Percentage Points of the χ² Distribution | 卡方分布的百分位点

📚 Percentage Points of the χ² Distribution | 卡方分布的百分位点

In A-Level Statistics, the chi-squared distribution is central to tests for categorical data. Understanding its percentage points is essential because these values determine the boundary between accepting and rejecting a null hypothesis. This article explains what percentage points are, how to read them from tables, and how to use them in goodness-of-fit and independence tests.

在 A-Level 统计中，卡方分布是分类数据检验的核心。理解其百分位点至关重要，因为这些值决定了接受与拒绝原假设之间的界限。本文解释什么是百分位点、如何从表中读取，以及如何在拟合优度检验和独立性检验中使用它们。

1. What is the Chi-Squared Distribution? | 什么是卡方分布？

Suppose Z₁, Z₂, …, Zν are independent standard normal variables. Then X² = Z₁² + Z₂² + … + Zν² has a chi-squared distribution with ν degrees of freedom.

设 Z₁、Z₂、…、Zν 是独立的标准正态变量。则 X² = Z₁² + Z₂² + … + Zν² 服从自由度为 ν 的卡方分布。

X² = Z₁² + Z₂² + … + Zν²

The χ² variable only takes non-negative values. Its curve starts at 0, rises to a mode when ν > 2, and is skewed to the right. As ν increases, the distribution becomes more symmetric and approximately normal.

卡方变量只取非负值。当 ν > 2 时，其曲线从 0 开始上升到一个众数，并向右偏斜。随着 ν 增大，分布变得更对称，并近似正态。

2. Defining Percentage Points | 百分位点的定义

A percentage point of the χ² distribution is the value that leaves a given proportion of probability in the upper tail. For example, the 5% percentage point, written χ²ν,0.05, satisfies P(X² > χ²ν,0.05) = 0.05.

卡方分布的百分位点是使上尾留有给定概率的数值。例如，5% 百分位点写作 χ²ν,0.05，满足 P(X² > χ²ν,0.05) = 0.05。

Some textbooks call these values critical values or upper-tail critical values. They are used to define the rejection region of a test.

一些教材称这些值为临界值或上尾临界值。它们用于定义检验的拒绝域。

Unlike the standard normal distribution, the χ² distribution is not symmetric, so we do not use ± critical values. Every percentage point is positive and depends on the degrees of freedom.

与标准正态分布不同，卡方分布不对称，因此我们不使用正负临界值。每个百分位点都是正数，并取决于自由度。

3. Upper-Tail and Lower-Tail Percentage Points | 上尾与下尾百分位点

Chi-squared tables conventionally list upper-tail percentage points because tests for association and goodness of fit reject only for large values of the test statistic. A lower-tail percentage point would satisfy P(X² < value) = α.

卡方表通常列出上尾百分位点，因为关联性检验和拟合优度检验仅在检验统计量较大时拒绝。下尾百分位点则满足 P(X² < 数值) = α。

If you need a lower-tail value, use the complement: the lower 5% point of χ²ν equals the upper 95% point of χ²ν. Because tables rarely give upper 95% directly, you may need software or interpolation.

如果需要下尾值，则使用互补关系：χ²ν 的下 5% 点等于 χ²ν 的上 95% 点。由于表格很少直接给出上 95% 点，可能需要软件或插值。

4. Degrees of Freedom in χ² Tests | 卡方检验中的自由度

The shape and percentage points of the χ² distribution depend on the number of degrees of freedom, ν. In goodness-of-fit tests, ν is usually numberOfCells – 1 – numberOfEstimatedParameters.

卡方分布的形状和百分位点取决于自由度 ν。在拟合优度检验中，ν 通常等于组数 − 1 − 估计参数的个数。

In a contingency table test for independence, ν = (r – 1

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）