📚 Scatter Graphs and Correlation | 散点图与相关性
A scatter graph is a diagram used to display the relationship between two continuous variables. Each point on the graph represents a pair of data values, with one variable measured on the horizontal x-axis and the other on the vertical y-axis.
散点图是用来表示两个连续变量之间关系的图表。图上的每个点都代表一对数据值，其中一个变量在水平 x 轴上测量，另一个变量在垂直 y 轴上测量。
Scatter graphs are especially useful in KS3 mathematics because they help you see whether there is a pattern, trend, or correlation between the two sets of data. Being able to describe correlation is a key skill in the Cambridge Checkpoint and later IGCSE courses.
散点图在 KS3 数学中特别有用，因为它可以帮助你看到两组数据之间是否存在模式、趋势或相关性。能够描述相关性是剑桥 Checkpoint 以及后续 IGCSE 课程中的一项关键技能。
1. What is a Scatter Graph? | 什么是散点图？
A scatter graph is drawn on a coordinate grid. The independent variable is usually placed on the x-axis, while the dependent variable is placed on the y-axis. For example, if you are investigating how outside temperature affects the number of ice creams sold, temperature is the independent variable and ice cream sales are the dependent variable.
散点图绘制在坐标网格上。自变量通常放在 x 轴上，因变量放在 y 轴上。例如，如果你正在研究室外温度如何影响冰淇淋的销量，那么温度是自变量，冰淇淋销量是因变量。
Each pair of data is plotted as a single point, such as (24 °C, 80 ice creams). The pattern formed by the points tells you whether the two variables are linked.
每对数据都绘制成一个点，例如 (24 °C, 80 个冰淇淋)。这些点形成的分布模式可以告诉你两个变量之间是否存在联系。
2. Plotting Points on a Coordinate Grid | 在坐标网格上描点
When plotting a scatter graph, you must read each axis carefully. Find the x-value first, move along the x-axis to that number, then go up or down to the y-value. Mark the point with a small cross or dot.
绘制散点图时，你必须仔细读取每个坐标轴。先找到 x 值，沿 x 轴移动到该数字，然后向上或向下到达 y 值的位置。用小十字或圆点标出该点。
For example, to plot the point (5, 12), move 5 units along the x-axis, then move 12 units up parallel to the y-axis. Make sure your scale is consistent on each axis.
例如，要绘制点 (5, 12)，沿 x 轴移动 5 个单位，然后平行于 y 轴向上移动 12 个单位。确保每个坐标轴上的刻度保持一致。
Always label your axes with the variable name and unit, such as Height (cm) or Test score (%). A good scatter graph has a title and a suitable scale that uses most of the grid.
始终用变量名称和单位标记坐标轴，例如身高 (cm) 或测试成绩 (%)。一幅好的散点图应有标题，并采用能充分利用网格的合适刻度。
| Point | x-value | y-value |
|---|---|---|
| A | 3 | 7 |
| B | 8 | 14 |
| C | 12 | 20 |
3. Types of Correlation | 相关性的类型
Correlation describes the trend shown by the points on a scatter graph. There are three main types of correlation: positive, negative, and no correlation.
相关性描述散点图上的点所显示的趋势。相关性主要有三种类型：正相关、负相关和无相关。
Positive correlation: as one variable increases, the other variable also increases. For example, height and shoe size often show positive correlation.
正相关：当一个变量增大时，另一个变量也增大。例如，身高和鞋码通常呈正相关。
Negative correlation: as one variable increases, the other variable decreases. For example, the number of hours spent watching TV and exam score may show negative correlation.
负相关：当一个变量增大时，另一个变量减小。例如，看电视的小时数和考试分数可能呈负相关。
No correlation: there is no clear pattern between the variables. For example, shoe size and the number of siblings a person has are unlikely to be related.
无相关：变量之间没有清晰的模式。例如，鞋码和一个人的兄弟姐妹数量不太可能相关。
4. Strength of Correlation | 相关性的强度
Correlation is not just positive, negative, or none. It can also be described as strong, moderate, or weak. If the points lie very close to a straight line, the correlation is strong. If the points are widely scattered but still show a general direction, the correlation is weak.
相关性不只是正相关、负相关或无相关。它还可以被描述为强、中或弱。如果点非常接近一条直线，则相关性很强。如果点分布较散但仍显示出总体方向，则相关性较弱。
For example, points tightly following an upward slope show strong positive correlation. Points loosely spread around a downward slope show weak negative correlation.
例如，紧密沿着上升趋势分布的点显示出强正相关。松散地分布在下降趋势周围的点显示出弱负相关。
In exam questions you may be asked to describe both the direction and the strength of the correlation, such as 'strong positive correlation' or 'weak negative correlation'.
在考试题目中，你可能会被要求同时描述相关性的方向和强度，例如“强正相关”或“弱负相关”。
5. Line of Best Fit | 最佳拟合线
When a scatter graph shows a clear correlation, you can draw a line of best fit. This is a straight line that passes through the middle of the points, with roughly equal numbers of points on each side of the line.
当散点图显示出明显的相关性时，你可以绘制一条最佳拟合线。这是一条穿过各点中间的直线，直线两侧的点数大致相等。
The line of best fit does not have to pass through the origin, and it does not have to pass through any of the plotted points exactly. It should summarise the overall trend of the data.
最佳拟合线不必经过原点，也不必精确经过任何一个已描出的点。它应当总结数据的总体趋势。
To draw the line, position a ruler so that the points are balanced above and below it. Ignore any anomalous points that are clearly far away from the main pattern.
绘制这条线时，将直尺放在使点上下平衡的位置。忽略那些明显远离主要分布模式的异常点。
6. Making Predictions | 进行预测
You can use a line of best fit to estimate an unknown value. If the unknown value lies within the range of the data, this is called interpolation. If it lies outside the range, this is called extrapolation.
你可以使用最佳拟合线来估计未知值。如果未知值位于数据范围内，这称为内插。如果它超出数据范围，则称为外推。
For example, if your data shows the height of a plant from week 1 to week 10, predicting the height at week 6 is interpolation. Predicting the height at week 15 is extrapolation.
例如，如果你的数据显示了植物从第 1 周到第 10 周的高度，那么预测第 6 周的高度是内插，预测第 15 周的高度是外推。
Extrapolation can be unreliable because the trend may not continue beyond the data you have collected. Always be cautious when making predictions outside the original range.
外推可能不可靠，因为趋势可能不会在你收集的数据范围之外继续下去。在对原始范围之外的数值进行预测时始终要谨慎。
7. Interpolation and Extrapolation | 内插与外推
Interpolation is generally considered more reliable than extrapolation because you are estimating within the region where data has already been observed. The pattern is supported by evidence from nearby points.
内插通常被认为比外推更可靠，因为你在已经观察到数据的区域内进行估计。这种模式受到附近点的证据支持。
Extrapolation assumes that the same trend continues beyond the data. This may be true in some cases, but in others the relationship may break down, so examiners expect you to mention the risk.
外推假设同样的趋势在数据之外继续存在。这在某些情况下可能是正确的，但在其他情况下这种关系可能会失效，因此考官希望你提到这种风险。
When asked to use a scatter graph for prediction, draw clear lines on the graph to show your working. For example, from an x-value of 45, go up to the line of best fit, then read across to the y-axis.
当被要求使用散点图进行预测时，请在图上画出清晰的线来展示你的解题过程。例如，从 x 值为 45 处向上画到最佳拟合线，然后水平画到 y 轴读数。
8. Correlation and Causation | 相关性与因果关系
A very common mistake is to think that correlation means one variable causes the other. Correlation does not imply causation. Two variables can be strongly correlated without one causing the other.
一个非常常见的错误是认为相关性意味着一个变量导致了另一个变量。相关性并不表示因果关系。两个变量可能强相关，但其中一个并不一定导致另一个。
For example, there may be a strong positive correlation between ice cream sales and drowning incidents. This does not mean ice cream causes drowning; both are linked to warmer weather.
例如，冰淇淋销量和溺水事件之间可能存在强正相关。这并不意味着冰淇淋导致溺水；两者都与较暖的天气有关。
Always consider whether there could be a third factor behind the trend. In exam questions, look out for statements that wrongly suggest one variable is the direct cause of another.
始终要考虑趋势背后是否存在第三个因素。在考试题目中，要留意那些错误地暗示一个变量是另一个变量的直接原因的陈述。
9. Worked Example | 例题解析
A teacher records the number of revision hours and the test mark for 10 students. The data shows that as revision hours increase, the test mark generally increases. The points form an upward pattern close to a straight line.
一位老师记录了 10 名学生的复习小时数和测试成绩。数据显示，随着复习小时数的增加，测试成绩通常会提高。这些点形成接近直线的向上分布模式。
This is described as strong positive correlation. You could draw a line of best fit to predict the test mark for a student who revised for 6.5 hours.
这可以描述为强正相关。你可以绘制一条最佳拟合线来预测复习 6.5 小时的学生的测试成绩。
Suppose the line of best fit has equation:
假设最佳拟合线的方程为：
y = 6x + 30
To predict the mark for 6.5 hours, substitute x = 6.5:
要预测 6.5 小时的成绩，代入 x = 6.5：
y = 6 × 6.5 + 30 = 39 + 30 = 69
Therefore, a student who revises for 6.5 hours is predicted to score 69 marks. This is an interpolation because 6.5 hours is within the given data range.
因此，复习 6.5 小时的学生预计得分 69 分。这是内插，因为 6.5 小时在给定的数据范围内。
10. Common Mistakes | 常见错误
One common mistake is drawing a line of best fit through the origin even when the data does not suggest this. Always let the pattern of the points decide where the line should go.
一个常见错误是即使数据不支持，也把最佳拟合线画过原点。始终让点的分布模式决定这条线应该画在哪里。
Another mistake is forcing the line through a particular point, such as the first or last point. The line of best fit should represent all the data, not just the ends.
另一个错误是强行让这条线经过某个特定的点，例如第一个或最后一个点。最佳拟合线应该代表所有数据，而不仅仅是端点。
Students also sometimes forget to label axes or use an inconsistent scale. This can make the scatter graph difficult to read and can lose marks in a written examination.
学生有时还会忘记标记坐标轴或使用不一致的刻度。这会使散点图难以阅读，并可能在书面考试中失分。
Finally, do not confuse positive and negative correlation. Check the general direction of the points: up to the right means positive, down to the right means negative.
最后，不要混淆正相关和负相关。检查点的总体方向：向右上方表示正相关，向右下方表示负相关。
11. Exam Tips | 考试技巧
When describing correlation, use the full phrase, such as 'strong positive correlation'. Do not just write 'positive' unless the question asks for the direction only.
在描述相关性时，请使用完整短语，例如“强正相关”。除非题目只要求方向，否则不要只写“正”。
For prediction questions, show the method clearly. Draw the line of best fit first, then draw horizontal and vertical lines to read the estimate from the graph.
对于预测题，请清晰地展示解题方法。先画出最佳拟合线，然后画水平线和垂直线，从图中读取估计值。
If the question asks whether a correlation is reliable, think about the sample size and the spread of the points. A small sample or a weak pattern makes any prediction less reliable.
如果题目问相关性是否可靠，请考虑样本大小和点的离散程度。小样本或较弱的模式会使任何预测都不可靠。
Always answer in the context of the problem. Instead of saying 'as x increases, y increases', say 'as revision hours increase, test marks tend to increase'.
始终结合题目背景作答。不要说“随着 x 增大，y 增大”，而要说“随着复习小时数增加，测试成绩往往提高”。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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