📚 Normal Distribution Key Topics for IB and Edexcel Maths | IB与Edexcel数学正态分布考点精讲
The normal distribution is one of the most important continuous probability distributions in statistics. In both IB Mathematics (Analysis & Approaches and Applications & Interpretation) and Edexcel A Level Mathematics, a solid understanding of the normal distribution is essential for success in exams. This article consolidates all key concepts, from the properties of the bell-shaped curve to Z-scores, probability calculations, inverse normal, and the normal approximation to the binomial, providing a bilingual revision guide.
正态分布是统计学中最重要的连续概率分布之一。在IB数学（分析与方法和应用与解释）以及Edexcel A Level数学中，透彻理解正态分布对考试成功至关重要。本文整合了所有核心概念，从钟形曲线的性质到Z分数、概率计算、逆正态以及二项分布的正态近似，提供了一份双语复习指南。
1. What is the Normal Distribution? | 什么是正态分布？
A continuous random variable X follows a normal distribution with mean μ and variance σ², written as X ~ N(μ, σ²). Its probability density function is a symmetric bell-shaped curve centred at μ, and the spread is determined by σ. The total area under the curve equals 1.
若连续随机变量X服从均值为μ、方差为σ²的正态分布，记作X ~ N(μ, σ²)。其概率密度函数是一条以μ为中心、对称的钟形曲线，离散程度由σ决定。曲线下的总面积等于1。
The normal distribution arises naturally in many real-world situations, such as measurement errors, heights, and test scores. It forms the foundation of statistical inference.
正态分布自然出现在许多现实情境中，如测量误差、身高和考试成绩。它构成了统计推断的基础。
2. Key Properties of the Normal Curve | 正态曲线的主要性质
The normal curve is symmetric about the mean, so the mean, median and mode coincide. The curve extends infinitely in both tails, approaching but never touching the horizontal axis (asymptotic). The location is determined by μ and the shape by σ.
正态曲线关于均值对称，因此均值、中位数和众数重合。曲线向两端无限延伸，渐近于横轴但永不触碰。位置由μ决定，形态由σ决定。
A larger σ flattens and widens the curve, while a smaller σ makes it taller and more concentrated around the mean. The area between the curve and the axis within any interval gives the probability.
较大的σ使曲线扁平、展宽，较小的σ则使其高耸并更集中于均值附近。任意区间内曲线与轴围成的面积即为概率。
3. The Empirical Rule (68–95–99.7) | 经验法则 (68-95-99.7)
For any normal distribution, approximately 68% of the data lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ. This rule provides quick probability estimates without tables.
对于任何正态分布，大约68%的数据落在μ ± σ内，95%落在μ ± 2σ内，99.7%落在μ ± 3σ内。该法则无需查表即可快速估计概率。
These percentages are derived from the standard normal distribution and are particularly useful for checking the reasonableness of answers.
这些百分比源自标准正态分布，对于检查答案的合理性特别有用。
4. Standard Normal Distribution and Z-Scores | 标准正态分布与Z分数
The standard normal distribution has mean 0 and standard deviation 1, written as Z ~ N(0, 1). Any normal variable X can be transformed to Z by subtracting μ and dividing by σ:
标准正态分布具有均值0和标准差1，记作Z ~ N(0, 1)。任何正态变量X都可以通过减去μ再除以σ转换为Z：
Z = (X − μ) / σ
The resulting Z-value, or Z-score, indicates how many standard deviations X is away from the mean. Standardisation lets us use a single probability table for all normal distributions.
得到的Z值（Z分数）表示X偏离均值几个标准差。标准化使得我们可以对所有正态分布使用同一个概率表。
5. Using the Standard Normal Table | 标准正态分布表的使用
Standard normal tables typically provide the cumulative probability Φ(z) = P(Z ＜ z) for z ≥ 0. To find Φ(1.25), locate the row for 1.2 and the column for 0.05; the intersection gives 0.8944.
标准正态分布表通常提供z ≥ 0时的累积概率Φ(z) = P(Z ＜ z)。要查找Φ(1.25)，找到1.2所在行和0.05所在列，交叉处为0.8944。
| z | Φ(z) = P(Z ＜ z) |
|---|---|
| 0.00 | 0.5000 |
| 0.50 | 0.6915 |
| 1.00 | 0.8413 |
| 1.25 | 0.8944 |
| 1.50 | 0.9332 |
| 2.00 | 0.9772 |
For negative Z-values, use symmetry: P(Z ＜ −z) = 1 − Φ(z). For instance, P(Z ＜ −1.25) = 1 − 0.8944 = 0.1056.
对于负Z值，可利用对称性：P(Z ＜ −z) = 1 − Φ(z)。例如P(Z ＜ −1.25) = 1 − 0.8944 = 0.1056。
6. Calculating Probabilities: Less Than, Greater Than, Between | 概率计算：小于、大于和区间概率
To find P(X ＜ a) for X ~ N(μ, σ²), compute z = (a − μ)/σ and read Φ(z) from the table. For P(X ＞ a), use 1 − Φ(z).
对于X ~ N(μ, σ²)，计算P(X ＜ a)时先求z = (a − μ)/σ，再查表得Φ(z)。P(X ＞ a)则用1 − Φ(z)。
Example: X ~ N(100, 15²). Find P(X ＞ 120).
z = (120 − 100)/15 = 1.33. Φ(1.33) ≈ 0.9082, so P(X ＞ 120) = 1 − 0.9082 = 0.0918.
例如：X ~ N(100, 15²)，求P(X ＞ 120)。
z = (120 − 100)/15 = 1.33，Φ(1.33) ≈ 0.9082，因此P(X ＞ 120) = 1 − 0.9082 = 0.0918。
For a between probability P(a ＜ X ＜ b), compute Z-values z₁ and z₂, then Φ(z₂) − Φ(z₁). This directly gives the area between the two bounds.
对于区间概率P(a ＜ X ＜ b)，计算两个Z值z₁、z₂，再用Φ(z₂) − Φ(z₁)，即得边界之间的面积。
7. The Inverse Normal: Finding X from a Probability | 逆正态：由概率求X值
When a left-tail probability p is given, we
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