📚 A-Level Further Mathematics: Variance of the Normal Distribution | A-Level进阶数学：正态分布的方差
The normal distribution is one of the most important probability models in A-Level Further Mathematics. Its variance, denoted σ², measures how widely data values are spread around the mean μ. In this article, we explore the definition, derivation, interpretation, and exam-style application of the variance of a normal distribution.
正态分布是 A-Level 进阶数学中最重要的概率模型之一。其方差记作 σ²，用于度量数据围绕均值 μ 的离散程度。本文将深入探讨正态分布方差的定义、推导、含义及考试应用。
1. Probability Density Function | 概率密度函数
ous random variable X follows a normal distribution with mean μ and variance σ², written X ~ N(μ, σ²). Its probability density function (PDF) is given by f(x) = (1 / (σ√2π)) · e^(−(x−μ)² / (2σ²)), defined for all real x.
连续型随机变量 X 服从均值为 μ、方差为 σ² 的正态分布，记作 X ~ N(μ, σ²)。其概率密度函数为 f(x) = (1 / (σ√2π)) · e^(−(x−μ)² / (2σ²))，对所有实数 x 有定义。
f(x) = (1 / (σ√2π)) · e^(−(x−μ)² / (2σ²))
The parameter μ determines the location of the curve, while σ determines its spread. A larger σ produces a shorter and wider curve; a smaller σ produces a taller and narrower curve.
参数 μ 决定曲线的位置，σ 决定曲线的离散程度。σ 越大，曲线越矮越宽；σ 越小，曲线越高越窄。
2. Definition of Variance | 方差的定义
variance is defined as Var(X) = ∫₋∞⁺∞ (x − μ)² f(x) dx, where μ = E(X) = ∫₋∞⁺∞ x f(x) dx.
对于具有概率密度函数 f(x) 的连续型随机变量，方差定义为 Var(X) = ∫₋∞⁺∞ (x − μ)² f(x) dx，其中 μ = E(X) = ∫₋∞⁺∞ x f(x) dx。
Var(X) = ∫₋∞⁺∞ (x − μ)² f(x) dx = E(X²) − (E(X))²
Equivalently, variance can be computed as E(X²) − μ². This identity often simplifies calculations because E(X²) is easier to evaluate directly through integration.
等价地，方差可表示为 E(X²) − μ²。这一恒等式常能简化计算，因为 E(X²) 通常更容易直接通过积分求得。
3. The Standard Normal Distribution | 标准正态分布
n μ = 0 and σ² = 1, the distribution is called the standard normal distribution, denoted Z ~ N(0, 1). Its PDF is φ(z) = (1 / √2π) · e−z²/2.
当 μ = 0 且 σ² = 1 时，称为标准正态分布，记作 Z ~ N(0, 1)。其概率密度函数为 φ(z) = (1 / √2π) · e−z²/2。
transformation preserves probabilities and is essential for using standard normal tables.
任何正态随机变量都可以通过 Z = (X − μ) / σ 转化为标准正态分布。该变换保持概率不变，是使用标准正态分布表的关键。
4. Deriving the Variance by Integration | 通过积分推导方差
ameter σ² in the PDF is indeed the variance, we evaluate Var(X) = ∫₋∞⁺∞ (x − μ)² · (1 / (σ√2π)) · e^(−(x−μ)² / (2σ²)) dx.
为了证明 PDF 中的参数 σ² 确实是方差，我们需要计算 Var(X) = ∫₋∞⁺∞ (x − μ)² · (1 / (σ√2π)) · e^(−(x−μ)² / (2σ²)) dx。
Substitute u = (x − μ) / σ, so dx = σ du. The integral becomes σ² · (1 / √2π) ∫₋∞⁺∞ u² e−u²/2 du.
令 u = (x − μ) / σ，则 dx = σ du。积分变为 σ² · (1 / √2π) ∫₋∞⁺∞ u² e−u²/2 du。
Using integration by parts or the known Gaussian integral, ∫₋∞⁺∞ u² e−u²/2 du = √2π, so Var(X) = σ² · (1 / √2π) · √2π = σ².
利用分部积分或已知的高斯积分，∫₋∞⁺∞ u² e−u²/2 du = √2π，因此 Var(X) = σ² · (1 / √2π) · √2π = σ²。
Var(X) = σ²
Thus the symbol σ² in N(μ, σ²) is not arbitrary; it is exactly the variance of the distribution.
因此，N(μ, σ²) 中的符号 σ² 并非随意选择；它恰好就是该分布的方差。
5. Properties of the Variance | 方差的性质
n is degenerate, meaning all probability is concentrated at one point. For a normal distribution, σ² > 0.
方差始终非负，且 Var(X) = 0 仅当分布退化时成立，即所有概率集中在一点。对于正态分布，σ² > 0。
Translation: Var(X + c) = Var(X), where c is a constant.
平移：Var(X + c) = Var(X)，其中 c 为常数。
Scaling: Var(aX) = a² Var(X), where a is a constant.
缩放：Var(aX) = a² Var(X)，其中 a 为常数。
Linear combination: If X and Y are independent, Var(aX + bY) = a² Var(X) + b² Var(Y).
线性组合：若 X 与 Y 独立，则 Var(aX + bY) = a² Var(X) + b² Var(Y)。
These properties are frequently tested in Edexcel A-Level Further Mathematics papers, especially in questions involving linear transformations of normal variables.
这些性质在 Edexcel 进阶数学考试中频繁出现，尤其是在涉及正态变量线性变换的题目中。
6. The Chi-Squared Connection | 与卡方分布的联系
If Z₁, Z₂, …, Zₙ are independent standard normal variables, then the sum of their squares follows a chi-squared distribution with n degrees of freedom: Σᵢ₌₁ⁿ Zᵢ² ~ χ²(n).
若 Z₁, Z₂, …, Zₙ 是独立的标准正态变量，则它们的平方和服从自由度为 n 的卡方分布：Σᵢ₌₁ⁿ Zᵢ² ~ χ²(n)。
This result is central to many statistical inference topics in Further Mathematics, including confidence intervals for variance and hypothesis testing.
这一结论是进阶数学中许多统计推断内容的核心，包括方差的置信区间和假设检验。
In particular, if X₁, X₂, …, Xₙ are independent N(μ, σ²) variables, then (n − 1)S² / σ² ~ χ²(n − 1), where S² is the sample variance.
特别地，若 X₁, X₂, …, Xₙ 是独立同分布 N(μ, σ²) 的变量，则 (n − 1)S² / σ² ~ χ²(n − 1)，其中 S² 是样本方差。
7. Worked Example: Calculating Variance from Data | 例题：由数据计算方差
Suppose a random sample of 5 observations is taken from a normal distribution: 2.1, 2.8, 3.0, 3.4, 4.2. Estimate the population variance.
假设从正态分布中抽取容量为 5 的随机样本：2.1, 2.8, 3.0, 3.4, 4.2。估计总体方差。
The sample mean is x̄ = (2.1 + 2.8 + 3.0 + 3.4 + 4.2) / 5 = 15.5 / 5 = 3.1.
样本均值为 x̄ = (2.1 + 2.8 + 3.0 + 3.4 + 4.2) / 5 = 15.5 / 5 = 3.1。
The sample variance is s² = Σ(xᵢ − x̄ )² + (−0.1)² + (0.3)² + (1.1)²] / 4 = (1 + 0.09 + 0.01 + 0.09 + 1.21) / 4 = 2.4 / 4 = 0.6.
样本方差为 s² = Σ(xᵢ − x̄ 3)² + (−0.1)² + (0.3)² + (1.1)²] / 4 = (1 + 0.09 + 0.01 + 0.09 + 1.21) / 4 = 2.4 / 4 = 0.6。
Thus the estimate of the population variance is 0.6. In exam questions, be careful to use n − 1 for sample variance unless the question specifies the population variance.
因此总体方差估计值为 0.6。在考试中，除非题目明确要求总体方差，否则计算样本方差时应使用 n − 1。
8. Worked Example: Inverse Problem | 例题：反求参数
A normal distribution has mean 50 and variance σ². Given that P(X > 55) = 0.1587, find σ².
某正态分布均值为 50，方差为 σ²。已知 P(X > 55) = 0.1587，求 σ²。
0) / σ) = 0.1587. From the standard normal table, P(Z > 1) = 0.1587, so (55 − 50) / σ = 1.
标准化：P(Z > (55 − 50) / σ) = 0.1587。由标准正态表可知 P(Z > 1) = 0.1587，因此 (55 − 50) / σ = 1。
Hence σ = 5 and σ² = 25.
因此 σ = 5，σ² = 25。
(55 − 50) / σ = 1 ⇒ σ = 5 ⇒ σ² = 25
This type of inverse problem tests whether you can connect probabilities to z-values and then solve for the variance.
这类反问题考查你是否能将概率与 z 值对应，并进一步解出方差。
9. Common Mistakes and Exam Tips | 常见错误与考试提示
One common mistake is confusing standard deviation σ with variance σ². Always check what the question asks for before giving a final answer.
常见错误之一是混淆标准差 σ 与方差 σ²。作答前务必看清题目要求的是哪一个。
Always state the distribution clearly: X ~ N(μ, σ²).
始终明确写出分布：X ~ N(μ, σ²)。
When using the standard normal table, remember that the table often gives P(Z < z). Use symmetry to find upper-tail probabilities.
使用标准正态表时，注意表通常给出 P(Z < z)。利用对称性计算上尾概率。
When transforming X to Z, use Z = (X − μ) / σ, not Z = (X − μ) / σ².
将 X 标准化时，应使用 Z = (X − μ) / σ，而不是 Z = (X − μ) / σ²。
For independent normal variables, variances add, not standard deviations.
对于独立正态变量，方差相加，而非标准差相加。
In Edexcel papers, show every step of the standardisation and state the probability relationship using Φ or standard normal notation where appropriate.
在 Edexcel 考试中，应写出标准化过程的每一步，并适当使用 Φ 或标准正态记号表示概率关系。
10. Conclusion | 总结
The variance of the normal distribution is not just a parameter; it is the precise measure of spread defined by the integral of (x − μ)² weighted by the PDF. Understanding its derivation, properties, and connection to the standard normal and chi-squared distributions gives you a strong foundation for Further Mathematics.
正态分布的方差不仅仅是一个参数；它是以概率密度为权重对 (x − μ)² 积分所得到的精确离散程度度量。理解其推导、性质以及与标准正态分布和卡方分布的联系，将为你的进阶数学学习打下坚实基础。
Mastering the variance concept will also help you succeed in topics such as confidence intervals, hypothesis testing, and linear regression, which frequently appear in the Edexcel A-Level Further Mathematics syllabus.
掌握方差概念还将帮助你在置信区间、假设检验和线性回归等主题中取得好成绩，这些内容在 Edexcel A-Level 进阶数学考纲中频繁出现。
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