Measuring Correlation | 测量相关性

📚 Measuring Correlation | 测量相关性

Correlation measures the strength and direction of a linear relationship between two variables. In Edexcel A-Level Mathematics, the product-moment correlation coefficient (PMCC), denoted by r, is the key tool. This article explains how to calculate r, interpret its value, and carry out hypothesis tests for correlation using critical values and p-values.

相关性衡量两个变量之间线性关系的强度和方向。在Edexcel A-Level数学中，积矩相关系数（PMCC）r是关键工具。本文将解释如何计算r、解读其数值，以及使用临界值和p值进行相关性的假设检验。

1. Scatter Diagrams and Linear Association | 散点图与线性关联

A scatter diagram plots paired observations (x, y). It reveals any underlying pattern of correlation. A linear trend suggests a linear relationship is worth investigating.

散点图绘制成对观测值 (x, y)，揭示潜在的相关模式。线性趋势表明值得进一步研究线性关系。

Points clustering around a straight line with a positive slope show positive correlation; a negative slope indicates negative correlation.

点围绕一条正斜率直线聚集表明正相关；负斜率则表明负相关。

Random scatter with no discernible pattern indicates zero linear correlation.

随机散点且无明显模式表示零线性相关。

2. Understanding Correlation | 理解相关性

Correlation coefficients always lie between -1 and +1. An r value of +1 means perfect positive linear correlation; -1 means perfect negative linear correlation; 0 means no linear correlation.

相关系数始终介于 -1 和 +1 之间。r = +1 表示完全正线性相关；r = -1 表示完全负线性相关；r = 0 表示无线性相关。

It is crucial to remember that correlation does not imply causation. Even a strong correlation may be caused by a lurking third variable.

务必记住相关并不意味着因果。即使强相关也可能由某个潜在第三变量所导致。

3. Covariance and Pearson’s PMCC | 协方差与皮尔逊PMCC

The PMCC r is derived from the idea of covariance. Sample covariance is given by s_xy = Σ(x – x̄)(y – ȳ) / (n – 1). Pearson’s r standardises this by the product of the sample standard deviations of x and y.

PMCC r 源于协方差的概念。样本协方差为 s_xy = Σ(x – x̄)(y – ȳ) / (n – 1)。皮尔逊 r 通过 x 和 y 的样本标准差乘积对该协方差进行标准化。

This leads to the defining formula:

由此得到定义公式：

r = Sxy / √(Sxx × Syy)

where Sxy = Σ(x – x̄)(y – ȳ), Sxx = Σ(x – x̄)², Syy = Σ(y – ȳ)², and the divisor (n – 1) cancels out.

其中 Sxy = Σ(x – x̄)(y – ȳ)，Sxx = Σ(x – x̄)²，Syy = Σ(y – ȳ)²，除数 (n – 1) 在化简时抵消。

4. Computational Formula for r | 计算 r 的简便公式

For ease of calculation, the formula can be rewritten using totals:

为便于计算，可将公式改写为基于总计的形式：

r = (n Σxy – Σx Σy) / √( [n Σx² – (Σx)²] [n Σy² – (Σy)²] )

This avoids calculating deviations from the mean and is ideal for use with calculators or spreadsheets.

这避免计算离差，非常适合计算器或电子表格使用。

5. Step-by-Step Manual Calculation | 手动逐步计算示例

Consider five pairs of data: (1, 2), (2, 4), (3, 5), (4, 4), (5, 7). The required sums are Σx = 15, Σy = 22, Σx² = 55, Σy² = 108, Σxy = 76, and n = 5.

考虑五对数据：(1, 2), (2, 4), (3, 5), (4, 4), (5, 7)。所需总和为 Σx = 15, Σy = 22, Σx² = 55, Σy² = 108, Σxy = 76, n = 5。

Substitute into the computational formula:

代入计算公式：

r = (5 × 76 – 15 × 22) / √( [5 × 55 – 15²] × [5 × 108 – 22²] )

r = (380 – 330) / √( (275 – 225) × (540 – 484) )

r = 50 / √(50 × 56) = 50 / √2800 ≈ 50 / 52.915 = 0.945

This yields r ≈ 0.94, indicating a strong positive linear correlation between x and y.

得到 r ≈ 0.94，表明 x 和 y 之间存在强正线性相关。

6. Using a Calculator to Find PMCC | 使用计算器求 PMCC

Most A-Level calculators, such as the Casio fx-991EX, can compute r directly. Enter paired data in statistics mode, choose linear regression (y = a + bx), and the calculator will display r as well as r².

大多数 A-Level 计算器（如 Casio fx-991EX）可直接计算 r。在统计模式下输入成对数据，选择线性回归（y = a + bx），计算器将显示 r 和 r²。

Always ensure the output is set to show the correlation coefficient r and not just the coefficient of determination r².

务必确保输出设置为显示相关系数 r，而不只是决定系数 r²。

7. Interpreting the Correlation Coefficient | 相关系数的解读

General guidelines suggest: |r| > 0.7 implies a strong correlation; 0.4 < |r| < 0.7 indicates a moderate correlation; |r| < 0.4 suggests a weak correlation. However, context and sample size must be considered.

一般准则提示：|r| > 0.7 表示强相关；0.4 < |r| < 0.7 为中等相关；|r| < 0.4 为弱相关。但必须结合具体情境和样本量。

A single outlier can drastically inflate or deflate r. Always inspect the scatter diagram before drawing conclusions.

单个异常值可能显著抬高或压低 r。下结论前务必检查散点图。

8. Hypothesis Testing for Zero Correlation | 零相关的假设检验

We often test whether the population correlation coefficient ρ is zero. The null hypothesis is H₀: ρ = 0. The alternative can be two-tailed (H₁: ρ ≠ 0) or one-tailed (H₁: ρ > 0 or H₁: ρ < 0).

我们常检验总体相关系数 ρ 是否为零。原假设为 H₀: ρ = 0。备择假设可以是双尾（H₁: ρ ≠ 0）或单尾（H₁: ρ > 0 或 H₁: ρ < 0）。

The test statistic is the sample PMCC r. It is compared with a critical value obtained from the Pearson correlation coefficient table, using n – 2 degrees of freedom.

检验统计量为样本 PMCC r。将其与 Pearson 相关系数表中查得的临界值进行比较，自由度为 n – 2。

9. Critical Values and Decision Making | 临界值与决策

For a sample of size n, degrees of freedom = n – 2. Choose a significance level α (commonly 0.05 or 0.01). Find the critical value for the given tail(s) and df in the PMCC table.

对于样本量 n，自由度 = n – 2。选择显著性水平 α（通常为 0.05 或 0.01）。在 PMCC 表中根据尾数和自由度查找临界值。

If |r| exceeds the critical value, reject H₀; there is sufficient evidence of linear correlation in the population.

若 |r| 超过临界值，则拒绝 H₀；有足够证据表明总体存在线性相关。

Modern calculators also provide a p-value. If the p-value < α, the null hypothesis is rejected.

现代计算器也可提供 p 值。若 p 值 < α，则拒绝原假设。

10. Assumptions and Limitations | 假设与局限性

Pearson’s r measures only linear relationships. Data should be on an interval or ratio scale, and the relationship should be reasonably linear when viewed on a scatter plot.

皮尔逊 r 仅衡量线性关系。数据应为定距或定比尺度，且从散点图来看关系应大致呈线性。

The coefficient is sensitive to outliers and is not suitable for ordinal data. If the relationship is monotonic but non-linear, Spearman’s rank correlation is a more appropriate measure (though it is not required in the standard Edexcel A-Level specification).

该系数对异常值敏感，不适用于有序数据。如果关系呈单调但非线性，斯皮尔曼秩相关系数更为合适（尽管标准 Edexcel A-Level 不要求掌握）。

Always plot the data before computing r. A single r value alone does not tell the full story of the relationship.

计算 r 前务必绘制数据图。单一的 r 值无法全面反映变量间的关系。

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Measuring Correlation | 测量相关性