📚 Year 12 Cambridge Statistics: Key Terms Quick Guide | 统计词汇术语速记指南
This guide is designed to help Year 12 Cambridge students quickly memorise essential statistical terminology. Mastering the language of statistics is the first step towards solving problems with confidence. Below, you will find core concepts grouped by topic, each explained concisely in both English and Chinese.
本指南旨在帮助 Year 12 剑桥学生快速记忆核心统计术语。掌握统计语言是自信解题的第一步。下面按主题分组介绍核心概念，每个概念均提供简洁的中英文解释。
1. Basic Concepts | 基本概念
A population is the entire set of individuals or items that we wish to study. A census collects data from every member of the population. A sample is a subset of the population selected to represent it, making data collection more practical and less costly.
总体是我们希望研究的全部个体或项目的集合。普查从总体的每个成员收集数据。样本是从总体中选出的、用于代表总体的子集，使得数据收集更加可行、成本更低。
A variable is any characteristic that can vary among individuals, such as height, test score, or eye colour. The values of variables form the raw data we analyse.
变量是可以在个体之间变化的各种特征，例如身高、测试成绩或眼睛颜色。变量的取值构成了我们分析的原始数据。
2. Types of Data | 数据类型
Qualitative (categorical) data describe qualities or categories, e.g. eye colour, blood type. They can be nominal (no natural order) or ordinal (ordered categories, like satisfaction ratings).
定性（分类）数据描述属性或类别，如眼睛颜色、血型。它们可以是名义型（无自然顺序）或顺序型（有序类别，如满意度评级）。
Quantitative (numerical) data arise from counting or measuring. Discrete data take only distinct, separate values (e.g. number of students). Continuous data can take any value within a range (e.g. time, mass).
定量（数值）数据来自计数或测量。离散型数据只取特定的、分离的数值（如学生人数）。连续型数据可以在一个区间内取任意值（如时间、质量）。
3. Measures of Central Tendency | 集中趋势的度量
The mean (x̄ for a sample, μ for a population) is the arithmetic average: sum all values and divide by the number of values. It uses every data point but is sensitive to extreme values.
均值（样本记作 x̄，总体记作 μ）是算术平均数：将所有数值相加后除以数据个数。它利用了每个数据点，但对极端值敏感。
The median is the middle value when the data are ordered. It is not affected by outliers and provides a better measure of centre for skewed distributions.
中位数是数据排序后位于中间的值。它不受异常值影响，对于偏态分布能提供更好的中心度量。
The mode is the most frequently occurring value. A data set may be unimodal, bimodal or multimodal. It is the simplest measure and can be used for qualitative data.
众数是出现频率最高的值。数据集可以是单峰、双峰或多峰的。它是最简单的度量，且可用于定性数据。
4. Measures of Spread | 离散程度的度量
The range is the difference between the largest and smallest values. It is easy to calculate but ignores the distribution of intermediate values.
极差是最大值与最小值之差。它计算简单，但忽略了中间值的分布情况。
The interquartile range (IQR) is Q₃ − Q₁, the range of the middle 50% of the data. It is resistant to outliers and gives a better picture of spread around the median.
四分位距 (IQR) 为 Q₃ − Q₁，即中间 50% 数据的范围。它抗异常值，能更好地描述中位数附近的散布情况。
The variance measures the average squared deviation from the mean. The standard deviation is the square root of variance, returning to the original units.
方差衡量各数据与均值偏差的平方的平均值。标准差是方差的平方根，恢复到原始单位。
σ² = Σ(x − μ)² / N | s² = Σ(x − x̄)² / (n − 1)
5. Probability Fundamentals | 概率基础
A random experiment is a process whose outcome cannot be predicted with certainty. An outcome is a single possible result, and the sample space (S) is the set of all possible outcomes.
随机试验是一个结果无法确定预测的过程。结果是单个可能的结果，样本空间 (S) 是所有可能结果的集合。
An event is a subset of the sample space. The probability of an event A is P(A) = n(A) / n(S), assuming equally likely outcomes. P(impossible event) = 0, P(certain event) = 1.
事件是样本空间的一个子集。事件 A 的概率为 P(A) = n(A) / n(S)，假设等可能结果。不可能事件的概率为 0，必然事件的概率为 1。
6. Probability Rules | 概率法则
The complement of A, denoted A’, is the event that A does not happen: P(A’) = 1 − P(A). The addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, they cannot occur together, so P(A ∩ B) = 0.
A 的补事件记作 A’，表示 A 不发生的事件：P(A’) = 1 − P(A)。加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若 A 与 B 互斥，它们不能同时发生，则 P(A ∩ B) = 0。
Two events are independent if the occurrence of one does not affect the probability of the other. Then P(A ∩ B) = P(A) × P(B). Conditional probability is P(A|B) = P(A ∩ B) / P(B), the probability of A given that B has occurred.
若一个事件的发生不影响另一个事件的概率，则两事件独立。此时 P(A ∩ B) = P(A) × P(B)。条件概率为 P(A|B) = P(A ∩ B) / P(B)，表示在 B 发生的条件下 A 发生的概率。
| English Term | 中文术语 | Formula / Note |
|---|---|---|
| Addition Rule | 加法法则 | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) |
| Multiplication Rule (independent) | 乘法法则（独立） | P(A ∩ B) = P(A) P(B) |
| Conditional Probability | 条件概率 | P(A|B) = P(A ∩ B) / P(B) |
7. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of possible values, each with a stated probability. The probability distribution is often displayed as a table. The sum of all probabilities must equal 1.
离散随机变量 X 取可数个可能值，每个值有一个确定的概率。概率分布常以表格形式呈现。所有概率之和必须等于 1。
The expected value E(X) = Σ x·p(x) is the theoretical long-run average. The variance Var(X) = E[(X − μ)²] = E(X²) − [E(X)]². For a linear transformation, E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).
期望值 E(X) = Σ x·p(x) 是理论上的长期平均值。方差 Var(X) = E[(X − μ)²] = E(X²) − [E(X)]²。对于线性变换，有 E(aX + b) = aE(X) + b，Var(aX + b) = a²Var(X)。
8. The Binomial Distribution | 二项分布
A binomial distribution models the number of successes in a fixed number of independent trials, n, each with the same probability of success, p. It is denoted X ~ B(n, p). The four conditions are: fixed n, two outcomes per trial, constant p, and independent trials.
二项分布描述了在固定次数 n 的独立试验中成功的次数，每次试验的成功概率 p 相同。记作 X ~ B(n, p)。四个条件是：固定 n、每次试验两种结果、p 恒定和试验独立。
The probability of exactly r successes is P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ, where C(n, r) is the binomial coefficient. The mean is E(X) = np, and the variance is Var(X) = np(1 − p).
恰好成功 r 次的概率为 P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ，其中 C(n, r) 为二项式系数。均值为 E(X) = np，方差为 Var(X) = np(1 − p)。
9. The Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. It is fully described by its mean μ and variance σ², written as X ~ N(μ, σ²). The total area under the curve equals 1.
正态分布是一种连续型概率分布，呈对称的钟形曲线。它完全由均值 μ 和方差 σ² 描述，记作 X ~ N(μ, σ²)。曲线下的总面积为 1。
To calculate probabilities, we standardise to the standard normal variable Z ~ N(0, 1) using z = (x − μ) / σ. The standard normal table or calculator then gives Φ(z) = P(Z < z). Key symmetry: Φ(−z) = 1 − Φ(z).
为计算概率，我们通过 z = (x − μ) / σ 转化为标准正态变量 Z ~ N(0, 1)。然后用标准正态表或计算器得到 Φ(z) = P(Z < z)。关键对称性：Φ(−z) = 1 − Φ(z)。
10. Sampling Techniques | 抽样方法
Simple random sampling gives every member of the population an equal chance of selection, with samples chosen without bias. Stratified sampling divides the population into distinct groups (strata) and randomly selects from each in proportion to its size.
简单随机抽样使总体中每个成员被选中的机会相等，样本选择无偏。分层抽样将总体分成不同的层，并按各层大小比例从中随机选取。
Systematic sampling selects every kth individual from an ordered list. Cluster sampling divides the population into clusters, then randomly selects whole clusters. Quota sampling (non-probability) selects a fixed number from each category, but is not truly random.
系统抽样从有序列表中每隔 k 个个体选一个。整群抽样将总体分成群，然后随机选取整个群。配额抽样（非概率抽样）从每个类别选取固定数量，但并非完全随机。
11. Representing Data | 数据表示
A histogram displays grouped continuous data. The area of each bar is proportional to frequency, so frequency density = frequency / class width. It is crucial for unequal class intervals.
直方图展示分组连续数据。每个直方的面积与频数成正比，因此频率密度 = 频数 ÷ 组距。这在组距不等时极为关键。
A cumulative frequency curve plots running total frequency against the upper class boundary. It can be used to estimate medians, quartiles, and percentiles graphically.
累积频率曲线将累计频数相对于组上界绘制。可用于从图形上估计中位数、四分位数和百分位数。
A box-and-whisker plot displays the minimum, Q₁, median, Q₃, and maximum. Outliers are typically defined as values < Q₁ − 1.5×IQR or > Q₃ + 1.5×IQR.
箱线图显示最小值、Q₁、中位数、Q₃ 和最大值。异常值通常定义为 < Q₁ − 1.5×IQR 或 > Q₃ + 1.5×IQR 的数值。
12. Correlation and Regression | 相关与回归
The scatter diagram shows the relationship between two variables. Correlation describes the strength and direction of a linear relationship. The product moment correlation coefficient (PMCC), r, satisfies −1 ≤ r ≤ 1.
散点图展示两个变量之间的关系。相关描述线性关系的强度与方向。积矩相关系数 (PMCC) r 满足 −1 ≤ r ≤ 1。
The regression line of y on x has equation y = a + bx, where b = Sxy / Sxx and a = ȳ − b x̄. It is used to predict y from x, but interpolation (within the data range) is reliable, while extrapolation (outside) can be misleading.
y 关于 x 的回归线方程为 y = a + bx，其中 b = Sxy / Sxx，a = ȳ − b x̄。它用于从 x 预测 y，但内插（在数据范围内）是可靠的，而外推（超出范围）可能产生误导。
Published by TutorHao | Statistics Revision Series | aleveler.com
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