📚 Degrees of Freedom and the Chi-Squared Family of Distributions | 自由度与卡方分布族
The chi-squared distribution plays a central role in A-Level Statistics, especially in hypothesis tests for goodness of fit and independence. To use it correctly, you must understand a key concept: degrees of freedom. This article explains what degrees of freedom mean, how the chi-squared family is built, and how to choose the correct member of the family in exam questions.
卡方分布在 A-Level 统计学中扮演着核心角色，尤其是在拟合优度检验和独立性检验中。要正确使用它，你必须理解一个关键概念：自由度。本文将解释自由度的含义、卡方分布族的构成方式，以及如何在考试题目中选择该分布族中正确的成员。
1. What is the Chi-Squared Distribution? | 什么是卡方分布？
The chi-squared distribution is a continuous probability distribution that only takes positive values. It is used to model the sum of squared standard normal variables and appears naturally when comparing observed and expected frequencies.
卡方分布是一种连续型概率分布，只取正值。它用于对标准正态变量的平方和进行建模，并在比较观测频数与期望频数时自然出现。
Unlike the normal distribution, the chi-squared distribution is not symmetric. Its shape is skewed to the right, especially when the number of degrees of freedom is small. As the degrees of freedom increase, the distribution becomes more symmetric and eventually resembles a normal curve.
与正态分布不同，卡方分布不是对称的。它的形状呈右偏，尤其是当自由度较小时。随着自由度增大，该分布变得越来越对称，最终趋近于正态曲线。
- Support: x > 0
- Parameter: degrees of freedom, often denoted by ν
- Shape: right-skewed for small ν, near-normal for large ν
- 支撑集：x > 0
- 参数：自由度，通常用 ν 表示
- 形状：ν 小时右偏，ν 大时近似正态
2. Degrees of Freedom: Core Idea | 自由度：核心思想
Degrees of freedom represent the number of independent pieces of information available when estimating a statistic. In the context of the chi-squared distribution, the degrees of freedom determine which specific chi-squared curve should be used.
自由度表示在估计统计量时可用的独立信息数量。在卡方分布的背景下，自由度决定了应使用哪一条具体的卡方曲线。
A common example is the sample variance. If you have n observations and you use the sample mean to estimate the population mean, only n − 1 of the deviations are free to vary. This is why the sample variance uses n − 1 in its denominator.
一个常见的例子是样本方差。如果你有 n 个观测值，并用样本均值估计总体均值，那么只有 n − 1 个偏差是自由变化的。这就是样本方差分母使用 n − 1 的原因。
For a chi-squared test, the degrees of freedom are not simply the sample size. They are calculated from the number of categories, constraints, and estimated parameters.
对于卡方检验，自由度并不是简单的样本容量。它是由类别数量、约束条件和已估计参数的数量计算得到的。
3. Definition via Squared Standard Normal Variables | 通过标准正态平方和定义
If Z₁, Z₂, …, Zₙ are independent standard normal variables, then the sum of their squares follows a chi-squared distribution with n degrees of freedom:
如果 Z₁、Z₂、…、Zₙ 是相互独立的标准正态变量，那么它们的平方和服从自由度为 n 的卡方分布：
Q = Z₁² + Z₂² + … + Zₙ² ~ χ²(n)
This definition explains why the chi-squared distribution is called a family. Each value of n gives a different distribution, and the shape changes as n changes.
这个定义解释了为什么卡方分布被称为一个分布族。每个 n 值都对应一个不同的分布，形状也随 n 的变化而变化。
For example, if n = 1, then Q = Z₁² follows χ²(1). If n = 2, then Q = Z₁² + Z₂² follows χ²(2). The degrees of freedom simply count the number of independent squared standard normal terms.
例如，当 n = 1 时，Q = Z₁² 服从 χ²(1)。当 n = 2 时，Q = Z₁² + Z₂² 服从 χ²(2)。自由度就是独立标准正态平方项的个数。
4. Probability Density Function and Its Shape | 概率密度函数与形状
The probability density function of a chi-squared distribution with ν degrees of freedom is given by:
自由度为 ν 的卡方分布的概率密度函数如下：
f(x) = [1 / (2^(ν/2) Γ(ν/2))] x^(ν/2 − 1) e^(−x/2), x > 0
Here Γ is the gamma function. You do not need to evaluate this density directly in Edexcel exams, but it helps to understand the shape. For ν = 1, the density tends to infinity as x approaches 0. For ν = 2, the distribution becomes an exponential distribution with rate 1/2. For ν ≥ 3, the curve starts at 0, rises to a peak, and then decreases.
这里 Γ 是伽马函数。在爱德思考试中你不需要直接计算这个密度，但它有助于理解分布形状。当 ν = 1 时，密度在 x 趋近于 0 时趋于无穷大。当 ν = 2 时，分布变成率为 1/2 的指数分布。当 ν ≥ 3 时，曲线从 0 开始，上升到峰值，然后下降。
The distribution is always skewed to the right, but the skewness decreases as ν increases. For large ν, the distribution is approximately normal with mean ν and variance 2ν.
该分布始终右偏，但偏度随 ν 增大而减小。当 ν 很大时，分布近似于均值为 ν、方差为 2ν 的正态分布。
5. Mean, Variance and Mode | 均值、方差与众数
For a chi-squared distribution with ν degrees of freedom, the key summary statistics are straightforward:
对于自由度为 ν 的卡方分布，主要的汇总统计量非常简单：
E(X) = ν, Var(X) = 2ν
The mean equals the degrees of freedom, and the variance is twice the degrees of freedom. This is a quick way to check whether a calculated test statistic is in a reasonable range.
均值等于自由度，方差是自由度的两倍。这是快速检验计算出的统计量是否处于合理范围的一种方法。
For ν > 2, the mode of the distribution is ν − 2. For ν = 1 or ν = 2, the mode is at 0 because the density either decreases from infinity or decays exponentially.
当 ν > 2 时，分布的众数为 ν − 2。当 ν = 1 或 ν = 2 时，众数位于 0，因为密度要么从无穷大处下降，要么按指数衰减。
6. How the Distribution Changes with Degrees of Freedom | 分布如何随自由度变化
As the degrees of freedom increase, several important changes occur. The mean moves to the right, the variance becomes larger, and the peak becomes less pronounced. The right tail remains, but the curve looks more symmetric.
随着自由度增大，会发生产生几个重要变化。均值向右移动，方差变大，峰值变得不那么明显。右尾仍然存在，但曲线看起来更加对称。
The table below summarises the mean, variance and mode for small values of ν:
下表总结了较小 ν 值下的均值、方差和众数：
| Degrees of freedom ν | Mean | Variance | Mode |
| 1 | 1 | 2 | 0 |
| 2 | 2 | 4 | 0 |
| 5 | 5 | 10 | 3 |
| 10 | 10 | 20 | 8 |
In exam questions, you may be asked to describe how the distribution shape changes with degrees of freedom. Remember: larger ν means more symmetric and less skewed.
在考试题目中，你可能会被要求描述分布形状如何随自由度变化。记住：ν 越大，分布越对称，偏度越小。
7. The Chi-Squared Family of Distributions | 卡方分布族
The term “chi-squared family” means that there is not just one chi-squared distribution, but infinitely many. Each positive integer degree of freedom gives one member of the family.
“卡方分布族” 意味着卡方分布不只是一个，而是无穷多个。每个正整数自由度对应分布族中的一个成员。
Special members include χ²(1), which is the square of one standard normal variable, and χ²(2), which is an exponential distribution with mean 2. These special cases help demonstrate the flexibility of the family.
特殊成员包括 χ²(1)，它是一个标准正态变量的平方；以及 χ²(2)，它是均值为 2 的指数分布。这些特殊情况有助于展示该分布族的灵活性。
In practice, you select the member of the family by calculating the correct degrees of freedom for your test. Using the wrong member leads to an incorrect critical value and an incorrect conclusion.
在实际中，你通过计算检验的正确自由度来选择分布族中的成员。使用错误的成员会导致错误的临界值和错误的结论。
8. Critical Values and Use of Tables | 临界值与查表
Chi-squared critical values are usually given for the upper tail of the distribution. You look up a value such as χ²_α(ν), where α is the significance level and ν is the degrees of freedom.
卡方临界值通常对应分布的上尾。你查找的值如 χ²_α(ν)，其中 α 是显著性水平，ν 是自由度。
For example, at the 5% significance level, the critical value for ν = 4 is 9.488. This means that P(X > 9.488) = 0.05. If a test statistic exceeds this value, you reject the null hypothesis.
例如，在 5% 显著性水平下，ν = 4 的临界值为 9.488。这意味着 P(X > 9.488) = 0.05。如果检验统计量超过该值，则拒绝原假设。
The table below gives some commonly used critical values:
下表给出一些常用的临界值：
| df ν | 5% critical value | 1% critical value |
| 1 | 3.841 | 6.635 |
| 2 | 5.991 | 9.210 |
| 3 | 7.815 | 11.345 |
| 4 | 9.488 | 13.277 |
| 5 | 11.070 | 15.086 |
Always check whether the table gives the upper-tail probability or the cumulative probability from the left. Edexcel tables typically give upper-tail critical values.
务必检查表格给出的是上尾概率还是从左
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply