📚 Scatter Graphs and Correlation for Cambridge KS3 Maths | 剑桥KS3数学：散点图与相关性
Scatter graphs are a key part of the Cambridge KS3 Mathematics curriculum. They help us display two sets of numerical data on the same diagram and show whether there is a relationship between the variables. In many exam papers, including page 221 question 2, you are asked to plot points, describe correlation, draw a line of best fit, and use it to make predictions. This article reviews the essential skills step by step.
散点图是剑桥KS3数学课程的重点之一。它们帮助我们在同一幅图中展示两组数值数据，并观察变量之间是否存在关系。在包括第221页第2题在内的许多试卷中，你都需要描点、描述相关性、画出最佳拟合线并利用它进行预测。本文将逐步复习这些核心技能。
1. What is a scatter graph? | 什么是散点图？
A scatter graph is a diagram that plots pairs of numerical data as points on a coordinate grid. Each point represents one observation. One variable is shown on the horizontal x-axis and the other is shown on the vertical y-axis.
散点图是一种将成对数值数据作为点绘制在坐标网格上的图。每个点表示一次观测。一个变量显示在水平 x 轴上，另一个变量显示在垂直 y 轴上。
Scatter graphs are used when we want to investigate a possible relationship between two continuous variables, such as height and arm span, temperature and ice cream sales, or revision time and test score. They are especially important in Cambridge Checkpoint data-handling questions.
当我们想要研究两个连续变量之间可能的关系时，会使用散点图，例如身高与臂展、温度与冰淇淋销量，或复习时间与考试成绩。它们在剑桥 Checkpoint 数据处理题中尤其重要。
2. Plotting points accurately | 准确描点
To plot a point, read each pair of values as (x, y). The first number tells you how far to move along the x-axis, and the second number tells you how far to move up the y-axis. Always check the scale on each axis before you start.
要描点，请按 (x, y) 读取每对数值。第一个数告诉你沿 x 轴移动多远，第二个数告诉你沿 y 轴向上移动多远。开始前务必检查每个轴上的刻度。
Use a sharp pencil and mark a small cross rather than a dot. A cross is easier to see and is accepted in Cambridge KS3 exams. Label each axis with the variable name and its unit, for example 'Height (cm)' or 'Test score (%)'.
使用削尖的铅笔并画小十字而不是圆点。十字更容易看清，并且在剑桥KS3考试中被接受。给每个坐标轴标上变量名称和单位，例如 '身高 (cm)' 或 '考试分数 (%)'。
3. Choosing a suitable scale | 选择合适的刻度
A good scale spreads the data points across most of the grid. If the scale is too small, all the points will be crowded into a corner. If the scale is too large, it will be difficult to read values accurately.
合适的刻度能让数据点分布在网格的大部分区域。如果刻度太小，所有点都会挤在一个角落里。如果刻度太大，就很难准确读取数值。
Use equal intervals on each axis, such as 2, 5, or 10, depending on the data. Do not use uneven intervals, because this distorts the pattern and can make weak correlation look strong.
每个轴上使用相等的间隔，例如 2、5 或 10，具体取决于数据。不要使用不均匀的间隔，因为这会扭曲分布模式，并可能使弱相关看起来很强。
4. Independent and dependent variables | 自变量与因变量
The independent variable is the one you control or change. It is usually plotted on the x-axis. The dependent variable is the one you measure or observe, and it is plotted on the y-axis.
自变量是你控制或改变的变量。它通常绘制在 x 轴上。因变量是你测量或观察的变量，它绘制在 y 轴上。
For example, if you investigate how test score depends on hours of revision, then hours of revision is the independent variable and test score is the dependent variable. In some questions the variables do not have a clear cause-and-effect relationship, but a convention still helps when drawing the graph.
例如，如果你研究考试成绩如何取决于复习小时数，那么复习小时数是自变量，考试成绩是因变量。在某些问题中，变量之间并没有明确的因果关系，但在绘制图形时，使用惯例仍然有帮助。
5. Describing correlation | 描述相关性
Correlation describes the pattern of points on a scatter graph. If the points tend to go upwards from left to right, the correlation is positive. If they tend to go downwards, the correlation is negative. If there is no clear pattern, there is no correlation.
相关性描述散点图上点的分布模式。如果点从左到右呈上升趋势，则为正相关。如果呈下降趋势，则为负相关。如果没有明显模式，则无相关。
You should also describe the strength of the correlation. If the points are close to a straight line, the correlation is strong. If they are widely spread but still show a trend, the correlation is weak or moderate.
你还应描述相关性的强度。如果点靠近一条直线，则相关性很强。如果点分布较散但仍显示出趋势，则相关性较弱或中等。
| Type of correlation | Pattern of points | Example |
|---|---|---|
| Positive | Rises from left to right | Temperature and cold drink sales |
| Negative | Falls from left to right | Age of car and value |
| No correlation | Random scatter | Shoe size and test score |
In Cambridge KS3 answers, write a sentence such as 'There is a strong positive correlation between temperature and ice cream sales.' Mention both direction and strength.
在剑桥KS3答案中，写出类似“温度与冰淇淋销量之间存在强正相关”的句子。要同时提到方向和强度。
6. Line of best fit | 最佳拟合线
When the scatter graph shows a linear correlation, we can draw a line of best fit. This line should follow the general trend of the points and pass through the middle of the data, with roughly the same number of points above and below the line.
当散点图显示线性相关时，我们可以画一条最佳拟合线。这条线应沿点的总体趋势，穿过数据中间，线上方和线下方的点数大致相同。
A line of best fit does not have to pass through the origin, and it does not have to pass through any specific point. Use a ruler and draw one straight line, not a curve. If the correlation is weak or there is no correlation, a line of best fit is not appropriate.
最佳拟合线不必经过原点，也不必经过任何特定点。使用直尺画一条直线，而不是曲线。如果相关性较弱或没有相关性，就不适合画最佳拟合线。
Think of the line as a summary of the linear trend. It helps you see the relationship more clearly and is the basis for estimating unknown values.
可以把这条线看作线性趋势的总结。它帮助你更清楚地看到关系，并且是估计未知值的基础。
7. Making predictions using the line of best fit | 利用最佳拟合线进行预测
Once you have drawn a line of best fit, you can use it to estimate an unknown value. Find one variable on an axis, move vertically or horizontally to the line, then read the corresponding value on the other axis.
画出最佳拟合线后，你可以用它来估计未知值。在一个轴上找到一个变量，向直线作垂直线或水平线，然后在另一个轴上读取相应的值。
Predictions within the range of the plotted data are called interpolation and are usually reliable. Predictions outside the range are called extrapolation and may be less reliable because the trend could change.
在已绘数据范围内的预测称为内插，通常比较可靠。超出数据范围的预测称为外推，可能不太可靠，因为趋势可能会改变。
In exam answers, state that the estimate is only approximate and based on the trend shown by the data. For example, write 'Using the line of best fit, a revised score of about 88 is expected for 6 hours of revision.'
在考试答案中，要说明估计值只是近似的，并且基于数据所显示的趋势。例如，写“根据最佳拟合线，复习6小时的考试分数预计约为88分”。
8. Identifying outliers | 识别异常值
An outlier is a point that lies far away from the overall pattern of the scatter graph. It does not fit the general trend and may be caused by a measurement error or an unusual result.
异常值是远离散点图整体模式的点。它不符合总体趋势，可能是由测量误差或异常结果造成的。
If you find an outlier, you should mention it when describing the correlation. For example, you can say 'Apart from one outlier, there is a positive correlation.' When drawing a line of best fit, an outlier can be ignored because it would pull the line away from the main trend.
如果发现异常值，在描述相关性时应提到它。例如，你可以说“除了一个异常值之外，存在正相关”。在画最佳拟合线时，可以忽略异常值，因为它会使直线偏离主要趋势。
Do not delete an outlier from the diagram unless the question asks you to. Instead, draw attention to it and explain its effect on the result.
除非题目要求，否则不要从图中删除异常值。相反，应指出它并解释它对结果的影响。
9. Common mistakes to avoid | 常见错误
A common mistake is drawing a line of best fit through the origin even when the points do not suggest that. Always let the data determine the position of the line.
一个常见错误是即使数据点并不表明如此，也把最佳拟合线画过原点。始终让数据决定直线的位置。
Another mistake is confusing positive and negative correlation. Check the direction of the slope: uphill is positive, downhill is negative. Also remember that a strong correlation means the points are close to the line, not that the line is steep.
另一个常见错误是混淆正相关和负相关。检查斜线的方向：向上为正，向下为负。还要记住，强相关意味着点靠近直线，而不是直线很陡。
Students sometimes forget to label the axes or use uneven scales. A scatter graph without clear labels or with a distorted scale can lead to wrong interpretations. Finally, do not connect the points with a line segment. A scatter graph uses a line of best fit, not a dot-to-dot drawing.
学生有时会忘记给坐标轴贴标签或使用不均匀的刻度。没有清晰标签或刻度失真的散点图会导致错误解读。最后，不要用线段连接各个点。散点图使用最佳拟合线，而不是逐点连线。
10. Worked example based on page 221 question 2 | 第221页第2题式例题
Question: The table shows the number of hours five students revised and their test scores. Plot the data, describe the correlation, draw a line of best fit, and estimate the score for a student who revised for 6 hours.
题目：表格显示五名学生的复习小时数和考试分数。绘制数据，描述相关性，画出最佳拟合线，并估计复习6小时的学生的分数。
Data: (1, 30), (2, 42), (3, 55), (4, 65), (5, 78). Plotting these points shows an upward trend, so there is a strong positive correlation.
数据：(1, 30)、(2, 42)、(3, 55)、(4, 65)、(5, 78)。描出这些点后呈现上升趋势，因此存在强正相关。
Draw a line of best fit through the middle of the points. To estimate the score for 6 hours, extend the line or read the trend; a value around 88 to 90 is reasonable. Show your working by drawing a dashed line from 6 on the x-axis to the line of best fit and then across to the y-axis.
穿过这些点的中间画一条最佳拟合线。要估计6小时的分数，延长直线或读取趋势；88到90左右的值是合理的。通过从 x 轴上的 6 向最佳拟合线画一条虚线，再延伸到 y 轴，展示你的解题过程。
This type of question tests plotting, interpretation, and prediction in one task, exactly like page 221 question 2 in many Cambridge KS3 practice resources.
这类题目像许多剑桥KS3练习资源中的第221页第2题一样，在一个任务中考查描图、解读和预测。
11. Practice questions | 练习题目
Try these quick questions to check your understanding. For each one, write a sentence describing the correlation.
尝试以下快速练习来检查你的理解。对于每一题，写一句话描述相关性。
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