📚 Percentage Points of Student’s t-Distribution | 学生 t 分布百分位点
In A-Level Statistics, Student’s t-distribution is essential when the population standard deviation is unknown and the sample size is small. This article explains percentage points, how to read t-tables, and how to use critical values in hypothesis tests and confidence intervals.
在 A-Level 统计学中，当总体标准差未知且样本量较小时，学生 t 分布非常重要。本文解释百分位点的含义、如何查阅 t 表，以及如何在假设检验和置信区间中使用临界值。
1. What is Student’s t-Distribution? | 什么是学生 t 分布？
Student’s t-distribution arises when we standardise a sample mean using the sample standard deviation instead of the population standard deviation. If X₁, X₂, …, Xₙ are independent normal observations with mean μ, then the statistic below follows a t-distribution with n − 1 degrees of freedom.
学生 t 分布产生于使用样本标准差而非总体标准差对样本均值进行标准化时。如果 X₁, X₂, …, Xₙ 是来自均值为 μ 的正态总体的一组独立观测值，则下面的统计量服从自由度为 n − 1 的 t 分布。
T = (x̄ − μ) / (s / √n) ∼ t(n − 1)
Here x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and μ is the population mean being tested.
其中 x̄ 是样本均值，s 是样本标准差，n 是样本量，μ 是需要检验的总体均值。
2. Degrees of Freedom and Shape | 自由度与分布形状
The shape of the t-distribution is controlled by its degrees of freedom, usually written as ν. For a single sample, ν = n − 1. When ν is small, the distribution has heavier tails than the normal distribution, meaning extreme values are more likely.
t 分布的形状由其自由度控制，自由度通常写作 ν。对于单样本情形，ν = n − 1。当 ν 很小时，分布比正态分布具有更厚的尾部，这意味着极端值更容易出现。
ν = n − 1
As the degrees of freedom increase, the t-distribution approaches the standard normal distribution N(0, 1). This is why t critical values get closer to z critical values as n grows.
随着自由度增大，t 分布逐渐接近标准正态分布 N(0, 1)。这就是为什么随着 n 增大，t 临界值越来越接近 z 临界值。
3. What Are Percentage Points? | 什么是百分位点？
A percentage point of the t-distribution is a critical value that cuts off a given upper-tail probability. For example, the 5% percentage point, written t(ν, 0.05), is the value such that 5% of the distribution lies to its right.
t 分布的百分位点是一个临界值，它切分出给定的上尾概率。例如，5% 百分位点写作 t(ν, 0.05)，表示分布右侧有 5% 的面积位于该值之外。
P(T > t(ν, p)) = p
In this notation, p is the upper-tail probability written as a decimal. The value t(ν, p) is called the upper-tail percentage point or critical value.
在这个记号中，p 是写成小数的上尾概率。数值 t(ν, p) 称为上尾百分位点或临界值。
4. One-Tailed and Two-Tailed Usage | 单尾与双尾用法
Percentage points are used differently for one-tailed and two-tailed tests. In a one-tailed test at significance level α, you compare the test statistic with t(ν, α). In a two-tailed test, you compare with t(ν, α/2) because the significance is split equally between the two tails.
百分位点在单尾和双尾检验中的使用方式不同。在显著性水平为 α 的单尾检验中，将检验统计量与 t(ν, α) 比较。在双尾检验中，则与 t(ν, α/2) 比较，因为显著性水平被均分到两个尾部。
Two-tailed: P(|T| > t(ν, α/2)) = α
For example, a two-tailed 5% test with ν = 10 uses t(10, 0.025) = 2.228, not t(10, 0.05).
例如，自由度为 10 的双尾 5% 检验使用 t(10, 0.025) = 2.228，而不是 t(10, 0.05)。
5. How to Read a t-Table | 如何查阅 t 表
A standard t-table lists degrees of freedom in the left column and upper-tail probabilities across the top row. To find a critical value, locate the row for ν and the column for the required p, then read off the intersection.
标准 t 表左侧列出行自由度，顶部行列出上尾概率。要查找临界值，先找到 ν 所在行和所需 p 所在列，然后读取交点处的数值。
Most tables only give upper-tail values, because lower-tail values are found by symmetry. For example, the lower 5% point is the negative of the upper 5% point.
大多数 t 表只给出上尾值，因为下尾值可以通过对称性得到。例如，下 5% 点是上 5% 点的相反数。
6. Table of Percentage Points | 百分位点表
The table below gives selected upper-tail percentage points t(ν, p) for Student’s t-distribution. The row with ν = ∞ corresponds to the standard normal distribution.
下表给出学生 t 分布的选定上尾百分位点 t(ν, p)。ν = ∞ 行对应标准正态分布。
| ν | p = 0.10 | p = 0.05 | p = 0.025 | p = 0.01 | p = 0.005 | p = 0.001 |
|---|---|---|---|---|---|---|
| 1 | 3.078 | 6.314 | 12.706 | 31.821 | 63.657 | 318.309 |
| 2 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 | 22.327 |
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