Degrees of Freedom | 自由度

📚 Degrees of Freedom | 自由度

Degrees of freedom are one of the most frequently examined ideas in Edexcel hypothesis tests: they control the shape of the t, chi-squared and F distributions and determine which critical value you use.

自由度是 Edexcel 假设检验中最常考的概念之一：它们决定了 t 分布、卡方分布和 F 分布的形状，并决定你使用哪个临界值。

This topic appears in A-Level Statistics and Further Statistics, especially in t-tests, chi-squared tests and variance comparison.

该主题出现在 A-Level 统计学和进阶统计学中，尤其是在 t 检验、卡方检验和方差比较中。

1. What Are Degrees of Freedom? | 什么是自由度

Degrees of freedom (df) count how many independent pieces of information are left after estimating parameters or applying constraints.

自由度（df）计算在估计参数或施加约束之后，还剩多少独立的信息。

Think of df as the number of values that are free to vary while still satisfying a fixed total or constraint.

可以把自由度想象成，在满足固定总数或约束的前提下，可以自由变动的数值个数。

2. A Simple Example: Four Numbers With a Fixed Mean | 一个简单例子：平均值固定的四个数

Suppose four observations are x₁, x₂, x₃, x₄ and the mean x̄ is already known. If you choose any three values, the fourth is forced: x₄ = 4x̄ − x₁ − x₂ − x₃.

假设有四个观测值 x₁、x₂、x₃、x₄，且平均值 x̄ 已知。若你任选三个值，第四个就被迫确定：x₄ = 4x̄ − x₁ − x₂ − x₃。

Therefore there are only 3 free choices, so df = 4 − 1 = 3.

因此只有 3 个自由选择，所以 df = 4 − 1 = 3。

3. Degrees of Freedom in Sample Variance | 样本方差中的自由度

The sample variance uses n − 1 in the denominator: s² = Σ(xᵢ − x̄)² / (n − 1).

样本方差分母使用 n − 1：s² = Σ(xᵢ − x̄)² / (n − 1)。

Once the sample mean x̄ has been calculated from the data, only n − 1 deviations are independent.

一旦样本均值 x̄ 由数据计算出来，只有 n − 1 个离差是独立的。

The constraint Σ(xᵢ − x̄) = 0 removes one degree of freedom, so df = n − 1.

约束 Σ(xᵢ − x̄) = 0 去除了一个自由度，因此 df = n − 1。

4. Why Dividing by n − 1 Matters | 为什么除以 n − 1 很重要

Dividing by n − 1 gives an unbiased estimator of the population variance σ²; dividing by n tends to underestimate the variance.

除以 n − 1 可以给出总体方差 σ² 的无偏估计；除以 n 会低估方差。

This is directly linked to losing one degree of freedom when estimating the population mean by x̄.

这直接与用 x̄ 估计总体均值时损失一个自由度相关。

5. Degrees of Freedom for the t-Distribution | t 分布的自由度

For a sample of size n from a normal population, the t-statistic is t = (x̄ − μ) / (s/√n), and its distribution has df = n − 1.

对于来自正态总体、容量为 n 的样本，t 统计量为 t = (x̄ − μ) / (s/√n)，其分布的自由度为 df = n − 1。

The t-distribution has heavier tails than the standard normal for small df, and approaches N(0, 1) as df → ∞.

当自由度较小时，t 分布比标准正态分布尾部更厚；当 df → ∞ 时，它趋近于 N(0, 1)。

Using the wrong df changes the critical value and therefore the conclusion of a hypothesis test.

使用错误的自由度会改变临界值，从而改变假设检验的结论。

6. Degrees of Freedom in Chi-Squared Goodness-of-Fit Tests | 卡方拟合优度检验中的自由度

For a goodness-of-fit test, the test statistic is χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ.

拟合优度检验的统计量为 χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ。

If there are k categories and no population parameter is estimated from the data, df = k − 1.

如果有 k 个类别，且没有从数据中估计总体参数，则 df = k − 1。

If p parameters are estimated, the rule becomes df = k − 1 − p.

如果估计了 p 个参数，规则变为 df = k − 1 − p。

For a binomial model where p is estimated from the sample, df = k − 2. For a Poisson model where λ is estimated, df = k − 2 as well.

对于从样本中估计 p 的二项分布模型，df = k − 2。对于估计 λ 的泊松模型，df 也等于 k − 2。

7. Degrees of Freedom in Contingency Tables | 列联表中的自由度

For a two-way table with r rows and c columns, df = (r − 1)(c − 1).

对于 r 行 c 列的双向列联表，df = (r − 1)(c − 1)。

This counts the number of expected cell counts that are free once row totals, column totals and the grand total are fixed.

这计算的是在行总数、列总数和总计固定的情况下，可以自由变动的期望频数个数。

For a 2 × 2 table, df = (2 − 1)(2 − 1) = 1, so the conclusions are often checked carefully in Edexcel questions.

对于 2 × 2 表，df = (2 − 1)(2 − 1) = 1，因此在 Edexcel 题目中通常会仔细检查结论。

8. Degrees of Freedom for the F-Distribution | F 分布的自由度

The F-distribution is used to compare two variances or to test groups in ANOVA; it has two degrees of freedom, df₁ and df₂.

F 分布用于比较两个方差或在 ANOVA 中检验组间差异；它有两个自由度，df₁ 和 df₂。

For two independent samples with sizes n₁ and n₂, the ratio F = S₁² / S₂² uses df₁ = n₁ − 1 and df₂ = n₂ − 1.

对于容量分别为 n₁ 和 n₂ 的两个独立样本，比值 F = S₁² / S₂² 使用 df₁ = n₁ − 1 和 df₂ = n₂ − 1。

The F-distribution is not symmetric; its shape depends on both numerator and denominator degrees of freedom.

F 分布不对称；其形状取决于分子自由度和分母自由度。

9. Degrees of Freedom in Regression Residuals | 回归残差中的自由度

In simple linear regression with n observations, estimating the intercept and slope uses 2 degrees of freedom, so the residual df = n − 2.

在包含 n 个观测值的简单线性回归中，估计截距和斜率需要 2 个自由度，因此残差 df = n − 2。

The residual mean square is s² = RSS / (n − 2), where RSS = Σ(yᵢ − ŷᵢ)².

残差均方为 s² = RSS / (n − 2)，其中 RSS = Σ(yᵢ − ŷᵢ)²。

This idea extends the sample variance rule: each estimated parameter removes one additional degree of freedom.

这一思想推广了样本方差的规则：每估计一个参数，就额外减少一个自由度。

10. Exam Tips and Common Errors | 考试提示与常见错误

For a one-sample t-test, write df = n − 1 on the answer line before looking up the critical value.

对于单样本 t 检验，在查临界值之前先在答题线上写出 df = n − 1。

For a chi-squared test, always check whether parameters have been estimated; this is a common trick in Edexcel papers.

对于卡方检验，始终检查是否估计了参数；这是 Edexcel 试卷中的常见陷阱。

For a contingency table, multiplying (r − 1) by (c − 1) is usually quicker than counting free cells.

对于列联表，将 (r − 1) 乘以 (c − 1) 通常比数自由格子更快。

Never use df = n for a sample variance or t-test unless the population mean is known, which is rare in exam questions.

除非总体均值已知，否则不要将样本方差或 t 检验的自由度写成 n；这在考试题目中很少见。

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Degrees of Freedom | 自由度