The Inverse Normal Distribution Function | 反正态分布函数

📚 The Inverse Normal Distribution Function | 反正态分布函数

The inverse normal distribution function reverses the usual normal probability calculation. Instead of starting with a value x and finding P(X ≤ x), you start with a known probability p and find the corresponding value x. In Edexcel A-Level Mathematics, this skill is tested on both the standard normal distribution Z ~ N(0, 1) and general normal distributions X ~ N(μ, σ²).

反正态分布函数将常见的正态概率计算反过来。不是从已知数值 x 出发求 P(X ≤ x)，而是从已知概率 p 出发，求出对应的 x 值。在 Edexcel A-Level 数学中，这一内容既考查标准正态分布 Z ~ N(0, 1)，也考查一般正态分布 X ~ N(μ, σ²)。

1. What Is the Inverse Normal Function? | 什么是反正态函数？

The ordinary cumulative normal function takes a value x and returns the probability P(X ≤ x). The inverse normal function does the reverse: for a given left-tail probability p, it returns the value x such that P(X ≤ x) = p. For the standard normal variable Z ~ N(0, 1), we write z = Φ⁻¹(p).

普通的正态累积函数接受一个数值 x，并返回概率 P(X ≤ x)。反正态函数则相反：对于给定的左尾概率 p，它返回满足 P(X ≤ x) = p 的 x 值。对于标准正态变量 Z ~ N(0, 1)，我们写作 z = Φ⁻¹(p)。

If P(X ≤ x) = p, then x = μ + σΦ⁻¹(p)

2. Notation and Calculator Syntax | 符号与计算器语法

In Edexcel exams you may see Φ⁻¹(p), invNorm, inverse normal, or N⁻¹. Calculator inputs are usually the left-tail area, the mean, and the standard deviation. For the standard normal distribution, enter mean 0 and standard deviation 1.

在 Edexcel 考试中，你可能会看到 Φ⁻¹(p)、invNorm、inverse normal 或 N⁻¹。计算器的输入通常是左尾面积、均值和标准差。对于标准正态分布，请输入均值 0 和标准差 1。

Standard normal: μ = 0, σ = 1 | 标准正态：μ = 0，σ = 1
General normal: μ and σ as given in the question | 一般正态：题目给出的 μ 和 σ
Area input must be a left-tail probability | 面积输入必须是左尾概率

3. From Probability to z-Score | 从概率到 z 分数

For Z ~ N(0, 1), the inverse normal value is the z-score that cuts off a given left-tail area. If p < 0.5, the z-score is negative because the cutoff lies below the mean. If p > 0.5, the z-score is positive. For example, Φ⁻¹(0.05) = −1.6449 and Φ⁻¹(0.95) = 1.6449.

对于 Z ~ N(0, 1)，反正态值就是截断给定左尾面积的 z 分数。如果 p < 0.5，z 分数为负，因为截断点位于均值以下。如果 p > 0.5，z 分数为正。例如，Φ⁻¹(0.05) = −1.6449，Φ⁻¹(0.95) = 1.6449。

z = Φ⁻¹(p)

4. From z-Score to x-Value | 从 z 分数到 x 值

After obtaining z, convert to the original variable using the standardisation equation reversed. Since z = (x − μ) ÷ σ, multiplying by σ and adding μ gives x = μ + σz. This formula connects inverse normal results on the standard scale to real data values.

得到 z 后，利用反向的标准化方程将其转换为原始变量。由于 z = (x − μ) ÷ σ，两边乘以 σ 再加上 μ，得到 x = μ + σz。这个公式将标准尺度上的反正态结果与真实数据值联系起来。

x = μ + σΦ⁻¹(p)

5. Left-Tail, Right-Tail and Central Problems | 左尾、右尾与中间区间问题

Inverse normal calculators default to a left-tail area. For a right-tail problem P(X > x) = α, replace α by 1 − α as the left-tail input. For a central interval with total probability 1 − α, use α/2 for the lower bound and 1 − α/2 for the upper bound.

反正态计算器默认使用左尾面积。对于右尾问题 P(X > x) = α，将左尾输入替换为 1 − α。对于总概率为 1 − α 的中间区间，下界使用 α/2，上界使用 1 − α/2。

Problem type \| 问题类型	Left-tail input \| 左尾输入
Left tail P(X ≤ x) = p	p
Right tail P(X > x) = p	1 − p
Central interval P(x₁ < X < x₂) = 1 − α	α/2 for x₁, 1 − α/2 for x₂

6. Standard Normal Critical Values | 标准正态临界值

Certain probabilities appear frequently in confidence intervals and hypothesis tests. You should be able to obtain these critical values quickly using the inverse normal function or recognise them from tables.

某些概率在置信区间和假设检验中经常出现。你应该能够使用反正态函数快速得到这些临界值，或者能够从表中识别它们。

p	z = Φ⁻¹(p)
0.90	1.2816
0.95	1.6449
0.975	1.9600
0.99	2.3263
0.995	2.5758

7. Worked Example: Finding a Cut-Off Value | 例题：求截断值

The weights of bags of flour are normally distributed with mean 500 g and standard deviation 20 g. Find the weight exceeded by 10% of bags.

某品牌面粉袋的重量服从正态分布，均值为 500 克，标准差为 20 克。求 10% 的面粉袋超过的重量。

First write the probability statement and convert the right-tail condition into a left-tail condition:

首先写出概率表达式，并将右尾条件转换为左尾条件：

P(X > x) = 0.10 ⇒ P(X ≤ x) = 0.90

Use the inverse normal function with μ = 500, σ = 20 and p = 0.90:

使用反正态函数，代入 μ = 500、σ = 20 和 p = 0.90：

z = Φ⁻¹(0.90) = 1.2816

x = 500 + 20 × 1.2816 = 525.63 g

So approximately 10% of bags weigh more than 525.6 g.

因此，大约 10% 的面粉袋重量超过 525.6 克。

8. Applications in Hypothesis Testing | 在假设检验中的应用

In hypothesis testing, critical values are inverse normal values. For an upper-tail 5% test, the critical z-value is Φ⁻¹(0.95) = 1.6449. For a lower-tail 5% test it is Φ⁻¹(0.05) = −1.6449. For a two-tailed 5% test, split 2.5% in each tail, giving ±1.9600.

在假设检验中，临界值就是反正态值。对于上尾 5% 检验，临界 z 值为 Φ⁻¹(0.95) = 1.6449。对于下尾 5% 检验，临界 z 值为 Φ⁻¹(0.05) = −1.6449。对于双尾 5% 检验，每个尾部各占 2.5%，得到 ±1.9600。

Two-tailed 5% critical value: z = ±Φ⁻¹(0.975) = ±1.9600

9. Common Errors to Avoid | 常见错误

Using the variance σ² instead of the standard deviation σ is one of the most common calculator mistakes. Always check whether the question gives variance or standard deviation.

将方差 σ² 当作标准差 σ 输入计算器是最常见的错误之一。务必检查题目给出的是方差还是标准差。

Another common mistake is entering a right-tail probability directly into a left-tail field. If P(X > x) = p, the left-tail input must be 1 − p.

另一个常见错误是将右尾概率直接输入到左尾字段。如果 P(X > x) = p，则左尾输入必须是 1 − p。

Also remember to convert the z-score back to the original x-value using x = μ + σz. Finding z alone does not answer a general normal question.

还要记住使用 x = μ + σz 将 z 分数转换回原始 x 值。对于一般正态分布问题，只求出 z 并不能得到最终答案。

10. Exam Strategy and Summary | 考试策略与总结

Draw a normal curve and shade the relevant probability before using the inverse normal function. Write P(X ≤ x) = p, identify μ and σ, find z = Φ⁻¹(p), then apply x = μ + σz. Check that your final value lies on the correct side of the mean.

在使用反正态函数之前，先画出正态曲线并标出相关概率。写出 P(X ≤ x) = p，确定 μ 和 σ，求出 z = Φ⁻¹(p)，然后应用 x = μ + σz。检查最终值是否位于均值的正确一侧。

The inverse normal function answers the question: given a probability, what value cuts off that probability? Master this carefully, and it becomes a reliable source of marks in Edexcel Statistics.

反正态函数回答的是：给定一个概率，哪个数值会截断这个概率？认真掌握这一内容，它将成为 Edexcel 统计考试中可靠的得分点。

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The Inverse Normal Distribution Function | 反正态分布函数