📚 The Normal Distribution | 正态分布
The normal distribution, often called the Gaussian distribution, is a cornerstone of probability and statistics. It models a continuous random variable whose probability density forms the familiar bell-shaped curve. Many natural phenomena – heights, test scores, measurement errors – follow an approximately normal pattern, making it an essential topic for IB Mathematics.
正态分布常被称为高斯分布，是概率与统计学的基石。它描述的是一个连续随机变量，其概率密度函数呈现出熟悉的钟形曲线。许多自然现象——如身高、考试成绩、测量误差——都近似服从正态分布，因此成为 IB 数学中不可或缺的课题。
1. What is a Normal Distribution? | 什么是正态分布？
A normal distribution is a continuous probability distribution defined by two parameters: the mean μ and the variance σ² (or standard deviation σ). The random variable X can take any real value, and the total area under its probability density curve equals 1. The notation X ~ N(μ, σ²) indicates that X follows a normal distribution with mean μ and variance σ².
正态分布是由两个参数——均值 μ 和方差 σ²（或标准差 σ）决定的一种连续概率分布。随机变量 X 可以取任意实数值，其概率密度曲线下的总面积等于 1。记号 X ~ N(μ, σ²) 表示 X 服从均值为 μ、方差为 σ² 的正态分布。
2. The Normal Probability Density Function | 正态概率密度函数
The probability density function (PDF) of a normal random variable X is given by:
正态随机变量 X 的概率密度函数（PDF）为：
f(x) = (1/(σ√2π)) e^[-(x - μ)² / (2σ²)]
Here, μ is the centre of symmetry and σ controls the spread. A larger σ produces a flatter, wider curve; a smaller σ yields a taller, narrower curve. The PDF itself does not give probabilities directly – probabilities correspond to areas under the curve between two values.
其中 μ 是对称中心，σ 控制分布的离散程度。σ 越大，曲线越扁宽；σ 越小，曲线越高窄。PDF 本身并不直接给出概率——概率对应曲线下某两点之间的面积。
3. Key Properties of the Normal Curve | 正态曲线的关键性质
The normal curve has several distinctive properties: it is symmetric about the mean μ, so the mean, median and mode all coincide. The curve is bell‑shaped, asymptotic to the horizontal axis, and the total area under the curve is exactly 1. There is no simple closed‑form antiderivative for the PDF, so probabilities are found using tables or technology.
正态曲线具有几个独特性质：它关于均值 μ 对称，因此均值、中位数和众数三者重合。曲线呈钟形，以水平轴为渐近线，曲线下的总面积恰好为 1。由于概率密度函数没有简单的闭式原函数，概率需借助表格或技术工具来求得。
Symmetry: P(X < μ - a) = P(X > μ + a) for any a > 0.
对称性：对任意 a > 0 有 P(X < μ - a) = P(X > μ + a)。
The curve never touches the horizontal axis but gets arbitrarily close.
曲线永远不触及横轴，但可无限接近。
4. The Standard Normal Distribution | 标准正态分布
e standard normal distribution is a special case with μ = 0 and σ = 1. A random variable Z ~ N(0, 1) is called a standard normal variable. Its probability density simplifies to:
标准正态分布是 μ = 0、σ = 1 的特殊情形。服从 Z ~ N(0, 1) 的随机变量称为标准正态变量。其概率密度简化为：
φ(z) = (1/√2π) e-z²/2
z). Standard normal tables provide Φ(z) for z ≥ 0, and symmetry handles negative z-values.
Z 的累积分布函数记作 Φ(z)，表示概率 P(Z ≤ z)。标准正态表通常给出 z ≥ 0 时的 Φ(z) 值，负 z 值可利用对称性处理。
5. Standardisation (Z‑scores) | 标准化（Z 分数）
Any normal variable X ~ N(μ, σ²) can be transformed into a standard normal Z by subtracting μ and dividing by σ:
任意正态变量 X ~ N(μ, σ²) 均可通过减去 μ 再除以 σ 转化为标准正态变量 Z：
Z = (X - μ) / σ
This process, called standardisation, shifts the distribution so that the mean becomes 0 and rescales it so that the variance becomes 1. The transformed value z is the number of standard deviations that x lies above or below μ.
这一过程称为标准化，它平移分布使均值变为 0，并重新放缩使方差变为 1。转换后的 z 值表示原始数据 x 偏离均值 μ 的标准差个数。
For example, if X ~ 15, then z = (115 - 100)/15 = 1, meaning the score is 1 standard deviation above the mean.
例如，若 X ~ N(100, 15²) 且 x = 115，则 z = (115 - 100)/15 = 1，意味着该分数高于均值 1 个标准差。
6. Using the Normal Distribution Table | 使用正态分布表
he symmetry property: Φ(-z) = 1 - Φ(z). To find P(a < Z < b), calculate Φ(b) - Φ(a).
标准正态表通常列出正 z 值对应的 Φ(z) = P(Z ≤ z)。对于负 z 利用对称性：Φ(-z) = 1 - Φ(z)。求 P(a < Z < b) 时计算 Φ(b) - Φ(a)。
| z | Φ(z) |
|---|---|
| 0.0 | 0.5000 |
| 1.0 | 0.8413 |
| 2.0 | 0.9772 |
| 3.0 | 0.9987 |
Always sketch the bell curve, shade the required region, and check whether you need a left‑tail, right‑tail, or interval probability before reading values from the table.
务必先画出钟形曲线，标示并涂阴影于所求区域，确认是需要左尾、右尾还是区间概率，再查表取值。
7. Calculating Probabilities for X ~ N(μ, σ²) | 计算一般正态分布的概率
To find probabilities for a general normal distribution, follow these steps:
计算一般正态分布的概率，遵循以下步骤：
State the distribution of X, e.g., X ~ N(150, 25²).
写出 X 的分布，例如 X ~ N(150, 25²)。
Standardise the bounds: z₁ = (x₁ - μ)/σ, z₂ = (x₂ - μ)/σ.
标准化界限：z₁ = (x₁ - μ)/σ，z₂ = (x₂ - μ)/σ。
Use the standard normal table to find Φ(z₁) and Φ(z₂).
利用标准正态表求出 Φ(z₁) 和 Φ(z₂)。
Compute the required probability: P(x₁ < X < x₂) = Φ(z₂) - Φ(z₁).
计算所求概率：P(x₁ < X < x₂) = Φ(z₂) - Φ(z₁)。
For right‑tail probabilities, use 1 - Φ(z).
右尾概率用 1 - Φ(z)。
cm, standardise: z = (185-170)/10 = 1.5. P(X > 185) = 1 - Φ(1.5) ≈ 1 - 0.9332 = 0.0668.
假设学生身高服从 N(170, 10²)。求高于 185 cm 的比例，标准化：z = (185-170)/10 = 1.5。P(X > 185) = 1 - Φ(1.5) ≈ 1 - 0.9332 = 0.0668。
8. Inverse Normal Calculations | 反查正态分布
he corresponding data value. For example, find the score x such that P(X ≤ x) = 0.95 for X ~ N(500, 100²).
有时我们知道概率，需要求出对应的数据值。例如，对 X ~ N(500, 100²)，求解满足 P(X ≤ x) = 0.95 的分数 x。
The procedure: first find the z‑ ≈ 1.645). Then solve z = (x - μ)/σ for x: x = μ + zσ. Here x = 500 + 1.645 × 100 = 664.5.
步骤：先由表查得满足 Φ(z) = 0.95 的 z 值（z ≈ 1.645），然后根据 z = (x - μ)/σ 求解 x：x = μ + zσ。此处 x = 500 + 1.645 × 100 = 664.5。
Your calculator’s inverse normal function can perform this directly, but the table method builds a solid conceptual understanding essential for IB exams.
计算器的逆正态功能可直接实现，但查表法能建立扎实的概念理解，这对应付 IB 考试至关重要。
9. The Empirical Rule (68–95–99.7) | 经验法则（68-95-99.7）
For any normal distribution, approximate probabilities can be remembered through the empirical rule:
对于任意正态分布，可以通过经验法则记住近似概率：
About 68% of data lie within 1 standard deviation of the mean: P(μ - σ < X < μ + σ) ≈ 0.6827.
约 68% 的数据落在均值左右 1 个标准差内：P(μ - σ < X < μ + σ) ≈ 0.6827。
About 95% lie within 2 standard deviations: P(μ - 2σ < X < μ + 2σ) ≈ 0.9545.
约 95% 落在 2 个标准差内：P(μ - 2σ < X < μ + 2σ) ≈ 0.9545。
About 99.7% lie within 3 standard deviations: P(μ - 3σ < X < μ + 3σ) ≈ 0.9973.
约 99.7% 落在 3 个标准差内：P(μ - 3σ < X < μ + 3σ) ≈ 0.9973。
This rule provides a quick sanity check and is heavily tested in IB problems, especially in modelling contexts where parameters must be estimated.
该法则提供了快速校验方法，在 IB 试题中频繁出现，尤其当需要估计参数时。
10. Applications and Tips | 应用与技巧
Normal distributions appear in countless real‑world contexts: IQ scores (often N(100, 15²)), manufacturing tolerances, and measurement errors. In IB questions, you may need to combine standardisation with solving for unknown μ or σ, requiring simultaneous equations.
正态分布广泛存在于现实场景：IQ 分数（通常为 N(100, 15²)）、工业公差、测量误差等。在 IB 题目中，可能需要将标准化与求解未知 μ 或 σ 结合，列方程组解决。
Common pitfalls to avoid: confusing variance σ² with standard deviation σ, forgetting to square σ when writing N(μ, σ²), and misreading the table by taking right‑tail probabilities when left‑tail values are given. Always define your variable, draw the curve, and check whether a continuity correction is needed when approximating a discrete distribution.
常见错误需避免：将方差 σ² 与标准差 σ 混淆，书写 N(μ, σ²) 时忘记对 σ 平方，误读表格把左尾值当作右尾概率。始终明确变量定义、画出曲线、并在近似离散分布时检查是否需要进行连续校正。
Mastering the normal distribution opens the door to statistical inference, hypothesis testing, and confidence intervals later in the IB course.
掌握正态分布为 IB 课程后续的统计推断、假设检验和置信区间打开了大门。
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