📚 Scatter Graphs and Correlation | 散点图与相关性
In this revision article, we explore the key concepts of scatter graphs and correlation, which appear in the Cambridge KS3 mathematics curriculum (see page 271 of your coursebook). A scatter graph is a powerful tool for displaying the relationship between two sets of data. By plotting paired values on a coordinate grid, you can quickly see whether an increase in one variable is associated with an increase or decrease in the other. This idea of 'correlation' helps you describe and interpret real-world patterns, from science experiments to business trends. We will guide you through plotting points, recognising positive, negative and zero correlation, drawing a line of best fit, and using the graph to make sensible predictions.
在这篇复习文章中，我们将深入探讨 Cambridge KS3 数学课程中出现的散点图与相关性核心概念（参考课本第 271 页）。散点图是展示两组数据之间关系的强大工具。通过将成对数值绘制在坐标网格上，你可以迅速判断一个变量的增加是否与另一个变量的增加或减少相关联。这种“相关性”概念能帮助你描述和解释从科学实验到商业趋势的各种现实模式。我们将带你逐步掌握描点、识别正相关、负相关和无相关、画出最佳拟合直线，并利用图形做出合理预测。
1. What is a Scatter Graph? | 什么是散点图？
A scatter graph, also called a scatter plot, is a diagram that uses Cartesian coordinates to display values for two variables from a data set. Each point on the graph represents a pair of data, with the first value plotted on the horizontal x-axis and the second on the vertical y-axis. Unlike a line graph, the points are not connected in order. Instead, the overall pattern or 'cloud' of points shows the relationship between the two variables. For example, you might plot the number of hours of revision against the test score for a group of students. The resulting scatter of points could reveal whether more revision tends to lead to higher scores.
散点图（又称散点分布图）是一种使用直角坐标系显示数据集中两个变量值的图表。图上的每一个点代表一对数据，第一个值标在横轴（x 轴）上，第二个值标在纵轴（y 轴）上。与折线图不同，这些点不按顺序连接。相反，点的整体形态或“云团”揭示出两个变量之间的关系。例如，你可以将一组学生的复习小时数与测验分数绘制出来。得到的点状分布可能揭示更多复习时间是否倾向于带来更高分数。
When constructing a scatter graph, it is important to choose suitable scales for both axes that make the data spread across the grid. The range of x-values and y-values should be covered, but you should also leave room to avoid crowding the points at the edges. The axes must be labelled clearly with the variable name and its units, and the graph should have a meaningful title.
在制作散点图时，为两个坐标轴选择合适的刻度非常重要，以使数据均匀分布在网格上。x 值和 y 值的范围要覆盖，但也要留出余量，避免点挤在图的边缘。坐标轴必须清晰标注变量名称和单位，图形也要有一个有意义的标题。
2. Plotting a Scatter Graph | 绘制散点图
To plot a scatter graph, you begin with a table of paired data. Suppose a teacher records the daily temperature (°C) and the number of ice creams sold at a kiosk. The data pairs might be (20, 45), (22, 50), (18, 40), (25, 60), etc. Take the first pair: 20 on the x‑axis and 45 on the y‑axis. Mark a small neat cross or dot at the intersection. Repeat for all data pairs. Use a sharp pencil and ensure the points are accurately placed. Common mistakes include misreading a value or swapping the x and y coordinates, so always double‑check.
绘制散点图时，你首先要有一张成对数据的表格。假设一位老师记录了每日温度（°C）和售货亭卖出冰淇淋的数量。数据对可能是 (20, 45)，(22, 50)，(18, 40)，(25, 60) 等。取第一对：x 轴为 20，y 轴为 45，在交叉处画一个小而整齐的叉号或圆点。对所有数据对重复此操作。使用尖细的铅笔并确保位置精确。常见的错误包括读错数值或交换 x 和 y 坐标，所以一定要细心检查。
Sometimes a pair of values is repeated, or two points land in exactly the same spot. To show this, you can draw the second point slightly offset, or simply note the repetition in your analysis. The finished scatter graph lets you see the shape of the data at a glance. Before moving on to describe correlation, always check that your points have been plotted correctly – one misplaced point can completely change the impression of a relationship.
有时一对数值会重复，或两个点恰好标在同一位置。为了显示这一点，可以将第二个点稍微偏移绘制，或在分析中注明重复。完成的散点图让你能一眼看出数据的形态。在进一步描述相关性之前，务必检查描点是否正确——一个放错位置的点可能完全改变关系给人的印象。
3. Understanding Correlation | 理解相关性
Correlation describes the type and strength of the relationship between two variables on a scatter graph. When the points show a clear trend, we say there is a correlation. If one variable tends to increase as the other increases, the correlation is positive. If one variable tends to decrease as the other increases, the correlation is negative. When the points are scattered randomly with no obvious pattern, there is zero correlation. It is essential to remember that correlation does not imply causation – just because two variables move together does not mean one causes the other.
相关性描述散点图上两个变量之间关系的类型与强度。当点呈现出清晰的趋势时，我们说存在相关性。如果一个变量随另一个增加而增加，则相关性为正相关。如果一个变量随另一个增加而减少，则相关性为负相关。如果点散乱分布，没有明显模式，就是无相关（零相关）。必须记住，相关性并不意味着因果关系——两个变量一起变化并不等于一个引起另一个。
In KS3 mathematics, you are expected to recognise these patterns and describe them using precise vocabulary: 'as the x value increases, the y value increases' for positive correlation, and 'as the x value increases, the y value decreases' for negative correlation. The concept of 'strength' will be refined later, but for now you can use words like strong, moderate or weak to describe how tightly the points cluster around an imaginary straight line.
在 KS3 数学中，你必须能够识别这些模式并使用准确的词汇来描述：正相关“随着 x 值增大，y 值也增大”，负相关“随着 x 值增大，y 值减小”。“强度”这一概念以后会进一步细化，但你在现阶段可以用强、中等或弱等词语来描述点围绕一条假想直线的紧密程度。
4. Positive Correlation | 正相关
A scatter graph shows positive correlation when the pattern of points slopes upwards from left to right. This means high values of one variable are associated with high values of the other, and low with low. A classic example is the relationship between a person's height and arm span: taller people generally have longer arm spans. If you were to plot a set of measurements, the points would drift upwards across the graph, forming a ‘cloud’ that stretches diagonally. The closer the points lie to a straight line, the stronger the positive correlation.
当点的形态从左向右上升时，散点图呈现正相关。这意味着一个变量的高值与另一个变量的高值相关联，低值与低值相关联。一个经典的例子是人的身高与臂展之间的关系：个子越高的人通常臂展越长。如果你把一组测量数据绘制出来，点会在图上向上延伸，形成一个沿对角线伸展的“云团”。点越靠近一条直线，正相关越强。
It is useful to sketch a straight line through the middle of the points to represent the trend – this is called a line of best fit (we will cover it later). For positive correlation, this line will have a positive gradient. In everyday language, you might say 'the more you study, the higher your likely score' – provided the points are positively correlated. However, always support such statements with evidence from the graph rather than assuming.
通过点的中间画一条直线来表示趋势会很有用——这条线称为最佳拟合直线（我们稍后会介绍）。对于正相关，这条线的斜率（梯度）为正。用日常语言可以说“学习时间越多，可能得到的分数越高”——前提是这些点呈正相关。但是，始终要用图上的证据来支持这类陈述，而非想当然。
5. Negative Correlation | 负相关
Negative correlation appears when the scatter graph slopes downwards from left to right. As one variable increases, the other tends to decrease. A familiar example is the sale of hot drinks against outside temperature: on hotter days, fewer warm drinks are sold. If you plot daily temperature against the number of teas sold, you would expect the points to fall as you move to the right. The line of best fit for negative correlation would have a negative gradient.
当散点图从左向右下降时，呈现负相关。随着一个变量增加，另一个变量趋于减少。常见的例子是热饮销量与外界气温：天气越热，热饮卖得越少。如果你把每日温度和热茶售出杯数绘制出来，你会看到点随着向右移动而下降。负相关的最佳拟合直线斜率为负。
Describing negative correlation on a graph provides a clear, visual story. In scientific experiments, negative correlation often indicates an inverse relationship. For KS3 investigations, you might measure the time taken for a candle to burn and the amount of wax used: as the burning time increases, the remaining wax decreases – a negative correlation. Always be careful to label axes with the appropriate independent variable (usually on the x‑axis) and dependent variable (on the y‑axis).
在图上描述负相关能清晰地讲述一个视觉故事。在科学实验中，负相关常表明一种反向关系。在 KS3 探究中，你可能测量蜡烛燃烧的时间与蜡的消耗量：随着燃烧时间增加，剩余蜡量减少——这就是负相关。务必注意用合适的自变量（通常放在 x 轴）和因变量（放在 y 轴）来标注坐标轴。
6. No Correlation | 无相关
When a scatter graph displays no clear pattern, the points are scattered in a random ‘cloud’ with no upward or downward trend. This indicates zero correlation. An example might be shoe size and mathematics test scores: there is no reason to believe these two variables are connected. Plotting such data would produce a shapeless cluster of points, and any attempt to draw a line of best fit would be meaningless.
当散点图没有清晰的模式时，点呈随机“云团”，没有上升或下降的趋势。这表示无相关。例如鞋码与数学测验分数之间就没有理由相信存在联系。绘制这类数据会得到一堆无形状的点，任何尝试画出最佳拟合直线都是无意义的。
It is important to recognise that 'no correlation' tells you just as much as finding a correlation. It means that knowing the value of one variable does not help you predict the other. In real life, you must be careful not to imagine a relationship where none exists – this is a common pitfall. Always look at the complete set of points before deciding whether a trend genuinely exists.
认识到“无相关”同样能提供信息和找到相关性是一样重要的。它意味着知道一个变量的值并不能帮你预测另一个变量。在现实生活中，你必须小心，不要凭空想象出一个本不存在的关系——这是一个常见的陷阱。在判断是否真的存在趋势之前，一定要观察全部的点。
7. Strength of Correlation | 相关性的强度
Beyond simply identifying the direction of correlation (positive or negative), we also describe its strength. If the points lie very close to an imaginary straight line, the correlation is strong. If they are widely spread but still show a general slope, the correlation is weak. A moderate correlation lies somewhere in between. Strength is judged by visual inspection, not by calculation, at KS3 level. The more tightly clustered the points, the stronger the relationship.
除了识别相关的方向（正或负），我们还要描述其强度。如果点非常靠近一条假想的直线，相关性强；如果点分布很宽但大体上仍有一定倾斜，相关性弱；中等相关则介于两者之间。在 KS3 阶段，强度是通过目测判断，不涉及计算。点越紧密聚集，关系就越强。
For example, a scatter graph comparing hours of exercise per week and resting heart rate might show a fairly strong negative correlation: the points form a narrow downward band. In contrast, a graph of hours spent watching TV against hair length would show a very weak or zero correlation. Using terms like 'strong positive correlation' or 'weak negative correlation' makes your descriptions more precise and is expected in tests.
例如，比较每周锻炼小时数与静息心率的散点图可能表现出相当强的负相关：点形成一条狭窄的下降带。相反，看电视的小时数与头发长度之间的关系则会呈现极弱或无相关。使用“强正相关”或“弱负相关”等术语会使你的描述更精确，这也是考试中期望的表达方式。
8. Outliers | 离群值
An outlier is a data point that lies far away from the general pattern of the scatter graph. It does not fit the trend shown by the rest of the points. Outliers can occur due to measurement errors, recording mistakes, or because one individual genuinely differs from the group. When you spot an outlier, you should investigate why it happened. In many KS3 problems, you may be asked to circle or describe the outlier and suggest a possible reason for it.
离群值是指远远偏离散点图总体模式的数据点。它与其余点呈现的趋势不符。离群值的出现可能是由于测量误差、记录错误，或者某一对象确实与众不同。当你发现离群值时，应该探究其产生的原因。在许多 KS3 问题中，可能会要求你圈出或描述离群值，并对其可能的原因提出看法。
Handling an outlier in a scatter graph requires care. If the outlier is a clear mistake, such as a student's height recorded as 200 cm when it was meant to be 160 cm, you might correct it. If it is genuine, you should note its effect: it can weaken the apparent correlation or even suggest a different trend. When drawing a line of best fit, you typically ignore the outlier so that the line represents the majority of the data.
处理散点图中的离群值需要谨慎。如果离群值明显是错误，比如某学生的身高本应记录为 160 cm 却写成了 200 cm，你可以改正。如果是真实数据，你应该注意其影响：它可能削弱表面的相关性，甚至暗示出不同的趋势。画最佳拟合直线时，通常会忽略这个离群值，使直线代表大多数数据。
9. Line of Best Fit | 最佳拟合直线
A line of best fit (also called a trend line) is a straight line drawn through the centre of the scatter graph to represent the relationship between the variables. It does not have to pass through all the points, and it should not be forced through the origin unless there is a valid reason. The goal is to have roughly the same number of points above and below the line, with the points as close to it as possible. To draw it, use a transparent ruler and adjust its angle and position until the points are balanced.
最佳拟合直线（也叫趋势线）是一条画在散点图中心的直线，用来表示变量之间的关系。它不一定通过所有的点，也不应勉强通过原点，除非有合理的理由。目标是使线上方和线下方的点数大致相同，并且各点尽可能靠近该线。作图时，用一把透明直尺调整角度和位置，直到各点达到平衡。
At KS3, the line of best fit is always drawn by eye (a ruler is essential). It is not calculated using a formula; you simply judge the best position. Once drawn, you can use the line to estimate unknown values. A well‑drawn line of best fit highlights the trend by smoothing out individual variations. However, avoid 'joining the dots' – the line should be one continuous straight line that captures the overall direction.
在 KS3 阶段，最佳拟合直线始终靠目测画出（直尺必不可少）。它不使用公式计算，而是靠你判断最佳位置。画好后，可以用它来估计未知值。一条画得好的最佳拟合直线通过抚平个别波动来突出趋势。但要避免“连点成线”——这条线应是一条捕捉总体方向的连续直线。
10. Making Predictions | 做出预测
Once a line of best fit has been drawn on a scatter graph, you can use it to make predictions. There are two types: interpolation and extrapolation. Interpolation is estimating a value within the range of the given data. For example, if you have data for temperatures from 10 °C to 30 °C and their corresponding ice cream sales, you can use the line to predict sales at 22 °C, which lies inside the data range. Extrapolation is estimating a value outside the given range, such as predicting sales at 5 °C or 40 °C. Extrapolation is less reliable because you are assuming the trend continues beyond the observed values.
在散点图上画出最佳拟合直线后，你可以用它进行预测。预测分为两种：内插和外推。内插是指在已有数据范围内进行估计。例如，如果你有从 10 °C 到 30 °C 的温度数据及其对应的冰淇淋销量，你就可以用这条线预测 22 °C 时的销量，这个值在数据范围之内。外推是估计数据范围之外的数值，比如预测 5 °C 或 40 °C 时的销量。外推的可靠性较低，因为你假设趋势在观测值之外依然延续。
To make a prediction, follow these steps: (1) locate the x‑value you want on the horizontal axis, (2) draw a vertical line up (or down) to meet the line of best fit, (3) from that intersection, draw a horizontal line across to the y‑axis, and (4) read off the estimated y‑value. Always show these construction lines if you are asked to work from the graph. During tests, you may need to describe how reliable your prediction is, especially if it comes from extrapolation.
得出预测值，遵循以下步骤：（1）在横轴上找到你所需的 x 值，（2）向上（或向下）画一条垂直线，与最佳拟合直线相交，（3）从该交点向右画一条水平线至 y 轴，（4）读出估计的 y 值。如果题目要求根据图形作答，一定要画出这些辅助线。在测验中，你可能需要描述预测值的可靠程度，尤其是如果它来自外推。
11. Real-Life Applications | 实际应用
Scatter graphs and correlation appear in many real-world situations. In science, researchers use them to see if changing the concentration of a reactant affects the rate of reaction. In business, companies plot advertising spend against sales revenue to decide whether increased marketing brings higher profit. In health, medical studies might examine the relationship between daily steps taken and body mass index. Each case requires careful plotting and analysis of trends, and often involves drawing a line of best fit to make forecasts.
散点图和相关性出现在许多现实情景中。在科学中，研究人员用它们来观察改变反应物浓度是否影响反应速率。在商业中，公司绘制广告支出与销售收入的关系图，以决定增加营销是否能带来更高利润。在健康领域，医学研究可能考察每日行走步数与体重指数之间的关系。每种情况都需要仔细描点和分析趋势，并且经常需要画出最佳拟合直线来做出预测。
Understanding correlation also helps you think critically about claims you see in the news. Headlines that say 'eating chocolate makes you smarter' might be based on a positive correlation, but as you now know, correlation does not prove causation. There could be a third factor involved. With your KS3 skills in interpreting scatter graphs, you can ask sensible questions about the data behind such claims and become a more informed citizen.
理解相关性还能帮助你批判性地看待新闻中的论断。诸如“吃巧克力让人更聪明”的标题可能是基于一种正相关，但正如你现在所知，相关并不证明因果关系。还可能涉及第三个因素。运用你在 KS3 阶段学到的解读散点图技能，你可以对此类论断背后的数据提出合理问题，从而成为更明辨事理的公民。
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