📚 Experimental Operation Guide in IGCSE Edexcel Mathematics | IGCSE Edexcel 数学实验操作指南
Mathematical experiments and investigations are a vital part of the IGCSE Edexcel Mathematics course. They encourage you to explore patterns, test hypotheses and handle real data, which builds strong problem-solving skills. This guide walks you through every stage of conducting a successful mathematical experiment, from planning to writing up your findings.
数学实验和探究是 IGCSE Edexcel 数学课程的重要组成部分。它们鼓励你探索规律、检验假设并处理真实数据，从而培养强大的问题解决能力。本指南将带你走过成功实施数学实验的每个阶段，从计划到撰写报告。
1. What Is a Mathematical Experiment? | 什么是数学实验？
In IGCSE Edexcel Mathematics, an experiment is a structured investigation where you collect data or simulate a situation to answer a mathematical question. Unlike a simple exercise, an experiment involves planning, trialling, recording and interpreting results. Common examples include rolling dice to study probability, measuring objects to find averages, or using random numbers to model real-life events.
在 IGCSE Edexcel 数学中，实验是指通过收集数据或模拟情境来回答某个数学问题的结构化探究。与简单的练习题不同，实验包含计划、试验、记录和解读结果。常见例子包括掷骰子研究概率、测量物体求平均值，或使用随机数模拟现实事件。
You will often work through a cycle: state the aim, make a prediction, design the method, carry out the experiment, analyse the data and reach a conclusion. This mirrors the statistical enquiry cycle used in the syllabus.
你通常会经历一个循环：陈述目标、做出预测、设计方法、实施实验、分析数据并得出结论。这对应了教学大纲中使用的统计探究循环。
The Edexcel IGCSE specification expects you to be able to comment on the reliability of your findings and suggest improvements. Therefore, treat every experiment as more than just number-crunching.
Edexcel IGCSE 大纲要求你能够评价结果的可靠性并提出改进建议。所以，要把每一次实验都看作远不止是数字运算。
2. Linking to the IGCSE Edexcel Syllabus | 对接 IGCSE Edexcel 大纲
Mathematical experiments appear most directly in the Statistics and Probability part of the specification. You need to be able to plan a data collection sheet, choose sensible class intervals, and construct diagrams like frequency polygons, cumulative frequency curves and histograms. Experimental probability relative frequency is a key concept.
数学实验最直接地出现在考试大纲的统计与概率部分。你需要能够设计数据收集表、选择合适的组距，并绘制频率多边形、累积频率曲线和直方图等图表。实验中的相对频率是一个关键概念。
In probability, you will carry out experiments to compare theoretical probability with experimental probability. For instance, tossing a fair coin 100 times and recording the proportion of heads shows how relative frequency approaches the theoretical value of ½.
在概率中，你将通过实验比较理论概率与实验概率。例如，抛一枚匀质硬币 100 次并记录正面的比例，以此展示相对频率如何接近理论值 ½。
Additionally, the syllabus includes ‘using ICT’ where appropriate. You can use spreadsheets or graphing tools to simulate experiments with large sample sizes, and this guide will cover such technological approaches.
此外，大纲包括在适当时候‘使用 ICT’。你可以用电子表格或绘图工具来模拟大规模实验，本指南也会涉及这些技术方法。
3. Planning and Designing Your Experiment | 规划与设计实验
A clear aim is the starting point. Turn a broad interest like ‘Is a dice fair?’ into a sharp question: ‘Does a six-sided dice produce each number with equal probability when rolled 120 times?’ This makes the hypothesis testable.
清晰的目标是起点。把‘骰子公平吗？’这样宽泛的兴趣点转化为尖锐的问题：‘一个六面骰子掷 120 次时每个数字出现的概率是否相等？’ 这使得假设可以检验。
Decide on the variables. In many investigations you will have an independent variable (what you change) and a dependent variable (what you measure). For a dice experiment the variable is the outcome, but you might also investigate how changing the surface or the throwing technique affects results.
确定变量。许多探究中会有自变量（你改变的量）和因变量（你测量的量）。对掷骰子实验，变量就是结果，但你也可以研究改变表面或投掷手法如何影响结果。
Write a step-by-step method that anyone can follow. For example: ‘Roll two dice simultaneously, record the sum, repeat 50 times.’ Specify how you will randomise trials, how you will record data, and how many repetitions are planned.
写出任何人都能遵循的分步方法。例如：‘同时掷两枚骰子，记录点数之和，重复 50 次。’ 说明你将如何随机化试验、如何记录数据以及计划重复的次数。
Always consider ethical or practical constraints. Mathematical experiments are usually safe, but if you involve people, ensure privacy and consent. A pilot trial can help check whether your method works before committing to a full run.
始终要考虑伦理或实际限制。数学实验通常很安全，但如果涉及他人，要确保隐私和同意。一次试运行可以帮你检查方法是否可行，然后再正式实施。
4. Data Collection Methods | 数据收集方法
Good data comes from careful recording. Use a tally chart to count the frequency of each outcome. For example, when rolling two dice and summing the faces, the sums range from 2 to 12. A tally chart with rows for each sum lets you quickly record ‘||||’ for each count and cross the fifth stroke to make groups of five.
好的数据来自细致的记录。使用计数表记录每个结果的频数。例如掷两枚骰子求和时，和的范围是 2 到 12。每个和对应一行的计数表可以让你快速用‘||||’记录每次计数，第五划横穿成为一组五计数。
For continuous data, such as measuring the lengths of leaves, a grouped frequency table is needed. Choose equal class widths if possible. A rule of thumb is to have about 5–10 groups, and ensure no gaps between them. Use inequalities to define intervals, e.g., 10 ≤ l < 15 cm.
对于连续数据，例如测量叶子长度，需要分组频率表。尽可能选择等宽的组距。经验法则是大约 5–10 组，并确保组间无间隙。用不等式定义区间，如 10 ≤ l < 15 cm。
Applications like Excel or Google Sheets can help you collect data directly. For a large simulation, you can generate random numbers using =RANDBETWEEN(1,6) to simulate a dice roll 1000 times. Learn to paste values only to freeze the random numbers, so your results remain unchanged.
像 Excel 或 Google Sheets 这样的软件可以帮助你直接收集数据。对于大规模模拟，你可以用 =RANDBETWEEN(1,6) 生成随机数，模拟掷骰子 1000 次。学会‘只粘贴数值’来冻结随机数，这样结果就不会变动。
5. Using Technology for Simulations | 使用技术进行模拟
When physical experiments are impractical, technology is your best tool. A spreadsheet can simulate thousands of trials in seconds. Besides RANDBETWEEN, you can use =RAND() to generate a random decimal between 0 and 1. To simulate a biased event, use =IF(RAND()<0.3, "Success", "Failure").
当实物实验不可行时，技术是你最好的工具。电子表格可以在几秒内模拟数千次试验。除了 RANDBETWEEN，你还可以用 =RAND() 生成 0 到 1 之间的随机小数。要模拟一个偏倚事件，可用 =IF(RAND()<0.3, "成功", "失败")。
Graphing software such as Desmos or GeoGebra can create dynamic models. For probability experiments, you can program a slider to run repetitions and instantly view a histogram of outcomes. This offers a visual understanding of the law of large numbers.
Desmos 或 GeoGebra 等绘图软件可以创建动态模型。对于概率实验，你可以编写一个滑块来运行重复试验，并即时查看结果的直方图。这能让你直观理解大数定律。
Be aware of exam expectations: you must be able to interpret output from technology. You should know how to calculate averages and measures of spread from frequency tables generated by software, and how to identify anomalies.
注意考试要求：你必须能够解读技术生成的输出结果。你应该知道如何从软件生成的频率表中计算平均数和离散程度，以及如何识别异常值。
Keep a record of the formulas and steps used. In your write-up, a screenshot or a clear description of the simulation method strengthens the reliability of your experiment.
保留所用公式和步骤的记录。在撰写报告时，一张截图或对模拟方法的清晰描述可以增强实验的可信度。
6. Probability Experiments in Detail | 概率实验详解
Probability experiments centre on the idea of relative frequency. If an event occurs r times in n trials, the experimental probability is r / n. The more trials you conduct, the closer this experimental value usually gets to the theoretical probability. This is known as the law of large numbers.
概率实验围绕相对频率的概念展开。若某事件在 n 次试验中发生 r 次，则实验概率为 r / n。你进行的试验次数越多，该实验值通常越接近理论概率。这被称为大数定律。
To test a dice fairly, you would roll it many times (e.g., 600) and find the relative frequency of each face. Then compare with the theoretical probability of 1/6 ≈ 0.167. A small discrepancy is normal, but a large, consistent bias may suggest the dice is unfair.
要检验骰子是否公平，你会掷很多次（例如 600 次），并找出每一面的相对频率。然后与理论概率 1/6 ≈ 0.167 比较。小的偏差很正常，但如果出现巨大且一致的偏倚就可能说明骰子不公平。
Combined events, such as ‘rolling a sum of 7 with two dice’, can be explored experimentally. Use a table of outcomes to find the theoretical probability (6/36 = 1/6) and then perform actual rolls. Plot a cumulative relative frequency graph to see convergence.
复合事件，比如‘两枚骰子之和为 7’，可通过实验探索。用结果表求出理论概率（6/36 = 1/6），然后进行真实投掷。绘制累积相对频率图来观察收敛情况。
Always discuss the limitations: a small sample size can give misleading results. Even a fair coin might give 8 heads in 10 tosses. That is why hypothesis testing needs a predetermined significance level, but at IGCSE level, you mainly comment on reliability.
始终讨论局限性：样本量小可能产生误导结果。即使是匀质硬币，也可能在 10 次抛掷中出现 8 次正面。这就是为什么假设检验需要预设显著性水平，但在 IGCSE 阶段，你主要是对可靠性进行评论。
7. Statistical Investigations | 统计调查
Statistical investigations involve collecting data from a sample and using it to make inferences about a population. You may be asked to compare two sets of data, such as heights of male and female students, and then comment on differences.
统计调查包括从样本中收集数据并用其推断总体。你可能会被要求比较两组数据，比如男女学生的身高，然后对差异进行评论。
Choose an appropriate sampling method. Simple random sampling avoids bias but can be impractical. Systematic sampling (e.g., every 10th person) is easier. Stratified sampling ensures subgroups are represented in proportion to their size in the population, which is often useful when comparing two groups.
选择合适的抽样方法。简单随机抽样能避免偏差，但可能不切实际。系统抽样（例如每第 10 个人）更简便。分层抽样确保子组按其在总体中的比例被代表，这在比较两组时常常很有用。
Summarise data using averages and measures of spread. For paired comparisons, box plots drawn from the same scale are excellent for visual comparison. The IGCSE syllabus expects you to compare medians and interquartile ranges (IQR) and to comment on skewness.
用平均数和离散程度指标概括数据。对于成对比较，相同尺度上绘制的箱线图非常利于视觉对比。IGCSE 大纲要求你能比较中位数和四分位距 (IQR) 并对偏斜度发表评论。
When presenting your investigation, state clearly what you notice and what it suggests. For instance, “The median height of females is 162 cm, while the median for males is 174 cm. This suggests that, in this sample, males tend to be taller.”
在展示你的调查时，清楚地说明你观察到的现象及其暗示的意义。例如，‘女生身高中位数为 162 cm，而男生中位数为 174 cm。这表明在该样本中，男生通常更高。’
8. Presenting Data: Graphs and Diagrams | 数据呈现：图形与图表
Visual displays make your findings accessible. The IGCSE Edexcel course covers bar charts, pie charts, histograms, frequency polygons, cumulative frequency diagrams and scatter graphs. Each has a specific purpose.
视觉展示让你的发现更容易理解。IGCSE Edexcel 课程涵盖条形图、饼图、直方图、频率多边形、累积频率图和散点图。每种图都有特定的用途。
For discrete data, a bar chart with equal widths is appropriate. Label axes clearly, with frequency on the vertical axis and the variable on the horizontal axis. Leave gaps between bars to show data is categorical or discrete.
对于离散数据，等宽的条形图是合适的。清晰标注坐标轴，纵轴为频率，横轴为变量。在条形之间留出空隙以表明数据是分类或离散的。
Histograms are used for continuous data. In IGCSE, if class widths are equal, the height of each bar represents frequency. If widths vary, you need to use frequency density = frequency ÷ class width, and then plot bars with area proportional to frequency. This is a common exam topic.
直方图用于连续数据。在 IGCSE 中，若组距相等，每一条的高度代表频数。若组距不等，你需要用频率密度 = 频数 ÷ 组距，然后以面积与频数成比例的方式绘制条形。这是一个常见的考试主题。
Cumulative frequency curves are handy for finding medians and quartiles. Plot cumulative frequency against the upper class boundary, join with a smooth curve, and then read off values. Always include a key or title.
累积频率曲线便于求中位数和四分位数。以累积频数对组上限值描点，用平滑曲线相连，然后读取数值。始终要附上图例或标题。
For bivariate data, scatter graphs show correlation. You can then add a line of best fit by eye and use it to make predictions. Be cautious: extrapolation outside the data range can be unreliable.
对于双变量数据，散点图能显示相关性。然后你可以用手工方式添加最佳拟合线并用它进行预测。要谨慎：在数据范围之外进行外推可能不可靠。
9. Analysing Results and Drawing Conclusions | 分析结果并得出结论
Analysis goes beyond description. Calculate the mean, median, mode and range, and pick the most relevant. In an experiment to compare two groups, you might say: “Group A has a higher mean, but also a larger range, indicating more variation.”
分析超越描述。计算平均数、中位数、众数和极差，并选择最相关的。在比较两组的实验中，你可能会说：‘A 组的平均数较高，但极差也更大，表明变异性更强。’
Consider the shape of distributions. Use terms like symmetrical, positively skewed (tail to the right) and negatively skewed (tail to the left). Skewness can affect which average is most representative.
考虑分布的形状。使用诸如对称、正偏斜（尾巴向右）和负偏斜（尾巴向左）等术语。偏斜度会影响哪一种平均数最具代表性。
Make a clear concluding statement that refers back to the aim. If you hypothesised that a dice is fair and experimental probabilities are close to 0.167, conclude that the evidence supports fairness. If not, suggest possible reasons.
写出清晰的结论句并回指实验目标。如果你假设骰子是公平的且实验概率接近 0.167，则结论可以是证据支持公平性。若不然，提出可能的原因。
Evaluate the method: was the sample size large enough? Were trials truly random? Could measurement errors have occurred? Always propose at least one specific improvement. For example, “Use a mechanical dice-roller to eliminate human bias.”
评价方法：样本量够大吗？试验是否真正随机？是否存在测量误差？务必提出至少一条具体的改进措施。例如，‘使用机械掷骰器以消除人为偏倚。’
10. Common Pitfalls and How to Avoid Them | 常见错误及避免方法
One frequent mistake is confusing experimental probability with theoretical probability. Remember that experimental results are estimates based on sample data; they are not exact truths. Always label your probabilities clearly.
一个常见错误是混淆实验概率与理论概率。请记住实验结果是基于样本数据的估计值，并非精确真理。务必清晰地标注你的概率。
Using inconsistent recording methods can ruin data. Decide in advance what you will count and how. For example, if measuring reaction times, define the start and end points clearly to avoid human error.
记录方法不一致会毁掉数据。预先决定你要统计什么以及如何统计。例如，若测量反应时间，明确定义起始与结束点以避免人为误差。
Many students forget to state the number of trials in a probability experiment. Without knowing n, the relative frequency r/n is meaningless. Always report both r and n, or at least r/n with a note about the sample size.
许多学生忘记在概率实验中陈述试验次数。如果不知道 n，相对频率 r/n 就毫无意义。始终同时报告 r 和 n，或者至少报告 r/n 并附注样本量。
When constructing graphs, avoid breaks in axes without clear indication, and choose sensible scales. Misleading scales can distort the visual message. Use rulers for straight lines and draw points neatly.
构建图表时，避免无明确标示地截断坐标轴，并选择合适的尺度。误导性尺度可能扭曲视觉信息。使用直尺画直线，并清晰地描点。
11. Example Investigation: Dice Sum Experiment | 示例探究：骰子点数之和实验
Aim: To investigate the experimental probability of obtaining each sum when two fair dice are rolled 150 times, and to compare with the theoretical probabilities.
目标：探究两枚匀质骰子掷 150 次时得到每个点数和实验概率，并与理论概率进行比较。
Method: I rolled two dice together and recorded the sum on a tally chart. I repeated this 150 times, ensuring each roll was independent. The sums range from 2 to 12.
方法：我同时掷两枚骰子，并在计数表上记录点数之和。重复此过程 150 次，确保每次掷骰独立。和的范围从 2 到 12。
Results: I found that sum 7 occurred 26 times, so experimental probability = 26/150 ≈ 0.173. The theoretical probability for sum 7 is 6/36 ≈ 0.167. Other sums are close to their theoretical values, with small differences.
结果：我发现点数之和为 7 出现了 26 次，因此实验概率 = 26/150 ≈ 0.173。和为 7 的理论概率为 6/36 ≈ 0.167。其他和的实验值也接近理论值，只有微小差异。
Conclusion: The experimental results are reasonably consistent with theoretical expectations. The slight differences can be explained by random variation. To improve reliability, I would increase the number of trials to at least 600.
结论：实验结果与理论预期基本一致。微小差异可以用随机变异来解释。为了提高可靠性，我会将试验次数增加到至少 600。
12. Writing Up Your Experiment for Assessment | 撰写实验报告以用于评估
A strong write-up follows a logical structure: Introduction (aim and prediction), Method, Results (tables and graphs), Analysis, Conclusion and Evaluation. This structure matches the scientific approach valued in exams.
一份出色的报告遵循逻辑结构：引言（目标和预测）、方法、结果（表格和图表）、分析、结论与评估。这一结构契合考试中重视的科学方法。
Use precise language and mathematical terms. Instead of “The numbers were spread out,” write “The data showed a wide range with a high standard deviation.” Quantitative statements are always better.
使用精确的语言和数学术语。与其说‘数字很分散’，不如写‘数据显示出大极差和高标准差’。定量陈述总是更好。
Include the raw data in a table, but don’t overcrowd the report. Summarised tables with frequencies and calculated probabilities are more effective. Label table rows and columns clearly.
在表格中包含原始数据，但不要让报告显得拥挤。列出频数和已计算概率的汇总表格更有效。清晰地标示表格的行和列。
Finally, reflect on what you learned and how the experiment could be extended. Could you investigate three dice? Could you examine the effect of a weighted dice? Showing curiosity is a hallmark of a high-level response.
最后，反思你学到了什么以及实验还可以如何扩展。你能探究三枚骰子吗？你能考察加重骰子的影响吗？展现好奇心是高阶答案的标志。
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