📚 Finding μ and σ in the Normal Distribution | 正态分布中求均值 μ 与标准差 σ
In Edexcel A-Level Mathematics, many normal distribution questions are worked in reverse: instead of finding a probability from a known mean and standard deviation, you are given a probability and asked to find the unknown parameter μ or σ. These problems are sometimes called inverse normal problems because you start with the area under the curve and work back to the parameter.
在 Edexcel A-Level 数学中，许多正态分布题目都是反向求解：不是已知均值和标准差去求概率，而是给定某个概率，要求你找出未知参数 μ 或 σ。这类问题有时被称为逆正态问题，因为你从曲线下的面积出发，反过来求参数。
be able to read z-values from tables or a calculator accurately. This article explains the main methods and common exam techniques.
要掌握这类题目，你必须对标准正态变换 z = (x − μ) / σ 非常熟悉，并且能够准确地从表格或计算器中读取 z 值。本文讲解主要方法和常见考试技巧。
1. The Role of μ and σ in a Normal Distribution | 正态分布中 μ 与 σ 的作用
A normal random variable X is written as X ~ N(μ, σ²), where μ is the population mean and σ is the standard deviation. The parameter μ determines the centre of the bell-shaped curve, while σ controls how spread out the curve is. A larger σ makes the curve flatter and wider; a smaller σ makes it taller and narrower.
正态随机变量 X 通常写作 X ~ N(μ, σ²)，其中 μ 是总体均值，σ 是标准差。参数 μ 决定钟形曲线的中心位置，而 σ 决定曲线的分散程度。σ 越大，曲线越扁平、越宽；σ 越小，曲线越高、越窄。
X ~ N(μ, σ²)
up an equation using the standard normal distribution and solve for the missing parameter.
在许多考试题中，μ 或 σ 或两者都是未知的。题目会给出诸如 P(X < a) = p 或 P(X > b) = q 的概率条件。你需要利用标准正态分布建立方程，并解出缺失的参数。
2. Standardising: The Z-Transformation | 标准化：Z 变换
For any normal variable X ~ rd normal tables give Φ(z) for positive z-values, and symmetry is used for negative z-values.
对于任意正态变量 X ~ N(μ, σ²)，标准化变量 Z = (X − μ) / σ 服从标准正态分布 N(0, 1)。Z 的累积分布函数通常记作 Φ(z) = P(Z ≤ z)。标准正态表给出正 z 值对应的 Φ(z)，负 z 值则利用对称性求得。
z = (x − μ) / σ
ains μ and σ. If one parameter is known, you can solve directly; if both are unknown, you need two separate probability statements to form two equations.
如果已知 P(X < x) = p，那么 P(Z < (x − μ) / σ) = p，因此标准化的值就是逆正态值：(x − μ) / σ = Φ⁻¹(p)。这个方程包含 μ 和 σ。如果其中一个参数已知，你可以直接求解；如果两个都未知，就需要两个独立的概率条件来建立两个方程。
3. Finding μ When σ Is Known | 已知 σ 求 μ
This is the most common single-unknown case. For example, suppose X ~ value that gives Φ(z) = 0.3085 is z = −0.5000, because P(Z < −0.5) = 0.3085. Therefore:
这是最常见的单未知数情形。例如，假设 X ~ N(μ, 5²)，且 P(X < 20) = 0.3085。首先写出标准化方程：z = (20 − μ) / 5。查标准正态表，满足 Φ(z) = 0.3085 的 z 值为 z = −0.5000，因为 P(Z < −0.5) = 0.3085。因此：
(20 − μ) / 5 = −0.5
normal curve and mark the given probability. This helps you check the sign of z: if the probability is below 0.5, z must be negative; if it is above 0.5, z must be positive.
两边同时乘以 5，得 20 − μ = −2.5，因此 μ = 22.5。始终画一个正态曲线草图并标出已知概率。这有助于你检查 z 的符号：如果概率小于 0.5，z 必须为负；如果概率大于 0.5，z 必须为正。
4. Finding σ When μ Is Known | 已知 μ 求 σ
Suppose X ~ e z-value corresponding to Φ(z) = 0.9772 is z = 2.00. Now use the standardisation formula:
假设 X ~ N(30, σ²)，且 P(X > 40) = 0.0228。标准正态表通常给出左尾概率，因此将该条件改写为 P(X < 40) = 1 − 0.0228 = 0.9772。查表得 Φ(z) = 0.9772 对应的 z 值为 z = 2.00。现在使用标准化公式：
(40 − 30) / σ = 2.00
or is using the right-tail probability directly as if it were a left-tail value. Always convert right-tail statements to left-tail form before reading Φ⁻¹ from tables or a calculator.
于是 10 / σ = 2，所以 σ = 5。一个非常常见的错误是把右尾概率直接当作左尾值来使用。在从表格或计算器读取 Φ⁻¹ 之前，务必先把右尾条件转换为左尾形式。
5. Finding Both μ and σ Using Two Conditions | 利用两个条件同时求 μ 和 σ
When both μ and σ are unknown, you need two independent probability statements. Each statement produces one z-equation. The two equations can then be solved simultaneously for μ and σ. This is a very popular A-Level exam style because it tests standardisation, inverse normal reading, and algebraic manipulation in one question.
当 μ 和 σ 都未知时，你需要两个独立的概率条件。每个条件产生一个 z 方程。然后可以联立求解 μ 和 σ。这是 A-Level 考试中非常常见的题型，因为它在一道题中同时考查标准化、逆正态查值以及代数运算。
Example: X ~ N(μ, σ²). Given P(X < 15) = 0.1587 and P(X > 25) = 0.0668, find μ and σ.
例题：已知 X ~ N(μ, σ²)，且 P(X < 15) = 0.1587，P(X > 25) = 0.0668，求 μ 和 σ。
25) = 0.9332, which gives z₂ = 1.50. Therefore (25 − μ) / σ = 1.5. Substitute μ = 15 + σ:
第一个条件：P(X < 15) = 0.1587 给出 z₁ = −1.00，所以 (15 − μ) / σ = −1。移项得 μ = 15 + σ。第二个条件：P(X > 25) = 0.0668，所以 P(X < 25) = 0.9332，查表得 z₂ = 1.50。因此 (25 − μ) / σ = 1.5。代入 μ = 15 + σ：
(25 − (15 + σ)) / σ = 1.5
1.5σ, giving 10 = 2.5σ and σ = 4. Then μ = 15 + 4 = 19. The final answer is μ = 19, σ = 4.
化简得 (10 − σ) / σ = 1.5，所以 10 − σ = 1.5σ，得到 10 = 2.5σ，因此 σ = 4。然后 μ = 15 + 4 = 19。最终答案为 μ = 19，σ = 4。
6. Using Symmetry to Read z-Values | 利用对称性读取 z 值
ry whenever the given probability is less than 0.5 or when the table only provides positive z-values.
标准正态曲线关于 0 对称。这给出两个非常有用的恒等式：Φ(−z) = 1 − Φ(z) 以及 P(Z > z) = 1 − Φ(z)。例如，若 P(X < a) = 0.95，则 z ≈ 1.6449；若 P(X < b) = 0.05，则 z ≈ −1.6449。当给定概率小于 0.5 或表格只提供正 z 值时，要利用对称性。
Several z-values appear so often in Edexcel exams that they are worth memorising:
在 Edexcel 考试中，有几个 z 值经常出现，值得记住：
| Left-tail probability | z-value |
|---|---|
| 0.975 | 1.960 |
| 0.950 | 1.6449 |
| 0.990 | 2.326 |
| 0.025 | −1.960 |
| 0.050 | −1.6449 |
7. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Many marks are lost in this topic through small slips. The following list summarises the mistakes seen most often, together with the correct approach.
在这个主题中，很多失分都来自小失误。下面的列表总结了最常见的错误以及正确的做法。
Mixing up μ and σ in the z formula. − mean) / standard deviation, never z = (mean − value) / σ or z = (value − μ) / variance.
在 z 公式中混淆 μ 和 σ。始终写作 z = (值 − 均值) / 标准差，切勿写成 z = (均值 − 值) / σ 或 z = (值 − μ) / 方差。
Reading the wrong tail. If P(X > a) = p, convert it to P(X < a) = 1 − p before finding the z-value.
读取错误的尾部。若 P(X > a) = p，应转换为 P(X < a) = 1 − p，再查 z 值。
Forgetting that σ² is the variance. If you solve for σ², take the square root to report σ unless the question explicitly asks for variance.
忘记 σ² 是方差。如果你解出的是 σ²，除非题目明确要求求方差，否则要开平方得到 σ。
Sign errors when z is negative. If x is below the mean, z must be negative. Check your sign by looking at the left-tail probability.
z 为负时符号出错。如果 x 低于均值，z 必须为负。通过查看左尾概率来检查符号。
Trying to find two unknowns with one equation. When both μ and σ are unknown, you must have two independent probability statements.
试图用一个方程求两个未知数。当 μ 和 σ 都未知时，必须有两个独立的概率条件。
8. Worked Example: Unknown Mean | 例题：未知均值
Question: X ~ N(μ, 8²). Given that P(X < 70) = 0.1056, find μ.
题目：X ~ N(μ, 8²)。已知 P(X < 70) = 0.1056，求 μ。
56. From tables, Φ(−1.25) = 0.1056, so z = −1.25. Then write the standardisation equation:
首先，查左尾概率为 0.1056 的标准正态 z 值。由表可知 Φ(−1.25) = 0.1056，所以 z = −1.25。然后写出标准化方程：
(70 − μ) / 8 = −1.25
e mean is 80. Notice that 70 is below the mean, which is consistent with the negative z-value.
两边同时乘以 8：70 − μ = −10。移项得 μ = 80。因此均值为 80。注意 70 低于均值，这与负的 z 值是一致的。
9. Worked Example: Unknown Standard Deviation | 例题：未知标准差
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