Percentage Points of the F-Distribution | F 分布的百分位点

📚 Percentage Points of the F-Distribution | F 分布的百分位点

Percentage points of the F-distribution are the critical values used in statistical inference when comparing two population variances. This revision note explains how to read, interpret and apply these values in Edexcel A-Level Statistics, especially in variance ratio tests and confidence intervals.

F 分布的百分位点是在比较两个总体方差时使用的临界值。本复习笔记解释如何在 Edexcel A-Level 统计学中读取、解释并应用这些值，特别是在方差比检验和置信区间中。

1. The F-Distribution and Its Role | F 分布及其作用

The F-distribution arises when we compare two independent estimates of population variance. It is used to test whether two sample variances are significantly different, for example when checking the assumption of equal variances before a two-sample t-test.

当比较两个独立的总体方差估计值时，就会产生 F 分布。它用于检验两个样本方差是否有显著差异，例如在进行双样本 t 检验之前检查方差相等这一假设。

Its percentage points allow us to decide whether an observed variance ratio is unusually large or unusually small. These percentage points are often called critical values or quantiles in exam papers.

它的百分位点使我们能够判断观测到的方差比是否异常大或异常小。这些百分位点在试卷中通常被称为临界值或分位数。

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2. Formal Definition via Chi-Squared Variables | 由卡方变量给出的正式定义

If U and V are independent chi-squared random variables with ν₁ and ν₂ degrees of freedom, written U ~ χ²(ν₁) and V ~ χ²(ν₂), then the ratio

如果 U 和 V 是独立的卡方随机变量，自由度分别为 ν₁ 和 ν₂，记作 U ~ χ²(ν₁) 和 V ~ χ²(ν₂)，那么比值

F = (U ÷ ν₁) ÷ (V ÷ ν₂)

follows an F-distribution with ν₁ numerator degrees of freedom and ν₂ denominator degrees of freedom, written F(ν₁, ν₂).

服从自由度为 ν₁ 的分子和自由度为 ν₂ 的分母的 F 分布，记作 F(ν₁, ν₂)。

The two degrees of freedom determine the shape of the distribution. In variance ratio tests, ν₁ = n₁ − 1 and ν₂ = n₂ − 1, where n₁ and n₂ are the two sample sizes.

这两个自由度决定了分布的形状。在方差比检验中，ν₁ = n₁ − 1，ν₂ = n₂ − 1，其中 n₁ 和 n₂ 是两个样本的大小。

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3. Shape and Properties | 形状与性质

F-distributions are continuous, non-negative and positively skewed. As both degrees of freedom become large, the distribution becomes more symmetric around 1, and in the limit it tends towards a chi-squared distribution shape scaled by the numerator df.

F 分布是连续的、非负的且正偏态。随着两个自由度增大，分布围绕 1 变得更加对称，在极限情况下它趋向于按分子自由度缩放的卡方分布形状。

For ν₂ > 2, the mean of F(ν₁, ν₂) is ν₂ ÷ (ν₂ − 2). Because the distribution is skewed, the mean is not usually equal to the upper percentage point shown in tables.

当 ν₂ > 2 时，F(ν₁, ν₂) 的均值是 ν₂ ÷ (ν₂ − 2)。由于分布是偏态的，均值通常不等于表中给出的上百分位点。

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4. Notation for Percentage Points | 百分位点的记号

The upper α percentage point of F(ν₁, ν₂) is denoted F(ν₁, ν₂; α) and satisfies

F(ν₁, ν₂) 的上 α 百分位点记作 F(ν₁, ν₂; α)，并满足

P(F > F(ν₁, ν₂; α)) = α

This means the area under the F(ν₁, ν₂) curve to the right of this value is exactly α. Tables usually give upper-tail percentage points for α = 0.05, 0.025, 0.01 and 0.001.

这意味着 F(ν₁, ν₂) 曲线下位于该值右侧的面积恰好为 α。表格通常给出 α = 0.05、0.025、0.01 和 0.001 的上尾百分位点。

In exam questions, you may also see the notation Fₐ(ν₁, ν₂) or simply F critical. Always check whether the paper is asking for an upper-tail or lower-tail point.

在考试题中，你还可能看到记号 Fₐ(ν₁, ν₂) 或简写 F critical。一定要检查题目要求的是上尾百分位点还是下尾百分位点。

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5. Reading an F-Table | 如何查 F 表

To read an F-table, locate the numerator degrees of freedom ν₁ across the top row, and the denominator degrees of freedom ν₂ down the left column. Choose the table page that corresponds to the required upper-tail probability α, then read off the critical value.

要查 F 表，先从顶部一行找到分子自由度 ν₁，再从左侧一列找到分母自由度 ν₂。选择与所需上尾概率 α 对应的表格页，然后读出临界值。

The table below shows some common upper 5% points, rounded to two decimal places.

下表列出了一些常见的上 5% 百分位点，保留两位小数。

ν₁	ν₂	Upper 5% point
1	1	161.45
5	5	5.05
10	10	2.98
20	20	2.12
∞	∞	1.00

Always use the exact values from your Edexcel statistical table booklet if a question requires more precision.

如果题目要求更高精度，请始终使用 Edexcel 统计表手册中的精确值。

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6. Reciprocal Property for Lower-Tail Points | 下尾百分位点的倒数关系

In practice, F-tables rarely list lower-tail percentage points. This is not a problem because of the reciprocal property: if X ~ F(ν₁, ν₂), then 1 ÷ X ~ F(ν₂, ν₁). Therefore

实际上，F 表很少列出下尾百分位点。这并不是问题，因为有一个倒数性质：如果 X ~ F(ν₁, ν₂)，那么 1 ÷ X ~ F(ν₂, ν₁)。因此

F(ν₁, ν₂; 1 − α) = 1 ÷ F(ν₂, ν₁; α)

This is used to obtain a lower-tail point from an upper-tail point with the degrees of freedom swapped. It is one of the most common tools required in A-Level questions involving F percentage points.

这用于通过互换自由度后的上尾百分位点来获得下尾百分位点。它是 A-Level 中涉及 F 百分位点的题目里最常用的工具之一。

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7. Two-Sample Variance Ratio Test | 两样本方差比检验

To test H₀: σ₁² = σ₂² against H₁: σ₁² ≠ σ₂², compute the test statistic

要检验 H₀：σ₁² = σ₂² 与 H₁：σ₁² ≠ σ₂²，计算检验统计量

F = s₁² ÷ s₂²

where s₁² and s₂² are the sample variances. In a two-tailed test, it is convenient to put the larger sample variance in the numerator, so F ≥ 1, and then compare with the upper α/2 percentage point of F(ν₁, ν₂).

其中 s₁² 和 s₂² 是样本方差。在双尾检验中，通常将较大的样本方差放在分子，这样 F ≥ 1，然后与 F(ν₁, ν₂) 的上 α/2 百分位点进行比较。

If F is larger than the critical value, reject H₀ and conclude that the population variances are significantly different. If not, there is insufficient evidence to reject equal variances.

如果 F 大于临界值，则拒绝 H₀，并认为总体方差有显著差异。如果不是，则没有足够证据拒绝方差相等。

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8. Worked Example: Upper Percentage Point | 例题：上百分位点

Find the upper 5% percentage point for an F-distribution with 6 numerator and 10 denominator degrees of freedom.

求分子自由度为 6、分母自由度为 10 的 F 分布的上 5% 百分位点。

From the F-table, F(6, 10; 0.05) =

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