📚 IGCSE CCEA Statistics: Core Knowledge Revision | IGCSE CCEA 统计：核心知识点梳理
This revision guide summarises the core knowledge required for the IGCSE CCEA Statistics course. It covers data handling, averages, spread, probability, bivariate data and key distributions in a clear bilingual format designed for quick revision and exam preparation.
本复习指南总结了 IGCSE CCEA 统计课程所需的核心知识点。它以清晰的中英双语形式涵盖数据处理、平均数、离散程度、概率、双变量数据以及重要分布，适合快速复习和备考。
1. Types of Data | 数据类型
In statistics, data are usually classified as qualitative or quantitative. Qualitative data describe categories or qualities, while quantitative data measure numerical amounts.
在统计学中，数据通常分为定性数据和定量数据。定性数据描述类别或性质，而定量数据测量数值大小。
Quantitative data can be further divided into discrete data, which take only exact separate values, and continuous data, which can take any value within a measured range.
定量数据可进一步分为离散数据和连续数据。离散数据只能取明确的分离数值，连续数据则可以在测量范围内取任意数值。
Qualitative data may also be nominal, where categories have no natural order, or ordinal, where categories can be ranked.
定性数据还可以是名义数据或有序数据。名义数据的类别没有自然顺序，有序数据的类别可以排序。
| Type | Description | Example |
| Qualitative nominal | Categories with no order | Eye colour, gender |
| Qualitative ordinal | Categories with a natural order | Survey rating: poor, fair, good |
| Quantitative discrete | Countable separate values | Number of students, goals scored |
| Quantitative continuous | Measured values on a scale | Height, time, temperature |
2. Data Collection and Sampling | 数据收集与抽样
A census collects data from every member of a population, while a sample collects data from only part of the population. A census gives fully representative results but is often too expensive or time-consuming.
普查从总体中的每一个成员收集数据，而抽样只从总体的一部分收集数据。普查能给出完全具有代表性的结果，但通常过于昂贵或耗时。
Common sampling methods include simple random sampling, stratified sampling, systematic sampling, quota sampling and convenience sampling. Each method has different strengths and weaknesses.
常见的抽样方法包括简单随机抽样、分层抽样、系统抽样、配额抽样和便利抽样。每种方法都有不同的优点和缺点。
Simple random sampling gives every member an equal chance of selection and removes bias, but it needs a complete sampling frame. Stratified sampling divides the population into relevant groups and takes a random sample from each group in proportion to its size, ensuring all groups are represented.
简单随机抽样使每个成员被选中的机会相等，并消除偏差，但需要完整的抽样框。分层抽样将总体分成相关小组，并按比例从每个小组中随机抽样，确保所有小组都得到代表。
Systematic sampling selects every kth item after a random start, which is quick and easy but can be biased if there is a hidden pattern in the list. Quota sampling is non-random and can introduce interviewer bias, while convenience sampling is fast but rarely representative.
系统抽样在随机起点之后每隔 k 个抽取一个，快速简便，但如果列表中存在隐藏模式则可能产生偏差。配额抽样是非随机的，可能引入访问者偏差；便利抽样虽然快捷，但很少具有代表性。
3. Presenting Data: Charts and Diagrams | 数据展示：图表
Bar charts are used for categorical or discrete data, with equal gaps between bars. Pie charts show proportions of a whole, where each sector angle is calculated as frequency divided by total multiplied by 360 degrees.
条形图用于分类数据或离散数据，条形之间有相等的间隙。饼图显示整体的比例，每个扇形的角度等于频数除以总数再乘以 360 度。
Histograms are used for continuous data. Unlike bar charts, there are no gaps between bars, and the area of each bar is proportional to the frequency, so frequency density must be used when class widths are unequal.
直方图用于连续数据。与条形图不同，直方图的条形之间没有间隙，每个条形的面积与频数成比例，因此当组距不相等时必须使用频率密度。
The formula for frequency density is frequency divided by class width. On a cumulative frequency curve, the median is read at half the total frequency, and the lower and upper quartiles are read at one quarter and three quarters of the total frequency.
频率密度的公式是频数除以组距。在累积频率曲线上，中位数在总频数的一半处读取，下四分位数在总频数的四分之一处读取，上四分位数在总频数的四分之三处读取。
A box plot, also called a box-and-whisker plot, displays the minimum, lower quartile, median, upper quartile and maximum. It is very useful for comparing two distributions and identifying skewness.
箱线图，也称为箱须图，展示最小值、下四分位数、中位数、上四分位数和最大值。它对于比较两个分布和识别偏态非常有用。
4. Measures of Central Tendency | 集中趋势度量
The three main measures of central tendency are the mean, median and mode. The mean is the arithmetic average, the median is the middle value when data are ordered, and the mode is the most frequent value.
三种主要的集中趋势度量是平均数、中位数和众数。平均数是算术平均值，中位数是数据排序后的中间值，众数是最常出现的值。
For a list of n values, the mean is calculated by adding all values and dividing by n. The symbol x̄ represents the sample mean.
对于 n 个数值，平均数的计算方法是将所有数值相加后除以 n。符号 x̄ 表示样本平均数。
Mean: x̄ = Σx / n
For grouped data, the exact mean cannot be found, so we estimate it by using the midpoint of each class. The estimated mean is Σfx divided by Σf, where x is the class midpoint and f is the frequency.
对于分组数据，无法求得精确平均数，因此我们使用每组的组中点来估计平均数。估计平均数为 Σfx 除以 Σf，其中 x 是组中点，f 是频数。
The median is less affected by extreme values than the mean, so it is often preferred for skewed distributions. The mode is most useful for categorical data or for finding the most common value.
中位数受极端值的影响比平均数小，因此对于偏态分布通常更适用。众数对于分类数据或寻找最常见的数值最为有用。
5. Measures of Dispersion | 离散程度度量
Dispersion measures how spread out the data are. The simplest measure is the range, which is the difference between the largest and smallest values.
离散程度衡量数据的分散程度。最简单的度量是极差，即最大值与最小值之差。
The interquartile range, or IQR, is the difference between the upper quartile and lower quartile. It measures the spread of the middle 50% of the data and is not affected by extreme values.
四分位距（IQR）是上四分位数与下四分位数之差。它衡量中间 50% 数据的分散程度，不受极端值影响。
Variance and standard deviation give a more precise measure of spread by using all data values. For a sample, the variance is the average of the squared deviations from the mean, with n minus 1 in the denominator.
方差和标准差通过使用所有数据值给出更精确的离散程度度量。对于样本，方差是与平均数偏差平方的平均值，分母为 n 减 1。
Variance: s² = Σ(x − x̄)² / (n − 1)
Standard deviation: s = √[ Σ(x − x̄)² / (n − 1) ]
as values lying more than 1.5 × IQR below the lower quartile or above the upper quartile.
标准差越大，数据越分散。离群值通常定义为低于下四分位数超过 1.5 × IQR 或高于上四分位数超过 1.5 × IQR 的数值。
6. Probability Basics | 概率基础
Probability measures how likely an event is to occur. It always lies between 0 and 1, where 0 means impossible and 1 means certain. The sample space lists all possible outcomes.
概率衡量事件发生的可能性。它始终在 0 到 1 之间，0 表示不可能，1 表示必然。样本空间列出所有可能的结果。
For equally likely outcomes, probability is the number of favourable outcomes divided by the total number of outcomes. The relative frequency of an event from an experiment can also be used to estimate probability.
对于等可能的结果，概率等于有利结果的数量除以结果的总数。实验中事件的相对频率也可用于估计概率。
P(A) = Number of favourable outcomes / Total number of outcomes
Mutually exclusive events cannot happen at the same time, so the probability of A or B occurring is P(A) + P(B). For non-mutually exclusive events, the intersection must be subtracted.
互斥事件不能同时发生，因此 A 或 B 发生的概率为 P(A) + P(B)。对于非互斥事件，必须减去交集。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
ing is P(A) × P(B). If events are not independent, conditional probabilities must be used.
独立事件互不影响，因此 A 和 B 同时发生的概率为 P(A) × P(B)。如果事件不独立，则必须使用条件概率。
7. Conditional Probability and Tree Diagrams | 条件概率与树形图
Conditional probability is the probability of event A occurring given that event B has already occurred. It is written as P(A|B) and calculated by dividing the probability of both events by the probability of the given event.
条件概率是在事件 B 已经发生的条件下事件 A 发生的概率。它写作 P(A|B)，计算方法是两个事件同时发生的概率除以给定事件的概率。
P(A|B) = P(A ∩ B) / P(B)
Tree diagrams are ideal for multi-stage probability problems. Along each branch, probabilities are written, and to find the probability of two events on a path, the branch probabilities are multiplied together.
树形图是解决多阶段概率问题的理想工具。每个分支上标有概率，要找到一条路径上两个事件的概率，需要将分支概率相乘。
If there are several possible paths leading to the same final outcome, the probabilities of those paths are added. For sampling without replacement, the probabilities on the second set of branches change, so the events are not independent.
如果有多条路径通向相同的最终结果，则需要将这些路径的概率相加。对于不放回抽样，第二组分支上的概率会发生变化，因此这些事件不是独立的。
8. Discrete Random Variables and Expectation | 离散随机变量与期望
A discrete random variable can take only a countable number of values, each with a given probability. A probability distribution lists all possible values and their associated probabilities, and the sum of all probabilities must equal 1.
离散随机变量只能取可数个值，每个值都有给定的概率。概率分布列出所有可能的取值及其相应概率，所有概率之和必须等于 1。
The expected value of a random variable, written E(X), is the long-run average value if the experiment is repeated many times. It is found by multiplying each value by its probability and adding the results.
随机变量的期望值，写作 E(X)，是实验重复多次后的长期平均值。计算方法是将每个值乘以其概率后相加。
E(X) = Σ x·P(X = x)
The variance of a random variable measures how far the values spread from the expectation. It can be calculated as E(X²) minus the square of E(X).
随机变量的方差衡量各值相对期望值的偏离程度。它可以计算为 E(X²) 减去 E(X) 的平方。
Var(X) = E(X²) − [E(X)]²
A common discrete distribution is the binomial distribution, which models the number of successes in a fixed number of independent trials. Each trial has only two outcomes: success or failure.
常见的离散分布是二项分布，它模拟在固定次数的独立试验中成功的次数。每次试验只有两个结果：成功或失败。
P(X = r) = C(n, r) × pʳ × (1 − p)ⁿ⁻ʳ
expectation and variance for a binomial distribution are E(X) = np and Var(X) = np(1 − p).
其中 n 是试验次数，p 是成功概率，r 是成功次数。二项分布的期望和方差分别为 E(X) = np 和 Var(X) = np(1 − p)。
9. Bivariate Data and Correlation | 双变量数据与相关性
Bivariate data involve two variables measured on the same individuals. A scatter diagram plots these pairs of values and helps to show whether there is a relationship between the variables.
双变量数据涉及在同一个体上测量的两个变量。散点图绘制这些成对数值，有助于显示变量之间是否存在关系。
If the points tend to rise from left to right, there is positive correlation. If the points tend to fall from left to right, there is negative correlation. If no clear pattern is seen, there is zero or weak correlation.
如果点从左到右趋于上升，则存在正相关。如果点从左到右趋于下降，则存在负相关。如果没有明显模式，则存在零相关或弱相关。
Correlation can be described as strong, moderate or weak depending on how closely the points follow a straight line. However, correlation does not imply causation; a strong relationship may be caused by a third hidden variable.
相关性的强弱取决于点接近直线的程度，可以描述为强相关、中等相关或弱相关。然而，相关性并不意味着因果性；强关系可能由第三个隐藏变量引起。
The product moment correlation coefficient, often denoted by r, gives a numerical measure between −1 and 1. Values close to 1 or −1 indicate strong linear correlation, while values near
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