📚 Year 12 Cambridge Statistics: Terminology Quick Reference Guide | Year 12 剑桥统计：词汇术语速记指南
A strong grasp of statistical terminology is the foundation of success in Year 12 Cambridge Statistics. Many exam errors stem from confusing terms like ‘parameter’ and ‘statistic’ or misunderstanding the precise definitions of ‘population’ and ‘sample’. This guide is designed to accelerate your memorisation by linking each concept with its exact meaning, notation and common pitfalls, helping you think, write and reason like a statistician.
牢固掌握统计术语是 Year 12 剑桥统计成功的基础。很多考试错误都源于术语混淆，例如分不清“参数”和“统计量”，或者对“总体”和“样本”的准确定义理解含糊。本指南旨在通过联系每个概念的确切含义、符号及常见陷阱，加速你的记忆，帮助你像统计学家一样思考、书写和推理。
1. Population vs Sample | 总体与样本
A population is the entire collection of individuals or items that we want to study. A sample is a subset of the population selected to represent it, usually because measuring the whole population is impractical.
总体是我们想研究的全部个体或项目的集合。样本是从总体中选出的一个子集，用以代表总体，通常因为测量整个总体不现实。
A parameter is a numerical summary that describes a characteristic of a population, such as the population mean μ or population variance σ². It is a fixed but often unknown value.
参数是描述总体特征的数值摘要，例如总体均值 μ 或总体方差 σ²。它是一个固定但通常未知的值。
A statistic is a numerical summary calculated from sample data, such as the sample mean x̄ or sample variance s². We use statistics to estimate unknown population parameters.
统计量是由样本数据计算得到的数值摘要，例如样本均值 x̄ 或样本方差 s²。我们使用统计量来估计未知的总体参数。
A census attempts to collect data from every member of the population. A sample survey collects data from only a part of the population, which is quicker and cheaper but introduces sampling error.
普查试图收集总体中每个成员的数据。抽样调查只收集部分总体的数据，更快、更省钱，但会引入抽样误差。
2. Data Types and Variables | 数据类型与变量
Categorical (qualitative) data can be divided into groups. Nominal data have labels with no intrinsic order (e.g. eye colour: blue, brown, green). Ordinal data have a clear order but the differences between levels are not consistent (e.g. satisfaction rating: poor, good, excellent).
分类（定性）数据可以分成不同的组。名义数据只有标签，没有内在顺序（如眼睛颜色：蓝、棕、绿）。有序数据有明确的顺序，但各等级之间的差值不一致（如满意度评分：差、好、优秀）。
Numerical (quantitative) data are expressed as numbers. Discrete data can only take distinct, separate values, often counts (e.g. number of cars per household). Continuous data can take any value within an interval and are usually measured (e.g. height, time, mass).
数值（定量）数据用数字表示。离散数据只能取特定的、分隔开的数值，通常为计数（如每户汽车数量）。连续数据可以在一个区间内取任意值，通常为测量值（如身高、时间、质量）。
A variable is any characteristic that can vary among units in a population. Understanding the type of variable guides the choice of graphical display and summary statistic.
变量是总体中各个单位之间可能变化的任何特征。理解变量类型有助于选择图形展示和汇总统计量。
3. 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. For a sample it is written x̄, and for a population it is μ. It uses every data point and is therefore sensitive to outliers.
均值（平均数）是所有观测值之和除以观测值个数。样本均值记作 x̄，总体均值记作 μ。它使用了每一个数据点，因此对异常值敏感。
The median is the middle value when the data are arranged in order. For an even number of observations, it is the average of the two middle values. The median is robust to outliers and is often preferred for skewed distributions.
中位数是数据按顺序排列后位于中间的值。若观测值个数为偶数，则是中间两个值的平均数。中位数对异常值稳健，对于偏态分布常被优先选用。
The mode is the value that occurs most frequently. A data set can have no mode, one mode (unimodal) or several modes (bimodal, multimodal). The mode is the only measure of central tendency that can be used with nominal data.
众数是出现频率最高的值。数据集可以没有众数，有一个众数（单峰），或有多个众数（双峰、多峰）。众数是唯一可用于名义数据的集中趋势度量。
When to use each: mean for symmetric distributions without outliers; median for skewed distributions or when outliers are present; mode for categorical data or when the most frequent category matters.
使用场合：均值适用于无异常值的对称分布；中位数用于偏态分布或存在异常值时；众数用于分类数据或关注最常见类别时。
4. Measures of Dispersion | 离散程度度量
The range is the difference between the maximum and minimum values. It gives a very basic sense of spread but is heavily affected by extreme values.
极差是最大值和最小值之差。它给出了最基础的离散程度，但极易受极端值的影响。
The interquartile range (IQR) is the difference between the upper quartile (Q3) and lower quartile (Q1). IQR covers the middle 50% of the data and is resistant to outliers, making it ideal for box plots.
四分位距（IQR）是上四分位数（Q3）与下四分位数（Q1）之差。IQR 覆盖了中间 50% 的数据，并且对异常值具有抵抗力，因此非常适合用于箱线图。
Variance 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). Dividing by (n − 1) gives an unbiased estimator of σ².
方差衡量的是与均值偏差的平方的平均值。总体方差：σ² = Σ(x − μ)² / N。样本方差：s² = Σ(x − x̄)² / (n − 1)。除以 (n − 1) 可得到 σ² 的无偏估计。
Standard deviation is the positive square root of variance (σ or s). It is expressed in the same units as the original data, making it more interpretable than variance for describing spread.
标准差是方差的正平方根（σ 或 s）。它使用与原数据相同的单位，因此在描述离散程度时比方差更易解读。
A small standard deviation relative to the mean indicates data are closely clustered around the centre; a large standard deviation signals wide spread. Always quote the standard deviation alongside the mean.
相对于均值而言较小的标准差表明数据紧密地聚集在中心周围；较大的标准差则表示分布广泛。引用标准差时务必同时给出均值。
5. Graphical Representations | 图形表示
A histogram displays continuous data grouped into intervals (bins). Crucially, the area of each bar is proportional to frequency. On a histogram, we plot frequency density = frequency ÷ class width, not raw frequency.
直方图展示分组后的连续型数据。关键在于，每个条形的面积与频数成正比。在直方图中，我们绘制的是频率密度 = 频数 ÷ 组距，而不是单纯的频数。
A box plot (box-and-whisker diagram) shows the minimum, Q1, median, Q3 and maximum. Potential outliers are marked as individual points beyond the whiskers, typically 1.5 × IQR beyond the quartiles.
箱线图（箱须图）显示最小值、Q1、中位数、Q3 和最大值。潜在的异常值通常标记为超出须线外的单独点，典型标准是超过四分位数 1.5×IQR 之外。
A cumulative frequency graph plots cumulative frequency against the upper class boundary. It is used to estimate quartiles, medians and percentiles. Drawing a smooth curve and reading values off the graph is a key skill.
累积频率图绘制的是累积频数对应组上限的图形。它用于估计四分位数、中位数和百分位数。绘制光滑曲线并从图上读取数值是一项关键技能。
Scatter diagrams display the relationship between two numerical variables. Each point represents a pair (x, y). The pattern of points helps visualise correlation before calculating a coefficient.
散点图展示两个数值变量之间的关系。每个点代表一对 (x, y)。点的分布模式有助于在计算相关系数之前直观地观察相关程度。
6. Probability Fundamentals | 概率基础
A random experiment produces outcomes that are not known in advance. The sample space (S) is the set of all possible outcomes. An event is any subset of the sample space.
随机试验产生的结果事先无法预知。样本空间 (S) 是所有可能结果的集合。事件是样本空间的任意子集。
Two events A and B are mutually exclusive if they cannot occur at the same time: P(A ∩ B) = 0. For mutually exclusive events, P(A ∪ B) = P(A) + P(B).
如果两个事件 A 与 B 不能同时发生，则它们互斥：P(A ∩ B) = 0。对于互斥事件，有 P(A ∪ B) = P(A) + P(B)。
Events A and B are independent if the occurrence of one does not affect the probability of the other: P(A ∩ B) = P(A) × P(B), or equivalently P(A | B) = P(A).
若一个事件的发生不影响另一个事件的概率，则事件 A 与 B 独立：P(A ∩ B) = P(A) × P(B)，或等价地 P(A | B) = P(A)。
Conditional probability P(A | B) is the probability that A occurs given that B has occurred. Formula: P(A | B) = P(A ∩ B) / P(B). This is essential for tree diagrams and contingency tables.
条件概率 P(A | B) 是在 B 已发生的条件下 A 发生的概率。公式：P(A | B) = P(A ∩ B) / P(B)。这对树状图和列联表至关重要。
The addition law: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). This prevents double counting the overlap and applies to any two events.
加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。这避免了重复计算交集部分，适用于任何两个事件。
7. Probability Distributions | 概率分布
A discrete random variable X takes a countable number of possible values. Its probability distribution lists each value x and the associated probability P(X = x). The sum of all probabilities must equal 1.
离散随机变量 X 取可数个可能值。其概率分布列出每个取值 x 以及对应的概率 P(X = x)。所有概率之和必须等于 1。
The expected value E(X) is the long‑run average of X: E(X) = Σ[x · P(X = x)]. It is often called the mean of the distribution, denoted by μ.
期望值 E(X) 是 X 的长期平均值：E(X) = Σ[x · P(X = x)]。它常被称为分布的均值，记作 μ。
The variance Var(X) measures the spread of the distribution: Var(X) = E[(X − μ)²] = E(X²) − [E(X)]². The standard deviation is √Var(X).
方差 Var(X) 衡量分布的离散程度：Var(X) = E[(X − μ)²] = E(X²) − [E(X)]²。标准差为 √Var(X)。
Knowledge of E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X) is crucial for transforming random variables. These linear transformation rules are frequently tested.
掌握 E(aX + b) = aE(X) + b 及 Var(aX + b) = a²Var(X) 对于随机变量的变换至关重要。这些线性变换公式经常出现在考题中。
8. The Binomial Distribution | 二项分布
A binomial distribution arises from a fixed number n of independent trials, each having the same two outcomes (success/failure) with constant probability of success p. We write X ~ B(n, p).
二项分布产生于固定次数 n 的独立试验，每次试验只有两种结果（成功/失败），成功的概率 p 恒定。记作 X ~ B(n, p)。
Formula: P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, where ⁿCᵣ = n! / [r!(n − r)!]. This term counts the number of ways r successes can occur in n trials.
公式：P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，其中 ⁿCᵣ = n! / [r!(n − r)!]。该项计算在 n 次试验中恰好出现 r 次成功的方式数。
For a binomial distribution: mean μ = np and variance σ² = np(1 − p). These allow quick calculations of expected success count and spread without working through the full distribution.
对于二项分布：均值 μ = np，方差 σ² = np(1 − p)。这些公式可快速计算期望成功数和离散程度，而不必完整构建分布列。
Cumulative binomial probabilities P(X ≤ r) are often found using tables or the calculator function. Remember the complementary rule: P(X ≥ r) = 1 − P(X ≤ r − 1).
累积二项概率 P(X ≤ r) 通常借助表格或计算器功能求出。记住互补规则：P(X ≥ r) = 1 − P(X ≤ r − 1)。
9. The Normal Distribution | 正态分布
The normal distribution is a continuous distribution with a symmetric bell-shaped curve. It is defined by its mean μ and standard deviation σ: X ~ N(μ, σ²).
正态分布是一种对称钟形曲线的连续分布。它由其均值 μ 和标准差 σ 定义：X ~ N(μ, σ²)。
The empirical rule (68–95–99.7) states that approximately 68% of data lie within 1σ of μ, 95% within 2σ, and 99.7% within 3σ. This helps quickly interpret standardised values.
经验法则（68–95–99.7）指出：约 68% 的数据落在 μ 的 ±1σ 内，95% 落在 ±2σ 内，99.7% 落在 ±3σ 内。这有助于快速解读标准化后的数值。
To find probabilities, we convert to the standard normal distribution Z ~ N(0, 1²) using z = (x − μ)/σ. Tables give Φ(z) = P(Z < z). Always sketch a diagram to visualise the required region.
为求概率，我们使用 z = (x − μ)/σ 转换为标准正态分布 Z ~ N(0, 1²)。查表给出 Φ(z) = P(Z < z)。务必画图示意所需区域。
The inverse normal is used when a probability is known and we need to find the corresponding x or z value. The key is to work with the left‑tail probability required by the table or calculator.
当概率已知而需要求对应的 x 或 z 值时，使用逆正态。关键是要处理好表或计算器所需的左尾概率。
Do not confuse the notation: σ² is variance, σ is standard deviation. Substituting variance for standard deviation is a frequent mistake. Always square or square‑root before input.
切勿混淆符号：σ² 是方差，σ 是标准差。将方差误作标准差代入公式是一个常见错误。代入前务必先平方或开平方。
10. Correlation and Regression | 相关与回归
Correlation measures the strength and direction of a linear relationship between two variables. The product moment correlation coefficient (r) takes values between −1 and +1. r = +1 indicates perfect positive linear correlation; r = −1 indicates perfect negative linear correlation; r = 0 suggests no linear correlation.
相关衡量两个变量之间线性关系的强度和方向。积矩相关系数 (r) 取 −1 到 +1 之间的值。r = +1 表示完全正线性相关；r = −1 表示完全负线性相关；r = 0 表明没有线性相关。
Correlation does not imply causation. A high r might be due to a lurking third variable. Always interpret correlation within the context of the data.
相关不代表因果。高 r 值可能源自某个潜藏第三变量。永远要结合数据背景来解释相关。
The regression line of y on x has equation y = a + bx, where b = Sxy / Sxx and a = ȳ − b x̄. Sxy = Σ(x − x̄)(y − ȳ) and Sxx = Σ(x − x̄)². This line minimises the sum of squared vertical residuals.
y 对 x 的回归线方程为 y = a + bx，其中 b = Sxy / Sxx，a = ȳ − b x̄。Sxy = Σ(x − x̄)(y − ȳ)，Sxx = Σ(x − x̄)²。这条线使竖直残差平方和最小。
A residual = observed y − predicted y. Residual plots are used to check the validity of the linear model: they should show no pattern and constant spread for a good fit.
残差 = 观测 y − 预测 y。残差图用于检验线性模型的有效性：良好的拟合应呈现无固定模式且散布均匀。
Spearman’s rank correlation coefficient (rₛ) assesses monotonic (not necessarily linear) relationships and is based on ranks rather than raw values. It is robust to outliers and non‑normality.
斯皮尔曼等级相关系数 (rₛ) 评估单调（不一定线性）关系，它基于等级而不是原始数值。它对异常值和非正态分布具有稳健性。
11. Hypothesis Testing Basics | 假设检验基础
The null hypothesis (H₀) is a statement of no effect or no difference that we assume to be true initially. The alternative hypothesis (H₁) represents what we want to prove or suspect is true, such as H₁: p > … or H₁: p ≠ … .
零假设 (H₀) 是一个关于无效应或无差异的陈述，我们最初假设其为真。备择假设 (H₁) 代表我们想证明或怀疑为真的情形，例如 H₁: p > … 或 H₁: p ≠ … 。
The significance level α is the probability of rejecting H₀ when it is actually true (Type I error). Common levels are 5% or 1%. A test is one‑tailed if H₁ specifies a direction (< or >); two‑tailed if H₁ is ≠.
显著性水平 α 是当 H₀ 实际为真时拒绝 H₀ 的概率（第一类错误）。常用水平为 5% 或 1%。若 H₁ 指明了方向（< 或 >），检验为单尾；若 H₁ 为 ≠，则为双尾。
The p‑value is the probability of obtaining a result at least as extreme as the one observed, assuming H₀ is true. If p‑value ≤ α, reject H₀; otherwise, do not reject H₀.
p 值是在假设 H₀ 为真的条件下，得到至少与实际观测结果一样极端的概率。若 p 值 ≤ α，拒绝 H₀；否则，不拒绝 H₀。
The critical region (or rejection region) contains the sample statistics that lead to rejecting H₀. The critical value(s) separate the critical region from the acceptance region based on α.
临界区域（拒绝域）包含导致拒绝 H₀ 的样本统计量。临界值是依据 α 划分临界区域和接受域的分界值。
Type II error occurs when H₀ is false but we fail to reject it. The power of a test is 1 − P(Type II error). Increasing sample size increases power and reduces both types of error in the long run.
第二类错误发生在 H₀ 为假但我们未能拒绝它时。检验功效为 1 − P(第二类错误)。增大样本量可提高功效，长期来看能减少两类错误。
12. Common Mistakes & Memory Aids | 常见错误与记忆口诀
Confusing parameter and statistic: remember ‘P for Population’ and ‘S for Sample’. Parameters are Greek (μ, σ², ρ); statistics are Roman (x̄, s², r).
混淆参数和统计量：记住“参数对总体，统计对样本”。参数用希腊字母（μ, σ², ρ）；统计量用罗马字母（x̄, s², r）。
Using the wrong denominator in variance: when calculating a sample variance to estimate the population variance, use (n − 1), not n. Think ‘sample standard deviation = s with (n − 1)’.
方差计算时分母错误：以样本方差估计总体方差时，分母用 (n − 1) 而不是 n。记住：样本标准差 s 的分母是 (n − 1)。
Treating ordinal data as continuous: Medians and IQR are appropriate for ordinal data; means and standard deviations are not. Match the measure to the data type.
将有序数据当做连续数据处理：中位数和 IQR 适用于有序数据；均值和标准差则不适用。统计量必须与数据类型匹配。
Forgetting to square standard deviation for variance: Var(X) = σ², so if σ = 4, then σ² = 16. A quick check: variance should always be non‑negative and, in the normal distribution, σ² is the second parameter.
忘记为得到方差而平方标准差：Var(X) = σ²，所以若 σ = 4，则 σ² = 16。快速检查：方差总为非负数，在正态分布中 σ² 是第二个参数。
Misapplying the normal distribution to discrete data: the normal distribution is continuous. Only apply continuity corrections when specifically required, and do not use normal approximations unless conditions (e.g. np > 5, nq > 5) are met.
将正态分布误用于离散数据：正态分布是连续分布。只有当明确要求时才能用连续性校正，并且仅在满足条件（如 np > 5, nq > 5）时才使用正态近似。
Memory phrase for correlation: ‘Correlates together does not mean one causes the other.’ Always write ‘no evidence to reject H₀’ rather than ‘accept H₀’, because hypothesis testing is about assessing evidence, not proving the null.
相关记忆口诀：“一起变化不代表一个引起另一个。” 写结论时永远用“没有证据拒绝 H₀”而不是“接受 H₀”，因为假设检验评估的是证据力度，而不是证明零假设。
Picturing the distribution: draw a quick sketch whenever using the normal distribution or finding a critical region. A visual check prevents errors like using the wrong tail or subtracting from 1 incorrectly.
想象分布形状：使用正态分布或求临界区域时，一定要快速画出示意图。视觉检查能避免诸如误用尾巴或错误地从 1 中减去等失误。
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