A-Level Mathematics: Measures of Correlation | A-Level 数学：相关性的度量方法

📚 A-Level Mathematics: Measures of Correlation | A-Level 数学：相关性的度量方法

Correlation is a statistical measure that describes whether two variables move together. In A-Level Mathematics, we move beyond scatter graphs and calculate numerical coefficients that give the strength and direction of a linear relationship.

相关性是描述两个变量是否一起变化的统计度量。在 A-Level 数学中，我们不止于绘制散点图，还会计算数值系数，用于给出线性关系的强度与方向。

1. Scatter Diagrams and Correlation | 散点图与相关性

A scatter diagram places each data pair (x, y) as a point on coordinate axes. The overall pattern reveals whether a linear relationship exists and helps us choose an appropriate correlation coefficient.

散点图把每对数据 (x, y) 作为坐标平面上的一个点。整体图形可以揭示是否存在线性关系，并帮助我们选择合适的相关系数。

When the points rise from left to right, the variables show positive correlation: larger x values tend to be paired with larger y values.

当点从左到右升高时，变量之间存在正相关：较大的 x 值往往对应较大的 y 值。

When the points fall from left to right, the variables show negative correlation: larger x values tend to be paired with smaller y values.

当点从左到右下降时，变量之间存在负相关：较大的 x 值往往对应较小的 y 值。

When the points form a shapeless cloud, there is little or no correlation, and a linear model will not be useful.

当点形成没有规律的云团时，相关性很弱或不存在，线性模型将没有用处。

2. Pearson’s Product-Moment Correlation Coefficient (PMCC) | 皮尔逊乘积矩相关系数（PMCC）

The Pearson product-moment correlation coefficient, denoted r, measures the strength and direction of a linear relationship between two quantitative variables.

皮尔逊乘积矩相关系数，记作 r，用于度量两个定量变量之间线性关系的强度与方向。

Let x̄ be the mean of the x-values and ȳ be the mean of the y-values. The formula for r is:

设 x̄ 为 x 值的平均数，ȳ 为 y 值的平均数。r 的计算公式为：

r = Σ(xᵢ − x̄)(yᵢ − ȳ) ÷ √[ Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)² ]

The same formula can be written using the summary sums Sxy, Sxx and Syy:

同样的公式也可以用汇总量 Sxy、Sxx 与 Syy 表示：

r = Sxy ÷ √( Sxx × Syy )

In the exam, you may be given Sxx, Syy and Sxy directly, so you only need to substitute correctly and evaluate the square root.

在考试中，题目可能直接给出 Sxx、Syy 和 Sxy，你只需要正确代入并求平方根。

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

The value of r always lies between -1 and 1. The sign of r tells us the direction of the relationship, while the absolute value tells us the strength.

r 的值总是在 -1 到 1 之间。r 的正负号告诉我们关系的方向，而绝对值告诉我们强度。

An r value close to 1 means a strong positive linear relationship. An r value close to -1 means a strong negative linear relationship. An r value close to 0 means a weak or no linear relationship.

r 接近 1 表示强的正线性关系；r 接近 -1 表示强的负线性关系；r 接近 0 表示弱的或不存在线性关系。

Value of r	Interpretation \| 解读
r = +1	Perfect positive linear correlation \| 完全正线性相关
r close to +1	Strong positive correlation \| 强正相关
r close to 0	Weak or no linear correlation \| 弱相关或无线性相关
r close to -1	Strong negative correlation \| 强负相关
r = -1	Perfect negative linear correlation \| 完全负线性相关

4. The Coefficient of Determination | 决定系数 r²

The square of the correlation coefficient, r², is often called the coefficient of determination.

相关系数的平方 r² 通常称为决定系数。

When multiplied by 100%, r² gives the percentage of the variation in y that is explained by the linear relationship with x.

将 r² 乘以 100%，就得到 y 的变化中可以被 x 的线性关系所解释的百分比。

For example, if r = 0.8, then r² = 0.64, so 64% of the variation in y is explained by x. The remaining 36% is due to other factors or random variation.

例如，若 r = 0.8，则 r² = 0.64，即 y 的变化中有 64% 可以由 x 解释，其余 36% 来自其他因素或随机变异。

Remember that r² is not a measure of the strength of correlation itself; it is a measure of explanatory power and should be interpreted alongside r.

注意 r² 本身不是相关强度的度量，而是解释力的度量，应与 r 一起解读。

5. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数

Spearman’s rank correlation coefficient, denoted r_s, is used when data are ranks or when the original data are not normally distributed. It measures the strength and direction of a monotonic relationship.

斯皮尔曼等级相关系数，记作 r_s，用于数据是等级形式，或者原始数据不服从正态分布的情形。它度量单调关系的强度与方向。

To calculate r_s, first rank the x-values and the y-values separately. Give rank 1 to the smallest value and rank n to the largest value.

计算 r_s 时，先把 x 值和 y 值分别排序。最小值给等级 1，最大值给等级 n。

Then calculate d, the difference between each pair of ranks:

然后计算 d，即每一对等级之差：

d = rank of x − rank of y

When there are no tied ranks, use the formula:

当没有并列等级时，使用公式：

r_s = 1 − 6Σdᵢ² ÷ [ n(n² − 1) ]

The interpretation of r_s is the same as r: values close to 1 indicate strong positive correlation, values close to -1 indicate strong negative correlation, and values close to 0 indicate weak or no correlation.

r_s 的解读与 r 相同：接近 1 表示强正相关，接近 -1 表示强负相关，接近 0 表示弱相关或无相关。

6. Tied Ranks and Formula Adjustments | 并列等级与公式调整

When two or more values are equal, they share the same average rank. For example, if the 3rd and 4th values are equal, both receive rank 3.5.

当两个或更多数值相等时，它们共用同一个平均等级。例如，若第 3 和第 4 个数值相等，则两者都取等级 3.5。

If there are tied ranks, the simplified formula with 6Σd² is no longer valid.

如果存在并列等级，使用 6Σd² 的简化公式就不再有效。

In such cases, calculate r_s by applying the Pearson formula to the ranks themselves:

此时，应把等级当作数值代入皮尔逊公式来计算 r_s：

r_s = [ Σ(xᵢ − x̄)(yᵢ − ȳ) ] ÷ √[ Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)² ]

This approach is always valid, so if you are unsure whether a simplified formula can be used, use the Pearson formula on the ranks.

这种方法总是有效的。因此，如果你不确定能否使用简化公式，就直接对等级使用皮尔逊公式。

7. Hypothesis Testing for Correlation | 相关性的假设检验

In A-Level Mathematics, you may be asked to test whether a sample correlation coefficient provides significant evidence of a relationship in the population.

在 A-Level 数学中，题目可能要求你检验样本相关系数是否提供总体中存在相关性的显著证据。

The null hypothesis is H₀: ρ = 0, meaning the population correlation coefficient is zero. The alternative hypothesis may be H₁: ρ ≠ 0 for a two-tailed test, or H₁: ρ > 0 or ρ < 0 for a one-tailed test.

原假设为 H₀：ρ = 0，即总体相关系数为零。备择假设可以是 H₁：ρ ≠ 0（双尾检验），或 H₁：ρ > 0 或 ρ < 0（单尾检验）。

Compare the calculated value of r or r_s with the critical value from the relevant table at the chosen significance level and sample size n.

将计算得到的 r 或 r_s 与相应表格中、给定显著性水平和样本量 n 下的临界值进行比较。

If the absolute value of the calculated coefficient is greater than the critical value, reject H₀ and conclude there is significant evidence of correlation.

如果计算得到的系数绝对值大于临界值，则拒绝 H₀，并认为存在相关性的显著证据。

If the absolute value is less than or equal to the critical value, there is insufficient evidence to reject H₀.

如果绝对值小于或等于临界值，则没有足够证据拒绝 H₀。

8. Correlation Does Not Imply Causation | 相关性不等于因果关系

A very common exam question asks you to explain why a high correlation coefficient does not prove that one variable causes the other.

一个非常常见的考试题目要求你解释为什么高的相关系数并不能证明一个变量导致另一个变量变化。

Correlation only shows that two variables tend to change together. It does not establish a cause-and-effect mechanism.

相关性只表明两个变量倾向于一起变化，并不能确立因果关系机制。

There may be a third variable, often called a confounding or lurking variable, that affects both x and y and creates the appearance of a relationship.

可能存在第三个变量，通常称为混杂变量或潜在变量，同时影响 x 和 y，从而造成表面上存在关系。

For example, ice cream sales and drowning incidents are positively correlated, but ice cream does not cause drowning. Both are related to warmer summer weather.

例如，冰淇淋销量与溺水事件呈正相关，但冰淇淋并不会导致溺水。两者都与夏季温暖天气有关。

You should always phrase conclusions carefully: say “there is evidence of an association” rather than “x causes y” unless a controlled experiment or strong theoretical reasoning supports causation.

下结论时应措辞谨慎：应说“存在关联的证据”，而不是“x 导致 y”，除非有受控实验或强理论支持因果关系。

9. Common Exam Worked Example | 常见考题示例

The following data show two variables x and y for five observations:

以下数据展示五个观测点的两个变量 x 和 y：

x	1	2	3	4	5
y	2	4	3	6	5

The sample means are x̄ = 3 and ȳ = 4. The required sums are Sxx = 10, Syy = 10 and Sxy = 8.

样本均值为 x̄ = 3，ȳ = 4。所需汇总量为 Sxx = 10，Syy = 10，Sxy = 8。

r = 8 ÷ √(10 × 10) = 8 ÷ 10 = 0.8

This value suggests a strong positive linear relationship between x and y.

该值表明 x 与 y 之间存在较强的正线性关系。

To find Spearman’s rank coefficient, rank both variables. The x ranks are 1, 2, 3, 4, 5 and the y ranks are 1, 3, 2, 5, 4.

为求斯皮尔曼等级系数，先对两个变量排序。x 的等级为 1, 2, 3, 4, 5，y 的等级为 1, 3, 2, 5, 4。

The rank differences are 0, -1, 1, -1, 1, so Σd² = 0 + 1 + 1 + 1 + 1 = 4. With n = 5:

等级差为 0, -1, 1, -1, 1，因此 Σd² = 0 + 1 + 1 + 1 + 1 = 4。由 n = 5：

r_s = 1 − (6 × 4) ÷ [5(25 − 1)] = 1 − 24 ÷ 120 = 1 − 0.2 = 0.8

Here both coefficients are equal because the relationship is completely monotonic. In other examples they may differ, so always calculate the coefficient that matches the question.

这里两个系数相等，因为关系是完全单调的。在其他例子中它们可能不同，因此请始终计算题目所要求的系数。

10. Summary and Revision Checklist | 总结与复习清单

Correlation coefficients are essential tools for summarising the relationship between two variables in A-Level Mathematics.

相关系数是 A-Level 数学中概括两个变量之间关系的重要工具。

Pearson’s r: measures linear correlation between quantitative variables. | 皮尔逊 r：度量定量变量之间的线性相关性。
Spearman’s r_s: measures monotonic correlation using ranks. | 斯皮尔曼 r_s：使用等级度量单调相关性。
Range: both coefficients lie between -1 and 1. | 取值范围：两个系数都在 -1 和 1 之间。
Sign: + means positive correlation, − means negative correlation. | 符号：+ 表示正相关，− 表示负相关。
r²: the coefficient of determination gives the proportion of variation explained. | r²：决定系数给出被解释的变化比例。
Hypothesis tests: compare r or r_s with critical values before concluding significance. | 假设检验：在得出结论前，将 r 或 r_s 与临界值比较。
Causation: correlation alone never proves causation. | 因果关系：仅有相关性永远不能证明因果关系。

In the exam, always state which coefficient you are using, show your substitution clearly, and interpret the result in the context of the question.

考试时，请说明你使用的是哪个系数，清楚展示代入过程，并结合题目背景解释结果。

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A-Level Mathematics: Measures of Correlation | A-Level 数学：相关性的度量方法