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The Normal Distribution | 正态分布

2026-08-29 20:23 · 本文部分内容由AI辅助创作

📚 The Normal Distribution | 正态分布

The normal distribution is one of the most important continuous probability distributions in A-Level Mathematics. It models continuous data that cluster symmetrically around a mean, and it is widely used in statistics, science and everyday measurements. This article covers the Edexcel A-Level normal distribution topic, including standardising, probabilities, inverse problems, sample means and the normal approximation to the binomial distribution, together with the calculator skills and exam technique needed for full marks.

正态分布是 A-Level 数学中最重要的连续型概率分布之一。它为围绕均值对称聚集的连续数据建模，广泛用于统计、科学和日常测量。本文涵盖 Edexcel A-Level 正态分布主题，包括标准化、概率计算、逆问题、样本均值以及二项分布的正态近似，并附上取得满分所需的计算器操作与应试技巧。

1. Definition and Shape of the Normal Distribution | 正态分布的定义与形状

ility density curve is a symmetrical bell-shaped curve centred on the mean μ. Because the variable is continuous, probabilities are found as areas under the curve rather than by adding up individual values, so P(X = x) = 0 for any single value x.

正态分布是一种连续型概率分布。若连续型随机变量 X 的概率密度曲线是一条关于均值 μ 对称的钟形曲线，则称 X 服从正态分布。由于变量是连续的，概率通过曲线下的面积而非逐个取值相加来求得，因此对任意单个取值 x 都有 P(X = x) = 0。

A larger standard deviation makes the curve wider and flatter, while a smaller standard deviation makes it narrower and taller. The area under the curve always remains 1.

标准差越大，曲线越宽越扁平；标准差越小，曲线越窄越高耸。无论形状如何变化，曲线下的面积始终为 1。

2. Notation, Mean, Variance and the Empirical Rule | 记号、均值、方差与经验法则

We write X ~ N(μ, σ²), where μ is the mean and σ² is the variance. The standard deviation is σ, so σ = √σ². The second parameter in the bracket is the variance, not the standard deviation, and mixing these up is one of the most common sources of lost marks.

我们写作 X ~ N(μ, σ²)，其中 μ 是均值，σ² 是方差。标准差为 σ，即 σ = √σ²。括号中第二个参数是方差而不是标准差，混淆二者是失分最常见的原因之一。

Changing μ shifts the whole curve left or right without altering its shape. Changing σ stretches or compresses the curve: it changes the spread but not the position of the centre. The distribution is completely determined by these two parameters.

改变 μ 会使整条曲线左右平移而不改变形状；改变 σ 则拉伸或压缩曲线，即改变离散程度而不改变中心位置。该分布完全由这两个参数确定。

For any normal distribution, the empirical rule (also called the 68-95-99.7 rule) gives useful benchmarks that let you check whether an answer is sensible.

对任何正态分布而言，经验法则（也称 68-95-99.7 法则）给出了有用的参考值，可用来判断答案是否合理。

; P(μ − 2σ < X < μ + 2σ) ≈ 0.9545    P(μ − 3σ < X < μ + 3σ) ≈ 0.9973

The probability density function itself is rarely needed at A-Level, but for completeness the curve for X ~ N(μ, σ²) has equation:

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概率密度函数本身在 A-Level 中很少需要用到，但为完整起见，X ~ N(μ, σ²) 的曲线方程为：

f(x) = (1 ÷ (σ√2π)) × e^(−(x − μ)² ÷ (2σ²))

3. The Standard Normal Distribution and Standardising | 标准正态分布与标准化

Because there are infinitely many normal distributions, one for each pair of values of μ and σ, we convert every problem to a single reference distribution: the standard normal distribution Z ~ N(0, 1), which has mean 0 and variance 1. Tables and calculators are built around this distribution.

由于均值与标准差每一组取值都对应一条正态曲线，这样的分布有无穷多个，因此我们把所有问题都转化为一个统一的参照分布：标准正态分布 Z ~ N(0, 1)，其均值为 0、方差为 1。分布表与计算器都是围绕它设计的。

The conversion is called standardising, and it measures how many standard deviations a value lies above or below the mean.

这一转化称为标准化，它衡量某个取值高于或低于均值多少个标准差。

Z = (X − μ) ÷ σ    so    z = (x − μ) ÷ σ

Standardising preserves probabilities: if X ~ e x greater than the mean gives a positive z, and a value below the mean gives a negative z.

标准化保持概率不变：若 X ~ N(μ, σ²) 且 Z = (X − μ) ÷ σ，则 P(X < x) = P(Z < z)，其中 z = (x − μ) ÷ σ。大于均值的取值对应正的 z，小于均值的取值对应负的 z。

written Φ(z) = P(Z ≤ z), and this is the quantity tabulated in the standard normal tables.

标准正态分布的累积概率记作 Φ(z) = P(Z ≤ z)，这正是标准正态分布表中所列出的量。

4. Reading Tables and Using Symmetry | 查表与对称性的运用

nly. To handle negative values and upper tails, use the symmetry of the curve about z = 0.

印刷的表格通常只给出 z ≥ 0 时 Φ(z) = P(Z ≤ z) 的值。要处理负值和上尾概率，需利用曲线关于 z = 0 的对称性。