📚 IGCSE WJEC Statistics: Core Knowledge Checklist | IGCSE WJEC 统计：核心知识点梳理
This revision guide summarises the core knowledge needed for the IGCSE WJEC Statistics specification. It covers data types, sampling, diagrams, averages, spread, probability, correlation, time series and index numbers. Each section pairs an English explanation with a Chinese translation to support bilingual learners and examination preparation.
本复习指南总结了 IGCSE WJEC 统计课程所需的核心知识，涵盖数据类型、抽样、图表、平均数、离散程度、概率、相关、时间序列和指数。每个小节都采用英文解释与中文翻译配对的方式，帮助双语学习者备考。
1. Types of Data | 数据类型
Statistics begins with recognising the type of data. Qualitative data are non-numerical categories such as eye colour or type of transport. Quantitative data are numerical and can be discrete or continuous. Discrete data can only take particular values, often whole numbers, such as number of cars; continuous data can take any value in a range, such as height or time.
统计学首先要识别数据类型。定性数据是非数值类别，例如眼睛颜色或交通方式。定量数据是数值型数据，可分为离散型和连续型。离散数据只能取特定值，通常是整数，如汽车数量；连续数据可以在一个范围内取任意值，如身高或时间。
Primary data are collected by the researcher for a specific purpose, while secondary data are obtained from existing sources such as government reports. Understanding the source helps assess bias, accuracy and relevance before drawing conclusions.
原始数据由研究者为特定目的收集，二手数据则来自现有来源，如政府报告。在得出结论之前，了解数据来源有助于评估偏差、准确性和相关性。
2. Sampling Methods | 抽样方法
A population is the entire group being studied. A census collects data from every member, but it is often expensive and time-consuming. A sample is a smaller selection of the population used to estimate population characteristics.
总体是所研究的全部对象。普查从每个成员收集数据，但通常昂贵且耗时。样本是从总体中选出的较小部分，用于估计总体特征。
Random sampling gives every member an equal chance of being chosen, reducing bias. Stratified sampling divides the population into groups and samples proportionally from each group. Systematic sampling selects members at regular intervals, while convenience sampling uses easily available people and may be biased.
随机抽样使每个成员被选中的机会均等，从而减少偏差。分层抽样将总体分组，并按比例从每组中抽样。系统抽样按固定间隔选择成员，而便利抽样使用容易接触的人群，可能产生偏差。
3. Charts and Diagrams | 图表与图示
Bar charts show categorical data with gaps between bars. Pie charts display proportions as sectors of a circle. Histograms show continuous data with no gaps between bars, and the area of each bar is proportional to frequency. Frequency polygons connect the midpoints of histogram bars.
条形图用有间隔的条形展示分类数据。饼图用圆的扇形显示比例。直方图展示连续数据，条形之间无间隔，且每个条形的面积与频率成正比。频数多边形连接直方图各条形中点。
Stem-and-leaf diagrams keep original data values visible while showing shape. Pictograms use symbols to represent frequency but need a clear key. Choosing the right diagram depends on the data type and what needs to be communicated.
茎叶图在展示分布形状的同时保留原始数据数值。象形图用符号表示频数，但需要清晰的图例。选择正确的图表取决于数据类型和需要传达的信息。
4. Measures of Central Tendency | 集中趋势度量
The mean is the arithmetic average found by adding all values and dividing by the number of values. The median is the middle value when data are ordered. The mode is the most frequently occurring value. Each average has strengths and weaknesses depending on the data.
平均数是将所有数值相加再除以数值个数得到的算术平均值。中位数是将数据排序后位于中间的值。众数是出现次数最多的值。根据数据情况，每种平均数都有优缺点。
Mean = Σx ÷ n
For grouped data, the mean is estimated using midpoints and frequencies. Find the midpoint of each class, multiply by frequency, total these products, then divide by total frequency.
对于分组数据，使用组中值和频数估计平均数。求出每组的组中值，乘以频数，将所有乘积相加，再除以总频数。
Estimated mean = Σfx ÷ Σf
Here f is the frequency of each group and x is the class midpoint. The median interval can be located using cumulative frequency.
其中 f 是每组的频数，x 是组中值。中位数所在区间可以通过累积频率确定。
5. Measures of Spread | 离散程度度量
The range is the simplest measure of spread: subtract the smallest value from the largest value. It is quick to calculate but easily affected by outliers. The interquartile range, IQR = Q3 – Q1, measures the spread of the middle 50 percent of data and is more resistant to outliers.
极差是最简单的离散程度度量：用最大值减去最小值。它计算简单，但容易受异常值影响。四分位距 IQR = Q3 – Q1，衡量中间 50% 数据的分散程度，对异常值更具抵抗力。
Standard deviation measures how far values are, on average, from the mean. A small standard deviation means data are clustered near the mean; a large standard deviation indicates greater spread. The calculation squares deviations, averages them, and then takes the square root.
标准差衡量数据值平均距离平均值的远近。标准差小表示数据聚集在平均值附近；标准差大表示离散程度较大。计算时先求偏差平方，再求平均，最后开平方。
Standard deviation = √(Σ(x – mean)² ÷ n)
Use spread together with an average to describe a distribution fully. Reporting only one measure can mislead the audience.
应同时使用离散程度和平均数来完整描述分布。只报告一个度量可能会误导读者。
6. Cumulative Frequency and Box Plots | 累积频率与箱线图
Cumulative frequency is a running total of frequencies. Plot the upper class boundary of each interval against its cumulative frequency to draw a cumulative frequency curve, sometimes called an ogive. From this curve, you can read off the median and quartiles.
累积频率是频数的逐项累加总和。将每个区间的组上界与累积频率对应绘制，即可画出累积频率曲线，有时称为尖形图。从这条曲线上可以读取中位数和四分位数。
A box plot, or box-and-whisker diagram, shows five key values: minimum, lower quartile, median, upper quartile and maximum. The box represents the middle 50 percent of data, and the whiskers extend to the extremes. Box plots are useful for showing skew and comparing distributions.
箱线图，又称盒须图，显示五个关键值：最小值、下四分位数、中位数、上四分位数和最大值。箱体代表数据中间 50%，须线延伸到极值。箱线图适合展示偏态和比较分布。
If the median is closer to the lower quartile than the upper quartile, the distribution is positively skewed; if closer to the upper quartile, it is negatively skewed. Whisker length can also indicate skew and outliers.
如果中位数更靠近下四分位数而不是上四分位数，分布为正偏态；如果更靠近上四分位数，则为负偏态。须线长度也能显示偏态和异常值。
7. Probability Basics | 概率基础
Probability is measured on a scale from 0 to 1, where 0 means impossible and 1 means certain. For equally likely outcomes, probability is the number of favourable outcomes divided by the total number of possible outcomes.
概率在 0 到 1 的范围内度量，0 表示不可能，1 表示必然发生。对于等可能的结果，概率等于有利结果数除以所有可能结果总数。
P(A) = number of favourable outcomes ÷ total outcomes
For mutually exclusive events, which cannot occur together, add probabilities: P(A or B) = P(A) + P(B). For independent events, which do not affect each other, multiply probabilities: P(A and B) = P(A) × P(B). Expected frequency is found by multiplying probability by the number of trials.
互斥事件不能同时发生，使用加法：P(A 或 B) = P(A) + P(B)。独立事件互不影响，使用乘法：P(A 和 B) = P(A) × P(B)。期望频率等于概率乘以试验次数。
Tree diagrams and Venn diagrams help list outcomes and handle conditional probabilities. Always check that probabilities sum to 1 in a complete set of outcomes.
树状图和维恩图有助于列出结果并处理条件概率。始终检查完整结果集的概率之和是否为 1。
8. Scatter Graphs and Correlation | 散点图与相关
A scatter graph displays paired data to show the relationship between two variables. Positive correlation means that as one variable increases, the other also increases. Negative correlation means that as one variable increases, the other decreases. No correlation means there is no clear pattern.
散点图展示成对数据，以显示两个变量之间的关系。正相关表示一个变量增大时另一个也增大。负相关表示一个变量增大时另一个减小。无相关则表示没有明确模式。
A line of best fit can be drawn by eye, balancing points above and below the line. Predictions should only be made within the range of the data, called interpolation; extending beyond this range is extrapolation and can be unreliable.
最佳拟合线可以通过观察画出，使线上方和下方的点数大致平衡。预测应只在数据范围内进行，称为内插；将范围外推称为外推，可能不可靠。
Correlation does not imply causation. A strong correlation can suggest a relationship, but other variables or chance may explain the pattern.
相关不意味着因果关系。强相关可以提示某种关系，但其他变量或偶然性也可能解释该模式。
9. Time Series and Moving Averages | 时间序列与移动平均
A time series records data values at regular intervals over time. The trend is the long-term movement of the series after short-term fluctuations are smoothed. Moving averages are used to identify this trend.
时间序列按固定时间间隔记录数据值。趋势是序列经过短期波动平滑后的长期运动。移动平均用于识别这一趋势。
3-point moving average = (value before + value + value after) ÷ 3
For a 3-point moving average, take the mean of three consecutive values. Seasonal variation is found by subtracting the moving average from the actual value. This helps analyse repeating patterns such as quarterly sales or monthly temperatures.
对于三点移动平均，取连续三个数值的平均数。季节变动等于实际值减去移动平均值。这有助于分析重复模式，例如季度销售或月度温度。
Plot moving averages on the time series graph to see the trend clearly. The trend line can then be used to make predictions, but caution is needed because past patterns may not continue.
在时间序列图上绘制移动平均，可以清楚地看到趋势。趋势线可用于预测，但需要谨慎，因为过去的模式可能不会延续。
10. Index Numbers | 指数
An index number compares a value with a base value, often expressed as 100. The base period or base value is given the index number 100, and all other values are compared with it.
指数将某个数值与基准值比较，通常以 100 表示。基准期或基准值被赋予指数 100，其他所有数值都与它进行比较。
Index number = (current value ÷ base value) × 100
Weighted index numbers use weights to reflect the importance of different items. For example, in a price index, items that consumers buy more often may receive higher weights. The Retail Price Index and Consumer Price Index are common real-world applications.
加权指数使用权数反映不同项目的重要性。例如，在价格指数中，消费者更常购买的商品可能获得更高的权数。零售价格指数和消费者价格指数是常见的实际应用。
If the index is above 100, the value has increased relative to the base; if below 100, it has decreased. Percentage change can be found by comparing the index number with 100.
如果指数高于 100，说明相对于基期有所增加；如果低于 100，则有所减少。百分比变化可以通过将指数与 100 比较得出。
11. Comparing Distributions | 分布比较
When comparing two or more datasets, always comment on a measure of location and a measure of spread. For example, state which dataset has a higher median and which has a larger interquartile range, and explain what that means in context.
比较两个或多个数据集时，始终要同时说明位置度量和离散度量。例如，说明哪个数据集的中位数更高，哪个四分位距更大，并结合实际背景解释含义。
Box plots are especially useful for comparing distributions because they show the median, quartiles and range together. They make skew and outliers easy to see. Avoid comparing only the range if outliers are present.
箱线图特别适合比较分布，因为它同时显示中位数、四分位数和极差。它使偏态和异常值一目了然。如果存在异常值，避免只比较极差。
Use exact data values from the diagram or table to support every comparison. A strong statistical comparison links the numbers back to the situation being studied.
利用图表或表格中的具体数据值支持每个比较。有力的统计比较应将数字与所研究的情境联系起来。
12. Statistical Enquiry Cycle | 统计探究循环
Statistical problem solving follows a cycle: plan, collect data, process and analyse, interpret and evaluate. Planning identifies the question, population and methodology. Data collection uses an appropriate sampling method to reduce bias.
统计问题解决遵循一个循环：计划、收集数据、处理与分析、解释与评价。计划阶段确定问题、总体和研究方法。数据收集使用适当的抽样方法以减少偏差。
Processing and analysis include producing tables, charts, averages and measures of spread. Interpretation explains what the results show. Evaluation considers limitations, bias and possible improvements to the investigation.
处理与分析包括制作表格、图表，计算平均数和离散程度。解释阶段说明结果的含义。评价阶段考虑局限性、偏差以及可能的改进方向。
A clear conclusion should answer the original question using evidence from the data. Showing awareness of the enquiry cycle is important for high-scoring answers in the examination.
清晰的结论应使用数据证据回答最初的问题。在考试中，展示对探究循环的认识对于获得高分非常重要。
Published by TutorHao | Statistics Revision Series | aleveler.com
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