📚 Correlation | 相关性
Correlation is one of the most useful ideas in A Level Mathematics: it measures how two quantitative variables move together in a linear way. In the Edexcel specification, you need to be able to draw and interpret scatter diagrams, calculate the product moment correlation coefficient (PMCC), and carry out a hypothesis test for zero correlation. This article breaks the whole topic into clear, exam-focused sections.
相关性是 A Level 数学中最有用的概念之一：它衡量两个定量变量之间线性相关的方向和强度。在 Edexcel 考试大纲中，你需要能够绘制并解释散点图、计算积矩相关系数，并对总体相关系数是否为零进行假设检验。本文把这个主题拆分成清晰、紧扣考点的若干部分。
1. What Is Correlation? | 什么是相关性
Correlation describes the strength and direction of a linear relationship between two variables. It is usually measured on a scale from -1 to 1. A positive correlation means that high values of one variable tend to occur with high values of the other; a negative correlation means that high values of one variable tend to occur with low values of the other. A correlation of 0 suggests no linear relationship, but there could still be a non-linear relationship.
相关性描述两个变量之间线性关系的强度和方向。它通常在 -1 到 1 之间取值。正相关意味着一个变量取较大值时，另一个变量也倾向于取较大值；负相关意味着一个变量取较大值时，另一个变量倾向于取较小值。相关系数为 0 表明不存在线性关系，但仍可能存在非线性关系。
2. Scatter Diagrams | 散点图
A scatter diagram is the first step in any correlation analysis. Each paired observation is plotted as a single point, with the explanatory variable on the horizontal axis and the response variable on the vertical axis. The pattern of points gives a quick visual impression of the direction, strength, and possible outliers in the data.
散点图是任何相关性分析的第一步。每对观测值都绘制为一个点，解释变量放在横轴，响应变量放在纵轴。点的分布模式可以快速直观地判断数据的方向、强度和可能存在的异常值。
In an exam, you may be given a scatter diagram and asked to describe the correlation. Use precise language: positive, negative, strong, weak, linear, non-linear, or no correlation. If the points follow a curved pattern, say that the relationship is non-linear, not simply that there is no correlation.
考试中可能会给你一个散点图并要求描述相关性。请使用准确的语言：正相关、负相关、强相关、弱相关、线性相关、非线性相关或无相关。如果点呈现曲线趋势，应说明关系是非线性的，而不是简单地说没有相关。
3. Types and Strength of Correlation | 相关性的类型与强度
of r = 0 means no linear correlation, though it does not rule out a non-linear relationship.
积矩相关系数通常记为 r，取值范围在 -1 到 1 之间。r = 1 表示完全正线性相关：所有点都恰好位于一条有正斜率的直线上。r = -1 表示完全负线性相关。r = 0 表示没有线性相关，但并不排除非线性关系的存在。
The table below gives a practical guide to interpreting the size of r in many A Level problems.
下表给出了许多 A Level 问题中解释 r 大小的实用参考。
| Value of r | Interpretation |
|---|---|
| r = 1 | Perfect positive linear correlation |
| 0.7 ≤ r < 1 | Strong positive linear correlation |
| 0.4 ≤ r < 0.7 | Moderate positive linear correlation |
| 0 < r < 0.4 | Weak positive linear correlation |
| r = 0 | No linear correlation |
| -0.4 < r < 0 | Weak negative linear correlation |
| -0.7 < r ≤ -0.4 | Moderate negative linear correlation |
| -1 < r ≤ -0.7 | Strong negative linear correlation |
| r = -1 | Perfect negative linear correlation |
4. Product Moment Correlation Coefficient (PMCC) | 积矩相关系数
The PMCC is also called Pearson's correlation coefficient. It measures only linear association, so it should not be used to judge curved relationships. The formula is:
积矩相关系数也称为 Pearson 相关系数。它只衡量线性相关，因此不应用于判断曲线关系。其公式为：
r = Sxy / √Sxx × Syy
where Sxx and Syy measure the variability in x and y separately, and Sxy measures the joint variability of x and y. The sign of r is the same as the sign of Sxy.
其中 Sxx 和 Syy 分别度量 x 和 y 的变异性，Sxy 度量 x 和 y 的共同变异性。r 的符号与 Sxy 的符号相同。
5. Calculating Sxx, Syy and Sxy | 计算 Sxx、Syy 和 Sxy
For a sample of n paired observations (x₁, y₁), (x₂, y₂), ..., (xₙ, yₙ), the key sums are calculated as follows:
对于 n 对观测值 (x₁, y₁)、(x₂, y₂)、……、(xₙ, yₙ)，关键和式按以下方法计算：
Sxx = Σx² − (Σx)² / n
Syy = Σy² − (Σy)² / n
Sxy = Σxy − (Σx)(Σy) / n
These corrected sums remove the effect of the means, so r is not affected by changing the units of measurement. In the exam, you often need to use these three values from a summary table.
这些校正和式去除了均值的影响，因此 r 不受测量单位变化的影响。考试中通常需要从汇总表中得到这三个值。
6. Worked Example: Finding r | 计算示例：求 r
Five students take a short test before and after revision. The pairs (x, y) are (1, 2), (2, 4), (3, 5), (4, 4) and (5, 7). Calculate the PMCC.
五名学生在复习前后分别完成一个简短测试，数据对 (x, y) 为 (1, 2)、(2, 4)、(3, 5)、(4, 4) 和 (5, 7)。计算积矩相关系数。
| x | y | xy | x² | y² |
|---|---|---|---|---|
| 1 | 2 | 2 | 1 | 4 |
| 2 | 4 | 8 | 4 | 16 |
| 3 | 5 | 15 | 9 | 25 |
| 4 | 4 | 16 | 16 | 16 |
| 5 | 7 | 35 | 25 | 49 |
Here n = 5, Σx = 15, Σy = 22, Σxy = 76, Σx² = 55 and Σy² = 110. Therefore:
这里 n = 5，Σx = 15，Σy = 22，Σxy = 76，Σx² = 55，Σy² = 110。因此：
Sxy = 76 − (15 × 22) / 5 = 76 − 66 = 10
Sxx = 55 − 15² / 5 = 55 − 45 = 10
Syy = 110 − 22² / 5 = 110 − 96.8 = 13.2
r = 10 / √10 × 13.2 = 10 / √132 ≈ 0.870
This value suggests a strong positive linear correlation between the two test scores.
该值表明两次测试成绩之间存在较强的正线性相关。
7. Hypothesis Test for Zero Correlation | 零相关的假设检验
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