📚 Year 11 CIE Statistics: Core Knowledge Review | Year 11 CIE 统计：核心知识点梳理
Welcome to your one‑stop revision guide for Year 11 CIE Statistics. This article gathers all the key knowledge points from the IGCSE syllabus, including data handling, probability, statistical measures, and real‑world applications. Mastering these core concepts will build a strong foundation for exam success and advanced studies.
欢迎阅读 Year 11 CIE 统计一站式复习指南。本文汇集了 IGCSE 大纲的所有核心知识点，涵盖数据处理、概率、统计量度和实际应用。掌握这些核心概念将为考试成功和进阶学习打下坚实基础。
1. Types of Data | 数据类型
Data is classified into two broad categories: qualitative (categorical) and quantitative (numerical). Qualitative data describes qualities or attributes, such as colours, makes of car, or survey responses like ‘Yes’/’No’.
数据分为两大类：定性（分类）数据和定量（数值）数据。定性数据描述性质或属性，如颜色、汽车品牌或问卷中’是’/’否’等回答。
Quantitative data consists of numbers. It can be discrete or continuous. Discrete data takes only specific, often counted values (e.g., number of students in a class, number of cars in a car park). Continuous data can take any value within a range and is usually measured (e.g., height, weight, time, temperature).
定量数据由数字组成。它可以是离散的或连续的。离散数据仅取特定值，通常是计数得到的（如班级学生数、停车场汽车数量）。连续数据可以在一个范围内取任意值，通常是测量得到的（如身高、体重、时间、温度）。
Identifying the data type is the first step in any statistical analysis because it determines which charts and summary statistics are appropriate.
在任何统计分析中，识别数据类型是第一步，因为它决定了哪些图表和汇总统计量是合适的。
2. Data Collection and Sampling Methods | 数据收集与抽样方法
Data can be primary (collected directly by the researcher for a specific purpose) or secondary (obtained from existing sources such as government publications, databases, or previous studies).
数据可以是初级数据（由研究者为特定目的直接收集）或次级数据（从现有资料获得，如政府出版物、数据库或先前的研究）。
Common sampling techniques include: simple random sampling, where every member of the population has an equal chance of selection; stratified sampling, which divides the population into distinct subgroups (strata) and samples proportionally from each; systematic sampling, where every k‑th item is chosen from a list; and quota sampling, a non‑random method that selects individuals to match pre‑defined characteristics.
常用的抽样方法包括：简单随机抽样，总体中每个成员都有相等被选中的机会；分层抽样，将总体分成不同的子群（层），并按比例从每层中抽样；系统抽样，从名单中每隔固定间隔选取一个个体；配额抽样，一种非随机方法，选取个体以匹配预先确定的特征。
Bias arises when a sample does not fairly represent the population. Careful design helps minimise under‑coverage, response bias, or measurement errors.
当样本不能公平地代表总体时，就会产生偏差。精心设计有助于减少覆盖不足、回答偏差或测量误差。
3. Displaying Data: Charts and Graphs | 数据展示：图表
Choosing the right visualisation is essential. For qualitative data, bar charts, pie charts and pictograms are common. For discrete quantitative data, bar charts and frequency polygons are often used.
选择合适的可视化方式至关重要。对于定性数据，常用条形图、饼图和象形图。对于离散定量数据，常使用条形图和频数多边形。
For continuous data, histograms are the standard. In a histogram, the area of each bar is proportional to the frequency. To achieve this, frequency density is calculated as frequency ÷ class width. A frequency polygon can be created by joining the midpoints of the tops of the histogram bars.
对于连续数据，直方图是标准工具。在直方图中，每个矩形的面积与频数成正比。为实现这一点，需要计算频率密度 = 频数 ÷ 组距。通过连接直方图条顶端的中点可以创建频数多边形。
Stem‑and‑leaf diagrams show the shape of the distribution while keeping all original data values. Box‑and‑whisker plots (box plots) display the minimum, lower quartile, median, upper quartile and maximum, offering a clear summary of spread.
茎叶图既显示分布形状，又保留了所有原始数据值。箱线图（盒形图）展示最小值、下四分位数、中位数、上四分位数和最大值，对数据散布程度进行了清晰概括。
4. Measures of Central Tendency | 集中趋势的度量
The three principal measures are the mean, median and mode. The mean (x̄) is the arithmetic average: sum of all values divided by the number of values. It is suitable for symmetrical data but is sensitive to outliers.
三个主要量度是平均数、中位数和众数。平均数 (x̄) 是算术平均值：所有数值之和除以数值的个数。它适用于对称数据，但对离群值敏感。
The median is the middle value when data are arranged in order. For grouped data, linear interpolation can estimate the median using cumulative frequency. The median is robust to outliers and is preferred for skewed distributions.
中位数是数据按顺序排列后位于中间的值。对于分组数据，可利用累积频率进行线性插值来估计中位数。中位数不易受离群值影响，是偏态分布的首选。
The mode is the most frequently occurring value or class. For skewed data, it can highlight the peak of the distribution. In a perfectly symmetrical distribution, mean ≈ median ≈ mode.
众数是出现次数最多的值或组。对于偏态数据，它可以突出分布的峰值。在完全对称的分布中，平均数 ≈ 中位数 ≈ 众数。
5. Measures of Dispersion | 离散程度的度量
Dispersion measures how spread out the data are. The range (max – min) is the simplest but is affected by extreme values. The interquartile range (IQR = Q₃ – Q₁) covers the middle 50% of data and is resistant to outliers.
离散程度衡量数据的分散程度。极差（最大值 – 最小值）最简单，但受极值影响。四分位距（IQR = Q₃ – Q₁）涵盖了中间 50% 的数据，并且不受离群值影响。
Variance and standard deviation measure variation around the mean. For a sample, variance s² = Σ(x – x̄)² / (n – 1). The standard deviation s is the square root of variance. A larger standard deviation indicates greater spread.
方差和标准差度量围绕平均数的变异。对于样本，方差 s² = Σ(x – x̄)² / (n – 1)。标准差 s 是方差的平方根。标准差越大表示分散程度越大。
Use IQR when the data is skewed or has outliers; use standard deviation together with the mean when the distribution is roughly symmetric and free of extreme values.
当数据偏斜或存在离群值时，使用 IQR；当分布大致对称且无极端值时，将标准差与平均数结合使用。
6. Cumulative Frequency and Box Plots | 累积频率与箱线图
A cumulative frequency table adds up frequencies as you move through classes. Plotting cumulative frequency against the upper class boundary produces an S‑shaped curve (ogive).
累积频率表在遍历各组时将频数逐组累加。将累积频率对组上界绘图，会得到一条 S 形曲线（累积频率曲线）。
From the graph, the median is the value at 50% of the total frequency. Q₁ is found at 25%, and Q₃ at 75%. The interquartile range is Q₃ – Q₁, which measures the spread of the central half of the data.
从图上可以找到中位数，即总频数 50% 处的值。Q₁ 在 25% 处，Q₃ 在 75% 处。四分位距为 Q₃ – Q₁，衡量数据中心一半的散布程度。
A box plot visualises the five‑number summary: minimum, Q₁, median, Q₃, maximum. Outliers can be identified using the 1.5×IQR rule: values below Q₁ – 1.5×IQR or above Q₃ + 1.5×IQR are potential outliers, often plotted as individual points.
箱线图将五数概括可视化：最小值、Q₁、中位数、Q₃、最大值。离群值可用 1.5×IQR 规则识别：低于 Q₁ – 1.5×IQR 或高于 Q₃ + 1.5×IQR 的值可能是离群值，通常绘制为单独的点。
7. Probability | 概率
Probability quantifies how likely an event is. For equally likely outcomes, P(A) = number of favourable outcomes / total number of outcomes. The probability of the complement is P(not A) = 1 – P(A).
概率量化事件发生的可能性。对于等可能结果，P(A) = 有利结果数 / 可能结果总数。对立事件的概率是 P(非A) = 1 – P(A)。
For combined events, the addition rule is P(A or B) = P(A) + P(B) – P(A and B). If A and B are mutually exclusive, P(A and B) = 0. Conditional probability P(A|B) = P(A and B) / P(B).
对于组合事件，加法规则为 P(A 或 B) = P(A) + P(B) – P(A 和 B)。如果 A 和 B 互斥，则 P(A 和 B) = 0。条件概率 P(A|B) = P(A 和 B) / P(B)。
Tree diagrams help break down multi‑stage experiments. Multiply probabilities along branches for ‘and’ scenarios; add probabilities of different branches for ‘or’ scenarios. Always ensure probabilities on branches from a single point sum to 1.
树形图有助于分解多阶段试验。对于’且’的情形，沿分支将概率相乘；对于’或’的情形，将不同分支的概率相加。务必确保从同一点出发的各分支概率之和为 1。
8. Binomial Distribution | 二项分布
A binomial distribution applies when there are a fixed number of independent trials n, each with two outcomes (success/failure), and a constant probability of success p (with q = 1 – p).
二项分布适用于以下情况：固定次数的独立试验 n，每次试验有两种结果（成功/失败），且每次成功的概率 p 恒定（q = 1 – p）。
The probability of obtaining exactly r successes is given by the binomial probability function:
恰好获得 r 次成功的概率由二项概率函数给出：
P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ
The mean (expected value) and variance of a binomial distribution are:
二项分布的均值（期望值）和方差为：
| Mean (μ) | μ = np |
| Variance (σ²) | σ² = npq |
You may be expected to calculate probabilities using the formula, tables, or technology. Binomial problems often appear in quality control and survey contexts.
你可能需要利用公式、查表或使用技术来计算概率。二项分布问题常出现在质量控制和调查分析的背景中。
9. Scatter Graphs and Correlation | 散点图与相关性
A scatter diagram displays the relationship between two variables. Look for patterns: positive correlation (as x increases, y increases), negative correlation (as x increases, y decreases), or no clear pattern.
散点图展示两个变量之间的关系。观察模式：正相关（x 增加，y 增加）、负相关（x 增加，y 减少），或无明显模式。
Pearson’s product‑moment
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