Critical Values for Correlation Coefficients | 相关系数临界值

📚 Critical Values for Correlation Coefficients | 相关系数临界值

When you calculate a correlation coefficient in A-Level Further Mathematics, the number you obtain on its own only tells you the strength of a linear relationship in your sample. To infer whether such a relationship exists in the wider population, you must compare your result with a critical value from a statistical table. This article explains what critical values are, how to find them, and how to use them correctly in hypothesis tests.

在进阶数学中，计算得到的相关系数本身只能说明样本内线性关系的强弱。若要推断总体中是否存在这种关系，你必须将计算结果与统计表中的临界值进行比较。本文解释什么是临界值、如何查找临界值，以及如何在假设检验中正确使用它们。

1. The Pearson Product-Moment Correlation Coefficient | 皮尔逊积矩相关系数

The Pearson product-moment correlation coefficient, usually denoted by r, measures the strength and direction of a linear relationship between two quantitative variables. Its value always lies between -1 and +1. A value near +1 indicates strong positive linear correlation, a value near -1 indicates strong negative linear correlation, and a value near 0 indicates little or no linear correlation.

皮尔逊积矩相关系数通常用 r 表示，用于衡量两个定量变量之间线性关系的强弱和方向。其取值始终介于 -1 和 +1 之间。接近 +1 表示强正线性相关，接近 -1 表示强负线性相关，接近 0 表示几乎不存在线性相关。

The sample correlation coefficient is calculated using the formula

r = Σ(xᵢ − x̄)(yᵢ − ȳ) / √(Σ(xᵢ − x̄)² Σ(yᵢ − ȳ)²)

where xᵢ and yᵢ are the paired sample values, and x̄ and ȳ are the sample means.

其中 xᵢ 和 yᵢ 是成对样本值，x̄ 和 ȳ 是样本均值。

2. Why Do We Need Critical Values? | 为什么需要临界值？

Suppose you collect a small random sample and find r = 0.65. Could this value have arisen purely by chance, even if no correlation exists in the population? To answer this, we set up a hypothesis test. The critical value is the threshold that the sample correlation coefficient must exceed in order to reject the null hypothesis. Without a critical value, you cannot judge whether your observed r is statistically significant.

假设你收集了一个小随机样本并计算得 r = 0.65。即使总体中真的不存在相关，这个值是否也可能纯属偶然出现？为了回答这个问题，我们需要建立假设检验。临界值是样本相关系数必须超过的阈值，只有超过它才能拒绝零假设。没有临界值，你就无法判断所观察到的 r 是否具有统计显著性。

3. The Null and Alternative Hypotheses | 零假设与备择假设

For a two-tailed test of correlation, the null hypothesis is usually written as

H₀: ρ = 0

meaning that there is no linear correlation in the population. The alternative hypothesis is

H₁: ρ ≠ 0

meaning that there is a significant linear correlation in either direction.

对于双尾相关检验，零假设通常写成 H₀: ρ = 0，表示总体中不存在线性相关。备择假设为 H₁: ρ ≠ 0，表示存在任一方向的显著线性相关。

You may also use one-tailed tests when you have a prior directional expectation, for example H₁: ρ > 0 or H₁: ρ < 0. The critical value you use will depend on whether your test is one-tailed or two-tailed.

如果你事先对方向有预期，也可以使用单尾检验，例如 H₁: ρ > 0 或 H₁: ρ < 0。你需要使用的临界值取决于检验是单尾还是双尾。

4. Significance Levels | 显著性水平

The significance level, often 5% or 1%, is the probability of rejecting H₀ when H₀ is actually true. A 5% significance level means that if there is truly no correlation, you would accept a 5% chance of wrongly concluding that there is correlation. The smaller the significance level, the stronger the evidence you require before rejecting the null hypothesis.

显著性水平通常为 5% 或 1%，是指在 H₀ 实际为真的情况下拒绝 H₀ 的概率。5% 的显著性水平意味着，如果总体中真的没有相关，你接受有 5% 的可能性错误地得出存在相关的结论。显著性水平越小，拒绝零假设所需要的证据就越强。

In a two-tailed test, the total probability of rejection is split equally into two tails. For a 5% significance level, each tail contains 2.5% of the distribution. For a one-tailed test, the entire 5% is placed in one tail.

在双尾检验中，拒绝域的总概率被平均分配到两个尾端。对于 5% 的显著性水平，每个尾端各占 2.5%。在单尾检验中，整个 5% 都放在一个尾端。

5. Structure of a Correlation Critical Value Table | 相关系数临界值表的结构

Most critical value tables for Pearson correlation are arranged with degrees of freedom in the first column and significance levels across the top row. The number of degrees of freedom for a sample of size n is n − 2. The table entry gives the minimum absolute value of r that is significant for that degrees of freedom and significance level.

大多数皮尔逊相关系数临界值表以自由度为第一列，顶部为显著性水平。样本容量为 n 时，自由度数为 n − 2。表中的数值表示在该自由度和显著性水平下，r 的绝对值至少需要达到多少才算显著。

A typical table for two-tailed tests looks like this (values are approximate):

双尾检验的典型临界值表如下（数值为近似值）：

n	df = n − 2	5% (0.05)	1% (0.01)
5	3	0.878	0.959
8	6	0.707	0.834
10	8	0.632	0.765
12	10	0.576	0.708
15	13	0.514	0.641

6. How to Use the Table in an Exam | 如何在考试中使用临界值表

In an Edexcel exam, you will usually be given the relevant critical value or a short table. The procedure is straightforward. First calculate the degrees of freedom, df = n − 2, where n is the number of data pairs. Next determine whether your test is one-tailed or two-tailed. Then select the correct significance level from the table. Finally compare the absolute value of your calculated r with the critical value.

在爱德思考试中，你通常会得到相关临界值或一段简表。步骤很简单。首先计算自由度 df = n − 2，其中 n 是数据对的数量。然后判断检验是单尾还是双尾。接着从表中选择正确的显著性水平。最后比较计算所得 r 的绝对值与临界值。

The decision rule is: if |r| > critical value, reject H₀. If |r| ≤ critical value, do not reject H₀. Note that in two-tailed tests you look at the absolute value because the correlation could be significant in either the positive or negative direction.

决策规则是：如果 |r| > 临界值，拒绝 H₀；如果 |r| ≤ 临界值，不能拒绝 H₀。注意在双尾检验中要使用绝对值，因为相关可能在正方向或负方向显著。

7. Worked Example | 例题演示

Ten A-Level students are sampled, and the Pearson correlation coefficient between their mock exam score and final exam score is calculated as r = 0.68. Test at the 5% significance level whether there is a significant positive correlation in the population.

抽取 10 名进阶数学学生，计算他们的模拟考试成绩与期末考试成绩之间的皮尔逊相关系数为 r = 0.68。在 5% 显著性水平下检验总体中是否存在显著正相关。

Here n = 10, so df = 10 − 2 = 8. The test is one-tailed with alternative H₁: ρ > 0. From the table, for df = 8 at the 5% one-tailed level, the critical value is generally 0.632 (the same as the two-tailed 10% value, but many tables provide one-tailed values directly). Since 0.68 > 0.632, we reject H₀ and conclude that there is significant positive correlation between mock and final scores at the 5% level.

这里 n = 10，所以 df = 10 − 2 = 8。检验是单尾，备择假设为 H₁: ρ > 0。查表可知，df = 8 时，5% 单尾显著性水平的临界值通常为 0.632（与双尾 10% 水平的值相同，但许多表会直接给出单尾值）。因为 0.68 > 0.632，我们拒绝 H₀，并得出结论：在 5% 水平下，模拟考分数与期末考分数之间存在显著正相关。

If you were testing a two-tailed alternative at the 5% level, the critical value for df = 8 would be approximately 0.632 as well, but in a full table you would read the column labelled ‘two-tailed significance 0.05’ rather than ‘one-tailed significance 0.05’.

如果你是在 5% 水平下进行双尾检验，df = 8 的临界值仍约为 0.632，但在完整表格中你应该读取标记为“双尾显著性 0.05”而不是“单尾显著性 0.05”的列。

8. One-Tailed vs Two-Tailed Tests | 单尾检验与双尾检验

The same significance level gives different critical values depending on the direction of the alternative hypothesis. For a given df, a one-tailed critical value with significance α is numerically smaller than a two-tailed critical value with significance α, because in the two-tailed case the total tail area α is divided between two ends.

在相同显著性水平下，备择假设的方向不同，临界值也不同。对于给定 df，单尾 α 显著性水平的临界值在数值上小于双尾 α 显著性水平的临界值，因为双尾检验中的总尾面积 α 被分在两端。

For example, with df = 8, the one-tailed 5% critical value is 0.632, while the two-tailed 5% critical value is also approximately 0.632 only because this table gives two-tailed values at 0.05. In a standard table you will see that the one-tailed 2.5% critical value equals the two-tailed 5% critical value. Always check whether your exam table separates these cases.

例如，df = 8 时，单尾 5% 的临界值为 0.632，而双尾 5% 的临界值也约为 0.632，这只是因为该表给出的双尾值恰好对应 0.05。在标准表中，单尾 2.5% 的临界值等于双尾 5% 的临界值。务必检查你的考试表是否区分了这两种情况。

9. Spearman’s Rank Correlation | 斯皮尔曼等级相关

The same hypothesis-testing procedure applies to Spearman’s rank correlation coefficient, denoted ρₛ or rₛ. You calculate the ranks of each variable, find the correlation of the ranks, and then compare your value with the critical value from a Spearman table. The degrees of freedom are still n − 2 in the approximation used by most A-Level tables, although some tables give critical values directly against n.

同样的假设检验步骤也适用于斯皮尔曼等级相关系数，记作 ρₛ 或 rₛ。你先对每个变量排名，再计算排名的相关系数，然后将该值与斯皮尔曼表中的临界值比较。大多数进阶数学表仍使用自由度 n − 2 进行近似，但有些表直接按 n 给出临界值。

Spearman’s test is useful when the data are ordinal or when a monotonic but not necessarily linear relationship is suspected. The interpretation of the critical value is identical: reject H₀ if |rₛ| exceeds the table value.

斯皮尔曼检验适用于定序数据，或当你怀疑存在单调但不一定是线性的关系时。临界值的解释完全类似：若 |rₛ| 超过表中数值，则拒绝 H₀。

10. Common Mistakes and Pitfalls | 常见错误与陷阱

One common error is forgetting to take the absolute value of r in a two-tailed test. A correlation of -0.8 may be just as significant as +0.8, and you must compare |r| with the critical value. Another error is using the wrong degrees of freedom. For Pearson correlation, df = n − 2, not n. Some students mistakenly use n − 1, which is used for single-sample t-tests.

一个常见错误是在双尾检验中忘记取 r 的绝对值。−0.8 的相关与 +0.8 同样可能显著，你必须用 |r| 与临界值比较。另一个错误是使用错误的自由度。皮尔逊相关的自由度是 n − 2 而不是 n。有些学生会误用 n − 1，那是单样本 t 检验的自由度。

A further pitfall is confusing the significance level with the confidence level. A 5% significance level corresponds to a 95% confidence level, but the critical value is determined by the significance level. Also, be careful about the direction of the test: writing H₁: ρ > 0 but using a two-tailed critical value is inconsistent.

还有一个陷阱是把显著性水平与置信水平混淆。5% 显著性水平对应 95% 置信水平，但临界值由显著性水平决定。同时要注意检验方向：写出 H₁: ρ > 0 却使用双尾临界值是不一致的。

11. Exam Strategy and Calculator Tips | 考试策略与计算器技巧

In the Edexcel Further Mathematics exam, you are expected to show your hypotheses and your conclusion clearly. Even if the critical value is given in the question, write it down explicitly. State whether you are using a one-tailed or two-tailed test, and quote the significance level. When you state your conclusion, use the wording ‘significant’ or ‘not significant at the 5% level’ rather than simply saying ‘reject’ or ‘accept’.

在爱德思进阶数学考试中，你需要清楚写出假设和结论。即使题目已经给出临界值，也要明确写出来。说明使用的是单尾还是双尾检验，并引用显著性水平。在写结论时，最好使用“在 5% 水平下显著/不显著”的说法，而不是简单地说“拒绝”或“接受”。

On a calculator, you can verify your r value using the linear regression or correlation mode. Most scientific calculators can compute Pearson’s r directly from data pairs. However, calculators do not give critical values, so you still need to compare your result with the table. For Spearman’s rank correlation, you may need to rank the data manually and then compute the Pearson correlation on the ranks.

在计算器上，你可以使用线性回归或相关模式验证 r 值。大多数科学计算器可以直接从数据对计算皮尔逊 r。但计算器不提供临界值，你仍然需要将结果与表值比较。对于斯皮尔曼等级相关，你可能需要手动排名，然后对排名计算皮尔逊相关。

12. Summary and Final Advice | 总结与最终建议

The critical value for a correlation coefficient is the boundary between statistical significance and insignificance. To use it correctly, always identify the sample size, calculate df = n − 2, determine the direction of the test, choose the significance level, read the correct critical value, and compare |r| with that value. A clear, systematic approach will earn you reliable marks in the Edexcel Further Mathematics hypothesis-testing questions.

相关系数的临界值是统计显著与不显著之间的分界点。要正确使用它，一定要确定样本容量、计算 df = n − 2、判断检验方向、选择显著性水平、读取正确的临界值，并将 |r| 与该值比较。一个清晰、系统的方法能帮助你在爱德思进阶数学的假设检验题目中获得稳定的分数。

Remember that correlation does not imply causation. Even if you reject the null hypothesis and conclude that a significant correlation exists, you have not proven that one variable causes changes in the other. The critical value test only tells you that the observed relationship is unlikely to be due to chance alone.

请记住，相关并不等于因果。即使你拒绝零假设并得出结论存在显著相关，也不能证明一个变量导致了另一个变量的变化。临界值检验只能告诉你，所观察到的关系不太可能仅仅是偶然造成的。

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Critical Values for Correlation Coefficients | 相关系数临界值