Measuring Correlation | 相关性的度量

📚 Measuring Correlation | 相关性的度量

In A-Level Mathematics, measuring correlation is about quantifying how strongly two variables are related in a linear way. Edexcel questions often ask you to calculate the product moment correlation coefficient (PMCC), interpret its value, and test whether a correlation is statistically significant.

在 A-Level 数学中，度量相关性就是量化两个变量之间线性关系的强度。Edexcel 试题经常要求你计算积矩相关系数（PMCC）、解释其数值，并检验相关性是否具有统计显著性。

1. What Is Correlation? | 什么是相关性？

Correlation measures the strength and direction of a linear relationship between two quantitative variables. It is usually denoted by r for the product moment correlation coefficient.

相关性衡量两个定量变量之间线性关系的强度和方向。对于积矩相关系数，它通常用 r 表示。

The value of r always lies between -1 and 1. The sign indicates direction, and the magnitude indicates strength.

r 的值始终在 -1 到 1 之间。符号表示方向，绝对值大小表示强度。

Correlation is a symmetric measure: the correlation of x with y is the same as the correlation of y with x.

相关性是对称的度量：x 与 y 的相关性和 y 与 x 的相关性相同。

2. Scatter Diagrams | 散点图

Before calculating any number, draw a scatter diagram by plotting each pair (x, y). This helps you see the pattern of the data.

在计算任何数值之前，先绘制散点图，标出每一对数据点 (x, y)。这有助于观察数据的整体模式。

If points lie exactly on an upward-sloping straight line, there is perfect positive correlation and r = 1.

如果所有点恰好落在一条向上的直线上，则存在完全正相关，r = 1。

If points lie exactly on a downward-sloping straight line, there is perfect negative correlation and r = -1.

如果所有点恰好落在一条向下的直线上，则存在完全负相关，r = -1。

If the points show no clear straight-line pattern, the linear correlation is near 0. However, a non-linear pattern may still be present.

如果数据点没有明显的直线模式，线性相关性接近 0。不过，仍可能存在非线性关系。

3. Direction, Form and Strength | 方向、形式和强度

Direction tells us whether y tends to increase or decrease as x increases. A positive direction means both variables move together; a negative direction means they move in opposite ways.

方向说明当 x 增大时，y 是倾向于增大还是减小。正方向意味着两个变量同向变化；负方向意味着它们反向变化。

Form describes whether the relationship is linear or non-linear. The PMCC is designed to measure linear form only.

形式描述关系是线性的还是非线性的。PMCC 仅用于度量线性形式。

Strength describes how closely the points fit that linear pattern. Strong correlation means points are tightly clustered around a straight line.

强度描述数据点与直线模式的贴合程度。强相关意味着点紧密地聚集在一条直线周围。

A strong quadratic relationship can give r close to 0 even though the variables are strongly related. This is why a scatter diagram is essential.

一个很强的二次关系可能使 r 接近 0，尽管变量之间有很强的关系。这就是散点图必不可少的原因。

4. Covariance and Why It Matters | 协方差及其重要性

Covariance captures how x and y vary together around their means. If values above the mean of x tend to occur with values above the mean of y, the covariance is positive.

协方差反映 x 和 y 如何围绕各自的均值共同变化。如果高于 x 均值的值倾向于与高于 y 均值的值同时出现，协方差为正。

If values above the mean of x tend to occur with values below the mean of y, the covariance is negative.

如果高于 x 均值的值倾向于与低于 y 均值的值同时出现，协方差为负。

The PMCC standardises covariance by dividing it by the product of the two sample standard deviations. This removes the units and gives a scale from -1 to 1.

PMCC 通过除以两个样本标准差的乘积，将协方差标准化。这消除了单位，并给出从 -1 到 1 的尺度。

5. Product Moment Correlation Coefficient | 积矩相关系数

For a sample of n paired observations (x, y), first compute the corrected sums of squares and products:

对于 n 对观测值 (x, y)，首先计算修正的平方和与乘积和：

Sxx = Σx² − (Σx)²/n

Syy = Σy² − (Σy)²/n

Sxy = Σxy − (ΣxΣy)/n

Then the product moment correlation coefficient is:

那么积矩相关系数为：

r = Sxy / √(Sxx Syy)

You need to calculate the five sums Σx, Σy, Σx², Σy² and Σxy accurately. A table of columns x, y, x², y² and xy is useful.

你需要准确计算五个和：Σx、Σy、Σx²、Σy² 和 Σxy。列出包含 x、y、x²、y² 和 xy 的表格会很有帮助。

If Sxx or Syy is zero, then all x-values or all y-values are identical, and r is not defined. In that case there is no meaningful linear correlation.

如果 Sxx 或 Syy 为零，则所有 x 值或所有 y 值都相同，r 无定义。在这种情况下不存在有意义的线性相关。

6. Worked Example: Calculating r | 计算 r 的示例

Find the PMCC for the data: (1, 2), (2, 3), (3, 5), (4, 4), (5, 7).

计算以下数据的积矩相关系数：(1, 2)、(2, 3)、(3, 5)、(4, 4)、(5, 7)。

Step 1: calculate the required sums.

第 1 步：计算所需的和。

Σx = 1 + 2 + 3 + 4 + 5 = 15

Σy = 2 + 3 + 5 + 4 + 7 = 21

Σx² = 1 + 4 + 9 + 16 + 25 = 55

Σy² = 4 + 9 + 25 + 16 + 49 = 103

Σxy = 2 + 6 + 15 + 16 + 35 = 74

Step 2: compute Sxx, Syy and Sxy with n = 5.

第 2 步：用 n = 5 计算 Sxx、Syy 和 Sxy。

Sxx = 55 − 15²/5 = 55 − 45 = 10

Syy = 103 − 21²/5 = 103 − 88.2 = 14.8

Sxy = 74 − (15 × 21)/5 = 74 − 63 = 11

Step 3: substitute into the formula.

第 3 步：代入公式。

r = 11 / √(10 × 14.8) = 11 / √148 ≈ 0.904

This indicates a strong positive linear correlation between x and y.

这表明 x 与 y 之间存在很强的正线性相关。

7. Interpreting PMCC Values | 解释 PMCC 的值

An r close to 1 indicates strong positive linear correlation; an r close to -1 indicates strong negative linear correlation; an r close to 0 indicates weak or no linear correlation.

r 接近 1 表示强正线性相关；r 接近 -1 表示强负线性相关；r 接近 0 表示弱线性相关或没有线性相关。

The interpretation must be made in context. For example, r = 0.8 may be strong in biology but weak in a precise physics experiment.

解释必须结合具体背景。例如，r = 0.8 在生物学中可能算强相关，但在精确物理实验中可能算弱相关。

A single outlier can drastically increase or decrease r, so you should always check the scatter diagram before trusting the value.

单个离群值可能急剧增大或减小 r，因此你应始终先检查散点图，再信任该值。

Restriction of range can also weaken the observed correlation. If you only measure a narrow interval of x-values, r may appear lower than the true population correlation.

范围受限也会削弱观测到的相关性。如果只测量很窄的 x 值区间，r 可能显得比总体真实的相关系数更小。

8. Spearman’s Rank Correlation Coefficient | 斯皮尔曼秩相关系数

Spearman’s rank correlation coefficient, denoted rs, is used when data are ordinal, ranked, or when the relationship is monotonic but not linear.

斯皮尔曼秩相关系数，记作 rs，用于有序数据、等级数据，或关系为单调但非线性的情形。

To compute it, rank the x-values and y-values separately from 1 to n. For each pair, calculate d = rank of x − rank of y.

计算时，将 x 值和 y 值分别从 1 到 n 排序。对每一对数据，计算 d = x 的秩 − y 的秩。

Then use the formula:

然后使用公式：

rs = 1 − (6Σd²)/(n(n² − 1))

If there are tied values, assign the average rank to the tied items. Edexcel questions may expect you to state this rule explicitly.

如果有并列值，则给并列项分配平均秩。Edexcel 试题可能要求你明确写出这一规则。

Spearman’s coefficient is less sensitive to outliers than the PMCC, and it can detect monotonic non-linear relationships more effectively.

斯皮尔曼系数对离群值的敏感性低于 PMCC，并且它能更有效地检测单调非线性关系。

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

In Edexcel A-Level Statistics, you may be asked to test whether a population PMCC is zero. The null hypothesis is usually H₀: ρ = 0, where ρ is the population correlation coefficient.

在 Edexcel A-Level 统计学中，你可能需要检验总体 PMCC 是否为零。原假设通常为 H₀: ρ = 0，其中 ρ 是总体相关系数。

The alternative hypothesis can be two-tailed, H₁: ρ ≠ 0, or one-tailed, H₁: ρ > 0 or H₁: ρ < 0, depending on the question.

备择假设可以是双侧的 H₁: ρ ≠ 0，也可以是单侧的 H₁: ρ > 0 或 H₁: ρ < 0，具体取决于题意。

If the sample comes from a bivariate normal distribution, the test statistic is:

如果样本来自二元正态分布，检验统计量为：

t = r√(n−2) / √(1−r²)

This follows a t-distribution with n − 2 degrees of freedom.

它服从自由度为 n − 2 的 t 分布。

Compare the calculated t-value with the critical value from the t-table, or compare r directly with a PMCC critical value table if provided.

将计算出的 t 值与 t 分布表中的临界值进行比较，或者如果题目提供了 PMCC 临界值表，也可以直接比较 r 与临界值。

Your conclusion must be written in context. For example: “Since 2.45 > 2.306, there is sufficient evidence at the 5% level to reject H₀ and conclude that there is significant positive correlation.”

你的结论必须结合背景来写。例如：“由于 2.45 > 2.306，在 5% 显著性水平下有足够证据拒绝 H₀，并得出结论：存在显著的正相关。”

10. Correlation Does Not Imply Causation | 相关不等于因果

A high value of r only shows an association between two variables. It does not prove that changing one variable causes the other to change.

r 值高仅表明两个变量之间存在关联。它并不能证明改变一个变量会导致另一个变量发生变化。

There may be a confounding variable that influences both variables. For example, ice cream sales and drowning incidents both rise in summer, but ice cream does not cause drowning.

可能存在同时影响两个变量的混杂变量。例如，冰淇淋销量和溺水事件在夏季都会上升，但冰淇淋并不会导致溺水。

There may also be reverse causation, where the supposed effect actually influences the supposed cause.

还可能存在反向因果，即被认为是结果的那个变量实际上影响了被认为是原因的那个变量。

In exam questions, always state that correlation shows association, not causation, unless the context provides a controlled experiment.

在考试中，除非题目背景是受控实验，否则应始终写明：相关性显示的是关联，而不是因果关系。

11. Common Exam Mistakes | 常见考试错误

One common mistake is concluding that r = 0 means there is no relationship at all. It only means there is no linear relationship; a strong curve may still exist.

一个常见错误是认为 r = 0 意味着完全没有关系。它只意味着没有线性关系；可能仍存在很强的曲线关系。

Another mistake is using the PMCC without plotting the data first. An outlier or a curved pattern can make r misleading.

另一个错误是未先绘制数据图就直接使用 PMCC。离群值或曲线模式可能使 r 产生误导。

Rounding too early in a calculation can lead to an inaccurate final answer. Keep full calculator accuracy until the final step.

计算过程中过早四舍五入可能导致最终答案不准确。在最后一步之前应保留计算器的完整精度。

In hypothesis testing, students sometimes confuse one-tailed and two-tailed critical values, leading to a wrong conclusion.

在假设检验中，学生有时会混淆单侧和双侧临界值，从而得出错误结论。

When using Spearman’s rank, forgetting to assign average ranks for tied values is a frequent error.

使用斯皮尔曼秩时，忘记给并列值分配平均秩是一个常见错误。

Finally, remember that r has no units. It is a pure number, because the units in Sxy and in √(Sxx Syy) cancel out.

最后，请记住 r 没有单位。它是一个纯数，因为 Sxy 和 √(Sxx Syy) 中的单位相互抵消。

12. Exam Tips and Summary | 考试技巧与小结

Always begin with a scatter diagram. Describe the direction, form, strength and any outliers before doing calculations.

始终先画散点图。在进行计算之前，描述方向、形式、强度和任何离群值。

When calculating the PMCC, show the values of Sxx, Syy and Sxy before substituting into the formula. This earns method marks even if the final answer is wrong.

计算 PMCC 时，先写出 Sxx、Syy 和 Sxy 的值，再代入公式。即使最终答案出错，这也能得到方法分。

Interpret any correlation value in the context of the question. Do not just write “positive correlation”; state what that means for the variables.

解释相关系数时要结合题意。不要只写“正相关”；要说明这对变量意味着什么。

For a hypothesis test, state the hypotheses, the test statistic, the degrees of freedom, the critical value and a contextual conclusion.

进行假设检验时，要写出假设、检验统计量、自由度、临界值和结合背景的结论。

Finally, remember that correlation measures linear association. When data are non-linear, ordinal or heavily influenced by outliers, consider Spearman’s rank coefficient.

最后，请记住相关性度量的是线性关联。当数据是非线性的、有序的或受离群值严重影响时，考虑使用斯皮尔曼秩相关系数。

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Measuring Correlation | 相关性的度量