positive square root of variance, bringing the measure back to the original units.
标准差 (σ 或 s) 是方差的正平方根，使度量恢复为原始单位。
s = √[ Σ(x – x̄)² / (n – 1) ]
A shortcut for variance: Var(X) = E(X²) – [E(X)]², which we will see again with random variables.
方差计算的简便公式：Var(X) = E(X²) – [E(X)]²，我们会在随机变量部分再次见到。
6. Probability Fundamentals: Random Experiment, Outcome, Event | 概率基础：随机试验、结果、事件
A random experiment is a process with unpredictable outcomes that are governed by chance. The sample space (S) is the set of all possible outcomes. An event is a subset of sample space outcomes.
随机试验是一个结果无法预测、由偶然性支配的过程。样本空间 (S) 是所有可能结果的集合。事件是样本空间结果的一个子集。
Probability of event A, P(A) = number of outcomes in A / total number of outcomes, provided all outcomes are equally likely. 0 ≤ P(A) ≤ 1.
事件 A 的概率 P(A) = A 包含的结果数 / 总的可能结果数，前提是所有结果等可能。任何概率都满足 0 ≤ P(A) ≤ 1。
The complement of A, denoted A’, is “not A”. P(A’) = 1 – P(A). The addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
A 的补事件，记作 A’，表示“非 A”，且 P(A’) = 1 – P(A)。加法公式：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。
7. Mutually Exclusive and Independent Events | 互斥事件与独立事件
Mutually exclusive events cannot happen at the same time; their intersection is empty. Therefore P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).
互斥事件不能同时发生，它们的交集为空。因此 P(A ∩ B) = 0，并且 P(A ∪ B) = P(A) + P(B)。
Independent events: one event occurring does not affect the probability of the other. This means P(A ∩ B) = P(A) × P(B) and also P(A|B) = P(A).
独立事件：一个事件的发生不影响另一个事件发生的概率。这意味着 P(A ∩ B) = P(A) × P(B) 且 P(A|B) = P(A)。
Common mistake: mutually exclusive events are not independent (unless one event has zero probability).
常见错误：互斥事件不是独立事件（除非其中一个事件的概率为零）。
8. Discrete Random Variables and Expectation | 离散随机变量与期望
A discrete random variable X takes a countable number of values with associated probabilities. The probability distribution must satisfy Σ P(X = x) = 1.
离散随机变量 X 取可数个数值，每个数值有相应的概率。概率分布必须满足 Σ P(X = x) = 1。
Expected value E(X) = Σ x · P(X = x), giving the long-run average. It acts like the mean of the random variable.
期望值 E(X) = Σ x · P(X = x)，代表长期平均值，相当于随机变量的均值。
Variance of X: Var(X) = E[(X – μ)²] = E(X²) – [E(X)]². Use the second form for quicker calculation.
X 的方差：Var(X) = E[(X – μ)²] = E(X²) – [E(X)]²。使用后一个公式计算更快。
E(aX + b) = a E(X) + b, Var(aX + b) = a² Var(X)
Linear transformations scale the mean by a and the variance by a², adding b shifts the mean but not the variance.
线性变换：均值被 a 缩放并平移 b，方差被 a² 缩放，而平移不会改变方差。
9. Binomial Distribution | 二项分布
A binomial distribution arises when we have a fixed number of trials n, each trial has two outcomes ‘success’ or ‘failure’, trials are independent, and the probability of success p is constant. We write X ~ B(n, p).
当试验次数 n 固定、每次试验只有“成功”或“失败”两种结果、各次试验相互独立、且每次成功的概率 p 保持不变时，就产生了二项分布。我们记作 X ~ B(n, p)。
The probability of obtaining exactly r successes is:
P(X = r) = ⁿCᵣ · pʳ · (1 – p)ⁿ⁻ʳ
其中 ⁿCᵣ = n! / [r! (n – r)!]，即二项式系数。
The expected value E(X) = n p, and variance Var(X) = n p (1 – p). These are standard results you can quote directly.
期望值 E(X) = n p，方差 Var(X) = n p (1 – p)。这些是标准结论，可以直接引用。
Memory: Binomial – ‘Bi’ means two outcomes; think of tossing a coin n times.
记忆：二项分布——“Bi”表示两种结果；想象抛一枚硬币 n 次。
10. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution with a bell-shaped curve, symmetric about the mean μ. Its spread is determined by the standard deviation σ. We write X ~ N(μ, σ²).
正态分布是一种连续型概率分布，概率密度曲线呈钟形，关于均值 μ 对称。其离散程度由标准差 σ 决定。我们记作 X ~ N(μ, σ²)。
The total area under the curve equals 1. Probabilities are found by standardising to the standard normal Z ~ N(0, 1) using z = (x – μ) / σ, then consulting normal distribution tables.
曲线下方的总面积等于 1。求概率时，先通过 z = (x – μ) / σ 转化为标准正态分布 Z ~ N(0, 1)，然后查正态分布表。
Key facts: about 68% within 1σ, 95% within 1.96σ, and 99.7% within 3σ. When using tables, always draw a sketch to avoid sign errors.
关键事实：大约 68% 的数据落在 μ ± σ 内，95% 落在 μ ± 1.96σ 内，99.7% 落在 μ ± 3σ 内。查表时，务必画简图避免符号错误。
Tip: ‘Zebra’ trick – z brings anything to the standard scale.
提示：记住“Z 变换”将任何正态变量拉回标准尺度。
11. Coding and Its Effects on Summary Statistics | 编码及其对汇总统计量的影响
Coding is used to simplify large or awkward numbers, e.g. y = (x – a)/b. If y = ax + b, the mean and variance transform predictably: ȳ = a x̄ + b and sᵧ² = a² sₓ².
编码用于简化庞大或不规整的数值，如 y = (x – a)/b。如果 y = ax + b，均值和方差按规律变换：ȳ = a x̄ + b 且 sᵧ² = a² sₓ²。
When coding back to find the original mean or standard deviation, reverse the operations: x = (y – b)/a. The standard deviation is always multiplied by |a|.
当需要还原原始均值或标准差时，要逆运算：x = (y – b)/a。标准差永远是乘以 |a|，不受平移影响。
Common coding in exam: time as ‘minutes after 10:00’. Subtract 600, divide by 60 – then decode carefully.
考试中常见编码：把时间表示为“10:00 后的分钟数”。减去 600，除以 60——然后仔细还原。
12. Quick Memorisation Tips and Common Pitfalls | 快速记忆技巧与常见错误
Use acronyms: ‘MIM’ for Measures – Mean, Median, Mode; ‘RIV’ for Range, IQR, Variance. For distributions remember ‘BiN’ – Binomial: n fixed, p constant; ‘NoS’ – Normal: symmetric, standardise.
使用缩略词记忆：“MIM”代表度量——均值、中位数、众数；“RIV”代表极差、四分位距、方差。对于分布，记住“BiN”——二项分布：n 固定，p 恒定；“NoS”——正态分布：对称，标准化。
Pitfall 1: confusing variance and standard deviation. Always square-root variance for SD. Pitfall 2: using raw frequencies not frequency density in histograms with unequal class widths. Pitfall 3: forgetting that mutually exclusive events cannot be independent unless P = 0.
常见误区 1：混淆方差和标准差。要求标准差记得对方差开平方。误区 2：在组距不等的直方图中使用原始频数而非频率密度。误区 3：忘记互斥事件不可能是独立事件（除非概率为零）。
Finally, when reading a question, underline the command words: ‘find’, ‘estimate’, ‘state’. Check if data is discrete or continuous, and whether you must work with sample or population statistics.
最后，审题时画出指令词：“求”、“估计”、“写出”。判断数据是离散还是连续，以及你需要用样本统计量还是总体统计量。
Published by TutorHao | Statistics Revision Series | aleveler.com
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