IGCSE Cambridge Statistics: Core Knowledge Review | IGCSE 剑桥统计：核心知识点梳理

📚 IGCSE Cambridge Statistics: Core Knowledge Review | IGCSE 剑桥统计：核心知识点梳理

This article summarises the essential knowledge areas tested in the Cambridge IGCSE Statistics syllabus (0479), providing clear explanations and exam-focused formulas.

本文梳理剑桥 IGCSE 统计课程（0479）的核心考点，提供清晰解释和考试常用公式。

1. Types of Data | 数据类型

Data can be classified as qualitative (categorical) or quantitative (numerical).

数据可分为定性（分类）数据或定量（数值）数据。

Quantitative data is further split into discrete data, which can only take specific values, and continuous data, which can take any value within a range.

定量数据进一步分为离散数据（只能取特定值）和连续数据（可在某一范围内取任意值）。

Type	Example / 例子
Qualitative / 定性	Favourite colour / 最喜欢的颜色
Quantitative discrete / 定量离散	Number of students / 学生人数
Quantitative continuous / 定量连续	Height in cm / 身高（厘米）

Choosing the correct type is essential because it determines which diagram and which average are appropriate.

选择正确的数据类型很重要，因为它决定了应使用哪种图表和哪种平均数。

2. Data Collection and Sampling | 数据收集与抽样

Primary data is collected by the investigator directly, while secondary data comes from existing sources such as government publications or online databases.

原始数据由调查者直接收集，二手数据来自现有来源，如政府出版物或在线数据库。

Sampling methods include random sampling, stratified sampling, systematic sampling and quota sampling.

抽样方法包括随机抽样、分层抽样、系统抽样和配额抽样。

In stratified sampling, the population is divided into groups called strata, and a random sample is taken from each group in proportion to its size.

在分层抽样中，总体被分成若干层，并按各层大小比例从每层中随机抽取样本。

A sample should be representative of the population so that valid conclusions can be drawn.

样本应具有总体代表性，以便得出有效结论。

3. Charts and Diagrams | 图表与图示

Common diagrams include bar charts, pie charts, histograms, frequency polygons, stem-and-leaf diagrams and pictograms.

常见图示包括条形图、饼图、直方图、频数多边形、茎叶图和象形图。

For grouped continuous data, a histogram uses area to represent frequency, so the vertical axis is frequency density.

对于分组连续数据，直方图用面积表示频数，因此纵轴为频数密度。

Frequency density = Frequency ÷ Class width

Stem-and-leaf diagrams keep the raw data visible while showing the shape of the distribution.

茎叶图在保留原始数据的同时展现分布形态。

A frequency polygon can be drawn by joining the midpoints of the tops of histogram bars.

频数多边形可通过连接直方图各柱顶部中点绘制。

4. Measures of Central Tendency | 集中趋势度量

The three main measures are mode, median and mean.

三个主要度量是众数、中位数和平均数。

The mode is the most frequent value; the median is the middle value when data are ordered; the mean is the sum divided by the number of values.

众数是出现频率最高的数值；中位数是数据排序后的中间值；平均数是总和除以数值个数。

For n ordered values, the median is at position (n + 1) ÷ 2.

对于 n 个已排序数值，中位数的位置是 (n + 1) ÷ 2。

For grouped data, the mean is estimated by Σfx ÷ Σf, where f is frequency and x is the class midpoint.

对于分组数据，平均数用 Σfx ÷ Σf 估计，其中 f 是频数，x 是组中值。

Mean = Σfx ÷ Σf

The median is often preferred when data contain outliers because it is not affected by extreme values.

当数据含有异常值时，通常优先使用中位数，因为它不受极端值影响。

5. Measures of Dispersion | 离散程度度量

Range is the difference between the largest and smallest values.

极差是最大值与最小值之差。

Interquartile range (IQR) = upper quartile – lower quartile, measuring the spread of the middle 50% of the data.

四分位距（IQR）= 上四分位数 – 下四分位数，衡量中间 50% 数据的离散情况。

Variance and standard deviation measure how far the values are from the mean.

方差和标准差衡量各数值与平均数的偏离程度。

For a population of size n, variance is σ² = Σ(x – μ)² ÷ n, and standard deviation is σ = √[Σ(x – μ)² ÷ n].

对于容量为 n 的总体，方差为 σ² = Σ(x – μ)² ÷ n，标准差为 σ = √[Σ(x – μ)² ÷ n]。

σ² = Σ(x – μ)² ÷ n

σ = √[Σ(x – μ)² ÷ n]

For a sample, the divisor is usually n – 1 when estimating the population variance.

对于样本，在估计总体方差时除数通常为 n – 1。

6. Cumulative Frequency and Box Plots | 累积频率与箱线图

Cumulative frequency is the running total of frequencies, used to estimate median, quartiles and percentiles.

累积频率是频数的累计总和，用于估计中位数、四分位数和百分位数。

A cumulative frequency curve, or ogive, is plotted using upper class boundaries against cumulative frequency.

累积频率曲线（尖顶图）以上组界为横坐标、累积频率为纵坐标绘制。

Box plots display minimum, lower quartile, median, upper quartile and maximum.

箱线图显示最小值、下四分位数、中位数、上四分位数和最大值。

Box plots are useful for comparing distributions and identifying outliers.

箱线图有助于比较分布和识别异常值。

7. Probability Rules | 概率规则

Probability measures the chance that an event occurs, with 0 ≤ P(A) ≤ 1.

概率度量事件发生的可能性，满足 0 ≤ P(A) ≤ 1。

If all outcomes are equally likely, P(A) = number of favourable outcomes ÷ total number of outcomes.

若所有结果等可能，则 P(A) = 有利结果数 ÷ 总结果数。

For any event A, P(A’) = 1 – P(A), where A’ is the complement of A.

对于任意事件 A，P(A’) = 1 – P(A)，其中 A’ 是 A 的补事件。

For any two events, P(A or B) = P(A) + P(B) – P(A and B).

对于任意两个事件，P(A 或 B) = P(A) + P(B) – P(A 与 B)。

If A and B are mutually exclusive, then P(A and B) = 0.

若 A 和 B 互斥，则 P(A 与 B) = 0。

If A and B are independent, then P(A and B) = P(A) × P(B).

若 A 和 B 独立，则 P(A 与 B) = P(A) × P(B)。

8. Probability Trees and Venn Diagrams | 概率树与维恩图

Tree diagrams multiply along branches for successive independent events and add for mutually exclusive paths.

树状图在连续独立事件中沿分支相乘，在互斥路径间相加。

Conditional probability is written P(A|B) = P(A and B) ÷ P(B).

条件概率写作 P(A|B) = P(A 与 B) ÷ P(B)。

Venn diagrams illustrate unions, intersections and complements visually.

维恩图直观展示并集、交集和补集。

When using a tree diagram, the probabilities on each set of branches must sum to 1.

使用树状图时，每组分支上的概率之和必须等于 1。

9. The Binomial Distribution | 二项分布

A binomial experiment has a fixed number n of independent trials, each with two outcomes labelled success (probability p) and failure (probability q = 1 – p).

二项试验有固定的 n 次独立试验，每次试验只有两个结果：成功（概率 p）和失败（概率 q = 1 – p）。

The probability of exactly r successes is:

恰好 r 次成功的概率为：

P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ

The mean of a binomial distribution is np, and the variance is npq.

二项分布的平均数为 np，方差为 npq。

Use the binomial formula when trials are independent and the probability of success remains constant.

当各次试验独立且成功概率保持不变时，使用二项公式。

10. Scatter Diagrams and Correlation | 散点图与相关性

A scatter diagram shows the relationship between two variables.

散点图展示两个变量之间的关系。

Correlation describes the strength and direction of a linear relationship: positive, negative or none.

相关性描述线性关系的强度和方向：正相关、负相关或无相关。

Spearman’s rank correlation coefficient rₛ measures monotonic correlation using ranks.

斯皮尔曼等级相关系数 rₛ 通过排序来衡量单调相关。

rₛ = 1 – (6Σd²) ÷ [n(n² – 1)]

Here d is the difference between the paired ranks, and n is the number of data pairs.

其中 d 是成对等级的差值，n 是数据对的数量。

A value close to +1 indicates strong positive correlation; a value close to -1 indicates strong negative correlation.

数值接近 +1 表示强正相关；接近 -1 表示强负相关。

11. Regression Lines | 回归直线

The line of best fit is drawn to model the linear relationship

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IGCSE Cambridge Statistics: Core Knowledge Review | IGCSE 剑桥统计：核心知识点梳理