📚 Year 12 CAIE Statistics: A Quick Guide to Key Vocabulary and Terms | CAIE 12年级统计：核心术语速记指南
Welcome to your all-in-one glossary for CAIE AS Statistics. Mastering the terminology is half the battle – each term is a tool for solving problems. Here we break down key vocabulary with memory hooks, clear explanations and exam-focused usage so you can recall them quickly in the exam hall.
欢迎来到 CAIE AS 统计的术语速记指南。掌握术语就是成功的一半，每一个术语都是解题的工具。我们将通过记忆技巧、清晰解释和考试导向的用法，帮你拆解核心词汇，让你在考场上迅速回想起来。
1. Population, Sample and Census | 总体、样本与普查
The population is the complete set of individuals or items you want to investigate. Think of it as the 'whole world' of your study. A sample is a smaller, manageable subset selected from that population.
总体是你想要研究的全部个体或项目的完整集合，把它想象成你研究中的 "全世界"。样本是从总体中选出的一个较小、便于处理的子集。
A census measures every single member of a population. It gives true values but is often impractical. A sample survey collects data from only part of the population and aims to estimate population characteristics.
普查会测量总体中的每一个成员，它能给出真实值，但常常难以实现。抽样调查只收集总体一部分的数据，目标是估计总体特征。
A sampling unit is a single individual or item being sampled, while the sampling frame is the list of all sampling units. If the frame is biased – for example, missing some groups – the sample will be unrepresentative.
抽样单位是被抽样的单个个体或物品，抽样框是所有抽样单位的清单。如果抽样框有偏差，比如遗漏了某些群体，样本就会失去代表性。
To remember the difference between census and sample: 'A census is a complete count, a sample is a sneak peek.'
记忆普查与样本的区别："普查是全面清查，样本只是管中窥豹。"
2. Types of Data and Variables | 数据类型与变量
Quantitative data consists of numbers representing counts or measurements. It splits further into discrete (countable, like number of students) and continuous (measurable, like height).
定量数据由代表计数或测量值的数字组成。它进一步分为离散型（可数的，如学生人数）和连续型（可测量的，如身高）。
Qualitative data describes categories or attributes that cannot be measured with a natural numerical scale, such as colour, eye colour or types of cuisine.
定性数据描述的是无法用自然数值尺度测量的类别或属性，比如颜色、眼睛颜色或菜系类型。
Think: 'Discrete data you can count on your fingers; continuous data you can measure with a tape or scale. Qualitative data answers "what kind?" not "how many?".'
可以这样想："离散数据你可以用手指来数，连续数据你可以用卷尺或秤来量。定性数据回答的是 '哪种类型'，而不是 '有多少'。"
A variable is discrete if it takes only isolated values, often integers. A variable is continuous if it can take any value within an interval.
如果变量只能取孤立的数值（通常是整数），它就是离散变量。如果变量可以在一个区间内取任意值，它就是连续变量。
3. Measures of Central Tendency | 集中趋势的度量
The mean (x̄ or μ) is the arithmetic average: sum all values and divide by the count. It uses every data point but is sensitive to extreme values.
均值（x̄ 或 μ）是算术平均数：将所有数值相加，再除以个数。它使用了每一个数据点，但对极端值非常敏感。
The median is the middle value when data are ordered. It is robust against outliers and often preferred for skewed distributions. If n is even, take the mean of the two middle numbers.
中位数是将数据排序后的中间值。它对离群值的抵抗力很强，常用于偏态分布。如果 n 为偶数，则取中间两个数的均值。
The mode is the value that occurs most frequently. A data set can have one mode (unimodal), two modes (bimodal) or no mode at all.
众数是出现频率最高的值。一个数据集可以有一个众数（单峰）、两个众数（双峰），或者根本没有众数。
Memory hook: 'Mean uses every piece, median splits the middle, mode votes for the most popular.'
记忆口诀："均值人人有份，中位一刀分中，众数票选最红。"
4. Measures of Dispersion | 离散程度的度量
The range is the simplest spread measure: largest value minus smallest. It is easily affected by one outlier.
极差是最简单的离散度量：最大值减最小值。它极易受单个离群值的影响。
The interquartile range (IQR) = Q₃ − Q₁. It covers the middle 50% of data and is resistant to outliers. Useful for comparing spreads alongside medians.
四分位距 (IQR) = Q₃ − Q₁。它涵盖中间 50% 的数据，抵抗离群值，适合与中位数一起比较离散程度。
Variance quantifies 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).
方差量化了离均差的平方的平均值。总体方差：σ² = Σ(x − μ)² / N；样本方差：s² = Σ(x − x̄)² / (n − 1)。
Standard deviation is the square root of variance, σ or s. It has the same units as the data and is the most commonly reported measure of spread.
标准差是方差的平方根，σ 或 s。它与数据的单位相同，是最常报告的离散度量。
Quick formula link:
σ² = Σ(x − μ)² / N, s² = Σ(x − x̄)² / (n − 1)
5. Presenting Data: Frequency Density and Cumulative Frequency | 数据呈现：频率密度与累积频率
When drawing a histogram with unequal class widths, we use frequency density = frequency ÷ class width. The area of each bar represents frequency.
在绘制组距不等的直方图时，我们使用频率密度 = 频数 ÷ 组距。每个直条的面积代表了频数。
A cumulative frequency graph (or ogive) plots the running total of frequencies against the upper class boundaries. The median, quartiles and percentiles can be read directly from the curve.
累积频率图（或称折线图）是将累积频率对应上组界绘制而成。中位数、四分位数和百分位数都可以直接从曲线上读取。
To find the median from a cumulative frequency graph, go to 50% of the total frequency on the vertical axis, draw across to the curve and drop down to the data axis.
要从累积频率图中找中位数，先纵轴上取总数频率的 50%，画水平线交曲线，再向下作垂线到数据轴。
Memory phrase: 'Frequency density means area, cumulative frequency means total so far.'
记忆语："频率密度意味着面积，累积频率意味着到目前为止的总数。"
6. Box-and-Whisker Plots and Outliers | 箱线图与异常值
A box-and-whisker plot displays the five-number summary: minimum, Q₁, median, Q₃, maximum. The box spans Q₁ to Q₃, with a line at the median.
箱线图展示了五数概括：最小值、Q₁、中位数、Q₃、最大值。箱子从 Q₁ 延伸到 Q₃，并在中位数处画一条线。
Outliers at a value is an outlier if it is less than Q₁ − 1.5 × IQR or greater than Q₃ + 1.5 × IQR.
异常值是远离其他数据的观测值。常用的判断规则是：小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的数值被视作异常值。
Outliers are marked as stars or points beyond the whiskers. Always check for outliers as they can distort calculations of the mean and range.
异常值会以星号或点的形式标在须线之外。一定要检查异常值，因为它们会扭曲均值和极差的计算。
Mnemo: '1.5 IQR is the fence; anything beyond is a suspected outlier.'
记忆法："1.5 倍 IQR 是栅栏，越过栅栏就是嫌疑离群值。"
7. Probability Fundamentals | 概率基础
Sample space is the set of all possible outcomes of an experiment. An event e. Probability of an event A is P(A) = number of outcomes in A ÷ total number of outcomes.
样本空间是实验所有可能结果的集合。事件是样本空间的子集。事件 A 的概率为 P(A) = A 中的结果个数 ÷ 总结果个数。
Two events are mutually exclusive if they cannot happen at the same time. In this case P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).
两个事件如果不可能同时发生，则为互斥事件。此时 P(A ∩ B) = 0，且 P(A ∪ B) = P(A) + P(B)。
Events are independent currence of one does not affect the probability of the other. For independent A and B, P(A ∩ B) = P(A) × P(B).
若一个事件的发生不影响另一个事件的概率，则它们是独立事件。对独立事件 A 和 B，有 P(A ∩ B) = P(A) × P(B)。
Conditional probability P(A | B) is the probability of A given that B has occurred. It is calculated as P(A | B) = P(A ∩ B) ÷ P(B). Use this with tree diagrams for multi-stage problems.
条件概率 P(A | B) 是在 B 已发生的条件下 A 发生的概率，计算公式为 P(A | B) = P(A ∩ B) ÷ P(B)。在多阶段问题中搭配树状图使用。
Quick check: 'Mutually exclusive means cannot share; independent means cannot influence.'
速查："互斥意味着不能共享，独立意味着互不影响。"
8. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of possible values, each with a probability. The sum of all probabilities in its distribution must equal 1.
离散随机变量 X 取可数的若干个值，每个值对应一个概率。其概率分布中所有概率之和必须等于 1。
The expected value · P(X = x)]. It represents the long-run average if the experiment is repeated many times.
X 的期望值（均值）为 E(X) = Σ [x · P(X = x)]，它代表实验重复多次后的长期平均值。
The variance Var(X) = E[(X − μ)²] = Σ (x − μ)² P(X = x). A faster formula is Var(X) = E(X²) − [E(X)]².
X 的方差为 Var(X) = E[(X − μ)²] = Σ (x − μ)² P(X = x)。一个更快的计算公式是 Var(X) = E(X²) − [E(X)]²。
Expectation is a weighted average: each x is weighted by its probability. Think: 'Expected value is the probability-weighted centre of mass.'
期望就是加权平均：每个 x 以其概率加权。可以这样想："期望值是概率加权的质心。"
9. Binomial Distribution | 二项分布
A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. We write X ~ B(n, p).
二项分布用于描述在固定次数的独立试验中成功的次数，每次试验的成功概率 p 相同。记为 X ~ B(n, p)。
Conditions: fixed number of trials n; two possible outcomes (success/failure); trials independent; constant p. If any condition fails, it is not binomial.
适用条件：试验次数 n 固定；每次试验只有两种可能结果（成功 / 失败）；试验相互独立；p 保持不变。任一条件不满足，就不是二项分布。
Probability of exactly r successes: P(X = r) = C(n, r) pʳ qⁿ⁻ʳ, where q = 1 − p.
恰好获得 r 次成功的概率：P(X = r) = C(n, r) pʳ qⁿ⁻ʳ，其中 q = 1 − p。
The mean is E(X) = np and variance is Var(X) = npq. These are useful shortcuts – you do not need to sum over all r.
均值为 E(X) = np，方差为 Var(X) = npq。这是很有用的捷径，不需要对所有 r 求和。
To decide if a situation is binomial, ask: 'Fixed n, independent trials, same p, success/failure only.'
判断一个情境是否为二项分布，就问自己："固定 n、独立试验、相同 p、只有成功失败。"
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. The spread is determined by the standard deviation σ.
正态分布是一种连续型概率分布，曲线呈钟形，关于均值 μ 对称。散布程度由标准差 σ 决定。
Roughly 68% of data lie within 1σ of μ, 95% within 2σ, 99.7% within 3σ. This empirical rule helps with visual estimation.
大约 68% 的数据落在 μ ± 1σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。这条经验法则有助于直观估计。
To find probabilities, we convert any normal variable X to the standard normal Z ~ N(0, 1) using Z = (X − μ) / σ. Then use normal distribution tables.
为求概率，我们通过 Z = (X − μ) / σ 将任意正态变量 X 转化为标准正态变量 Z ~ N(0, 1)，然后查正态分布表。
The z-value tells you how many standard deviations X is above or below the mean. A negative z means below the mean.
z 值表示 X 比均值高出或低出多少个标准差。z 为负，表示低于均值。
When using the table, remember the table usually gives P(Z < z). For 'greater than' probabilities, use 1 − table value.
查表时要记住表格通常给出 P(Z < z)。对于 "大于" 的概率，用 1 减去表值。
Key formula:
Z = (X − μ) ÷ σ
11. Quick Mnemonic Tips & Recap | 快速记忆技巧与回顾
For central tendency: 'Mean is fussy (affected by outliers), Median is tough, Mode is popular.'
记忆集中趋势："均值很娇气（受离群值影响），中位数很坚强，众数很网红。"
Spread measures: 'Range gives the full stretch, IQR gives the sturdy middle, Standard deviation gives the typical wobble.'
离散度量："极差看全距，IQR 看坚固的中段，标准差看典型波动。"
Probability junction: 'Mutually exclusive? Can't happen together. Independent? One doesn't care about the other.'
概率岔路口："互斥就是不能同框，独立就是各自为王。"
Binomial vs normal: 'Binomial counts successes in n tries; normal measures how far from average on a smooth curve.'
二项与正态："二项分布数成功次数，正态分布量波纹高低
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