📚 IGCSE WJEC Maths: Statistics Revision Essentials | IGCSE WJEC 数学：统计考点精讲
Statistics is a core topic in the IGCSE WJEC Mathematics syllabus, testing your ability to collect, represent, analyse and interpret data. This guide breaks down every essential concept, from types of data to probability, with clear bilingual explanations and worked examples tailored to the WJEC exam style. Mastering these ideas will not only boost your confidence but also secure vital marks on both calculator and non-calculator papers.
统计是 IGCSE WJEC 数学教学大纲中的核心主题，考查学生收集、表示、分析和解释数据的能力。本指南将每个基本概念从数据类型到概率逐一分解，配以双语讲解和符合 WJEC 考试风格的示例。掌握这些知识点不仅能增强信心，还能在计算器与非计算器试卷中稳拿关键分数。
1. Types of Data | 数据类型
Data can be classified as qualitative (categorical) or quantitative (numerical). Qualitative data describes qualities, such as eye colour or types of pet, and cannot be measured numerically. Quantitative data involves numbers and is further split into discrete data, which can only take specific values (e.g. number of students), and continuous data, which can take any value within a range (e.g. height, time).
数据可分为定性（分类）数据和定量（数值）数据。定性数据描述性质，如眼睛颜色或宠物类型，无法用数字度量。定量数据涉及数字，可进一步分为离散数据（只能取特定值，如学生人数）和连续数据（可取范围内任意值，如身高、时间）。
Recognising the data type is crucial because it determines which diagrams and statistical measures are appropriate. For WJEC questions, you are often asked to identify whether data is discrete or continuous and to justify your choice of graph.
识别数据类型至关重要，因为它决定了适合的图表和统计指标。在 WJEC 试题中，经常要求你判断数据是离散还是连续，并解释选取某种图表的理由。
- Qualitative: favourite colour, gender, car brands
- Quantitative discrete: number of goals scored, shoe size
- Quantitative continuous: weight, temperature, distance
- 定性数据：最喜欢的颜色、性别、汽车品牌
- 定量离散数据：进球数、鞋码
- 定量连续数据：体重、温度、距离
2. Collecting and Sampling | 数据收集与抽样
Data collection must be unbiased and representative. A population is the entire group you want to study, while a sample is a subset of the population. WJEC expects you to understand random sampling, stratified sampling, and systematic sampling, and to spot bias in survey methods.
数据收集必须无偏且具有代表性。总体是你想研究的整个群体，样本则是总体的一个子集。WJEC 要求你理解随机抽样、分层抽样和系统抽样，并能识别调查方法中的偏差。
A simple random sample gives every member an equal chance of being chosen, often using random number generators. Stratified sampling divides the population into groups (strata) and takes a proportional number from each group. This ensures fair representation of all subgroups, which is particularly useful when the population has obvious strata like year groups in a school.
简单随机抽样让每个成员有均等的被选中机会，常借助随机数生成器。分层抽样将总体分成若干层，然后从每层按比例抽取样本。这能保证所有子群公平呈现，当总体存在明显分层（如学校的年级）时尤其有用。
| Sampling Method | Description |
|---|---|
| Random | Every individual has an equal chance |
| Stratified | Population split into groups, proportionally sampled |
| Systematic | Choosing every nth individual from a list |
| 抽样方法 | 说明 |
|---|---|
| 随机抽样 | 每个个体被选中的机会均等 |
| 分层抽样 | 总体分群，按比例抽取 |
| 系统抽样 | 从名单中每隔一定数量选取一个 |
3. Frequency Tables and Diagrams | 频率表与频率图
Organising data into frequency tables is the first step in analysis. Tally charts help count occurrences, and the frequency column shows how many times each value appears. For grouped continuous data, class intervals such as 0 ≤ h < 10 are used, and you must ensure there are no gaps or overlaps.
将数据整理成频率表是分析的第一步。划记表有助于计数，频率列显示每个数值出现的次数。对于分组连续数据，使用形如 0 ≤ h < 10 的组距，需要确保没有间断或重叠。
A frequency diagram for discrete data uses vertical line charts or bar charts where bars are separate. For continuous data, you must use a frequency polygon or a histogram. When drawing a frequency polygon, plot the midpoints of each class interval against frequency and join them with straight lines.
离散数据的频率图使用垂线图或条形图，条形之间分开。连续数据则必须使用频率折线图或直方图。绘制频率折线图时，以每个组距的中点对频率描点，再用直线连接。
Always label axes clearly, give the chart a title, and use a sensible scale. WJEC examiners will deduct marks for missing labels or inconsistent scaling.
务必清晰标记坐标轴，给图表加上标题，并使用合理的刻度。WJEC 阅卷人会因缺失标签或比例不一致而扣分。
4. Bar Charts and Pie Charts | 条形图与饼图
Bar charts represent categorical or discrete data with rectangular bars. The height of each bar corresponds to its frequency, and bars are drawn with equal width and gaps between them. A dual or composite bar chart can compare two or more data sets side by side.
条形图用矩形条表示分类数据或离散数据。每个条形的高度对应其频率，条的宽度相等且彼此间留有间隙。双柱图或堆叠条形图可并列比较两套或更多数据。
Pie charts display proportions of a whole. To calculate the angle for each category, use the formula: (frequency / total frequency) × 360°. A protractor is needed to draw the sectors accurately. Include a key or labels, and make sure the angles sum to 360°.
饼图展示各部分占整体的比例。计算每个类别的圆心角公式： （频率/总频率）× 360°。绘制扇形需要使用量角器确保精确。要添加图例或标签，并确认所有角度之和为 360°。
Typical WJEC questions ask you to construct a pie chart from a frequency table or to interpret a given pie chart by calculating frequencies from angles.
典型的 WJEC 题目会要求根据频率表绘制饼图，或者通过给定饼图的角度推算频率。
5. Histograms | 直方图
A histogram is used for continuous data grouped into class intervals. Unlike a bar chart, there are no gaps between bars, and the area of each bar is proportional to the frequency. When class widths are unequal, you must calculate frequency density = frequency / class width and plot this on the vertical axis.
直方图用于按组距分组的连续数据。与条形图不同，直方图条形之间没有间隙，且每个条形的面积与频率成正比。当组距宽度不相等时，必须计算频率密度 = 频率 / 组距宽度，并将其绘制在纵轴上。
Understanding frequency density is a key WJEC requirement. For example, an interval 0 ≤ x < 20 with frequency 30 has frequency density 1.5, whereas an interval 20 ≤ x < 50 with frequency 60 also has density 2.0. Always use the formula to construct and interpret histograms correctly.
理解频率密度是 WJEC 的关键要求。例如，区间 0 ≤ x < 20、频率为 30 时，频率密度为 1.5；而区间 20 ≤ x < 50、频率为 60 时，密度为 2.0。始终使用该公式正确构建和解读直方图。
When estimating frequencies from a histogram, multiply the class width by the frequency density. Exam questions often give a partially completed histogram and ask you to complete it or to estimate the number of items in a specific interval.
当从直方图估算频率时，需将组距宽度乘以频率密度。考题经常给出部分完成的直方图，要求补充完整或估算特定区间内的项目数。
6. Cumulative Frequency and Box Plots | 累积频率与箱线图
Cumulative frequency is the running total of frequencies. To construct a cumulative frequency table, add each frequency to the sum of previous frequencies. Plot cumulative frequency against the upper boundary of each class interval and join the points with a smooth curve, not straight lines.
累积频率是频率的累加总和。构建累积频率表时，将每个频率加上之前所有频率的和。以每个组距的上限对累积频率描点，并用平滑曲线连接各点，而非直线。
From the cumulative frequency curve, you can estimate the median, quartiles, and inter-percentile ranges. The lower quartile Q₁ is at 25% of the total frequency, the median Q₂ at 50%, and the upper quartile Q₃ at 75%. Read these values from the horizontal axis at the corresponding cumulative frequency.
通过累积频率曲线可以估算中位数、四分位数和百分位数间距。下四分位数 Q₁ 位于总频率的 25% 处，中位数 Q₂ 位于 50% 处，上四分位数 Q₃ 位于 75% 处。在相应的累积频率位置从横轴读取这些值。
A box plot (box-and-whisker diagram) displays the five-number summary: minimum, Q₁, median, Q₃, and maximum. Draw a box from Q₁ to Q₃ with a line at the median, and whiskers extending to the minimum and maximum values, provided there are no outliers.
箱线图（盒须图）显示五数概括：最小值、Q₁、中位数、Q₃ 和最大值。从 Q₁ 到 Q₃ 画一个盒子，在中位数处画一条线，触须延伸至最小值和最大值（假定没有异常值）。
7. Measures of Central Tendency: Mean, Median, Mode | 集中趋势度量：平均数、中位数、众数
The three principal averages are the mean, median, and mode. The mean is calculated by summing all data values and dividing by the number of values. It is the most widely used average but can be distorted by extreme outliers.
三个主要的平均数是平均数、中位数和众数。平均数通过将所有数据值求和再除以数据个数得出。它是最常用的平均数，但可能被极端异常值扭曲。
Mean = (Σx) / n
平均数 = (Σx) / n
The median is the middle value when data is ordered. If there are an even number of data points, the median is the mean of the two middle values. The median is unaffected by outliers, making it useful for skewed data.
中位数是将数据排序后的中间值。若数据点个数为偶数，则中位数是中间两个值的平均数。中位数不受异常值影响，因此适用于偏斜数据。
The mode is the value that occurs most frequently. There can be one mode (unimodal), two modes (bimodal), or more. For grouped data, the modal class is the class interval with the highest frequency.
众数是出现频率最高的值。可以有一个众数（单峰）、两个众数（双峰）或更多。对于分组数据，众数所在组是频率最高的组距。
8. Measures of Spread: Range and Interquartile Range | 离散程度：极差与四分位距
Spread tells you how varied the data is. The range is the simplest measure: maximum value minus minimum value. It gives a quick sense of dispersion but is heavily influenced by outliers.
离散程度反映数据的变异大小。极差是最简单的度量：最大值减最小值。它迅速给出离散度的感觉，但极易受异常值影响。
The interquartile range (IQR) is the difference between the upper and lower quartiles: IQR = Q₃ – Q₁. It covers the middle 50% of the data and is more robust than the range because it ignores extreme values.
四分位距 (IQR) 是上四分位数与下四分位数之差：IQR = Q₃ – Q₁。它包含中间 50% 的数据，由于忽略了极端值，比极差更稳健。
WJEC questions frequently require you to compare data sets using median and IQR. Stating that one set has a higher median indicates a higher typical value; a smaller IQR signals less variability.
WJEC 试题经常要求使用中位数和 IQR 比较数据集。说一个数据集的中位数更高表明其典型值更高；IQR 更小则表明变异性更小。
9. Standard Deviation | 标准差
Standard deviation measures the average distance of each data point from the mean. It is the most comprehensive measure of spread for symmetric distributions. The formula for the standard deviation of a population is:
标准差衡量各数据点与平均数之间的平均距离。对于对称分布，它是最全面的离散度量。总体标准差的公式为：
σ = √(Σ(x – μ)² / n)
σ = √(Σ(x – μ)² / n)
Here μ is the mean and n is the number of data points. In WJEC IGCSE, you will typically use this formula with calculators allowed. A small standard deviation means data points are tightly clustered around the mean; a large one indicates they are spread out.
其中 μ 是平均数，n 是数据点个数。在 WJEC IGCSE 考试中，通常允许使用计算器套用此公式。标准差小意味着数据点紧密聚集在平均数周围；标准差大则表示数据点分散。
When calculating manually, create a table with columns for x, x – μ, (x – μ)², then sum the squared deviations, divide by n, and take the square root. Always check your rounding – final answers should be given to 3 significant figures unless otherwise stated.
手动计算时，可创建表格，列出 x、x – μ、(x – μ)² 三列，然后求和平方偏差、除以 n，最后取平方根。务必检查舍入——除非另有说明，最终答案应保留三位有效数字。
10. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph is used to investigate the relationship between two continuous variables. Each point represents a pair of values (x, y). If the points show an upward trend, there is positive correlation; a downward trend indicates negative correlation. If no pattern is visible, there is zero or no correlation.
散点图用于考察两个连续变量之间的关系。每个点代表一对数值 (x, y)。若点呈上升趋势，则为正相关；下降趋势表示负相关；若无明显模式，则为零相关或无相关。
Correlation can be strong, moderate, or weak depending on how closely the points follow a straight line. A line of best fit is drawn by eye, passing as near as possible to all points with roughly equal numbers of points above and below the line. This line can be used to estimate unknown values.
相关性的强弱取决于点簇沿直线的紧密程度。最佳拟合线凭视觉画出，尽量贴近所有点，并使线上方和下方的点数大致相等。该直线可用于估算未知数值。
WJEC expects you to describe the correlation clearly (e.g., ‘strong positive correlation’) and to interpret the gradient and y-intercept of the line of best fit if the variables have real-world meaning.
WJEC 要求你清晰描述相关性（如“强正相关”），并在变量具有实际意义时解释最佳拟合线的斜率和 y 轴截距。
11. Probability Basics | 概率基础
Probability is a measure of how likely an event is to happen. It is expressed as a fraction, decimal, or percentage between 0 (impossible) and 1 (certain). The probability of an event A is given by: P(A) = number of favourable outcomes / total number of possible outcomes, provided all outcomes are equally likely.
概率是事件发生可能性大小的度量，用介于 0（不可能）和 1（必然）之间的分数、小数或百分数表示。假设所有结果等可能，事件 A 的概率公式为：P(A) = 有利结果的数量 / 所有可能结果的总数。
The probability that an event does not happen is 1 – P(A). For mutually exclusive events, which cannot happen at the same time, the probability of either occurring is the sum of their individual probabilities: P(A or B) = P(A) + P(B).
事件不发生的概率是 1 – P(A)。对于互斥事件（不能同时发生），任一件事发生的概率为各自概率之和：P(A 或 B) = P(A) + P(B)。
Sample space diagrams and tree diagrams are essential tools for listing outcomes and calculating combined probabilities. Tree diagrams show probabilities along branches; multiply along branches for ‘and’ probabilities, and add the final probabilities of different branches for ‘or’ situations. Always check that probabilities on branches from the same point sum to 1.
样本空间图和树状图是列举结果和计算组合概率的基本工具。树状图沿分支显示概率；对于“且”的情况，沿分支相乘概率；对于“或”的情况，将不同分支的最终概率相加。务必检查同一点出发的分支概率之和是否为 1。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Read the question carefully and identify what type of data is involved before choosing a diagram. Students often lose marks by drawing a bar chart for continuous data or a histogram for discrete data. Always label axes and use a ruler and pencil for graphs.
仔细读题，在选图前先辨别数据类型。学生常因在连续数据上画条形图或在离散数据上画直方图而丢分。务必标注坐标轴，使用直尺和铅笔绘制图表。
When calculating mean from a frequency table, remember to multiply each value by its frequency and sum these products before dividing by the total frequency. For grouped data, use midpoints. A common mistake is to divide the sum of midpoints by the number of classes.
当利用频率表计算平均数时，记得先将每个值乘以其频率，再将乘积求和，然后除以总频率。对于分组数据，使用组距中点。常犯的错误是将各中点之和除以组数。
For cumulative frequency, always plot points at the end of the interval, not the midpoint. When reading off quartiles, be exact – draw horizontal lines from the cumulative frequency axis to the curve and then vertical lines down to the scale. Check that your IQR makes sense in the context of the data range.
绘制累积频率图时，始终在区间末端描点，而非中点。读取四分位数时要精确——从累积频率轴引水平线至曲线，再垂直下引至刻度。并检查计算出的 IQR 是否在数据范围内合理。
Finally, manage your time wisely. Statistics questions can be long, but method marks are awarded for correct working even if the final answer is wrong. Show all your steps clearly, and double-check arithmetic under pressure.
最后，合理分配时间。统计题可能较长，但即使最终答案错误，正确解题步骤也能获得方法分。清晰展示所有步骤，并在压力下复核算术运算。
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