Percentage Points of the χ² Distribution | 卡方分布的百分点

📚 Percentage Points of the χ² Distribution | 卡方分布的百分点

The chi-squared (χ²) distribution is one of the most widely used probability distributions in A-Level Statistics, especially for hypothesis tests involving categorical data. Understanding its percentage points – the critical values that mark the boundaries of extreme outcomes – is essential for making correct decisions in goodness-of-fit tests, tests for independence and other inference procedures. This article walks you through the definition, interpretation and practical use of χ² percentage points, with a strong emphasis on the Edexcel A-Level Statistics specification.

卡方分布（χ² 分布）是 A-Level 统计学中最常用的概率分布之一，特别是在处理分类数据的假设检验中。理解它的百分点——也就是标记极端结果边界的临界值——对于正确进行拟合优度检验、独立性检验和其他推断过程至关重要。本文将系统地介绍 χ² 百分点的定义、含义和实际使用方法，紧扣 Edexcel A-Level 数学大纲要求。

1. The χ² Distribution | 卡方分布

The χ² distribution arises naturally when independent standard normal variables are squared and summed. If Z₁, Z₂, … , Z_ν n the sum of their squares follows a chi-squared distribution with ν degrees of freedom: ∑^ν_i=1 Z_i² ~ χ²_ν. Because it is built from squares, the distribution is non-negative and skewed to the right, becoming more symmetric as ν increases.

卡方分布是由独立标准正态变量的平方和自然产生的。如果 Z₁, Z₂, … , Z_ν 是独立的 N(0,1) 观测值，那么它们的平方和就服从自由度为 ν 的卡方分布：∑^ν_i=1 Z_i² ~ χ²_ν。由于分布由平方项构成，因此取值非负且呈现右偏形态，随着自由度增大分布逐渐趋向对称。

2. Degrees of Freedom and Shape | 自由度与形状

The single parameter ν (degrees of freedom) controls the shape of the χ² normal distribution. In A-Level hypothesis tests, ν is determined by the number of categories or cells minus constraints – for example, ν = k – 1 in a simple goodness-of-fit test when no parameters are estimated.

参数 ν（自由度）控制着卡方密度曲线的形状。ν = 1 时分布高度右偏，众数为 0。随着 ν 增加，均值变为 ν，方差为 2ν，曲线逐渐接近正态分布。在 A-Level 的假设检验中，ν 通常由类别数减去约束条件得到——例如，在没有参数估计的简单拟合优度检验中，ν = k – 1。

The right-skewed nature means that a large proportion of the probability lies in the upper tail when ν is small. This directly affects the magnitude of upper-tail percentage points, which we use as critical values in most χ² tests in the Edexcel specification. In contrast, lower-tail points are rarely needed but can be derived from the symmetry of complementary probabilities.

右偏的性质意味着当 ν 较小时，较大部分概率位于上尾。这直接影响了上尾百分点的大小，而我们在 Edexcel 规范下的多数卡方检验中正是使用这些上尾值作为临界值。相比之下，下尾点极少直接用到，但可以通过互补概率的对称性推算出来。

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3. What Are Percentage Points? | 百分点是什么？

A percentage point of a distribution is the value that cuts off a specified percentage of the area under the probability density curve. For the χ² distribution, the upper 100α% point, written χ²_ν(α), satisfies P(χ² > χ²_ν(α)) = α. For instance, the upper 5% point χ²_ν(0.05) leaves exactly 5% of the probability in the right-hand tail. These points are the critical values we compare test statistics against.

分布的百分点是指能够切掉概率密度曲线下指定面积比例的数值。对于卡方分布，上侧 100α% 百分点记作 χ²_ν(α)，满足 P(χ² > χ²_ν(α)) = α。例如，上侧 5% 百分点 χ²_ν(0.05) 会在右尾截掉恰好 5% 的概率。这些百分点正是我们用来与检验统计量作比较的临界值。

In A-Level work, percentage points are usually obtained from printed statistical tables (provided in the formula booklet) rather than by direct integration. The table typically gives χ²_ν(p) for selected values of p (such as 0.995, 0.99, 0.975, 0.95, 0.10, 0.05, 0.025, 0.01, 0.005) and a range of degrees of freedom. Students must be able to read the correct value given ν and the required significance level.

在 A-Level 学习中，百分点通常从打印的统计表（在公式手册中提供）中查取，而不是通过直接积分计算。表格一般会列出选定 p 值（如 0.995、0.99、0.975、0.95、0.10、0.05、0.025、0.01、0.005）和一系列自由度下的 χ²_ν(p)。学生必须能够根据给定的 ν 和显著性水平读出正确的数值。

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4. Upper-Tail Percentage Points | 上尾百分点

Almost all χ² hypothesis tests covered at A-Level use an upper-tail rejection region. The null hypothesis is rejected for large values of the test statistic, i.e. when X² > χ²_ν(α). For a test at the 5% significance level, α = 0.05, so we look up the upper 5% point. For a 1% test, we use χ²_ν(0.01). The critical region is always one-sided on the right because the test statistic measures discrepancy: the larger the discrepancy, the larger the χ² value.

A-Level 中几乎所有的 χ² 假设检验都采用上尾拒绝域。当检验统计量的值很大时，即 X² > χ²_ν(α)，拒绝原假设。对于 5% 显著性水平的检验，α = 0.05，因此查上侧 5% 点；对于 1% 检验，使用 χ²_ν(0.01)。临界区域总是右侧单边的，因为检验统计量衡量的是偏差：偏差越大，χ² 值就越大。

Occasionally, a two-tailed approach may be hinted at when constructing confidence intervals for variance, but this is beyond the core Edexcel Statistics modules. Students should focus on the fact that typical χ² tests – goodness of fit, independence, homogeneity – are all right-tailed, so they only need the upper percentage points from tables.

在为方差构造置信区间时，偶尔会涉及双尾的思路，但这超出了 Edexcel 核心统计模块的范围。同学们只需记住典型的 χ² 检验——拟合优度、独立性、齐性——都是右尾检验，因此只需要从表格中查找上尾百分点即可。

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5. Lower-Tail Percentage Points | 下尾百分点

Lower-tail points satisfy P(χ² < χ²_ν(1– n A-Level hypothesis tests, they occasionally appear when you need to find a critical value for the left tail or when interpreting a two-sided confidence interval for a population variance. In the standard χ² table, lower-tail points are obtained by reading the value for probability 1–α. For example, the lower 5% point χ²_ν(0.95) is the value that leaves 5% in the left tail.

下尾百分点满足 P(χ² < χ²_ν(1–α)) = α。虽然在 A-Level 假设检验中并不常用，但在需要寻找左尾临界值或理解总体方差的双侧置信区间时偶尔会涉及。在标准的 χ² 分布表中，下尾点可以通过查阅概率 1–α 的值得到。比如，下侧 5% 点 χ²_ν(0.95) 就是左尾截掉 5% 面积的数值。

It is important not to confuse 'percentage points' with 'percentiles'. At this level, the term 'percentage point' explicitly refers to the tail area above or below a given value, directly linked to the α notation used in hypothesis testing. The table in the Edexcel formula booklet typically lists upper-tail probabilities, so be careful to match your α with the correct column.

注意不要把“百分点”和“百分位数”混淆。在 A-Level 水平上，“百分点”这一术语特指尾部面积对应的临界值，与假设检验中的 α 记号直接关联。Edexcel 公式手册中的表格通常列出上尾概率，因此要确保将 α 与正确的列匹配。

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6. Reading Percentage Points from Tables | 从表格中读取百分点

ing to ν = 5. The columns are headed by the upper-tail probability p. Typical entries are:

我们模拟一下自由度为 5 时的查表过程。在 Edexcel 统计表中，找到 ν = 5 这一行。列标题为上尾概率 p。典型的数值如下表所示：

p	0.995	0.99	0.975	0.95	0.10	0.05	0.025	0.01	0.005
ν=5	0.412	0.554	0.831	1.145	9.236	11.070	12.833	15.086	16.750

For a 5% significance test, we take χ²₅(0.05) = 11.070. This means if our test statistic exceeds 11.070, we reject H₀ at the 5% level. Notice how the critical value increases as the tail probability gets smaller – you need a more extreme result to reject at a stricter significance level.

对于 5% 显著性检验，我们取 χ²₅(0.05) = 11.070。这意味着如果检验统计量超过 11.070，就在 5% 水平上拒绝 H₀。注意临界值随着尾部概率变小而增大——要在更严格的显著性水平下拒绝，需要更极端的结果。

If a question asks for the critical value at the 1% level, you would use χ²₅ rect degrees of freedom and probability column; a common mistake is reading the wrong column or confusing one-tailed and two-tailed scenarios.

如果题目要求在 1% 水平下的临界值，你应该使用 χ²₅(0.01) = 15.086。一定要仔细核对正确的自由度和概率列；常见的错误是读错列，或者混淆单尾与双尾的情况。

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7. Critical Values and Significance Levels | 临界值与显著性水平

The connection between significance level α and the percentage point is direct: χ²_ν(α) is the critical value that defines the rejection region. In the Edexcel examinations, you may be given the significance level (e.g. 5%) and asked to find the critical value, or you may be given the critical value and asked to state the conclusion. The decision rule is: reject H₀ if X² > χ²_ν(α); otherwise do not reject.

显著性水平 α 与百分点的联系是直接对应：χ²_ν(α) 就是界定拒绝域的临界值。在 Edexcel 考试中，可能会给定显著性水平（如 5%）要求找出临界值，也可能给定临界值要求说明结论。决策规则是：若 X² > χ²_ν(α) 则拒绝 H₀；否则不拒绝。

The p-value approach is also accepted: the p-value is the probability, under H₀, of obtaining a statistic at least as extreme as the observed one. If p-value < α, we reject H₀. In A-Level exams, however, direct comparison of the test statistic with the tabulated percentage point is the most common method because tables are provided.

p 值方法同样可以使用：p 值是在 H₀ 下获得至少和观测值一样极端的统计量的概率。若 p 值 < α，则拒绝 H₀。不过在 A-Level 考试中，由于提供了统计表，最常见的方法还是将检验统计量与查表得到的百分点直接比较。

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8. χ² Tests for Goodness of Fit | 卡方拟合优度检验

In a goodness-of-fit test, we compare observed frequencies O_i with expected frequencies E_i under a hypothesised distribution. The test statistic is X² = Σ (O_i – E_i)² / E_i. Under H₀, X² approximately follows a χ²_ν distribution with ν = number of categories – 1 (or fewer if parameters are estimated). We then compare X² with the upper percentage point χ²_ν(α).

在拟合优度检验中，我们比较实际观测频数 O_i 与在假设分布下的期望频数 E_i。检验统计量为 X² = Σ (O_i – E_i)² / E_i。在原假设下，X² 近似服从自由度为 ν = 类别数 – 1（若估计了参数则更少）的卡方分布。然后将 X² 与上尾百分点 χ²_ν(α) 比较得出结论。

imes and ask whether it is fair (i.e. uniform distribution). With 6 faces, ν = 5. At the 5% level, the critical value is 11.07. If the computed X² is, say, 13.2, we reject fairness because 13.2 > 11.07.

典型的考题可能会给出一个骰子投掷 120 次的观测频数，要求检验其是否均匀。因为有 6 个面，ν = 5。在 5% 水平下，临界值为 11.07。如果计算出的 X² 为 13.2，因为 13.2 > 11.07，我们拒绝均匀性的原假设。

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9. χ² Test for Independence | 卡方独立性检验

Contingency tables test whether two categorical variables are associated. The statistic is the same form: X² = Σ (O_ij – E_ij)² / E_ij ted under the assumption of independence. Degrees of freedom for an r × c table are ν = (r – 1)(c – 1). The critical value is still an upper percentage point χ²_ν(α).

列联表检验用于判断两个分类变量是否相关。统计量形式相同：X² = Σ (O_ij – E_ij)² / E_ij，其中期望频数基于独立性假设计算得出。对于 r × c 的表格，自由度为 ν = (r – 1)(c – 1)。临界值同样是上尾百分点 χ²_ν(α)。

For example, a 2 × 2 table gives ν = 1. At α = 0.05, the upper 5% point χ²₁ ears in many contexts, including the normal approximation to a 2×2 independence test. If X² > 3.841, we reject the hypothesis of independence at the 5% level.

例如，一个 2×2 的表格给出 ν = 1。在 α = 0.05 时，上侧 5% 点 χ²₁(0.05) = 3.841。这个著名的临界值出现在许多场合，包括 2×2 独立性检验的正态近似。如果 X² > 3.841，我们在 5% 水平下拒绝独立性假设。

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10. Worked Example Using Upper 5% Point | 上侧 5% 点应用实例

Suppose a teacher records the grades (A, B, C, D, E) of 100 students and wants to test if they follow the distribution 10%, 20%, 40%, 20%, 10%. Observed counts: 8, 18, 42, 22, 10. Calculate E_i: 10, 20, 40, 20, 10. Then X² = (8–10)²/10 + (18–20)²/20 + (42–40)²/40 + (22–20)²/20 + (10–10)²/10 = 0.4 + 0.2 + 0.1 + 0.2 + 0 = 0.9. Degrees of freedom ν = 4. At α = 0.05, χ²₄ 9 < 9.488, we do not reject H₀: the data are consistent with the hypothesised percentages.

假设一位教师记录了 100 名学生的成绩等级（A、B、C、D、E），想检验它们是否服从比例 10%、20%、40%、20%、10%。观测频数：8, 18, 42, 22, 10。计算期望频数 E_i：10, 20, 40, 20, 10。那么 X² = (8–10)²/10 + (18–20)²/20 + (42–40)²/40 + (22–20)²/20 + (10–10)²/10 = 0.4 + 0.2 + 0.1 + 0.2 + 0 = 0.9。自由度 ν = 4。在 α = 0.05 下，χ²₄(0.05) = 9.488。因为 0.9 < 9.488，我们不拒绝 H₀：数据与假设的百分比一致。

This example highlights how the percentage point acts as a threshold. Even with some deviation, the test statistic is far below the critical value, so the variation can be attributed to random sampling error. Always remember to state the conclusion in the context of the problem.

这个例子凸显了百分点如何充当阈值。即使存在一些偏差，检验统计量仍远低于临界值，因此这种变异可归因于随机抽样误差。务必记得在题目背景下给出结论。

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11. Approximation and Large ν | 近似与大的自由度

When degrees of freedom are large (not covered in the core A-Level, but useful to know), the χ²_ν distribution can be approximated by a normal distribution with mean ν and variance 2ν. A continuity correction may be applied. The percentage points then relate to normal quantiles: χ²_ν(α)

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