A-Level Further Mathematics: Percentage Points of the Chi-Squared Distribution | A-Level进阶数学：卡方分布百分位点

📚 A-Level Further Mathematics: Percentage Points of the Chi-Squared Distribution | A-Level进阶数学：卡方分布百分位点

The chi-squared distribution is one of the most important sampling distributions in A-Level Further Mathematics. Its percentage points — often called critical values — are used in hypothesis tests, goodness-of-fit tests, and confidence intervals for a population variance. This article explains what percentage points mean, how to read the chi-squared table, and how to apply them correctly in exam-style problems.

卡方分布是 A-Level 进阶数学中最重要的抽样分布之一。它的百分位点——通常称为临界值——用于假设检验、拟合优度检验以及总体方差的置信区间。本文将解释百分位点的含义、如何查卡方分布表，以及如何在考试题型中正确应用。

1. The Chi-Squared Distribution | 卡方分布的定义

If \(Z_1, Z_2, \dots, Z_n\) are independent standard normal random variables, then the sum of their squares follows a chi-squared distribution with \(n\) degrees of freedom:

若 \(Z_1, Z_2, \dots, Z_n\) 是相互独立的标准正态随机变量，则它们的平方和服从自由度为 \(n\) 的卡方分布：

X² = Z₁² + Z₂² + … + Zₙ² ～ χ²(n)

The chi-squared distribution is positively skewed, and its shape depends entirely on the degrees of freedom, often denoted \(
u\). For small \(\
u\) the curve is highly skewed to the right; as \(\
u\) increases, the distribution becomes more symmetric and approaches a normal shape.

卡方分布是正偏态分布，其形状完全取决于自由度，通常记为 \(
u\)。自由度较小时，曲线右偏非常明显；随着自由度增大，分布逐渐对称，并趋近正态形态。

2. Degrees of Freedom | 自由度

Degrees of freedom (\(
u\)) count the number of independent pieces of information available after estimating parameters. In a chi-squared test, \(\
u\) is usually the number of categories minus one, then minus the number of parameters estimated from the data.

自由度（\(
u\)）表示在估计参数后剩余的独立信息数量。在卡方检验中，自由度通常等于类别数减一，再减去从数据中估计的参数个数。

Goodness of fit: \(\
u = k – 1 – m\), where \(k\) is the number of categories and \(m\) is the number of parameters estimated from the data.

拟合优度检验：\(\
u = k – 1 – m\)，其中 \(k\) 为类别数，\(m\) 为从数据中估计的参数个数。
Contingency tables: \(\
u = (r – 1)(c – 1)\), where \(r\) is the number of rows and \(c\) the number of columns.

列联表：\(\
u = (r – 1)(c – 1)\)，其中 \(r\) 为行数，\(c\) 为列数。
Variance test from a sample: \(\
u = n – 1\), where \(n\) is the sample size.

由样本检验方差：\(\
u = n – 1\)，其中 \(n\) 为样本容量。

3. What Is a Percentage Point? | 什么是百分位点？

For a chi-squared random variable \(X\) with \(\
u\) degrees of freedom, the percentage point \(\chi^2_{\alpha}( \
u)\) is the value such that the probability of exceeding it is \(\alpha\):

对于自由度为 \(\
u\) 的卡方随机变量 \(X\)，百分位点 \(\chi^2_{\alpha}( \
u)\) 是满足“超过它的概率为 \(\alpha\)”的那个值：

P(X > χ²_α(ν)) = α

Equivalently, \(\chi^2_{\alpha}( \
u)\) is the upper \(\alpha\) quantile of the distribution. Most exam tables provide upper-tail percentage points for common values of \(\alpha\) such as 0.10, 0.05, 0.025, 0.01 and \(\
u\) from 1 to 30 or more.

等价地说，\(\chi^2_{\alpha}( \
u)\) 是该分布的上 \(\alpha\) 分位数。大多数考试用表给出常见 \(\alpha\) 值（如 0.10、0.05、0.025、0.01）以及 \(\
u\) 从 1 到 30 或更大的上尾百分位点。

4. Reading the Chi-Squared Table | 查卡方分布表

A typical chi-squared table has rows labelled by degrees of freedom and columns labelled by the tail probability \(\alpha\). The entry at the intersection is \(\chi^2_{\alpha}( \
u)\).

典型的卡方分布表以自由度为行，以上尾概率 \(\alpha\) 为列。行列交叉处的数值就是 \(\chi^2_{\alpha}( \
u)\)。

For example, with \(\
u = 5\) and \(\alpha = 0.05\), the table gives \(\chi^2_{0.05}(5) = 11.070\). This means that only 5% of the distribution lies to the right of 11.070.

例如，当 \(\
u = 5\)、\(\alpha = 0.05\) 时，查表得 \(\chi^2_{0.05}(5) = 11.070\)。这意味着该分布只有 5% 的面积位于 11.070 的右侧。

\( u\) \ \(\alpha\)	0.10	0.05	0.025	0.01
1	2.706	3.841	5.024	6.635
2	4.605	5.991	7.378	9.210
3	6.251	7.815	9.348	11.345
4	7.779	9.488	11.143	13.277
5	9.236	11.070	12.833	15.086

Notice that the percentage point increases with both \(\
u\) and decreasing \(\alpha\). A smaller tail probability requires a larger cut-off value.

注意：百分位点随自由度增大而增大，也随 \(\alpha\) 减小而增大。尾概率越小，临界值越大。

5. Critical Values in Hypothesis Testing | 假设检验中的临界值

In a chi-squared hypothesis test, you compare the test statistic with the percentage point for the chosen significance level \(\alpha\) and the correct degrees of freedom.

在卡方假设检验中，你需要将检验统计量与给定显著性水平 \(\alpha\) 和正确自由度下的百分位点进行比较。

If the test statistic > \(\chi^2_{\alpha}( \
u)\), reject the null hypothesis.

若检验统计量 > \(\chi^2_{\alpha}( \
u)\)，则拒绝原假设。
If the test statistic ≤ \(\chi^2_{\alpha}( \
u)\), do not reject the null hypothesis.

若检验统计量 ≤ \(\chi^2_{\alpha}( \
u)\)，则不拒绝原假设。

For a one-tailed test at significance level \(\alpha\), the critical region is the upper tail beyond \(\chi^2_{\alpha}( \
u)\). For a two-tailed test of a variance, you need both lower and upper percentage points.

对于显著性水平为 \(\alpha\) 的单尾检验，拒绝域是 \(\chi^2_{\alpha}( \
u)\) 右侧的上尾区域。对于方差的单样本检验，如果使用双边检验，则需要同时考虑下尾和上尾的百分位点。

6. Goodness of Fit Test | 拟合优度检验

In a goodness-of-fit test, the observed frequencies are compared with the expected frequencies under a proposed model. The test statistic is:

在拟合优度检验中，将观测频数与假设模型下的期望频数进行比较。检验统计量为：

X² = Σ (Oᵢ − Eᵢ)² / Eᵢ

where \(O_i\) is the observed frequency and \(E_i\) the expected frequency for category \(i\). The sum is taken over all categories.

其中 \(O_i\) 为第 \(i\) 类的观测频数，\(E_i\) 为期望频数，对所有类别求和。

After calculating \(\chi^2\), compare it with \(\chi^2_{0.05}( \
u)\). You must also check that all expected frequencies are at least 5; otherwise adjacent categories should be combined.

计算出 \(\chi^2\) 后，与 \(\chi^2_{0.05}( \
u)\) 比较。你还必须检查所有期望频数是否至少为 5；若不满足，应合并相邻类别。

7. Contingency Tables | 列联表检验

For a contingency table with \(r\) rows and \(c\) columns, the expected frequency in each cell is:

对于 \(r\) 行 \(c\) 列的列联表，每个单元格的期望频数为：

Eᵢⱼ = (row total × column total) / grand total

The test statistic is again \(\sum (O – E)^2 / E\), now summed over all cells. The degrees of freedom are \((r – 1)(c – 1)\).

检验统计量仍然为 \(\sum (O – E)^2 / E\)，对所有单元格求和。自由度为 \((r – 1)(c – 1)\)。

For a \(2 \times 2\) table, \(\
u = 1\). The critical value at the 5% level is \(\chi^2_{0.05}(1) = 3.841\). If the calculated statistic exceeds this, there is evidence of association between the two variables.

对于 \(2 \times 2\) 列联表，\(\
u = 1\)。在 5% 显著性水平下，临界值为 \(\chi^2_{0.05}(1) = 3.841\)。若计算出的统计量超过此值，则有证据表明两个变量之间存在关联。

8. Confidence Interval for a Population Variance | 总体方差的置信区间

If a random sample of size \(n\) is drawn from a normal population, the quantity

若从正态总体中抽取容量为 \(n\) 的随机样本，则统计量

(n − 1)S² / σ² ～ χ²(n − 1)

can be used to construct a confidence interval for the population variance \(\sigma^2\). For a \(100(1 − \alpha)\%\) confidence interval, you need both \(\chi^2_{\alpha/2}(n−1)\) and \(\chi^2_{1−\alpha/2}(n−1)\).

可用于构造总体方差 \(\sigma^2\) 的置信区间。对于 \(100(1 − \alpha)\%\) 置信区间，你需要同时使用 \(\chi^2_{\alpha/2}(n−1)\) 和 \(\chi^2_{1−\alpha/2}(n−1)\)。

((n − 1)S²) / χ²_{α/2} ≤ σ² ≤ ((n − 1)S²) / χ²_{1−α/2}

Note the lower percentage point \(\chi^2_{1−\alpha/2}\) is small, and it appears in the denominator for the upper bound. The interval is not symmetric.

注意下尾百分位点 \(\chi^2_{1−\alpha/2}\) 较小，它出现在上界的分母中。该区间并不对称。

9. Inverse Use: Finding Approximate p-Values | 反查：近似 p 值

Sometimes your test statistic falls between two tabulated percentage points. In that case you can state that the p-value lies between the corresponding tail probabilities.

有时你的检验统计量落在两个列出的百分位点之间。此时你可以说明 p 值介于对应的尾概率之间。

For example, with \(\
u = 5\), suppose \(\chi^2 = 12.0\). Comparing with the table:

例如，当 \(\
u = 5\) 时，假设 \(\chi^2 = 12.0\)。与表比较：

\(\chi^2_{0.05}(5) = 11.070\) and \(\chi^2_{0.025}(5) = 12.833\). Since 11.070 < 12.0 < 12.833, the p-value satisfies 0.025 < p < 0.05.

\(\chi^2_{0.05}(5) = 11.070\)、\(\chi^2_{0.025}(5) = 12.833\)。由于 11.070 < 12.0 < 12.833，因此 p 值满足 0.025 < p < 0.05。

Hence the result is significant at the 5% level but not at the 2.5% level.

因此结果在 5% 水平下显著，但在 2.5% 水平下不显著。

10. Common Pitfalls | 常见易错点

Several mistakes appear frequently in exams. Avoid them by checking these details.

考试中有几个常见错误。通过检查以下细节来避免它们。

Using the wrong degrees of freedom. Always recalculate \(\
u\) from the number of categories or rows and columns, and subtract the number of estimated parameters.

使用错误的自由度。务必根据类别数或行列数重新计算 \(\
u\)，并减去估计的参数个数。
Confusing the lower and upper percentage points. For a confidence interval, the lower tail uses \(\chi^2_{1-\alpha/2}\) and the upper tail uses \(\chi^2_{\alpha/2}\).

混淆下尾和上尾百分位点。在置信区间中，下尾使用 \(\chi^2_{1-\alpha/2}\)，上尾使用 \(\chi^2_{\alpha/2}\)。
Forgetting to combine categories when expected frequencies are below 5.

当期望频数低于 5 时忘记合并类别。
Using expected frequencies as integers. They may be fractional; do not round them to whole numbers before calculating.

将期望频数当作整数。期望频数可以是小数；计算前不要四舍五入成整数。

11. Worked Example | 完整例题

A die is tossed 120 times. The observed frequencies are: 1: 25, 2: 18, 3: 20, 4: 22, 5: 17, 6: 18. Test at the 5% level whether the die is fair.

一枚骰子被投掷 120 次。观测频数为：1: 25，2: 18，3: 20，4: 22，5: 17，6: 18。在 5% 显著性水平下检验骰子是否公平。

If the die is fair, each expected frequency is \(120 \div 6 = 20\). Calculate:

若骰子公平，每个期望频数为 \(120 \div 6 = 20\)。计算：

X² = (25−20)²/20 + (18−20)²/20 + (20−20)²/20 + (22−20)²/20 + (17−20)²/20 + (18−20)²/20

Thus \(X^2 = 1.25 + 0.2 + 0 + 0.2 + 0.45 + 0.2 = 2.30\). Degrees of freedom: \(\
u = 6 – 1 = 5\). The critical value is \(\chi^2_{0.05}(5) = 11.070\). Since \(2.30 < 11.070\), we do not reject the null hypothesis; there is insufficient evidence that the die is unfair.

因此 \(X^2 = 1.25 + 0.2 + 0 + 0.2 + 0.45 + 0.2 = 2.30\)。自由度：\(\
u = 6 – 1 = 5\)。临界值为 \(\chi^2_{0.05}(5) = 11.070\)。由于 \(2.30 < 11.070\)，我们不拒绝原假设；没有充分证据表明骰子不公平。

12. Exam Strategy | 考试策略

Always write down the null and alternative hypotheses, state the test statistic formula, calculate the expected frequencies carefully, and state the degrees of freedom. Then quote the percentage point from the table and make a clear conclusion in context.

务必写出原假设和备择假设，列出检验统计量公式，细心计算期望频数，并说明自由度。然后从表中引用百分位点，并结合实际问题给出明确结论。

When using the chi-squared table, locate the correct row for \(\
u\) and the correct column for the required tail probability. For a two-tailed variance test, remember to halve the significance level before using each tail. Precision in reading the table is just as important as the calculation itself.

使用卡方分布表时，先找到对应 \(\
u\) 的行，再找到所需尾概率对应的列。对于方差的双边检验，记得先对显著性水平减半，再分别使用两个尾部。查表的准确性同样重要，与计算本身同等关键。

Understanding percentage points is not merely a table-reading exercise. It strengthens your grasp of tail probabilities, critical regions, and the logic of hypothesis testing — all of which recur across Edexcel A-Level Further Mathematics papers.

理解百分位点不只是查表练习。它能加深你对尾概率、拒绝域和假设检验逻辑的理解——这些内容在 Edexcel A-Level 进阶数学试卷中反复出现。

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A-Level Further Mathematics: Percentage Points of the Chi-Squared Distribution | A-Level进阶数学：卡方分布百分位点