📚 A-Level CIE Maths: Experimental Operations Guide | A-Level CIE 数学：实验操作指南
Experimental work in A-Level CIE Mathematics often refers to the practical aspects of statistics, including designing surveys, conducting simulations, and performing hypothesis tests. This guide breaks down the key experimental operations you need to master for Papers 5 and 6 (Probability & Statistics), ensuring you can handle data collection, randomisation, and simulation with confidence.
A-Level CIE 数学中的实验操作通常指统计学的实践环节，包括设计调查、进行模拟和实施假设检验。本指南将逐一解析你在试卷5和试卷6（概率与统计）中需要掌握的关键实验操作，确保你能自信地处理数据收集、随机化和模拟问题。
1. Understanding Experimental Context in CIE Maths | 理解CIE数学中的实验背景
In the CIE syllabus, an ‘experiment’ can mean a planned activity to collect data under controlled conditions, or a simulation using random numbers to model real-world situations. Typical contexts include testing a new teaching method, comparing two medical treatments, or estimating a population proportion.
在CIE课程大纲中，“实验”可以指在受控条件下收集数据的有计划活动，也可以指使用随机数来模拟现实情境。典型的情境包括测试一种新教学方法、比较两种医疗方法或估计总体比例。
You will often be asked to describe how to carry out an experiment, identify the population and sample, and explain the importance of randomisation. These questions assess your understanding of statistical principles, not laboratory skills.
你经常会被要求描述如何实施一项实验、识别总体和样本，并解释随机化的重要性。这些问题考查的是你对统计原则的理解，而非实验室操作技能。
2. Designing a Statistical Experiment | 设计统计实验
A well-designed experiment clearly defines the objective, the population of interest, and the variables to be measured. For example, if we want to test whether a new revision app improves exam scores, the objective is to compare mean scores with and without the app.
一个设计良好的实验会明确界定目标、感兴趣的总体以及需要测量的变量。例如，如果我们想测试一款新的复习应用程序是否能提高考试成绩，目标就是比较使用和不使用该应用程序时的平均分数。
Identify the treatment group (receives the app) and control group (does not receive it). Both groups should be as similar as possible in all other respects, so that any difference in scores can be attributed to the app. This is the core of a comparative experiment.
确定实验组（使用应用程序）和对照组（不使用）。两组在其他所有方面应尽可能相似，这样分数上的任何差异都可归因于该应用程序。这是比较实验的核心。
3. Randomisation and Control | 随机化与控制
Randomisation is the process of assigning subjects to groups purely by chance. It helps eliminate selection bias and balances out unknown confounding variables. In an exam, you might say: ‘Use a random number generator to assign each student a number, then allocate odd numbers to the treatment group and even numbers to the control group.’
随机化是通过纯随机的方式将受试者分配到各组的过程。它有助于消除选择偏倚，并平衡未知的混杂变量。在考试中你可以这样说：“使用随机数生成器给每名学生分配一个编号，然后将奇数编号的学生分入实验组，偶数编号分入对照组。”
Control refers to keeping other conditions fixed. For instance, both groups should have the same amount of study time, similar prior attainment, and take the same test. You may also need to use a placebo if the experiment involves a psychological effect.
控制是指保持其他条件不变。例如，两组应有相同的学习时间、相似的基础水平，并参加同一测试。如果实验涉及心理效应，可能还需要使用安慰剂。
4. Sampling Methods | 抽样方法
When you cannot experiment on an entire population, you take a sample. Common sampling methods in CIE Maths include simple random sampling, stratified sampling, systematic sampling, and quota sampling. Each has advantages and disadvantages regarding bias and practicality.
当无法对整个总体进行实验时，就需要抽取样本。CIE数学中常见的抽样方法包括简单随机抽样、分层抽样、系统抽样和配额抽样。每种方法在偏倚和可行性方面各有优缺点。
For a simple random sample, every member of the population has an equal chance of being selected. This can be achieved using random number tables or a calculator’s random function. Stratified sampling ensures proportional representation from different subgroups, reducing variability.
对于简单随机样本，总体中的每个成员都有相等的被选中机会。这可以通过随机数表或计算器的随机功能来实现。分层抽样确保不同子群体按比例被代表，从而减少变异性。
| Sampling Method | Description | 抽样方法 | 描述 |
|---|---|---|---|
| Simple Random | Each member equally likely | 简单随机 | 每个成员被选中的概率相等 |
| Stratified | Divide into groups, sample proportionally | 分层 | 分组后按比例抽样 |
| Systematic | Select every kth member | 系统 | 每隔k个选取一个 |
| Quota | Choose a fixed number from each category | 配额 | 从每类中选取固定数量 |
5. Data Collection Techniques | 数据收集技巧
Data can be collected through observation, surveys, questionnaires, or automated recording. When designing a questionnaire, avoid leading questions, ensure clarity, and offer exhaustive response options. For continuous data, decide on measurement units and precision.
数据可以通过观察、调查、问卷或自动记录收集。设计问卷时，要避免诱导性问题，确保语言清晰，并提供穷尽的答案选项。对于连续型数据，要确定测量单位和精度。
In experimental contexts, you must also plan for data recording: create a table with columns for group, subject ID, and outcomes. Timing of measurements is critical, especially in before-and-after studies. State how you will ensure consistent conditions for all subjects.
在实验情境中，还必须规划数据记录方式：创建一个包含分组、受试者编号和结果的表格。测量的时机至关重要，特别是在前后对照研究中。说明如何确保所有受试者的条件一致。
6. Using Random Number Tables and Generators | 使用随机数表与生成器
Random number tables are printed lists of digits that are statistically random. To use them, you assign numbers to your population, read off digits in groups of the required length, and ignore repeats or numbers outside your range. The CIE exam may provide a small table and ask you to select a sample.
随机数表是统计上具有随机性的数字列表。使用时，先给总体中的每个成员分配数字，然后按所需长度分组读取数字，忽略重复值或超出范围的数字。CIE考试可能会提供一个小型随机数表，让你据此抽取样本。
Alternatively, a calculator’s ‘RAND’ or ‘RANDOM’ function generates pseudo-random numbers. For instance, to pick 5 students from a list of 80, you could use 80 × RAND + 1, round down to integers, and re-run if duplicates occur. This simulates simple random sampling.
也可以使用计算器的“RAND”或“RANDOM”函数生成伪随机数。例如，要从80名学生中选出5名，可使用80×RAND+1，向下取整，若出现重复则重新生成。这模拟了简单随机抽样。
7. Simulating Probability Experiments | 模拟概率实验
Simulation is a powerful tool when actual experiments are impractical. Suppose you want to estimate the probability that in a group of 30 people, at least two share a birthday. You can assign numbers 1 to 365 to birthdays and use random digits to generate 30-number sets repeatedly.
当实际实验不可行时，模拟是一种强大的工具。假设你想估计在30人的群体中至少有两人生日相同的概率，你可以为生日分配数字1到365，并用随机数字反复生成30个数字的集合。
Track the number of trials where a match occurs, then divide by the total number of trials to estimate the probability. The more trials you run, the closer your estimate gets to the theoretical value. CIE questions often ask you to describe such a simulation procedure step by step.
记录出现匹配的试验次数，再除以总试验次数，即可估计该概率。运行的试验次数越多，估计值就越接近理论值。CIE的问题经常要求你按步骤描述此类模拟过程。
8. Hypothesis Testing as an Experiment | 假设检验作为实验
Hypothesis testing is a formal statistical experiment. You formulate null (H₀) and alternative (H₁) hypotheses, collect sample data, and calculate a test statistic and p-value. The decision to reject H₀ depends on the significance level α, usually 5% or 1%.
假设检验是一种正式的统计实验。你先陈述原假设（H₀）和备择假设（H₁），收集样本数据，计算检验统计量和p值。拒绝H₀的决策取决于显著性水平α，通常为5%或1%。
For a binomial test, you compare the observed number of successes to the critical region. In a normal test, you standardise the sample mean. Always state your conclusion in the context of the problem, avoiding the phrase ‘accept H₀’ — you only ‘do not reject H₀’.
对于二项检验，你将观测到的成功次数与临界域进行比较。在正态检验中，需对样本均值进行标准化。结论务必结合问题背景陈述，避免使用“接受H₀”的说法——你只能说“不拒绝H₀”。
9. Common Pitfalls and How to Avoid Them | 常见陷阱与避免方法
One frequent error is confusing a sample with the population. If you are testing a drug on 200 patients, the population is all potential patients, not just the 200. Another pitfall is ignoring the need for randomisation — without it, results may be biased and invalid.
一个常见的错误是混淆样本与总体。如果你在200名患者身上测试药物，总体是所有潜在患者，而不仅仅是这200人。另一个陷阱是忽视随机化的必要性——没有随机化，结果可能带有偏倚且无效。
When carrying out a simulation, students often forget to define the stopping rule clearly. For instance, ‘continue generating numbers until you have 30 distinct dates’. Also, avoid using small trial numbers in simulations, as the estimate will be unreliable. Aim for at least 1000 trials if describing a simulation.
实施模拟时，学生常常忘记清晰地定义停止规则。例如，“继续生成数字，直到获得30个不同的日期”。此外，模拟时避免使用过少的试验次数，因为这样得出的估计不可靠。描述模拟时，至少应进行1000次试验。
10. Calculator and Software Tools | 计算器与软件工具
The CIE syllabus expects you to be proficient with a scientific calculator that has statistical functions. Learn how to generate random numbers, compute binomial and normal probabilities directly, and find critical values. For example, on many models, the ‘STAT’ mode allows you to enter grouped data and obtain mean and standard deviation.
CIE大纲要求你熟练使用具备统计功能的科学计算器。学会如何生成随机数、直接计算二项和正态概率以及查找临界值。例如，在许多型号上，“STAT”模式可以让你输入分组数据并得出均值和标准差。
Although not required for the exam, you may use spreadsheet software for practice. Functions like RANDBETWEEN, AVERAGE, STDEV.S, and BINOM.DIST can reinforce concepts. However, your exam answers must describe manual or calculator-based methods explicitly as requested.
虽然考试不要求使用电子表格软件，但你可以用它来练习。RANDBETWEEN、AVERAGE、STDEV.S和BINOM.DIST等函数可以巩固概念。但你的考试答案必须按要求明确描述手动方法或基于计算器的方法。
11. Presenting Experimental Findings | 呈现实验结果
Clear presentation is essential. Summarise raw data using frequency tables, diagrams (bar charts, histograms, box plots), and numerical measures (mean, median, standard deviation). When comparing two groups, use side-by-side box plots or back-to-back stem-and-leaf diagrams.
清晰的呈现至关重要。使用频数表、图示（条形图、直方图、箱线图）和数值测量（均值、中位数、标准差）来汇总原始数据。比较两组时，使用并排箱线图或背靠背茎叶图。
Your written description should refer to central tendency and spread. For instance: ‘The treatment group had a higher median score (78) compared to the control group (65), but the interquartile range was also larger, indicating more variability.’ Always link the statistics to the experimental context.
书面描述应涉及集中趋势和离散程度。例如：“实验组的中位数分数（78分）高于对照组（65分），但四分位距也更大，表明变数更大。”始终将统计数据与实验背景联系起来。
12. Exam Tips for Experimental Questions | 实验题的考试技巧
Experimental design questions are often marked for the clarity of your procedure. Use numbered steps: (1) State the population and how the sample is chosen. (2) Explain randomisation clearly. (3) Describe what data will be recorded and how it will be analysed. (4) Mention how to compare the results.
实验设计题通常按步骤的清晰程度评分。使用编号步骤：(1) 陈述总体以及如何选择样本。(2) 清晰地解释随机化过程。(3) 描述将记录哪些数据以及如何分析。(4) 说明如何比较结果。
If a question asks for a simulation, be specific about the digits or numbers used, how a trial is defined, and how the estimated probability is calculated. Use the formula: Estimated probability = (Number of successful trials) / (Total number of trials). Always relate your conclusion back to the problem statement.
如果问题是要求设计模拟，要具体说明使用的数字、如何定义一次试验以及如何计算估计概率。使用公式：估计概率 = (成功试验次数) / (总试验次数)。始终将你的结论与问题陈述相联系。
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