📚 Critical Values for Correlation Coefficients | 相关系数的临界值
In A-Level Mathematics, the product moment correlation coefficient (PMCC) measures the strength and direction of a linear relationship between two variables. Once you have calculated the PMCC for a sample, you often need to decide whether the observed correlation is strong enough to suggest a genuine linear association in the population. This is where hypothesis testing and critical values come into play. By comparing your sample PMCC with the appropriate critical value from statistical tables, you can draw a conclusion at a given significance level. Mastering the use of critical values for correlation coefficients is essential for success in the Edexcel Statistics component, and this article will guide you through every step.
在A-Level数学中，积矩相关系数（PMCC）衡量两个变量之间线性关系的强度和方向。计算出样本的PMCC后，你通常需要判断观察到的相关是否足够强，以表明总体中存在真实的线性关联。这就是假设检验和临界值发挥作用的地方。通过将样本PMCC与统计表中对应的临界值进行比较，你可以在给定的显著性水平下得出结论。掌握相关系数临界值的运用对于Edexcel统计学部分的成功至关重要，本文将逐步带你了解每一个环节。
1. Product Moment Correlation Coefficient (PMCC) – A Refresher | 积矩相关系数（PMCC）回顾
The product moment correlation coefficient, denoted by r, quantifies how close data points lie to a straight line. Its value ranges from –1 to +1. A value of +1 indicates a perfect positive linear correlation, –1 indicates a perfect negative linear correlation, and 0 indicates no linear correlation. In Edexcel exams, you may be given raw data and asked to compute r using a calculator or summary statistics. Remember that r is a sample statistic, meaning it varies from sample to sample, while the population correlation coefficient is denoted by the Greek letter rho (ρ). When we test for correlation, we are essentially making an inference about ρ based on the observed r.
积矩相关系数，记作r，量化了数据点接近一条直线的程度。它的取值范围从–1到+1。+1表示完全正线性相关，–1表示完全负线性相关，0表示没有线性相关。在Edexcel考试中，你可能会得到原始数据，要求使用计算器或汇总统计量计算r。请记住，r是一个样本统计量，这意味着它随样本的不同而变化，而总体相关系数用希腊字母ρ表示。当我们检验相关性时，本质上是在基于观察到的r对ρ进行推断。
2. Setting Up a Hypothesis Test for Correlation | 建立相关性假设检验
H₁. The default assumption under H₀ is that there is no linear correlation in the population, i.e. ρ = 0. The alternative hypothesis specifies what you suspect or want to prove: ρ ≠ 0 for a two-tailed test, or ρ > 0 / ρ < 0 for a one-tailed test. The choice between one‑tailed and two‑tailed depends on the context – estions often provide a statement that allows you to determine the appropriate hypothesis.
每个相关系数的假设检验都从两个对立的陈述开始：原假设H₀和备择假设H₁。H₀的默认假设是总体中没有线性相关，即ρ = 0。备择假设则指明你所怀疑或想要证明的：对于双尾检验，ρ ≠ 0；对于单尾检验，ρ > 0或ρ < 0。选择单尾还是双尾取决于情境——你是寻找任何相关，还是心中已有特定的方向。Edexcel的题目通常会提供一个陈述，让你确定合适的假设。
Once the hypotheses are stated, the test is carried out by comparing the absolute value of the sample PMCC, |r|, with a critical value obtained from tables. If |r| is sufficiently large, we reject H₀ in favour of H₁.
一旦陈述了假设，检验就通过比较样本PMCC的绝对值|r|与从表中获得的临界值来进行。如果|r|足够大，我们就拒绝H₀，转而支持H₁。
3. What Are Critical Values? | 什么是临界值？
A critical value is a threshold that separates the rejection region from the non‑ pends on the chosen significance level (α), the sample size n, and whether the test is one‑tailed or two‑tailed. If the test statistic (here, |r| . The critical value is derived from the sampling distribution of r when ρ = 0, which, for large n, approximates a normal distribution, but for smaller samples relies on a transformation to the t‑distribution.
临界值是一个阈值，它在原假设下将检验统计量的拒绝域与非拒绝域分隔开来。对于相关性检验，临界值取决于所选的显著性水平（α）、样本量n以及检验是单尾还是双尾。如果检验统计量（这里是|r|）超出了临界值，结果就具有统计显著性，我们拒绝H₀。否则，我们没有足够的证据拒绝H₀。临界值来自当ρ = 0时r的抽样分布，对于大样本，该分布近似于正态分布，但对于较小样本则依赖于向t分布的转换。
In Edexcel examinations, you do not need to derive the critical values – they are provided in the formula booklet. The table is titled 'Critical Values for the Product Moment Correlation Coefficient' and lists the minimum absolute value of r needed for significance at various levels.
在Edexcel考试中，你无需推导临界值——公式手册中会提供这些值。该表标题为“积矩相关系数的临界值”，列出了在各显著性水平下达到显著所需的最小|r|值。
4. One-Tailed and Two-Tailed Tests Explained | 单尾与双尾检验解释
The distinction between one‑tailed and two‑tailed tests is crucial because it changes the critical value. A two‑tailed test is used when the alternative hypothesis is ρ ≠ 0, meaning you are testing for the existence of any linear correlation, positive or negative. A one‑tailed test is appropriate when the alternative states a direction: H₁: ρ > 0 (positive correlation) or H₁: ρ < 0 (negative correlation). In the PMCC critical values table, columns are labelled for 'One Tail' and 'Two Tail' at common significance levels. For example, a two‑tailed test at the 5% significance level uses the critical value from the 'Two Tail 0.05' column, while a one‑tailed test at the 5% level uses 'One Tail 0.05' (which corresponds to a 10% two‑tail probability). Always read the table carefully and match your test type.
单尾与双尾检验的区别至关重要，因为它会改变临界值。当备择假设为ρ ≠ 0时使用双尾检验，意味着你在检验任何线性相关的存在性，无论是正还是负。当备择假设指明了方向时适用单尾检验：H₁: ρ > 0（正相关）或H₁: ρ < 0（负相关）。在PMCC临界值表中，各列被标记为常见显著性水平下的“单尾”和“双尾”。例如，在5%显著性水平下的双尾检验使用“双尾 0.05”列的临界值，而5%水平下的单尾检验则使用“单尾 0.05”列（该列对应于10%的双尾概率）。务必仔细阅读表格，并匹配你的检验类型。
5. Using the PMCC Critical Values Table | 使用PMCC临界值表
ly starting at n = 3 and going up to n = 30 or beyond. Columns provide critical values for different significance levels, usually: One Tail 0.05 (Two Tail 0.10), One Tail 0.025 (Two Tail 0.05), One Tail 0.01 (Two Tail 0.02) and One Tail 0.005 (Two Tail 0.01). To find the required critical value, locate the row corresponding to your sample size n, then read across to the column that matches your significance level and tail setup. For instance, if n = 5 and you are conducting a two‑tailed test at the 5% level, the critical value from the table is 0.878. Always double‑check that you are using the correct row and column.
Edexcel公式手册中有一张具体的积矩相关系数表。行标有样本量n，通常从n = 3开始，一直到n = 30或更大。各列提供了不同显著性水平下的临界值，通常包括：单尾 0.05（双尾 0.10）、单尾 0.025（双尾 0.05）、单尾 0.01（双尾 0.02）以及单尾 0.005（双尾 0.01）。要找到所需的临界值，先找到与你样本量n对应的行，然后横向查阅与你显著性水平和尾部设定相匹配的列。例如，若n = 5且你正在进行5%水平下的双尾检验，表中的临界值为0.878。务必再三确认你使用的是正确的行和列。
6. Significance Levels: 5%, 1%, and More | 显著性水平：5%、1%等
ing a true null hypothesis (Type I error). Edexcel questions most commonly use α = 0.05 (5%) or α = 0.01 (1%). A 5% significance level means there is a 5% chance of concluding there is a correlation when, in fact, there is none. A 1% es it harder to reject H₀, requiring stronger sample evidence. The critical value at the 1% level is always more extreme (further from zero) than at the 5% level. For example, with n = 5, the two‑tailed 5% critical value is 0.878, while the two‑tailed 1% critical value is 0.959. Thus, a sample r of 0.92 would be significant at the 5% level but not at the 1% level.
显著性水平α是当原假设为真时拒绝它的概率（第一类错误）。Edexcel题目最常用α = 0.05（5%）或α = 0.01（1%）。5%的显著性水平意味着有5%的机率在实际上没有相关的情况下得出有相关的结论。1%的水平则使拒绝H₀更难，需要更强的样本证据。1%水平下的临界值总是比5%水平下的更极端（离零更远）。例如，当n = 5时，双尾5%的临界值为0.878，而双尾1%的临界值为0.959。因此，样本r = 0.92在5%水平下是显著的，但在1%水平下则不显著。
7. Step-by-Step Example: Two-Tailed Test | 逐步示例：双尾检验
iables on a sample of five subjects and calculates the PMCC to be r = 0.92. Test, at the 5% nificance level, whether this provides evidence of a linear correlation in the population.
一位研究人员测量了五个个体样本的两个变量，并计算出PMCC为r = 0.92。在5%显著性水平下检验这是否为总体中存在线性相关提供了证据。
Step 1 – Hypotheses: H₀: ρ = 0, H₁: ρ ≠ 0 (two‑tailed).
步骤1 – 假设： H₀: ρ = 0, H₁: ρ ≠ 0（双尾）。
Step 2 – Significance level and sample size: α = 0.05, n = 5.
步骤2 – 显著性水平与样本量： α = 0.05, n = 5。
Step 3 – Critical value: From the Edexcel table, two‑tailed 5% critical value for n = 5 is 0.878.
步骤3 – 临界值： 查Edexcel表，n = 5时双尾5%的临界值为0.878。
Step 4 – Compare: |r| = 0.92 > 0.878, so the test statistic falls in the rejection region.
步骤4 – 比较： |r| = 0.92 > 0.878，因此检验统计量落在拒绝域。
Step 5 – Conclusion: Reject H₀. There is sufficient evidence at the 5% level to suggest a linear correlation exists between the two variables in the population.
步骤5 – 结论： 拒绝H₀。在5%显著性水平下有足够证据表明总体中这两个变量之间存在线性相关。
Notice that the conclusion mentions the significance level and does not claim “proof” of a correlation, only evidence.
注意，结论提到了显著性水平，并且没有声称“证明”了相关性，只是提供了证据。
8. Step-by-Step Example: One-Tailed Test | 逐步示例：单尾检验
correlated. A random sample of 5 athletes yields a sample PMCC of r = 0.81. Perform a one‑tailed test at the 5% significance level.
一位体育科学家认为训练时间与成绩得分之间存在正相关。一个包含5名运动员的随机样本给出的样本PMCC为r = 0.81。在5%显著性水平下进行单尾检验。
Step 1 – Hypotheses: H₀: ρ = 0, H₁: ρ > 0 (one‑tailed).
步骤1 – 假设： H₀: ρ = 0, H₁: ρ > 0（单尾）。
Step 2 – Significance level and sample size: α = 0.05 (one‑tailed), n = 5.
步骤2 – 显著性水平与样本量： α = 0.05（单尾），n = 5。
Step 3 – Critical value: From the table, the one‑tailed 5% alue for n = 5 is 0.805 (found under One Tail 0.05, which is equivalent to Two Tail 0.10).
步骤3 – 临界值： 查表，n = 5时单尾5%的临界值为0.805（查阅单尾0.05列，相当于双尾0.10）。
Step 4 – Compare: r = 0.81 > 0.805, so we reject H₀.
步骤4 – 比较： r = 0.81 > 0.805，因此我们拒绝H₀。
Step 5 – Conclusion: There is sufficient evidence at the 5% level to support the researcher's belief that training hours and performance are positively correlated.
步骤5 – 结论： 在5%显著性水平下有足够证据支持研究者关于训练时间与成绩正相关的看法。
If r had been negative while testing for positive correlation, the test statistic would not be in the rejection region, regardless of magnitude.
如果在检验正相关时r为负值，那么无论其大小如何，检验统计量都不会落在拒绝域内。
9. Interpreting the Results and Drawing Conclusions | 解释结果并得出结论
A well‑ o the significance level, and express the result in the context of the original problem. Avoid phrasing such as 'accept H₀' – instead say 'there is insufficient evidence to reject H₀' or 'H₀ is not rejected' ns the sample did not provide strong enough evidence of a correlation. In Edexcel mark schemes, marks are awarded for a conclusion that is contextualised and correctly references the significance level.
一个结构良好的结论应说明H₀是否被拒绝，提及显著性水平，并在原始问题的情境下表述结果。避免使用“接受H₀”这样的措辞——应说“没有足够证据拒绝H₀”或“H₀未被拒绝”。同时记住，未能拒绝H₀并不证明ρ = 0；它仅仅意味着样本未能提供足够强的相关证据。在Edexcel的评分标准中，条理清晰且正确引用显著性水平的情境化结论可以获得分数。
For instance, 'Since 0.92 > 0.878, we reject the null hypothesis and conclude that there is evidence, at the 5% level, of a linear relationship between X and Y' is a perfect concluding sentence.
例如，“由于0.92 > 0.878，我们拒绝原假设，并得出结论：在5%显著性水平下有证据表明X与Y之间存在线性关系”就是一个完美的结论句。
10. The Impact of Sample Size on Critical Values | 样本量对临界值的影响
As the sample size n increases, the critical values become smaller (closer to zero). This reflects the fact that larger samples provide more precise estimates, so a weaker observed correlation can still be statistically significant. Below is a simplified comparison of two‑tailed 5% critical values for various sample sizes from the Edexcel table:
随着样本量n的增加，临界值变小（更接近零）。这反映出大样本能提供更精确的估计，因此即使观察到的相关较弱，也可能具有统计显著性。下面是从Edexcel表中选出的一些样本量在双尾5%显著性水平下的临界值比较：
| Sample size n | Two‑tailed 5% critical value |
|---|---|
| 5 | 0.878 |
| 10 | 0.576 |
| 20 | 0.423 |
| 30 | 0.349 |
Notice how the required strength of correlation drops markedly as n grows. In the exam, always be aware that a seemingly modest r (e.g. 0.4) may be significant with a large sample, while a high r (e.g. 0.9) might not be significant with a very small sample. This demonstrates why you must always consult the table and not rely solely on the magnitude of r.
请注意，随着n的增长，所需的相关强度显著下降。在考试中，一定要意识到，一个看似中等大小的r（例如0.4）在大样本下可能是显著的，而一个很高的r（例如0.9）在非常小的样本下可能并不显著。这也说明了为什么你必须始终查阅表格，而不能仅凭r的大小来判断。
11. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧
of the most frequent mistakes is using the wrong tail. Always read the question carefully – words like 'improve', 'increase' or 'positive' suggest a one‑tailed test, while 'any', 'whether' or 'is there a correlation' usually imply a two‑tailed test. Another pitfall is confusing significance levels: a 1% one‑tailed test does not use the same critical value as a 1% two‑ ence. Finally, remember to state both the critical value and the comparison in your working. Edexcel examiners expect to see 'critical value = …' and a clear inequality such as '0.92 > 0.878'.
最常见的错误之一是使用了错误的尾部。务必仔细读题——像“提升”、“增加”或“正向”这样的词暗示单尾检验，而“任何”、“是否”或“是否存在相关”通常意味着双尾检验。另一个陷阱是混淆显著性水平：1%单尾检验与1%双尾检验使用的临界值并不相同。此外，绝不要将样本PMCC当作总体相关的决定性度量；始终从证据的角度来组织你的结论。最后，记得在解题过程中既写出临界值，也写出比较过程。E
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