📚 IB Mathematics: The Normal Distribution — Definition and Properties | IB数学：正态分布定义与特征
The normal distribution is one of the most important probability distributions in IB Mathematics: Analysis and Approaches (AA) and Applications and Interpretation (AI). Its characteristic bell-shaped curve appears throughout statistics, from natural phenomena to measurement errors. This article provides a complete, syllabus-aligned review of its definition, properties, and key formulas.
正态分布是 IB 数学（分析与方法 AA 以及应用与解释 AI）中最重要的概率分布之一。它那标志性的钟形曲线广泛出现在从自然现象到测量误差的各类统计场景中。本文将完整、紧扣考纲地梳理其定义、特征与核心公式。
1. Definition of the Normal Distribution | 正态分布的定义
A continuous random variable X is said to follow a normal distribution with mean μ and standard deviation σ if its probability density function (pdf) is given by:
f(x) = (1 / (σ√2π)) · e^(−(x−μ)²/(2σ²))
where x is any real number, μ is the mean (location parameter), and σ is the standard deviation (spread parameter), with σ > 0. This is denoted as X ~ N(μ, σ²).
设连续型随机变量 X 服从均值为 μ、标准差为 σ 的正态分布，其概率密度函数（pdf）为：
f(x) = (1 / (σ√2π)) · e^(−(x−μ)²/(2σ²))
其中 x 为任意实数，μ 为均值（位置参数），σ 为标准差（尺度参数），且 σ > 0。记为 X ~ N(μ, σ²)。
2. The Bell-Shaped Curve | 钟形曲线
the vertical line x = μ, reaching its maximum at x = μ. The maximum height is f(μ) = 1/(σ√2π). The curve approaches the x-axis but never touches it, forming the classical 'bell' shape.
正态分布曲线关于 x = μ 对称，在 x = μ 处达到最大值，最大高度为 f(μ) = 1/(σ√2π)。曲线向 x 轴渐近，但永不相交，形成经典的“钟形”轮廓。
Symmetry / 对称性: f(μ + a) = f(μ − a) for any a. / 对任意 a，有 f(μ + a) = f(μ − a)。
Unimodal / 单峰: The curve has exactly one peak. / 曲线只有一个峰值。
Asymptotic / 渐近性: The tails extend indefinitely in both directions without touching the horizontal axis. / 两尾向无穷远处延伸，且不触及水平轴。
3. Parameters: Mean and Standard Deviation | 参数：均值与标准差
The mean μ determines the centre of the distribution. Changing μ shifts the entire curve left or right without changing its shape. The standard deviation σ determines the spread: larger σ gives a flatter, wider curve; smaller σ gives a taller, narrower curve.
均值 μ 决定分布的中心位置。改变 μ 会使整条曲线左右平移而不改变形状。标准差 σ 决定分布的离散程度：σ 越大，曲线越扁平、越宽；σ 越小，曲线越高、越窄。
| Parameter / 参数 | Effect / 影响 |
| μ increases / μ 增大 | Curve shifts right / 曲线右移 |
| σ increases / σ 增大 | Curve becomes wider and flatter / 曲线变宽、变平 |
| σ decreases / σ 减小 | Curve becomes narrower and taller / 曲线变窄、变高 |
4. Total Area Under the Curve | 曲线下总面积
For any continuous probability distribution, the total area under the pdf curve equals 1. For the normal distribution, this means:
∫₋∞⁺∞ f(x) dx = 1
Consequently, the probability that X lies between two values a and b is the area under the curve from a to b:
P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx
对于任何连续型概率分布，曲线下的总面积为 1。对于正态分布，这意味着：
∫₋∞⁺∞ f(x) dx = 1
因此，随机变量 X 落在 a 和 b 之间的概率等于曲线下从 a 到 b 的面积：
P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx
5. The 68-95-99.7 Rule | 68-95-99.7 法则
One of the most useful empirical properties of the normal distribution is the approximate proportion of data within k standard deviations of the mean:
正态分布最实用的经验法则之一，是均值附近 k 个标准差范围内的大致数据比例：
68% of values lie within μ ± σ. / 约 68% 的数据落在 μ ± σ 之间。
95% of values lie within μ ± 2σ. / 约 95% 的数据落在 μ ± 2σ 之间。
99.7% of values lie within μ ± 3σ. / 约 99.7% 的数据落在 μ ± 3σ 之间。
This rule is also called the empirical rule. It is especially useful for quick estimates and for checking whether a dataset is approximately normal.
该法则也称为经验法则，特别适用于快速估算，以及检验数据集是否近似服从正态分布。
6. Standard Normal Distribution | 标准正态分布
n μ = 0 and standard deviation σ = 1 is called the standard normal distribution, denoted Z ~ N(0, 1). Its pdf simplifies to:
φ(z) = (1/√2π) · e−z²/2
均值为 μ = 0、标准差为 σ = 1 的正态分布称为标准正态分布，记为 Z ~ N(0, 1)。其概率密度函数简化为：
φ(z) = (1/√2π) · e−z²/2
Any normal variable X can be standardized using the z-score transformation:
z = (x − μ) / σ
This transformation converts X ~ N(μ, σ²) into Z ~ N(0, 1), allowing probabilities to be looked up in standard normal tables.
任何正态变量 X 都可以通过 z 分数变换进行标准化：
z = (x − μ) / σ
该变换将 X ~ N(μ, σ²) 转化为 Z ~ N(0, 1)，从而可以在标准正态表中查找概率。
7. Key Properties of the Standard Normal Distribution | 标准正态分布的重要性质
Mean = 0, Median = 0, Mode = 0: all three measures of central tendency coincide. / 均值、中位数、众数均为 0，三者重合。
Variance = 1: the standard deviation equals 1. / 方差为 1，标准差等于 1。
Symmetric about z = 0: φ(−z) = φ(z). / 关于 z = 0 对称：φ(−z) = φ(z)。
Tail behaviour: P(Z < −a) = P(Z > a) for any a > 0. / 尾部行为：对任意 a > 0，有 P(Z < −a) = P(Z > a)。
Cumulative function: Φ(z) = P(Z ≤ z), with Φ(−z) = 1 − Φ(z). / 累积分布函数：Φ(z) = P(Z ≤ z)，且 Φ(−z) = 1 − Φ(z)。
8. Probability Calculations | 概率计算
To find probabilities for a general normal distribution X ~ N(μ, σ²), convert to Z and use the standard normal table or a GDC:
要对一般正态分布 X ~ N(μ, σ²) 求概率，可将其转化为 Z 并使用标准正态表或图形计算器（GDC）：
P(X ≤ x) = Φ((x − μ) / σ)
For inequalities involving intervals, use the difference:
P(a ≤ X ≤ b) = Φ((b − μ)/σ) − Φ((a − μ)/σ)
对于涉及区间的概率，使用差值计算：
P(a ≤ X ≤ b) = Φ((b − μ)/σ) − Φ((a − μ)/σ)
In IB exams, a GDC (e.g. TI-Nspire or Casio fx-CG) can directly compute normal probabilities, but understanding the z-score method is essential for non-calculator questions and for conceptual clarity.
在 IB 考试中，可以使用图形计算器（如 TI-Nspire 或 Casio fx-CG）直接计算正态分布概率，但理解 z 分数方法对于无计算器的题目以及概念的清晰掌握至关重要。
9. Inverse Normal Calculations | 逆正态计算
Sometimes we are given a probability and need to find the corresponding value of x. This is called an inverse normal calculation. For a given cumulative probability p, we find the value zp such that P(Z ≤ zp) = p, then convert back:
x = μ + zp · σ
有时我们已知概率，需要反求对应的 x 值，这称为逆正态计算。对于给定的累积概率 p，我们找到满足 P(Z ≤ zp) = p 的值 zp，再转换回去：
x = μ + zp · σ
For example, to find the 90th percentile of X ~ , first find z₀.₉₀ ≈ 1.2816, then x = 50 + 1.2816 × 10 = 62.816. Many GDCs have a built-in 'invNorm' function for this purpose.
例如，要求 X ~ N(50, 10²) 的 90 百分位数，先找到 z₀.₉₀ ≈ 1.2816，则 x = 50 + 1.2816 × 10 = 62.816。许多图形计算器都内置了“invNorm”函数来处理这类问题。
10. Normal Approximation to the Binomial Distribution | 正态近似二项分布
ion is good when np ≥ 5 and n(1 − p) ≥ 5 (or np ≥ 10 and n(1 − p) ≥ 10 in some textbooks).
当 n 较大时，二项分布 B(n, p) 可以用正态分布近似，其均值 μ = np，方差 σ² = np(1 − p)。常用的经验法则是：当 np ≥ 5 且 n(1 − p) ≥ 5 时（部分教材要求 np ≥ 10 且 n(1 − p) ≥ 10），近似效果良好。
For continuity correction, adjust the discrete binomial endpoints by 0.5 before using normal probabilities:
P(X ≤ k) ≈ P(Z ≤ (k + 0.5 − np) / √(np(1−p)))
使用正态近似时需要进行连续性修正，将离散二项分布的端点调整 0.5：
P(X ≤ k) ≈ P(Z ≤ (k + 0.5 − np) / √(np(1−p)))
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Forgetting to standardise: Always convert to Z before using standard normal tables unless your GDC handles it directly. / 忘记标准化：使用标准正态表前务必转换为 Z，除非计算器直接处理。
Misreading symbols: σ² is variance, σ is standard deviation. The N(μ, σ²) notation often causes errors. / 符号误读：σ² 是方差，σ 是标准差。N(μ, σ²) 的记号经常引起错误。
Ignoring continuity correction: In binomial-to-normal approximation, always add or subtract 0.5 as appropriate. / 忽略连续性修正：在二项分布的正态近似中，务必根据情况加减 0.5。
Using wrong tail: For P(X > a), write 1 − P(X ≤ a), not P(X ≤ a) directly. / 用错尾部：求 P(X > a) 时，用 1 − P(X ≤ a)，而不是直接使用 P(X ≤ a)。
Rounding too early: Keep at least 3 significant figures during intermediate steps to avoid losing marks in the final answer. / 过早四舍五入：中间步骤至少保留 3 位有效数字，以免影响最终答案的得分。
12. Summary and Real-World Relevance | 总结与现实意义
The normal distribution is fully characterised by its mean and standard deviation. Its symmetry, asymptotic tails, and the 68-95-99.7 rule provide a robust framework for statistical inference. Understanding how to standardise, calculate probabilities, and perform inverse normal calculations is essential for IB Paper 2 and the IA (for AI students in particular).
正态分布完全由其均值和标准差决定。其对称性、渐近尾部和 68-95-99.7 法则为统计推断提供了坚实的框架。理解标准化、概率计算和逆正态计算，对于 IB 试卷二以及内部评估（尤其是 AI 学生）至关重要。
Real-world applications include quality control (specification limits), finance (returns modelling), biology (height and measurement distributions), and standardised testing (e.g. IQ scores). Mastering these concepts ensures a strong performance in the probability and statistics strand of IB Mathematics.
现实应用包括质量控制（规格界限）、金融（收益率建模）、生物学（身高和测量分布）以及标准化考试（如 IQ 分数）。掌握这些概念，能帮助你在 IB 数学的概率与统计部分取得优异成绩。
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