📚 Scatter Graphs and Correlation | 散点图与相关性
Scatter graphs are one of the most powerful tools for spotting patterns in bivariate data. Whether you are checking if taller people really have longer arms or if studying longer leads to higher marks, a scatter graph gives you a clear visual story. In this article, we will explore how to plot, interpret, and draw conclusions from scatter graphs, covering positive and negative correlation, lines of best fit, outliers, and common exam pitfalls.
散点图是发现双变量数据模式的最有力工具之一。无论你是想验证个子高的人手臂是否真的更长，还是学习时间更长是否带来更高分数，散点图都能给你一个清晰的视觉故事。在本文中，我们将探讨如何绘制、解读散点图并得出结论，涵盖正相关、负相关、最佳拟合线、异常值以及常见考试陷阱。
1. What is a Scatter Graph? | 什么是散点图？
A scatter graph (or scatter plot) is a type of diagram that uses dots to represent values for two different numeric variables. The position of each dot on the horizontal axis (x-axis) and vertical axis (y-axis) shows an individual data pair.
散点图（或称散点图）是一种用点来表示两个不同数值变量值的图表类型。每个点在横轴（x轴）和纵轴（y轴）上的位置代表一个单独的数据对。
Scatter graphs help us see if there is a relationship, called correlation, between the two sets of data. They are especially useful when you have many data points and want to detect a trend.
散点图帮助我们观察两组数据之间是否存在称为相关性的关系。当你有许多数据点并希望发现趋势时，它们尤其有用。
Unlike bar charts or line graphs, the data in a scatter graph are not in a regular sequence; the order of plotting is not important as long as each pair is correctly represented.
与条形图或折线图不同，散点图中的数据没有固定的顺序；只要每对数据正确表示，点的绘制顺序并不重要。
2. Plotting Points | 绘制数据点
To plot a scatter graph, first decide which variable will go on the x-axis. Usually, the independent variable (the one you control or change) is placed on the x-axis, and the dependent variable (the one you measure) goes on the y-axis.
要绘制散点图，首先决定哪个变量放在x轴上。通常将自变量（你控制或改变的变量）放在x轴，因变量（你测量的变量）放在y轴。
For example, if you are investigating the link between height and arm span, you could put 'Height (cm)' on the x-axis and 'Arm span (cm)' on the y-axis. Then choose a suitable scale that uses most of the graph paper and label both axes clearly with the variable names and units.
例如，如果你在调查身高与臂展之间的联系，你可以把“身高（cm）”放在x轴，“臂展（cm）”放在y轴。然后选择合适的刻度，利用大部分图纸空间，并清楚标注轴名和单位。
Suppose you collected these measurements:
假设你收集了以下测量数据：
| Height (cm) | Arm Span (cm) |
|---|---|
| 150 | 148 |
| 160 | 162 |
| 170 | 171 |
| 180 | 179 |
| 165 | 163 |
Each row is plotted as a coordinate: (150, 148), (160, 162), and so on. Notice how the points rise roughly together, hinting at a positive relationship.
每一行作为一个坐标绘制：(150, 148)、(160, 162)等。注意这些点大致一起上升，暗示着正相关关系。
3. Understanding Correlation | 理解相关性
Correlation describes the strength and direction of the relationship between two variables. When the points on a scatter graph follow a clear pattern, we say there is a correlation.
相关性描述了两个变量之间关系的强度和方向。当散点图上的点呈现出清晰模式时，我们就说存在相关性。
Three main types of correlation appear at KS3: positive, negative, and no correlation. The table below summarises them.
KS3阶段主要出现三种相关性类型：正相关、负相关和无相关。下表对这些类型做了总结。
| Correlation Type | English Description | 中文描述 |
|---|---|---|
| Positive (正相关) | As one variable increases, the other tends to increase. The points slope upwards from left to right. | 当一个变量增加时，另一个变量往往也增加。各点从左到右向上倾斜。 |
| Negative (负相关) | As one variable increases, the other tends to decrease. The points slope downwards from left to right. | 当一个变量增加时，另一个变量往往减少。各点从左到右向下倾斜。 |
| No correlation (无相关) | No clear pattern; the points appear scattered randomly. | 没有明显模式；点看起来随机分布。 |
Correlation can also be described as strong or weak depending on how tightly the points cluster around an imaginary straight line.
根据点围绕一条虚拟直线的紧密程度，相关性也可描述为强或弱。
4. Positive Correlation in Detail | 详述正相关
In positive correlation, higher values of one variable are associated with higher values of the other. A classic example is the relationship between the number of hours a student studies and the score they achieve on a test: generally, more study time leads to a higher score.
在正相关中，一个变量的较高值与另一个变量的较高值相关联。一个经典例子是学生学习的小时数与测验得分之间的关系：通常学习时间越长，得分越高。
Other real-life positive correlations include: the age of a tree and its trunk diameter, a car's engine size and its fuel consumption (usually), and a person's height and shoe size. Notice that the points rise together but are rarely in a perfect straight line.
其他现实中的正相关例子包括：树的年龄与树干直径、汽车发动机排量与油耗（通常如此），以及人的身高与鞋码。注意这些点一起上升，但很少形成完美的直线。
When describing a positive correlation, use phrases like 'as x increases, y tends to increase' or 'there is a positive relationship between x and y'.
描述正相关时，请使用“随着x增大，y往往增大”或“x与y之间存在正关系”这类表述。
5. Negative Correlation in Detail | 详述负相关
Negative correlation means that as one variable goes up, the other tends to go down. For instance, the more hours a person exercises, the lower their resting heart rate might be. Another common example is the value of a second-hand car and its age: older cars are typically worth less.
负相关意味着随着一个变量上升，另一个变量往往下降。例如，一个人锻炼的时间越长，其静息心率可能越低。另一个常见例子是二手车的价值与其车龄：车龄越老，价值通常越低。
In a scatter graph showing negative correlation, the plotted points slope downwards from left to right. The slope can be steep or shallow depending on the strength of the relationship.
在显示负相关的散点图中，绘制的点从左到右向下倾斜。斜率的陡峭或平缓取决于关系的强度。
Use language such as 'as x increases, y tends to decrease' when interpreting such graphs in exams.
在考试中解读这类图表时，请使用“随着x增大，y往往减小”这样的语言。
6. No Correlation and Zero Correlation | 无相关与零相关
When there is no correlation, the data points are scattered all over the graph with no discernible pattern. This indicates that the two variables are not linked. For example, there is no relationship between a student's shoe size and their score in a mathematics test.
当无相关时，数据点散布在整个图上，没有可辨识的模式。这表明两个变量之间没有联系。例如，学生的鞋码与其数学测验成绩之间没有关系。
Some textbooks distinguish between 'no correlation' and 'zero correlation', but at KS3 level, the key idea is simply that knowing the value of one variable does not help you predict the other.
有些教材区分“无相关”与“零相关”，但在KS3层面，核心概念仅仅是知道一个变量的值无助于预测另一个变量。
In such graphs, drawing a line of best fit is meaningless because there is no trend to capture.
在这类图中绘制最佳拟合线没有意义，因为不存在趋势可捕捉。
7. Linear vs Non-Linear Relationships | 线性与非线性关系
Most scatter graphs you see at KS3 show a straight-line (linear) pattern, but some relationships are curved. For example, the relationship between the temperature of a cup of tea and the time since it was poured follows a curved cooling pattern, not a straight line.
在KS3阶段见到的大多数散点图显示出直线（线性）模式，但有些关系是曲线形的。例如，一杯茶的温度与倒出后时间的关系遵循一条冷却曲线，而非直线。
Another non-linear example is the relationship between a car's speed and the distance it takes to stop: the stopping distance increases more rapidly at higher speeds, creating an upward curve.
另一个非线性例子是汽车速度与制动距离之间的关系：速度越高，制动距离增加
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
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