← 返回资讯屏轩教育 · 行业资讯

The Product Moment Correlation Coefficient | 积矩相关系数

2026-08-02 17:02 · 本文部分内容由AI辅助创作

📚 The Product Moment Correlation Coefficient | 积矩相关系数

In statistics, the product moment correlation coefficient, often denoted by r, is a measure of the strength and direction of a linear relationship between two variables. It is a core topic in Edexcel A-Level Mathematics, frequently appearing in both pure statistics questions and in combination with regression analysis. Mastering its calculation and interpretation is essential for success.

在统计学中，积矩相关系数（通常记作 r）用于衡量两个变量之间线性关系的强度和方向。它是 Edexcel A-Level 数学的核心课题，经常出现在纯统计题以及与回归分析相结合的题目中。掌握其计算和解读对取得好成绩至关重要。

1. What is the Product Moment Correlation Coefficient? | 什么是积矩相关系数？

The product moment correlation coefficient (PMCC), also known as Pearson's r, was developed by Karl Pearson. It quantifies the degree to which two continuous variables are linearly related. The value of r always lies between -1 and +1 inclusive. A value of +1 indicates a perfect positive linear correlation, -1 a perfect negative linear correlation, and 0 means there is no linear correlation. In practice, real data will give r values somewhere between these extremes.

积矩相关系数 (PMCC)，也称为皮尔逊相关系数，由卡尔·皮尔逊提出。它量化了两个连续变量之间线性相关的程度。r 的值始终介于 -1 和 +1 之间（含端点）。+1 表示完全正线性相关，-1 表示完全负线性相关，0 表示没有线性相关。在实际中，真实数据给出的 r 值通常会落在这两个极值之间的某处。

2. The PMCC Formula | PMCC 公式

For a set of n paired observations (xᵢ, yᵢ), the product moment correlation coefficient is given by the formula:

对于 n 组成对观测值 (xᵢ, yᵢ)，积矩相关系数的公式如下：

r = Sₓᵧ / √Sₓₓ × Sᵧᵧ

where:
Sₓᵧ = Σxᵢyᵢ – (Σxᵢ)(Σyᵢ)/n
Sₓₓ = Σxᵢ² – (Σxᵢ)²/n
Sᵧᵧ = Σyᵢ² – (Σyᵢ)²/n

其中：
Sₓᵧ = Σxᵢyᵢ – (Σxᵢ)(Σyᵢ)/n
Sₓₓ = Σxᵢ² – (Σxᵢ)²/n
Sᵧᵧ = Σyᵢ² – (Σyᵢ)²/n

These are the corrected sums of squares and cross-products. They measure how much the values vary about their respective means.

这些是经过均值校正后的平方和与乘积和，它们衡量了变量围绕各自均值的变异程度。

广告 / Sponsored

🛒
CASIO fx-991CN X 科学计算器 经济统计
¥168
📱 去京东购买
京东精选

3. Calculating r Step by Step | 逐步计算 r

When computing r manually from raw data, a systematic approach avoids mistakes:

当从原始数据手动计算 r 时，采用系统的方法可避免错误：

1. Tabulate the values of x, y, x², y², and xy for each data pair.

1. 为每一对数据分别列出 x、y、x²、y² 和 xy 的值。

2. Calculate the column totals: Σx, Σy, Σx², Σy², Σxy, and note n.

2. 计算各列的总和：Σx、Σy、Σx²、Σy²、Σxy，并记下 n。

3. Compute Sₓₓ, Sᵧᵧ, and Sₓᵧ using the formulas above.

3. 利用上述公式计算 Sₓₓ、Sᵧᵧ 和 Sₓᵧ。

4. Evaluate √Sₓₓ × Sᵧᵧ and then find r = Sₓᵧ / √Sₓₓ × Sᵧᵧ.

4. 计算 √Sₓₓ × Sᵧᵧ，然后求 r = Sₓᵧ / √Sₓₓ × Sᵧᵧ。

5. Round the final value of r to a sensible precision, typically three decimal places.

5. 将最终 r 值舍入到合理精度，通常为三位小数。

4. Using Summary Statistics | 利用汇总统计量

Many Edexcel exam questions provide the summary quantities Σx, Σy, Σx², Σy², Σxy and n directly. In such cases, you can skip the tedious tabulation and simply plug the numbers into the S formulas. Make sure you use the correct n and do not confuse (Σx)²/n with Σ(x²)/n.

许多 Edexcel 考题会直接给出汇总量 Σx、Σy、Σx²、Σy²、Σxy 和 n。遇到这种情况，你可以跳过繁琐的表格，直接将数值代入 S 公式。务必使用正确的 n，并注意区分 (Σx)²/n 与 Σ(x²)/n。

Always store intermediate results in your calculator's memory to retain full accuracy.

始终将中间结果存入计算器的存储器中，以保留完整精度。

5. Interpreting the Value of r | r 值的解读

The absolute value of r, |r|, indicates the strength of the linear association. While there are no absolute rules, the following guidelines are commonly used in A-Level contexts:

r 的绝对值 |r| 表明了线性关联的强度。虽然没有绝对的规则，但在 A-Level 情境中通常使用以下参考准则：