Advanced Further Mathematics: Product-Moment Correlation Coefficient Calculation and Application | 进阶数学：积矩相关系数计算与应用

📚 Advanced Further Mathematics: Product-Moment Correlation Coefficient Calculation and Application | 进阶数学：积矩相关系数计算与应用

In A-Level Further Mathematics, the product-moment correlation coefficient, often abbreviated as PMCC, measures the strength and direction of a linear relationship between two variables. It is one of the most important ideas in statistics, and Edexcel expects students to be able to calculate it, interpret it and use it in hypothesis testing.

在 A-Level 进阶数学中，积矩相关系数（常简写为 PMCC）用于度量两个变量之间线性关系的强度与方向。这是统计学中最重要的概念之一，Edexcel 考试要求学生能够计算它、解释它，并将其用于假设检验。

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

The product-moment correlation coefficient is a value, usually denoted by r, that summarises how closely two numerical variables follow a straight-line pattern on a scatter graph. The word ‘product-moment’ is used because the coefficient is based on the sum of the products of the deviations from the mean.

积矩相关系数是一个通常用 r 表示的数值，用来概括两个数值变量在散点图上遵循直线模式的程度。“积矩”这个名称之所以出现，是因为该系数基于各偏差乘积之和。

If the data set is a sample, the sample coefficient is written as r. The population coefficient is written using the Greek letter ρ (rho). In A-Level work, you will almost always calculate r from a sample of n paired observations.

如果数据来自样本，样本相关系数记为 r。总体相关系数则用希腊字母 ρ（rho）表示。在 A-Level 的学习中，你通常需要由 n 组成对观测数据计算 r。

The value of r always lies between −1 and +1, inclusive. A value near +1 indicates a strong positive linear relationship, a value near −1 indicates a strong negative linear relationship, and a value near 0 indicates little or no linear relationship.

r 的值总是在 −1 和 +1 之间（含端点）。接近 +1 表示强正线性关系，接近 −1 表示强负线性关系，接近 0 表示几乎没有线性关系。

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2. The Formula and Notation | 公式与记号

For n pairs of data (x₁, y₁), (x₂, y₂), …, (xₙ, yₙ), the PMCC is calculated using the sums of the x-values, y-values, squares and products. The most convenient computational formula for Edexcel is based on Sxx, Syy and Sxy.

对于 n 对数据 (x₁, y₁), (x₂, y₂), …, (xₙ, yₙ)，PMCC 的计算需要使用 x 值之和、y 值之和、平方和与乘积和。对 Edexcel 考试而言，最方便的计算公式基于 Sxx、Syy 和 Sxy。

Sxx = Σx² − (Σx)²/n

Syy = Σy² − (Σy)²/n

Sxy = Σxy − (Σx)(Σy)/n

These quantities can also be written in deviation form: Sxx = Σ(x − x̄)², Syy = Σ(y − ȳ)² and Sxy = Σ(x − x̄)(y − ȳ). The computational forms are usually quicker to use with a calculator or a spreadsheet.

这些量也可以写成离差形式：Sxx = Σ(x − x̄)²，Syy = Σ(y − ȳ)²，Sxy = Σ(x − x̄)(y − ȳ)。计算形式通常在使用计算器或电子表格时更快。

r = Sxy / √(Sxx Syy)

The denominator √(SxxSyy) is always positive, as long as the x-values are not all identical and the y-values are not all identical. If Sxx = 0 or Syy = 0, the correlation coefficient is undefined because there is no linear spread in one of the variables.

只要 x 值不完全相同且 y 值不完全相同，分母 √(SxxSyy) 就恒为正。如果 Sxx = 0 或 Syy = 0，则相关系数没有定义，因为其中一个变量不存在线性离散程度。

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3. Calculating PMCC Step by Step | 分步计算 PMCC

Before doing the algebra, always create a clear table of working. This reduces careless errors and helps the examiner follow your method.

在进行代数运算之前，一定要制一张清晰的计算表。这能减少粗心错误，并帮助考官看清你的方法。

Step 1: List the data as pairs and record x, y, x², y² and xy in a table.

第一步：将数据按成对列出，并在表中记录 x、y、x²、y² 和 xy。

Step 2: Add each column to find Σx, Σy, Σx², Σy² and Σxy.

第二步：对每一列求和，得到 Σx、Σy、Σx²、Σy² 和 Σxy。

Step 3: Substitute these totals into Sxx = Σx² − (Σx)²/n, Syy = Σy² − (Σy)²/n and Sxy = Σxy − (Σx)(Σy)/n.

第三步：将这些总和代入 Sxx = Σx² − (Σx)²/n，Syy = Σy² − (Σy)²/n 和 Sxy = Σxy − (Σx)(Σy)/n。

Step 4: Calculate r = Sxy / √(SxxSyy), and round your final answer sensibly, usually to 3 or 4 decimal places.

第四步：计算 r = Sxy / √(SxxSyy)，并合理四舍五入，通常保留 3 或 4 位小数。

It is important not to round the values of Sxx, Syy and Sxy before the final division. Keep extra precision in your calculator or use exact fractions until the end.

在最后一步相除之前，不要对 Sxx、Syy 和 Sxy 提前四舍五入。应尽量在计算器中保留足够精度，或到最后一步再处理。

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4. Worked Example | 计算示例

Consider the following five pairs of data: x = 1, 2, 3, 4, 5 and y = 2, 4, 5, 4, 6. We will calculate the PMCC for these data.

考虑以下五对数据：x = 1, 2, 3, 4, 5，y = 2, 4, 5, 4, 6。我们来计算这些数据的 PMCC。

x	1	2	3	4	5
y	2	4	5	4	6
x²	1	4	9	16	25
y²	4	16	25	16	36
xy	2	8	15	16	30

Adding the columns gives Σx = 15, Σy = 21, Σx² = 55, Σy² = 97 and Σxy = 71. Since n = 5, we obtain:

对各列求和得到 Σx = 15，Σy = 21，Σx² = 55，Σy² = 97，Σxy = 71。由于 n = 5，可得：

Sxx = 55 − 15²/5 = 55 − 45 = 10

Syy = 97 − 21²/5 = 97 − 88.2 = 8.8

Sxy = 71 − (15)(21)/5 = 71 − 63 = 8

Therefore r = 8 / √(10 × 8.8) = 8 / √88 ≈ 0.8528. This suggests a fairly strong positive linear relationship between x and y.

因此 r = 8 / √(10 × 8.8) = 8 / √88 ≈ 0.8528。这说明 x 与 y 之间存在较强的正线性关系。

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5. Interpreting r: Strength and Direction | 解读 r：强度与方向

When interpreting r, you should always describe both the direction and the strength. Direction is positive or negative; strength is often described as weak, moderate, strong, or perfect.

在解读 r 时，你应该同时说明方向和强度。方向分为正或负；强度通常描述为弱、中等、强或完全线性。

• If r = 1, the points lie exactly on a line with positive slope.

  若 r = 1，则所有点恰好位于一条斜率为正的直线上。

• If r = −1, the points lie exactly on a line with negative slope.

  若 r = −1，则所有点恰好位于一条斜率为负的直线上。

• If r = 0, there is no linear correlation, although a non-linear pattern may still exist.

  若 r = 0，则不存在线性相关，但仍可能存在非线性模式。

There is no universal rule for ‘strong’ versus ‘weak’, but in Edexcel questions a sensible description is expected. For example, r = 0.85 is usually described as strong positive correlation, while r = 0.3 is weak positive correlation.

对于“强”与“弱”并没有统一标准，但 Edexcel 题目要求合理描述。例如，r = 0.85 通常称为强正相关，而 r = 0.3 则称为弱正相关。

You should also remember that r is dimensionless. Changing the units of measurement, such as converting from centimetres to metres, does not change r because the formula uses standardised sums.

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