📚 Scatter Graphs and Correlation | 散点图与相关性
Scatter graphs are a powerful visual tool used in statistics to explore the relationship between two sets of data. In the Cambridge KS3 Mathematics syllabus, page 265 introduces how to plot scatter graphs, describe correlation, and interpret trends. This article breaks down every key concept, from plotting points to drawing lines of best fit, so you can master this topic with confidence.
散点图是统计学中用于探索两组数据之间关系的强大可视化工具。在剑桥 KS3 数学大纲中，第 265 页介绍了如何绘制散点图、描述相关性以及解读趋势。本文分解了从描点到绘制最佳拟合线的每一个关键概念，帮助你自信掌握这一主题。
1. What are Scatter Graphs? | 什么是散点图？
A scatter graph (or scatter plot) is a type of diagram that uses Cartesian coordinates to display values for two variables from a data set. Each point on the graph represents a single observation. One variable is plotted on the horizontal x‑axis and the other on the vertical y‑axis. Scatter graphs do not join the dots with lines like line graphs; instead, they show the distribution of points, which helps to spot patterns and relationships.
散点图（或称散布图）是一种利用笛卡尔坐标展示数据集中两个变量数值的图表。图上的每一个点代表一次观测。一个变量画在水平 x 轴上，另一个画在垂直 y 轴上。散点图不像折线图那样用线段连接各点，而是展示点的分布，这有助于发现规律和关系。
For example, if you want to see whether there is a link between the number of hours students revise and their test scores, you could plot each student's revision time (hours) on the x‑axis and their test score (out of 100) on the y‑axis. The resulting scatter graph would reveal any possible connection.
例如，如果你想看看学生复习时数与考试成绩之间是否存在关联，可以把每个学生的复习时间（小时）画在 x 轴上，考试成绩（百分制）画在 y 轴上。得到的散点图会揭示出任何可能的联系。
2. How to Plot a Scatter Graph | 如何绘制散点图
To plot a scatter graph, start by drawing two perpendicular axes on graph paper or using digital tools. Label the x‑axis with the independent variable (the one you control or that comes first) and the y‑axis with the dependent variable (the one you measure or observe). Choose sensible scales that allow all data pairs to fit clearly. Then, for each pair of values (x, y), place a small cross or dot at the intersection of the corresponding grid lines. Add a title that explains what the graph shows.
绘制散点图时，先在坐标纸或数字工具上画出两条垂直的坐标轴。用自变量（你控制或先发生的变量）标记 x 轴，用因变量（你测量或观察的变量）标记 y 轴。选择合适的刻度，让所有数据对都能清晰显示。然后，对每一组数值 (x, y)，在相应网格线的交点处画一个小叉或点。最后添加能说明图表内容的标题。
The table below illustrates a typical data set ready for plotting:
下表展示了可以用于绘图的典型数据集：
| Student | Revision (hours) | Test score |
|---|---|---|
| A | 2 | 35 |
| B | 4 | 55 |
| C | 6 | 65 |
| D | 8 | 80 |
| E | 10 | 90 |
3. Types of Correlation | 相关性的类型
Correlation describes the direction of the relationship between the two variables shown on a scatter graph. There are three main types: positive correlation, negative correlation, and no correlation. In positive correlation, as one variable increases, the other tends to increase as well. The points slope upwards from left to right. In negative correlation, as one variable increases, the other tends to decrease; the points slope downwards. If there is no correlation, the points are scattered randomly with no clear upward or downward pattern.
相关性描述散点图中两个变量之间关系的方向。主要有三种类型：正相关、负相关和无相关。在正相关中，当一个变量增加时，另一个变量也趋向于增加，点从左到右呈上升趋势。在负相关中，当一个变量增加时，另一个趋于减少，点呈下降趋势。如果无相关，点随机散布，没有明显的上升或下降模式。
It is essential to remember that correlation does not imply causation. Just because two variables show a strong correlation does not mean one causes the other. There may be a third hidden factor influencing both.
必须记住，相关性并不意味着因果关系。两个变量呈现强相关，并不代表一个导致了另一个，可能存在第三个隐藏因素同时影响两者。
4. Strength of Correlation | 相关性的强度
Beyond direction, correlation also has a strength. The strength describes how closely the points follow a straight line. A strong correlation means the points lie very close to a straight line, either positive or negative. A weak correlation means the points are more loosely spread around a possible trend line. A moderate correlation falls somewhere in between. The strength can be judged visually by how narrow the point cluster is.
除了方向，相关性还有强度。强度描述点沿一条直线的紧密程度。强相关意味着点非常靠近一条直线，无论是正向还是负向。弱相关意味着点在一条可能的趋势线周围分布得更松散。中等相关则介于两者之间。强度可以通过点群的狭窄程度进行视觉判断。
When describing correlation in the exam, you must mention both direction and strength. For example: 'The scatter graph shows a strong positive correlation between temperature and ice cream sales.'
在考试中描述相关性时，必须同时提及方向和强度。例如：“散点图显示温度与冰淇淋销量之间存在强正相关。”
5. Interpreting Scatter Graphs | 解读散点图
Interpreting a scatter graph involves more than just stating the correlation. You need to think about what the pattern means in the real‑world context. Ask yourself: Does the trend make sense? Are there any points that stand out as unusual? What does the graph suggest about the relationship between the variables? Being able to write a sentence describing the overall story of the graph is a key skill in KS3 mathematics.
解读散点图不仅仅是说出相关性，还需要思考这种模式在现实情境中的含义。问问自己：这个趋势合理吗？有没有哪些点显得异常？这张图对变量之间的关系有何提示？能够用一句话写出图表所反映的整体情况，是 KS3 数学的关键技能之一。
For instance, if a scatter graph shows a negative correlation between the number of hours spent watching TV and reading scores, you might suggest that more TV time could be associated with less time for reading practice, but you must also consider other factors.
例如，如果散点图显示看电视的时数与阅读分数之间存在负相关，你可以推测更多的看电视时间可能与更少的阅读练习时间有关，但必须同时考虑其他因素。
6. Drawing a Line of Best Fit | 绘制最佳拟合线
When a scatter graph shows a moderate or strong correlation, it is useful to draw a line of best fit (also called a trend line). This line should be drawn so that it passes through the centre of the data points, with as many points above the line as below. It does not have to go through the origin, and it should be a straight line drawn with a ruler. The line helps to model the relationship and can be used to make predictions.
当散点图呈现中等或强相关时，绘制一条最佳拟合线（也称为趋势线）很有用。这条线应当穿过数据点的中心，其上下的点数大致相等。它不必经过原点，必须用直尺画成直线。这条线有助于建立关系模型，并可用于预测。
Follow these steps to draw the line: identify the general trend, place your ruler so that it balances the points, and draw a single, continuous line across the graph. Do not simply connect the first and last points – the line should represent all the data.
按以下步骤画线：确定总体趋势，放置直尺使各点平衡，画出一条贯穿全图的单一直线。不要只是连接首尾两点——这条线应代表所有数据。
7. Making Predictions from the Line of Best Fit | 利用最佳拟合线进行预测
Once the line of best fit is drawn, you can estimate unknown values. There are two types of predictions: interpolation and extrapolation. Interpolation means predicting a value within the range of the given data. This is reliable because it stays inside the known data pattern. Extrapolation means predicting a value outside the existing data range, extending the trend line beyond the original points. Extrapolation can be less reliable because the relationship may not continue linearly past the observed interval.
画好最佳拟合线后，你可以估算未知数值。预测分为两种：内推法和外推法。内推法指在已有数据范围内进行预测。这比较可靠，因为它停留在已知的数据模式内。外推法则是在已有数据范围之外进行预测，将趋势线延伸到原始点之外。外推法可能不太可靠，因为关系在观测区间之外可能不继续保持线性。
To predict, locate the x‑value on the horizontal axis, go up to the line of best fit, then read across to find the corresponding y‑value. Always state clearly whether your prediction is by interpolation or extrapolation.
进行预测时，在水平轴上找到 x 值，向上追溯到最佳拟合线，再横向读取相应的 y 值。务必清楚说明你的预测是内推还是外推。
8. Outliers | 异常值
An outlier is a data point that lies far away from the general pattern of the other points on a scatter graph. Outliers can occur because of measurement errors, recording mistakes, or a genuinely unusual observation. Identifying outliers is important because they can affect the position of the line of best fit and lead to incorrect conclusions.
异常值是指散点图上远离其他点总体形态的数据点。异常值可能由于测量误差、记录失误或真正异常的观测而产生。识别异常值很重要，因为它们可能影响最佳拟合线的位置，从而导致错误的结论。
When you find an outlier, you should mention it in your interpretation. Sometimes analysts will draw the line of best fit ignoring the outlier, but they must explain why they have done so. In your KS3 work, simply note the presence of an outlier and suggest a possible reason.
当你发现异常值时，应在解读中提及。有时分析者会忽略异常值来绘制最佳拟合线，但他们必须解释这样做的原因。在 KS3 的学习中，只需指出异常值的存在并给出可能的原因即可。
9. Common Mistakes to Avoid | 常见错误与避免方法
Many students lose marks on scatter graph questions due to avoidable errors. One common mistake is drawing a line of best fit that goes through the origin when the data does not support it. Another is forcing the line to pass through a specific point instead of balancing all points. Students also forget to label axes with units or choose inconsistent scales, making the graph difficult to read. Finally, some confuse correlation type and strength, and simply write 'positive' without mentioning whether it is strong, moderate, or weak.
许多学生在散点图题目上因可避免的错误而失分。一个常见错误是当数据不支持时，仍让最佳拟合线经过原点。另一个错误是强行使线经过某个特定点，而不是平衡所有点。学生还会忘记在坐标轴上标注单位，或选择不一致的刻度，导致图表难以阅读。最后，有些人混淆了相关性的类型和强度，只写“正相关”而未说明是强、中等还是弱。
Always check your graph: axes labelled, title given, scale consistent, points plotted accurately, and line of best fit straight and well‑positioned. Double‑check your description to include both direction and strength.
务必检查你的图形：坐标轴已标记，标题已给出，刻度一致，点描画准确，最佳拟合线为直线且位置恰当。再次确认你的描述包含了方向和强度。
10. Real‑life Applications | 实际生活中的应用
Scatter graphs are not just confined to the classroom; they are used widely in science, business, and social studies. Scientists use scatter graphs to examine links between temperature and reaction rate. Market researchers analyse advertising spend against sales figures. Health professionals study the relationship between exercise and heart rate. Understanding scatter graphs allows you to analyse data critically and make informed predictions in everyday life.
散点图不仅限于课堂，还广泛用于科学、商业和社会研究中。科学家利用散点图研究温度与反应速率之间的关系。市场研究人员分析广告支出与销售额数据。健康专业人士研究运动与心率的关系。理解散点图使你能够批判性地分析数据，并在日常生活中做出明智的预测。
When you look at news articles showing a correlation between two factors, you can now look at the scatter plot behind the headline and ask: Is the correlation strong or weak? Is the line of best fit reasonable? Are there outliers affecting the story?
当你看到新闻文章展示两个因素之间的相关性时，你现在可以看看标题背后的散点图并提出问题：相关性强吗？最佳拟合线合理吗？是否有异常值影响了结论？
11. Practice Problem Set | 配套练习题
Test your understanding with these quick questions. Answers are provided in a separate revision pack on aleveler.com.
- Plot the points from the table in Section 2. Describe the type and strength of correlation.
- A scatter graph shows a moderate negative correlation between the number of absent days and final grades. Sketch a likely scatter graph and draw a line of best fit.
- A line of best fit passes through (2, 30) and (8, 70). Estimate the y‑value when x = 5. Is this interpolation or extrapolation?
通过以下快速问题检验你的理解。答案可在 aleveler.com 的独立复习资料包中查看。
- 描出第 2 节表格中的点。描述相关性的类型和强度。
- 一张散点图显示缺勤天数与期末成绩之间呈中等负相关。画出可能的散点图草稿并画出最佳拟合线。
- 一条最佳拟合线经过 (2, 30) 和 (8, 70)。估计当 x = 5 时的 y 值。这是内推还是外推？
12. Summary and Key Takeaways | 总结与核心要点
Scatter graphs help us visualise bivariate data and understand relationships. Always remember to plot points accurately, describe correlation using both direction and strength, and draw the line of best fit only when it is appropriate. Distinguish between interpolation and extrapolation, and watch out for outliers. Mastering these skills will strengthen your data handling abilities and prepare you well for Checkpoint assessments.
散点图帮助我们可视化双变量数据并理解关系。请始终记住准确描点，使用方向和强度来描述相关性，并仅在合适时绘制最佳拟合线。区分内推和外推，并留意异常值。掌握这些技能将增强你的数据处理能力，为 Checkpoint 评估做好充分准备。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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