📚 Pre-U CIE Statistics: A Rapid Memory Guide to Key Terms and Vocabulary | Pre-U CIE 统计：关键术语速记指南
Mastering statistical vocabulary is the first step to excelling in Pre-U CIE Statistics. This guide presents core terms in logically grouped clusters, each accompanied by a simple memory hook, so you can recall definitions swiftly and apply them accurately in exams.
掌握统计词汇是在 Pre-U CIE 统计学中取得优异成绩的第一步。本指南将核心术语按逻辑分组成簇，每个术语都配有简单的记忆钩子，帮助你快速回忆定义，并在考试中准确应用。
1. Population and Sample | 总体与样本
The 'population' is the entire group we want information about, whereas a 'sample' is a subset selected to represent that population. Think P for Parent (all), S for Selection (part).
“总体”是我们想要获取信息的完整组群，而“样本”是从中选出的一个子集，用以代表总体。记住 P 就像 Parent（父母，涵盖一切），S 就像 Selection（选择，只取一部分）。
A 'parameter' is a numerical summary describing a population (e.g., population mean μ), while a 'statistic' is a numerical summary describing a sample (e.g., sample mean x̄). Link them by the first letters: P-parameter goes with P-population, S-statistic goes with S-sample.
“参数”是描述总体的数值概括（如总体均值 μ），而“统计量”是描述样本的数值概括（如样本均值 x̄）。用首字母连接： P-参数 对应 P-总体，S-统计量 对应 S-样本。
A 'census' collects data from every member of the population. It is often contrasted with a sample survey. The word census contains us, which reminds you that all of us are included.
“普查”从总体中的每一个成员收集数据。它常与抽样调查相对。单词 census 中含有 us，提醒你 “我们所有人” 都被包括了。
2. Measures of Central Tendency | 集中趋势度量
The 'mean' (average) is the sum of all observations divided by the number of observations. Its notation x̄ for a sample and μ for a population is key; recall bar for sample — a sample often comes in bars or groups.
“均值”（平均数）是所有观测值的总和除以观测值的个数。其符号是样本均值 x̄，总体均值 μ；记住 短横代表样本——样本常以条形（bars）或组形式出现。
The 'median' is the middle value when data are ordered. Median sounds like 'middle'; and if you imagine a road median strip, it sits in the centre.
“中位数”是数据排序后中间的值。Median 发音像 'middle'（中间）；想象道路中央分隔带（median strip），它就位于中心。
The 'mode' is the value that appears most frequently. Mode sounds like 'most' — it is the most common observation.
“众数”是出现频率最高的值。Mode 发音像 'most'（最多）——它是最常见的观测值。
- Mean — uses all values, sensitive to outliers.
- Median — robust to outliers.
- Mode — useful for categorical data.
- 均值 ——利用所有数值，对异常值敏感。
- 中位数 ——对异常值稳健。
- 众数 ——适用于分类数据。
3. Measures of Dispersion | 离散程度度量
The 'range' is the difference between the maximum and minimum values. Range = reach; it tells you how far the data stretch.
“极差”是最大值与最小值之差。Range = 范围；它告诉你数据延伸了多远。
The 'interquartile range' (IQR) is Q₃ - Q₁, the spread of the middle 50% of data. IQR focuses on the inter (inside) quarter region, discarding extremes.
“四分位距” (IQR) 是 Q₃ - Q₁，即中间 50% 数据的散布范围。IQR 关注 inter（内部）四分之一区域，舍弃极端值。
'Variance' measures the average squared deviation from the mean. For a population, it is σ²; for a sample, s². Variance = variation + distance². Remember squaring makes units squared (e.g., kg²).
“方差”度量了观测值与均值之差的平方的平均值。总体方差记为 σ²，样本方差记为 s²。方差 = 差异的平方。记住平方导致单位变为平方（如 kg²）。
'Standard deviation' is the square root of variance, restoring original units. Its symbol σ (population) or s (sample). Standard deviation = square root of variance — back to Earth units.
“标准差”是方差的平方根，使单位回到原始量纲。总体符号 σ，样本符号 s。标准差 = 方差的平方根——回到原单位。
4. Probability Fundamentals | 概率基础
An 'experiment' is any process that generates well-defined outcomes. A 'trial' is a single performance of an experiment. Think of experiment as the whole recipe, trial as one taste.
“试验”（experiment）是产生明确定义结果的过程。“一次尝试”（trial）是试验的单次执行。想象为 experiment 是整个配方，trial 是一次品尝。
The 'sample space' (S) is the set of all possible outcomes. Sample space = the whole menu.
“样本空间” (S) 是所有可能结果的集合。样本空间 = 整份菜单。
An 'event' a subset of the sample space. For a fair die, event A = {2,4,6} is getting an even number.
“事件”是样本空间的子集。对于一枚公平骰子，事件 A = {2,4,6} 表示掷出偶数。
Probability axioms: (1) 0 ≤ P(A) ≤ 1, (2) P(S) = 1, (3) for mutually exclusive events, P(A ∪ B) = P(A) + P(B). Think 0 is impossible, 1 is certain.
概率公理：(1) 0 ≤ P(A) ≤ 1，(2) P(S) = 1，(3) 对于互斥事件，P(A ∪ B) = P(A) + P(B)。想象 0 是不可能，1 是必然。
'Complement' of event A, denoted A' or Aᶜ, is 'not A'. P(A') = 1 - P(A). Complement completes the pie.
事件 A 的“补集”，记为 A' 或 Aᶜ，表示“非 A”。P(A') = 1 - P(A)。补集补全整个圆饼。
5. Conditional Probability and Independence | 条件概率与独立性
Conditional probability P(A|B) is read 'probability of A given B'. Formula: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. The given event shrinks the sample space to B.
条件概率 P(A|B) 读作“在 B 发生的条件下 A 的概率”。公式：P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0。给定的事件把样本空间缩小到 B。
Events A and B are 'independent' if P(A ∩ B) = P(A) × P(B). Equivalently, P(A|B) = P(A). Independent = no influence; the outcome of B does not affect A.
如果 P(A ∩ B) = P(A) × P(B)，则事件 A 和 B“独立”。等价于 P(A|B) = P(A)。独立 = 无影响；B 的结果不影响 A。
Mutually exclusive events cannot occur simultaneously, so P(A ∩ For mutually exclusive, independence is impossible unless one event has probability zero. Exclusive = cannot sit together; Independent = don't care about each other.
互斥事件不能同时发生，因此 P(A ∩ B) = 0。对于互斥事件，除非某个事件的概率为 0，否则不可能独立。互斥 = 不能共存；独立 = 互不关心。
6. Random Variables and Expectation | 随机变量与期望
A 'random variable' (r.v.) is a function that assigns a numerical value to each outcome in a sample space. A 'discrete' r.v. takes countable values; a 'continuous' r.v. takes values in an interval. Discrete = distinct dots; Continuous = connected curve.
“随机变量” (r.v.) 是将样本空间中的每个结果赋以一个数值的函数。“离散”随机变量取可数个值；“连续”随机变量在某个区间内取值。离散 = 分离的点；连续 = 连接的曲线。
The 'probability distribution' of a discrete r.v. lists its possible values and associated probabilities. The sum of all probabilities equals 1.
离散随机变量的“概率分布”列出其可能取值及相关概率。所有概率之和等于 1。
'Expectation' (X) is the long-run average value of a random variable. For discrete X, E(X) = Σ x·P(X=x). Expectation sounds like 'expected average'.
“期望” E(X) 是随机变量在长期重复试验中的平均取值。对于离散 X，E(X) = Σ x·P(X=x)。期望听起来是“预期的平均”。
'Variance' of a random variable is Var(X) = E[(X - μ)²] = E(X²) - [E(X)]². Same logic as data variance, but for a theoretical distribution.
随机变量的“方差”是 Var(X) = E[(X - μ)²] = E(X²) - [E(X)]²。与数据方差的逻辑一致，但用于理论分布。
7. Common Discrete Distributions | 常见离散分布
The 'binomial distribution' models the number of successes in a fixed number n of independent trials, each with success probability p. Notation: X ~ B(n, p). Conditions: fixed n, independent trials, two outcomes, constant p. Bi- means two outcomes; think coin-tossing.
“二项分布”模拟在固定的 n 次独立试验中成功的次数，每次成功的概率为 p。记作：X ~ B(n, p)。条件：固定 n，独立性，两种结果，恒定的 p。Bi- 意为两个结果；想象抛硬币。
The 'Poisson distribution' models the number of events occurring in a fixed interval of time or space, with events occurring independently at a constant mean rate λ. Notation: X ~ Po(λ). Poisson looks like 'poisson' fish in a net — random arrivals in a fixed area.
“泊松分布”模拟在固定的时间或空间区间内事件发生的次数，事件独立发生且平均发生率为 λ。记作：X ~ Po(λ)。Poisson 看起来像 'poisson'（法语鱼）落入网中——在固定区域内随机到来。
th large n and small p, when np stays moderate, Poisson(λ = np) approximates the binomial. When n is huge and p tiny, B(n,p) ≈ Po(np).
当 n 很大且 p 很小，并且 np 保持中等时，Poisson(λ = np) 近似于二项分布。当 n 巨大、p 微小时，B(n,p) ≈ Po(np)。
8. Normal Distribution | 正态分布
The 'normal distribution' is a continuous distribution with a symmetric bell-shaped curve defined by mean μ and standard deviation σ. Notation: X ~ N(μ, σ²). Bell-shaped, symmetrical, area under curve = 1.
“正态分布”是一种连续分布，具有由均值 μ 和标准差 σ 确定的对称钟形曲线。记作：X ~ N(μ, σ²)。钟形，对称，曲线下面积为 1。
The 'standard normal distribution' andom variable is usually denoted Z. Any normal X can be standardised using z = (x - μ)/σ. Z distribution is the ruler we measure deviations with.
“标准正态分布”具有 μ = 0 和 σ = 1。其随机变量通常记为 Z。任何正态 X 都可以通过 z = (x - μ)/σ 进行标准化。Z 分布是我们度量偏差的尺子。
The empirical rule (68-95-99.7 rule): about 68% of data fall within 1σ of μ, 95% within 2σ, and 99.7% within 3σ. Picture the bell: 1σ captures bulk, 2σ captures most, 3σ almost all.
经验法则（68-95-99.7 法则）：大约 68% 的数据落在 μ ± 1σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。想象大钟：1σ 覆盖主体，2σ 覆盖大多数，3σ 覆盖几乎全部。
9. Sampling Distributions and Central Limit Theorem | 抽样分布与中心极限定理
A 'sampling distribution' is the probability distribution of a statistic (e.g., sample mean) obtained from all possible samples of a fixed size. Not the data distribution, but the distribution of the summary number.
“抽样分布”是从固定大小的所有可能样本中得到的统计量（如样本均值）的概率分布。不是数据本身的分布，而是汇总数值的分布。
The 'Central Limit Theorem' e sampling distribution of the sample mean is approximately normal, regardless of the original population's shape, with mean μ and standard error σ/√n. CLT is the magic that turns any shape into a bell for means.
“中心极限定理” (CLT) 指出，对于足够大的样本量（通常 n ≥ 30），样本均值的抽样分布近似正态，无论原始总体的形状如何，其均值为 μ，标准误为 σ/√n。CLT 是将任何形状的分布变成均值之钟形曲线的魔法。
'Standard error' is the standard deviation of a sampling distribution. For the sample mean, SE = σ/√n. Standard error = uncertainty of the statistic; smaller error with larger sample — dividing by √n shrinks it.
“标准误”是抽样分布的标准差。对于样本均值，SE = σ/√n。标准误 = 统计量的不确定性；样本量越大误差越小——除以 √n 使其缩小。
10. Confidence Intervals | 置信区间
A 'confidence interval' gives a range of plausible values for an unknown population parameter, based on sample data. The confidence level (e.g., 95%) expresses how confident we are that the interval captures the parameter. Interval = inclusion range; confident = how sure.
“置信区间”基于样本数据给出未知总体参数的可能取值范围。置信水平（如 95%）表示我们对区间包含参数的信心程度。区间 = 包容范围；置信 = 有多肯定。
For a population mean with known σ, a 95% confidence interval is x̄ ± 1.96 × σ/√n. The value 1.96 is the critical z-value. 1.96 is the number that gives central 95% in standard normal.
对于已知 σ 的总体均值，95% 置信区间为 x̄ ± 1.96 × σ/√n。数值 1.96 是临界 z 值。1.96 是标准正态中给出中央 95% 的数字。
unknown, we use the t-distribution and sample standard deviation s. The interval becomes x̄ ± t* × s/√n, with degrees of freedom n-1. t fills in when σ is missing.
当 σ 未知时，我们使用 t 分布和样本标准差 s。区间变为 x̄ ± t* × s/√n，自由度为 n-1。t 在 σ 缺失时挺身而出。
11. Hypothesis Testing | 假设检验
A 'null hypothesis' (H₀) is a statement of no effect, no difference, or status quo. The 'alternative hypothesis' (H₁ or Hₐ) is what we hope to support. Null = no news; Alternative = action.
“零假设” (H₀) 是关于无效应、无差异或现状的陈述。“备择假设” (H₁ 或 Hₐ) 是我们希望得到支持的陈述。零假设 = 没有变化；备择假设 = 有作用。
The 'p-value' is the probability, assuming H₀ is true, of obtaining a test statistic at least as extreme as the one observed. p small → surprising under H₀ → reject H₀.
“p 值”是在假定 H₀ 为真的情况下，获得至少与观测值一样极端的检验统计量的概率。p 很小 → 在 H₀ 下令人惊讶 → 拒绝 H₀。
'Significance level' en it is actually true (Type I error). Common choices: 0.05, 0.01. If p ≤ α, we reject H₀. α is the cut-off line; keep it small to avoid false alarms.
“显著性水平” α 是在 H₀ 实际为真时拒绝 H₀ 的概率（第 I 类错误）。常见选择：0.05, 0.01。如果 p ≤ α，则拒绝 H₀。α 是截止线；保持它很小以避免误报。
A 'Type II error' (β) occurs when H₀ is false but we fail to reject it. The power of a test is 1 - β. Type II = missed opportunity; Power = ability to detect an effect.
“第 II 类错误” (β) 发生在 H₀ 为假但我们未能拒绝它时。检验的功效是 1 - β。第 II 类 = 错失良机；功效 = 检出效应的能力。
12. Correlation and Regression | 相关与回归
The 'product-moment correlation coefficient' (Pearson's r) measures the strength and direction of a linear relationship between two variables. Values range from -1 to +1. r close to ±1: strong linear; r close to 0: no linear pattern.
“积矩相关系数”（皮尔逊 r）度量了两个变量之间线性关系的强度和方向。取值范围从 -1 到 +1。r 接近 ±1：强线性；r 接近 0：无线性格局。
'Regression line' (least squares line) of y on x is y = a + bx, where b = Sxy / Sxx. Sxy is co-variation, Sxx is variation in x; slope b tells how much y changes per unit x.
y 对 x 的“回归线”（最小二乘线）为 y = a + bx，其中斜率 b = Sxy / Sxx。Sxy 是共同变异，Sxx 是 x 的变异；斜率 b 表示 x 每增加一个单位 y 的变化量。
'Residual' is the vertical difference between an observed y-value and the value predicted by the regression line. Residual = leftover error; smaller residuals mean better fit.
“残差”是观测的 y 值与回归线预测值之间的垂直差。残差 = 剩余误差；残差越小拟合越好。
'Coefficient of determination' r² is the proportion of the variation in y that is explained by the linear relationship with x. r² = explained share; if r² = 0.81, 81% of variability is accounted for.
“决定系数” r² 是 y 的变异可由与 x 的线性关系解释的比例。r² = 已解释的部分；如果 r² = 0.81，则 81% 的变异性得以解释。
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