📚 3: The F-distribution | F 分布
In A-Level Further Statistics, the F-distribution is used mainly for comparing two variances and for analysis of variance (ANOVA). It is named after Sir Ronald Fisher and is a continuous distribution with two degrees of freedom: one for the numerator and one for the denominator. This article covers the definition, shape, critical values, hypothesis tests, and worked examples relevant to Edexcel examinations.
在 A-Level 进阶统计中，F 分布主要用于比较两个方差以及进行方差分析(ANOVA)。它由罗纳德·费舍尔爵士命名，是一种具有两个自由度的连续型分布：一个用于分子，一个用于分母。本文介绍其定义、形状、临界值、假设检验以及与 Edexcel 考试相关的例题。
1. Definition and parameters | 定义与参数
If U ~ χ²(ν₁) and V ~ χ²(ν₂) are independent chi-squared random variables, then the ratio W = (U/ν₁) / (V/ν₂) follows an F-distribution with ν₁ numerator degrees of freedom and ν₂ denominator degrees of freedom. We write W ~ F(ν₁, ν₂). The parameters ν₁ and ν₂ are usually positive integers arising from sample sizes.
如果 U ~ χ²(ν₁) 和 V ~ χ²(ν₂) 是相互独立的卡方随机变量，那么比值 W = (U/ν₁) / (V/ν₂) 服从 F 分布，具有 ν₁ 个分子自由度和 ν₂ 个分母自由度。我们记作 W ~ F(ν₁, ν₂)。参数 ν₁ 和 ν₂ 通常是由样本容量得到的正整数。
For two independent normal samples with population variances σ₁² and σ₂², the sample variances S₁² and S₂² satisfy the ratio (S₁²/σ₁²) / (S₂²/σ₂²) ~ F(n₁ – 1, n₂ – 1). This ratio is the foundation of variance comparison tests.
对于来自正态总体的两个独立样本，若总体方差为 σ₁² 和 σ₂²，样本方差为 S₁² 和 S₂²，则满足 (S₁²/σ₁²) / (S₂²/σ₂²) ~ F(n₁ – 1, n₂ – 1)。这个比值是方差比较检验的基础。
2. Relationship with chi-squared and t distributions | 与卡方分布和 t 分布的关系
The F-distribution is built from two independent chi-squared variables. Also, if T ~ t(ν), then T² ~ F(1, ν). This relationship explains why squaring a t-statistic gives an F-statistic with 1 numerator degree of freedom, which is used in ANOVA and regression tests.
F 分布由两个独立的卡方变量构成。此外，如果 T ~ t(ν)，那么 T² ~ F(1, ν)。这种关系解释了为什么将 t 统计量平方后会得到一个分子自由度为 1 的 F 统计量，这在方差分析和回归检验中会被用到。
Because a chi-squared variable is a special gamma variable, the F-distribution can also be viewed as the ratio of two independent gamma variables scaled by their degrees of freedom. This connection helps when proving distributional results in S3.
由于卡方变量是一种特殊的伽马变量，F 分布也可以看作两个独立的伽马变量分别除以其自由度后的比值。这一联系有助于在 S3 中证明分布性质。
3. Probability density function | 概率密度函数
If X ~ F(ν₁, ν₂), its probability density function is defined for x ≥ 0 and is zero otherwise. The complete form uses the gamma function Γ and depends only on the two degrees of freedom.
如果 X ~ F(ν₁, ν₂)，其概率密度函数定义在 x ≥ 0 上，其余处为零。完整形式使用伽马函数 Γ，并且只依赖于两个自由度。
f(x) = [ Γ((ν₁ + ν₂) / 2) × (ν₁ / ν₂)^(ν₁ / 2) × x^(ν₁ / 2 – 1) ] / [ Γ(ν₁ / 2) × Γ(ν₂ / 2) × (1 + (ν₁ / ν₂)x)^((ν₁ + ν₂) / 2) ]
Although Edexcel questions rarely require you to substitute into this density, understanding that f(x) is only positive on x ≥ 0 explains why all F critical values are positive and why the distribution is right-skewed.
虽然 Edexcel 试题很少要求代入这个密度函数，但理解 f(x) 只在 x ≥ 0 上为正，可以解释为什么所有 F 临界值都是正的，以及为什么该分布是右偏的。
4. Shape and skewness | 形状与偏度
The F-distribution is skewed to the right. Its support is (0, ∞), so the random variable can never be negative. The shape depends on both degrees of freedom: for small denominator degrees of freedom the right tail is very heavy, while as ν₂ increases the distribution becomes more concentrated near 1.
F 分布是右偏的。它的支撑集是 (0, ∞)，因此随机变量不可能为负。形状取决于两个自由度：当分母自由度较小时，右尾非常厚重；随着 ν₂ 增大，分布会越来越集中在 1 附近。
As both ν₁ and ν₂ become large, the F-distribution becomes less skewed and approaches a normal distribution in a limiting sense. However, for typical sample sizes used in exams, the right skew is still noticeable.
当 ν₁ 和 ν₂ 都变大时，F 分布的偏度减小，并在极限意义下趋近于正态分布。但在考试中常见的样本容量下，右偏仍然很明显。
5. Mean and variance | 均值与方差
If X ~ F(ν₁, ν₂), then the mean exists only when ν₂ > 2 and is given by E(X) = ν₂ / (ν₂ – 2). Notice that the mean does not depend on the numerator degrees of freedom ν₁, and it is always greater than 1 for finite ν₂.
如果 X ~ F(ν₁, ν₂)，则均值仅当 ν₂ > 2 时存在，且 E(X) = ν₂ / (ν₂ – 2)。注意均值不依赖于分子自由度 ν₁，并且对于有限的 ν₂ 总是大于 1。
The variance exists only when ν₂ > 4 and is Var(X) = [ 2ν₂² (ν₁ + ν₂ – 2) ] / [ ν₁ (ν₂ – 2)² (ν₂ – 4) ]. Because these moments can be large, F-distributions often have high variability with small degrees of freedom.
方差仅当 ν₂ > 4 时存在，且 Var(X) = [ 2ν₂² (ν₁ + ν₂ – 2) ] / [ ν₁ (ν₂ – 2)² (ν₂ – 4) ]。由于这些矩可能会很大，在小自由度下 F 分布通常具有较高的变异性。
6. Critical values and F-tables | 临界值与 F 分布表
F-distribution tables usually give upper-tail critical values F(α; ν₁, ν₂) such that P(F ≥ F(α; ν₁, ν₂)) = α. Common significance levels are 0.05, 0.025, and 0.01. In Edexcel exams you may be given a table or the specific values needed.
F 分布表通常给出上尾临界值 F(α; ν₁, ν₂)，使得 P(F ≥ F(α; ν₁, ν₂)) = α。常见的显著性水平有 0.05、0.025 和 0.01。在 Edexcel 考试中，你可能会得到一张表或所需的具体值。
When reading an F-table, the columns usually correspond to numerator degrees of freedom ν₁ and the rows to denominator degrees of freedom ν₂. Always double-check which direction the table uses before reading off a value.
查 F 分布表时，列通常对应分子自由度 ν₁，行对应分母自由度 ν₂。在读取数值之前，一定要先确认表格使用的是哪个方向。
7. Reciprocal property | 倒数性质
Because the F-distribution is not symmetric, lower-tail critical values are not simply the negative of upper-tail values. Instead, use the reciprocal identity: F(1 – α; ν₁, ν₂) = 1 / F(α; ν₂, ν₁). This identity is essential for two-tailed tests and confidence intervals.
由于 F 分布不对称，下尾临界值并不是上尾临界值的相反数。相反，应使用倒数恒等式：F(1 – α; ν₁, ν₂) = 1 / F(α; ν₂, ν₁)。这个恒等式对于双尾检验和置信区间至关重要。
For example, if you know F(0.05; 5, 8) = 3.69, then F(0.95; 8,
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