📚 IGCSE WJEC Statistics: Core Concepts Summary | IGCSE WJEC 统计：核心知识点梳理
Welcome to our comprehensive revision guide for the IGCSE WJEC Statistics syllabus. This article distills the essential topics you need to master, from data types and sampling methods to probability distributions and the statistical enquiry cycle. Whether you are preparing for an exam or consolidating your understanding, these core concepts will support your success.
欢迎阅读我们为 IGCSE WJEC 统计课程整理的全面复习指南。本文提炼了您需要掌握的核心主题，从数据类型和抽样方法到概率分布和统计探究周期。无论您是为考试做准备还是巩固理解，这些核心知识点都将助您成功。
1. Types of Data | 数据类型
Data can be classified as qualitative (categorical) or quantitative (numerical). Quantitative data can be discrete, taking only separate distinct values (e.g., number of students), or continuous, which can take any value within a given range (e.g., height). It is also crucial to distinguish between primary data, which you collect yourself through surveys or experiments, and secondary data, which has been gathered by someone else.
数据可分为定性（分类）数据和定量（数值）数据。定量数据可以是离散的，只能取互不相交的值（例如学生人数），也可以是连续的，可以在给定范围内取任意值（例如身高）。区分一手数据（您自己通过调查或实验收集的）和二手数据（由他人收集的）也十分关键。
Understanding these categories helps you select appropriate methods for display and analysis. Qualitative data lends itself to bar charts and pie charts, while quantitative data is often displayed using histograms or scatter graphs.
理解这些类别有助于您选择合适的展示和分析方法。定性数据适合用条形图或饼图显示，而定量数据常用直方图或散点图展示。
2. Data Collection and Sampling | 数据收集与抽样
Data can be collected through controlled experiments, observational studies, or well-designed questionnaires. A sample is a subset of the population, and the way it is chosen influences the reliability of your conclusions. Simple random sampling gives every member an equal chance of being selected, reducing bias. Stratified sampling divides the population into distinct groups (strata) and then randomly selects from each, ensuring representation of key characteristics.
数据可以通过受控实验、观测研究或精心设计的问卷收集。样本是总体的一个子集，其选取方式会影响结论的可靠性。简单随机抽样让每个个体有均等的被选机会，可减少偏差。分层抽样先将总体划分为不同的组（层），然后从每层中随机选取，确保关键特征的涵盖。
Other methods include systematic sampling (e.g., selecting every 10th item), cluster sampling (sampling entire groups), and quota sampling (non-random selection to fill pre-set quotas). Beware of bias arising from an unrepresentative sample or poorly phrased questionnaire questions.
其他方法包括系统抽样（例如每第10个选取）、整群抽样（整组抽取）和配额抽样（非随机选择以满足预设配额）。要警惕因样本不具代表性或问卷措辞不当而产生的偏差。
3. Presenting Data: Charts and Diagrams | 数据展示：图表
Visual representations make data easier to interpret. Bar charts are used for discrete or categorical data with gaps between bars. Pie charts show proportions. Histograms display the distribution of continuous data, with the area of each bar proportional to frequency; for unequal class widths, frequency density (frequency ÷ class width) must be used. Cumulative frequency curves (ogives) allow you to read off medians and quartiles. Box-and-whisker plots illustrate the five-number summary: minimum, lower quartile, median, upper quartile, and maximum.
可视化表示使数据更易于解读。条形图用于离散或分类数据，条间有空隙。饼图展示比例。直方图显示连续数据的分布，每个条形的面积与频率成正比；对于不等组距的直方图，必须使用频率密度（频率 ÷ 组距）。累积频率曲线（ogive）可以读取中位数和四分位数。箱线图展示五数概括：最小值、下四分位数、中位数、上四分位数和最大值。
Choosing the right diagram depends on the data type and the message you wish to convey. Always label axes and provide a title.
选择正确的图表取决于数据类型和您希望传达的信息。务必标注坐标轴并给出标题。
4. Measures of Central Tendency | 集中趋势度量
The mean (average) is calculated by summing all values and dividing by the number of values. For grouped data, use midpoints of intervals. The median is the middle value when data are ordered; for grouped data it may be estimated from a cumulative frequency graph. The mode is the most frequent value or class.
均值（平均数）通过所有数值求和后除以数值个数计算得出。对于分组数据，使用区间的组中值。中位数是数据排序后的中间值；对于分组数据，可从累积频率图中估算。众数是出现频率最高的值或组。
The mean is sensitive to extreme values, while the median and mode are more robust. In a symmetrical distribution, the mean, median and mode coincide. In a skewed distribution, the mean is pulled towards the tail.
均值对极端值敏感，而中位数和众数更为稳健。在对称分布中，均值、中位数和众数重合。在偏态分布中，均值会被拉向尾部。
5. Measures of Spread | 离散程度度量
The range is the simplest measure of spread: maximum minus minimum. The interquartile range (IQR = Q₃ − Q₁) gives the range of the middle 50% of the data and is not affected by outliers. The standard deviation measures how far values deviate from the mean.
极差是最简单的离散程度度量：最大值减最小值。四分位距（IQR = Q₃ − Q₁）给出中间50%数据的范围，不受异常值影响。标准差测量各值与均值的偏差大小。
The standard deviation for a sample can be calculated using the formula:
s = √[ Σ(x − x̄)² / (n − 1) ]
Notice that the sample variance is s². For a population, divide by n instead. A larger standard deviation indicates greater variability.
样本标准差可以使用以下公式计算：
s = √[ Σ(x − x̄)² / (n − 1) ]
注意样本方差是 s²。对于总体，用 n 代替 (n−1) 作分母。标准差越大表示变异性越大。
6. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph shows the relationship between two variables. If points slope upwards, there is a positive correlation; if downwards, negative. The strength can be assessed by the closeness of points to a line of best fit. You can draw a line of best fit by eye, passing through the mean point (x̄, ȳ). The correlation coefficient r (or Spearman’s rank correlation coefficient ρ) quantifies the strength and direction: r = 1 is perfect positive, r = −1 perfect negative, r = 0 no linear correlation.
散点图显示两个变量之间的关系。若点呈上升趋势，则存在正相关；若下降，则为负相关。相关程度可通过点与最佳拟合线的接近程度评估。您可以通过目测画出最佳拟合线，使其通过均值点 (x̄, ȳ)。相关系数 r（或斯皮尔曼等级相关系数 ρ）量化了强度和方向：r = 1 为完全正相关，r = −1 为完全负相关，r = 0 为无线性相关。
Be careful not to assume causation from correlation. An observed correlation may be due to a third factor or coincidence.
切勿从相关性推断因果性。观察到的相关性可能由第三个因素或巧合引起。
7. Probability Basics | 概率基础
Probability is a measure of how likely an event is to occur, ranging from 0 (impossible) to 1 (certain). The sum of probabilities of all mutually exclusive outcomes in a sample space equals 1. For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B). More generally:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
概率是事件发生可能性的度量，范围从0（不可能）到1（必然）。样本空间中所有互斥结果的概率之和等于1。对于互斥事件 A 和 B，P(A ∪ B) = P(A) + P(B)。一般形式：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If two events are independent, the probability of both occurring is the product of their probabilities: P(A ∩ B) = P(A) × P(B). Conditional probability P(A|B) = P(A ∩ B) / P(B), where P(B) > 0. Tree diagrams help visualise multi-stage experiments.
如果两个事件独立，两者同时发生的概率为其各自概率的
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