📚 Critical Values for Correlation Coefficients | 相关系数的临界值
Hypothesis testing with the product moment correlation coefficient (PMCC) is a core part of the Edexcel A Level Mathematics statistics strand. In this topic, you calculate Pearson’s r from bivariate data and decide whether the linear association is statistically significant. The decision depends on comparing the observed r with a critical value obtained from a table for a given sample size and significance level.
积矩相关系数（PMCC）的假设检验是 Edexcel A Level 数学统计部分的核心内容。在这一主题中，你需要从双变量数据计算 Pearson 相关系数 r，并判断线性关联是否具有统计显著性。该判断依赖于将观测到的 r 与根据样本量和显著性水平从表中查得的临界值进行比较。
1. Correlation and the PMCC | 相关性与积矩相关系数
The product moment correlation coefficient, denoted r, measures the strength and direction of a linear relationship between two 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.
积矩相关系数记作 r，用于衡量两个变量之间线性关系的强度和方向。它的取值始终在 -1 到 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 表示几乎没有线性相关。
In Edexcel exams, the PMCC is usually calculated using the summary statistics formula:
在 Edexcel 考试中，PMCC 通常使用汇总统计量公式计算：
r = Sxy ÷ √(Sxx × Syy)
Sxy = Σxy – (Σx)(Σy)/n, Sxx = Σx² – (Σx)²/n, Syy = Σy² – (Σy)²/n
Here n is the number of paired observations. The formula is given in the formula booklet, so the main skill is using it accurately and then carrying out a hypothesis test with the correct critical value.
其中 n 是成对观测值的数量。该公式在公式表中给出，因此重点技能是准确使用它，然后使用正确的临界值进行假设检验。
2. Why Critical Values Are Needed | 为什么需要临界值
A sample correlation coefficient r is almost never exactly zero, even if there is no true linear correlation in the population. This is because random sampling variation can produce a small nonzero r by chance. We therefore need a formal test to decide whether the observed r is large enough to be considered statistically significant.
即使总体中没有真实的线性相关，样本相关系数 r 也几乎不会恰好为零。这是因为随机抽样波动可能偶然产生一个非零的 r。因此，我们需要一个正式的检验来判断观测到的 r 是否大到足以被认为具有统计显著性。
A critical value is the boundary between the rejection region and the acceptance region for the null hypothesis. If the observed test statistic is more extreme than the critical value, we reject the null hypothesis. For correlation, this usually means comparing the absolute value of r with the critical value from a table.
临界值是原假设拒绝域与接受域之间的分界值。如果观测到的检验统计量比临界值更极端，我们就拒绝原假设。对于相关性检验，这通常意味着将 r 的绝对值与表中的临界值进行比较。
3. Hypotheses for Correlation Tests | 相关性检验的假设
The population correlation coefficient is denoted by the Greek letter ρ (rho). The standard null hypothesis is that there is no linear correlation in the population:
总体相关系数用希腊字母 ρ（rho）表示。标准的原假设是总体中不存在线性相关：
H₀: ρ = 0
The alternative hypothesis depends on the question. For a two-tailed test, the alternative is simply that some linear correlation exists, positive or negative:
备择假设取决于题目要求。对于双尾检验，备择假设只是存在某种线性相关，无论是正还是负：
H₁: ρ ≠ 0
For a one-tailed test, the alternative specifies the direction. If the question asks whether there is positive correlation, use:
对于单尾检验，备择假设需指明方向。如果题目问是否存在正相关，则使用：
H₁: ρ > 0
If it asks whether there is negative correlation, use:
如果题目问是否存在负相关，则使用：
H₁: ρ < 0
The choice of alternative hypothesis directly affects which critical value column you use in the table, so it must be decided before looking at the data.
备择假设的选择直接影响你使用表中的哪一列临界值，因此必须在查看数据之前确定。
4. One-Tailed and Two-Tailed Tests | 单尾与双尾检验
In a two-tailed test, the rejection region is split between both tails of the distribution. At a 5% significance level, this means 2.5% is in each tail. The critical value for a two-tailed test at 5% is therefore the value that cuts off 2.5% in one tail, because the distribution of r under H₀ is symmetric.
在双尾检验中，拒绝域分布在分布的两端。在 5% 显著性水平下，这意味着每一端占 2.5%。因此，双尾 5% 检验的临界值就是在单尾截断 2.5% 的值，因为 r 在原假设下的分布是对称的。
In a one-tailed test, all of the significance level is placed in one tail. For example, a one-tailed test at the 5% level uses the column that cuts off 5% in that single direction. This makes it easier to reject H₀ than a two-tailed test at the same significance level, provided the direction of r matches the alternative hypothesis.
在单尾检验中，所有显著性水平都放在一个尾部。例如，5% 水平的单尾检验使用在单一方向上截断 5% 的列。只要 r 的方向与备择假设一致，这比相同显著性水平的双尾检验更容易拒绝原假设。
Edexcel questions will usually state clearly whether the test is one-tailed or two-tailed. Look for phrases such as ‘test for positive correlation’, which indicates a one-tailed test, or ‘test for correlation’, which usually indicates a two-tailed test.
Edexcel 题目通常会明确说明检验是单尾还是双尾。注意诸如“检验正相关”这样的表述，它表示单尾检验；而“检验相关性”通常表示双尾检验。
5. Using the Critical Values Table | 使用临界值表
The Edexcel formulae and statistical tables provide critical values for the product moment correlation coefficient. The table is organised by sample size n, not by degrees of freedom. You must find the row that corresponds to the number of paired observations in your data set.
Edexcel 公式和统计表提供了积矩相关系数的临界值表。该表按样本量 n 排列，而不是按自由度排列。你必须找到与数据集中成对观测值数量相对应的行。
The columns are determined by the test type and significance level. For a one-tailed test at 5%, use the 5% one-tailed column. For a two-tailed test at 5%, use the 2.5% one-tailed column, because the total significance is split between the two tails. The table usually labels columns in terms of tail probabilities, so check the header carefully.
列由检验类型和显著性水平决定。对于 5% 的单尾检验，使用 5% 单尾列。对于 5% 的双尾检验，使用 2.5% 单尾列，因为总显著性被拆分到两个尾部。表格通常按尾部概率标注列，因此要仔细查看表头。
Once you have the critical value c, compute |r| and compare. The rejection rule is:
得到临界值 c 后，计算 |r| 并进行比较。拒绝规则是：
Reject H₀ if |r| > c
如果 |r| > c，则拒绝 H₀
For example, typical critical values for a two-tailed 5% test are approximately:
例如，双尾 5% 检验的典型临界值大约为：
| n | 5% two-tailed critical value | 1% two-tailed critical value |
|---|---|---|
| 10 | 0.632 | 0.765 |
| 20 | 0.444 | 0.561 |
| 30 | 0.361 | 0.463 |
These values are indicative; in the exam you should always use the exact values from the Edexcel table provided in the booklet or insert.
这些数值仅供参考；考试中应始终使用 Edexcel 小册子或插页中提供的精确数值。
6. Worked Example: Two-Tailed Test | 例题：双尾检验
A researcher collects a sample of n = 18 paired observations and calculates r = 0.531. Test at the 5% significance level whether there is evidence of linear correlation in the population. No direction is specified, so a two-tailed test is appropriate.
一位研究人员收集了 n = 18 对观测值，并计算出 r = 0.531。在 5% 显著性水平下检验总体中是否存在线性相关。由于没有指定方向，因此使用双尾检验。
First state the hypotheses:
首先写出假设：
H₀: ρ = 0 H₁: ρ ≠ 0
From the Edexcel table, for n = 18 and a two-tailed 5% test, the critical value is approximately c = 0.468. The test statistic is |r| = 0.531. Since 0.531 > 0.468, the observed r lies in the rejection region.
查 Edexcel 表，对于 n = 18
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