📚 The Product Moment Correlation Coefficient | 乘积矩相关系数
In A-Level Mathematics, the product moment correlation coefficient (PMCC) is one of the most important tools for measuring the strength and direction of a linear relationship between two variables. On the Edexcel specification you must be able to calculate it, interpret its value in context, and carry out a hypothesis test for correlation using critical values from statistical tables.
在 A-Level 数学中，乘积矩相关系数（PMCC）是衡量两个变量之间线性关系强度和方向的最重要工具之一。在 Edexcel 考试大纲中，你需要能够计算 PMCC、解释它的取值，并利用统计表中的临界值对相关关系进行假设检验。
This guide takes you from the basic definition and formula all the way to a full hypothesis-testing procedure, worked examples, and the small details that decide whether you score full marks.
本文从基本定义和公式讲起，一直讲到完整的假设检验流程、例题演示，以及决定你能否拿满分的那些细节。
1. What Is the Product Moment Correlation Coefficient? | 什么是乘积矩相关系数？
The product moment correlation coefficient, usually denoted by r, was developed by Karl Pearson. It measures how close the data points lie to a straight line. If one variable tends to increase as the other increases, r is positive; if one increases while the other decreases, r is negative.
乘积矩相关系数通常记作 r，由卡尔·皮尔逊提出。它衡量数据点与一条直线的接近程度。如果一个变量随另一个变量增加而增加，则 r 为正；如果一个变量增加而另一个变量减少，则 r 为负。
Formally, r is a standardised measure of the linear association between two random variables X and Y. It is built from the covariance of X and Y divided by the product of their standard deviations, which is why it has no units and can be compared across completely different contexts.
严格来说，r 是两个随机变量 X 与 Y 之间线性关联的标准化度量。它由 X 与 Y 的协方差除以两者标准差的乘积构成，因此它没有单位，可以跨越完全不同的情境进行比较。
ans they lie exactly on a straight line with negative gradient, and r = 0 means there is no linear relationship at all.
r 的取值始终在 −1 与 1 之间（含端点）。r = 1 表示所有点恰好落在一条正斜率的直线上，r = −1 表示所有点恰好落在一条负斜率的直线上，而 r = 0 表示完全不存在线性关系。
On the Edexcel specification, the required skills are:
在 Edexcel 考试大纲中，需要掌握的能力包括：
- Understand and use the product moment correlation coefficient and its properties.
- 理解并使用乘积矩相关系数及其性质。
- Calculate r from raw data or from summary statistics.
- 根据原始数据或汇总统计量计算 r。
- Interpret the value of r in the context of the question.
- 结合题目情境解释 r 的取值。
- Carry out a hypothesis test for zero population correlation using tables of critical values.
- 利用临界值表对总体零相关进行假设检验。
- State conclusions clearly, referring to the original variables.
- 清晰陈述结论，并回扣原题中的变量。
2. The Formula and How to Compute r | 公式与计算方法
The most efficient way to compute r by hand is to work with the three summary quantities Sₓₓ, Sᵧᵧ and Sₓᵧ. These are the corrected sums of squares and the corrected sum of products.
手算 r 最有效的方法是先算出三个汇总量：Sₓₓ、Sᵧᵧ 和 Sₓᵧ，即校正后的平方和与乘积和。
Sₓₓ = Σxᵢ² − (Σxᵢ)² ÷ n
Sᵧᵧ = Σyᵢ² − (Σyᵢ)² ÷ n
Sₓᵧ = Σxᵢyᵢ − (Σxᵢ)(Σyᵢ) ÷ n
Once these three quantities are known, the product moment correlation coefficient is simply:
得到这三个量后，乘积矩相关系数就简单地写成：
r = Sₓᵧ ÷ √Sₓₓ × Sᵧᵧ
The equivalent formula written entirely in terms of raw sums is useful when the question gives you the sums directly rather than the data:
当题目直接给出各个求和量而不是原始数据时，用下面这个完全由原始和构成的等价公式更方便：
r = [Σxᵢyᵢ − (Σxᵢ)(Σyᵢ) ÷ n] ÷ √([Σxᵢ² − (Σxᵢ)² ÷ n] × [Σyᵢ² − (Σyᵢ)² ÷ n])
A reliable step-by-step method is:
一套稳妥的分步做法是：
