📚 CIE A Level Statistics 1 Coursebook | CIE A Level 统计学1 核心课程精讲
The CIE A Level Mathematics (9709) Statistics 1 paper is often seen as the most accessible applied module in AS and A Level. It introduces the core language of data, probability and random variables, and it rewards clear notation, organised working and a solid grasp of calculator use. This article follows the official S1 coursebook structure, condensing every major topic into paired English-Chinese revision notes with worked-style explanations.
CIE A Level 数学（9709）统计学1 常被视为 AS 和 A Level 中较容易上手的应用模块。它介绍了数据、概率和随机变量的核心语言，并且强调清晰记号、有序书写和对计算器的熟练使用。本文沿用官方 S1 教材的结构，将每个主要考点浓缩为中英对照的复习笔记，并用示例式讲解展开。
1. The S1 Exam at a Glance | 统计学1考试概览
Paper 5 (S1) lasts 1 hour 15 minutes and carries about 50 marks. Questions are compulsory. You need to know how to summarise data, calculate probabilities, use combinatorics, and work with distributions.
第 5 卷（S1）考试时间为 1 小时 15 分钟，分值约 50 分，全部为必答题。你需要掌握数据摘要、概率计算、排列组合以及分布的应用。
The exam will typically include one large data-based question, a probability or arrangements question, and at least one question on the binomial or normal distribution.
试卷通常包含一道较大的数据题、一道概率或排列组合题，以及至少一道二项分布或正态分布题。
Formula sheets are not provided, so you must memorise key formulas. However, statistical tables for the normal distribution are included.
考试不提供公式表，因此你必须熟记关键公式。但正态分布表会随试卷提供。
2. Representing Data | 数据表示
You must be able to read and construct stem-and-leaf diagrams, box-and-whisker plots, histograms and cumulative frequency diagrams. Each representation has a specific purpose: stem plots preserve raw values, box plots show spread and outliers, and histograms model frequency density.
你必须能够阅读并绘制茎叶图、箱线图、直方图和累积频率图。每种图都有独特作用：茎叶图保留原始数据，箱线图展示离散程度和离群值，直方图则用频率密度建模。
For a histogram with unequal class widths, the vertical axis is frequency density, calculated as frequency ÷ class width.
在组距不等的直方图中，纵轴为频率密度，计算公式为 频率 ÷ 组距。
A cumulative frequency graph is used to estimate the median, quartiles and percentiles by reading off the value at half, quarter or other fractions of total frequency.
累积频率图可用于估计中位数、四分位数和百分位数，方法是在总频率的一半、四分之一或其他比例处读取对应值。
3. Measures of Central Tendency | 集中趋势度量
The three central measures are the mean, median and mode. For ungrouped data, the mean is x̄ = Σx / n. For grouped data, use midpoints: x̄ = Σfx / Σf.
三种集中趋势度量是平均数、中位数和众数。对于未分组数据，平均数公式为 x̄ = Σx / n。对于分组数据，使用组中点：x̄ = Σfx / Σf。
The median is the middle value when data are ordered. If n is odd, it is the (n+1)/2 th value; if n is even, it is the average of the n/2 th and (n/2 + 1) th values.
中位数是数据排序后的中间值。若 n 为奇数，则为第 (n+1)/2 个值；若 n 为偶数，则为第 n/2 个和第 (n/2 + 1) 个值的平均数。
The mode is the most frequent value. In a grouped frequency table, the modal class is the class with the highest frequency density, not necessarily the highest frequency.
众数是出现频率最高的值。在分组频率表中，众数所在的组应是频率密度最高的组，而不一定是频率最高的组。
A key exam skill is choosing the best measure: use the median when data are skewed or contain outliers; use the mean when data are roughly symmetric because it uses all values.
考试中的关键能力是选择最佳度量：当数据偏斜或含有离群值时用中位数；当数据大致对称时用平均数，因为它利用了所有数值。
4. Measures of Spread | 离散程度度量
mple measures of spread. IQR = Q₃ − Q₁, where Q₁ and Q₃ are the lower and upper quartiles.
极差和四分位距（IQR）是简单的离散程度度量。IQR = Q₃ − Q₁，其中 Q₁ 和 Q₃ 分别是下四分位数和上四分位数。
Variance and standard deviation are more useful because they use every data value. For ungrouped data, variance is s² = Σ(x − x̄)² / n, equivalent to Σx² / n − x̄².
方差和标准差更有用，因为它们使用了每一个数据值。对于未分组数据，方差为 s² = Σ(x − x̄)² / n，也等价于 Σx² / n − x̄²。
For grouped data, replace x by class midpoints.
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