📚 Percentage Points of the Normal Distribution | 正态分布的百分位点
The normal distribution underpins much of statistical inference at A-Level. When you move beyond simply finding probabilities and need to determine critical values, the concept of percentage points becomes essential. These points, also called quantiles or inverse normal values, allow you to set up rejection regions for hypothesis tests and construct confidence intervals. In this article, we will break down what percentage points are, how to read Edexcel's provided tables, and how to apply them accurately in both one-tailed and two-tailed scenarios.
正态分布是A-Level统计推断的基础。当你不再局限于求概率，而需要确定临界值时，百分位点的概念就变得至关重要。这些点也被称为分位数或逆正态值，它们让你能够为假设检验设定拒绝域并构造置信区间。本文将详细解析百分位点是什么、如何读Edexcel提供的表格，以及如何在单尾和双尾情形中准确应用它们。
1. Introduction to the Normal Distribution | 正态分布简介
measurements, such as heights or test scores, follow an approximately normal pattern. In A-Level mathematics, we frequently standardise a normal variable X ~ ) by computing z = (x - μ) / σ, transforming it into a standard normal variable Z ~ N(0, 1). This standardisation allows us to use universal probability tables.
正态分布是一种连续的、对称的钟形分布，由均值 μ 和方差 σ² 决定。许多自然测量值，如身高或考试成绩，都近似服从正态分布。在A-Level数学中，我们经常通过计算 z = (x - μ) / σ 将正态变量 X ~ N(μ, σ²) 标准化为标准正态变量 Z ~ N(0, 1)。这种标准化使我们能够使用通用的概率表。
2. Standard Normal Distribution | 标准正态分布
z). Standard normal tables typically give values of Φ(z) for different z, letting you find the probability up to a given z-value. For instance, Φ(1) ≈ 0.8413, meaning about 84.13% e: given a probability, find the corresponding z. That is where percentage points come in.
标准正态分布的均值为0，标准差为1。其累积分布函数 (CDF) 记作 Φ(z) = P(Z ≤ z)。标准正态表通常给出不同 z 值对应的 Φ(z)，让你能够求低于某个 z 值的概率。例如，Φ(1) ≈ 0.8413，意味着约84.13%的数据在1以下。然而，许多问题需要反向操作：给定一个概率，求对应的 z 值。这就是百分位点的用武之地。
3. Understanding Percentage Points | 理解百分位点
il equals a specified probability p. In symbols, P(Z > z) = p. For example, the upper 5% ou compare a test statistic against a threshold, and in constructing confidence intervals.
标准正态分布的百分位点（或称临界值）是指使得右尾面积等于给定概率 p 的 z 值。用符号表示为 P(Z > z) = p。例如，上侧5%的点满足 P(Z > z) = 0.05。这些值在假设检验（比较检验统计量与阈值）和构造置信区间中不可或缺。
Unlike the forward-use cumulative table, which gives Φ(z) for an input z, the percentage points table gives z for an input tail probability. This 'inverse' operation is essential because we often know the acceptable error probability (α) and need to find the boundary of the critical region.
与正向使用的累积表给出输入 z 对应的 Φ(z) 不同，百分位点表给出输入尾部概率对应的 z 值。这种“逆向”操作至关重要，因为我们经常已知可接受的错误概率 (α)，然后需要找出拒绝域的边界。
4. Notation for Percentage Points | 百分位点的记号
There are several common notations. The upper p point is often written as zp or zα, where α is the significance level. In Edexcel specifications, you will encounter z0.05 for the right-tail 5% point. For a two-tailed test, the critical values are ±zα/2, such as ±z0.025. Another notation is the inverse CDF: z = Φ−1(1 − p), giving the value such that P(Z ≤ z) = 1 − p. For example, z0.05 = Φ−1(0.95).
有几种常见的记号。上侧 p 点常写作 zp 或 zα，其中 α 是显著性水平。在 Edexcel 考纲中，你会遇到 z0.05 表示右尾5%的点。对于双尾检验，临界值为 ±zα/2，例如 ±z0.025。另一种记法是逆 CDF：z = Φ−1(1 − p)，给出满足 P(Z ≤ z) = 1 − p 的值。例如，z0.05 = Φ−1(0.95)。
5. Using the Normal Distribution Table | 使用正态分布表
of p, typically from 0.25 down to 0.0005. Crucially, the header states that the table gives the value of z such that the probability in the upper tail is p. Always take a moment to confirm you are using the correct table: the cumulative table gives left-tail areas, whereas the percentage points table works directly with right-tail probabilities.
Edexcel 提供专门的正态分布百分位点表。该表列出满足 P(Z > z) = p 的一系列 z 值，p 的范围通常从0.25到0.0005。关键在于，表头说明该表给出的是上尾概率为 p 的 z 值。一定要花时间确认自己用的是正确的表：累积表给出左尾面积，而百分位点表直接处理右尾概率。
Below is a condensed version showing the most commonly used percentage points:
以下是常用百分位点的简明版本：
| p (Upper-tail probability) | z |
|---|---|
| 0.25 | 0.674 |
| 0.20 | 0.842 |
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