Normal Distribution | 正态分布考点精讲

📚 Normal Distribution | 正态分布考点精讲

The normal distribution is one of the most powerful and widely used continuous probability distributions in A-level mathematics. It models natural phenomena such as heights, weights, and measurement errors, and forms the foundation of many statistical inference procedures. For AQA A-Level Mathematics, you are expected to understand the properties of the normal curve, perform calculations using the standard normal table, standardise variables, find probabilities and quantiles, handle unknown parameters, and apply the normal approximation to the binomial distribution with a continuity correction.

正态分布是 A-Level 数学中最强大、应用最广泛的连续型概率分布之一。它模拟身高、体重、测量误差等自然现象，构成许多统计推断方法的基础。针对 AQA A-Level 数学的考试要求，你需要掌握正态曲线的性质、使用标准正态分布表进行计算、对变量进行标准化、求概率和分位数、处理未知参数，以及结合连续性校正用正态分布近似二项分布。

1. Introduction to the Normal Distribution | 正态分布简介

A continuous random variable X is said to follow a normal distribution with mean μ and variance σ², written as X ~ N(μ, σ²), if its probability density function is symmetric and bell-shaped. The curve is centred at the mean, and the spread is determined by the standard deviation σ. Key properties include: the total area under the curve equals 1; the mean, median and mode are equal; the curve is asymptotic to the horizontal axis; about 68% of data lies within 1σ of μ, 95% within 2σ, and 99.7% within 3σ.

如果连续型随机变量 X 的概率密度函数呈对称的钟形曲线，则称 X 服从均值为 μ、方差为 σ² 的正态分布，记作 X ~ N(μ, σ²)。曲线以均值为中心，分散程度由标准差 σ 决定。主要性质包括：曲线下总面积为1；均值、中位数和众数相等；曲线以水平轴为渐近线；约 68% 的数据落在 μ±1σ 范围内，95% 落在 μ±2σ 内，99.7% 落在 μ±3σ 内。

2. The Standard Normal Distribution | 标准正态分布

The standard normal distribution is a special case with mean 0 and variance 1, denoted Z ~ N(0, 1). Its cumulative distribution function Φ(z) = P(Z ⩽ z) gives the area to the left of z under the standard normal curve. The symmetry property Φ(-z) = 1 – Φ(z) is extremely useful for calculating probabilities for negative z-values using table values for positive z only.

标准正态分布是均值为 0、方差为 1 的特殊情形，记作 Z ~ N(0, 1)。其累积分布函数 Φ(z) = P(Z ⩽ z) 给出标准正态曲线下在 z 左侧的面积。对称性 Φ(-z) = 1 – Φ(z) 非常有用，可通过仅列出正 z 值的表格计算负 z 值的概率。

3. Using the Standard Normal Table | 标准正态分布表的使用

A typical normal distribution table provides values of Φ(z) for z from 0.00 to about 3.99 to four decimal places. For example, Φ(0.55) = 0.7088 means P(Z < 0.55) = 0.7088. To find P(Z > 0.55), use P(Z > 0.55) = 1 – Φ(0.55) = 0.2912. For P(-0.55 < Z < 0.55), symmetry gives Φ(0.55) - Φ(-0.55) = Φ(0.55) - [1 - Φ(0.55)] = 2Φ(0.55) - 1 = 0.4176.

典型的正态分布表提供 z 从 0.00 到约 3.99 的 Φ(z) 值，保留四位小数。例如，Φ(0.55) = 0.7088 表示 P(Z < 0.55) = 0.7088。求 P(Z > 0.55) 使用 P(Z > 0.55) = 1 – Φ(0.55) = 0.2912。求 P(-0.55 < Z < 0.55)，利用对称性得 Φ(0.55) - Φ(-0.55) = Φ(0.55) - [1 - Φ(0.55)] = 2Φ(0.55) - 1 = 0.4176。

4. Standardising a Normal Variable | 正态变量的标准化

Any normal variable X ~ N(μ, σ²) can be transformed into the standard normal Z using the formula Z = (X – μ)/σ. If we need P(X < a), we calculate z = (a - μ)/σ and then read Φ(z) from the table. Standardisation moves the mean to 0 and scales the standard deviation to 1, making probability calculations independent of the original units.

任何正态变量 X ~ N(μ, σ²) 均可通过公式 Z = (X – μ)/σ 转化为标准正态变量 Z。若需要求 P(X < a)，先计算 z = (a - μ)/σ，再查表得到 Φ(z)。标准化的过程将均值平移至 0，并将标准差缩放至 1，使概率计算与原始单位无关。

5. Finding Probabilities for Normal Distributions | 正态分布的概率计算

For X ~ N(μ, σ²), the probability P(a < X < b) is found by standardising both boundaries: P((a-μ)/σ < Z < (b-μ)/σ) = Φ((b-μ)/σ) - Φ((a-μ)/σ). If only one boundary is given, treat the missing infinity appropriately: P(X < b) = Φ((b-μ)/σ), P(X > a) = 1 – Φ((a-μ)/σ). Always sketch the bell curve to visualise the required area.

对于 X ~ N(μ, σ²)，求概率 P(a < X < b) 需对两个边界进行标准化：P((a-μ)/σ < Z < (b-μ)/σ) = Φ((b-μ)/σ) - Φ((a-μ)/σ)。若只给出一个边界，则相应地处理无穷大：P(X < b) = Φ((b-μ)/σ)，P(X > a) = 1 – Φ((a-μ)/σ)。始终画出钟形曲线草图，直观显示所求面积。

6. Inverse Normal Distribution Calculations | 逆正态分布计算

In inverse problems, we are given a probability p and need to find x such that P(X < x) = p. First find z from the table where Φ(z) = p, then transform back to X using x = μ + zσ. If the probability given is a right-tail probability P(X > x) = p, then Φ(z) = 1 – p. For symmetric two-tail problems such as P(-z < Z < z) = p, use Φ(z) = (1 + p)/2.

在逆运算中，给定概率 p 需要求满足 P(X < x) = p 的 x 值。首先从表中找到使 Φ(z) = p 的 z，然后通过 x = μ + zσ 反变换得到 X。如果给出的概率是右尾概率 P(X > x) = p，则 Φ(z) = 1 – p。对于对称双尾问题如 P(-z < Z < z) = p，使用 Φ(z) = (1 + p)/2。

7. Finding Unknown Mean or Standard Deviation | 求未知的均值或标准差

A typical AQA exam question provides two probability statements about a normal variable X with unknown μ and/or σ, e.g. P(X < 30) = 0.1 and P(X > 70) = 0.15. Standardise each to obtain two equations in μ and σ: (30 – μ)/σ = z₁ and (70 – μ)/σ = z₂, where z₁ and z₂ are the corresponding z-values from the table. Solve simultaneously to find μ and σ.

典型的 AQA 考题会给出关于正态变量 X 的两个概率条件，而均值 μ 和/或标准差 σ 未知，例如 P(X < 30) = 0.1 和 P(X > 70) = 0.15。将每个条件标准化，得到关于 μ 和 σ 的两个方程：(30 – μ)/σ = z₁ 和 (70 – μ)/σ = z₂，其中 z₁ 和 z₂ 是从表中查得的相应 z 值。联立求解即可得到 μ 和 σ。

8. The Normal Approximation to the Binomial Distribution | 正态分布近似二项分布

When a binomial random variable X ~ B(n, p) has large n and p is not too close to 0 or 1, the distribution can be approximated by a normal distribution with μ = np and σ² = np(1-p), i.e. X ≈ N(np, np(1-p)). The usual rule of thumb for AQA is that np > 5 and n(1-p) > 5. This approximation simplifies the calculation of binomial probabilities, especially when using the standard normal table.

当二项随机变量 X ~ B(n, p) 的 n 很大且 p 不太接近 0 或 1 时，其分布可用均值为 μ = np、方差为 σ² = np(1-p) 的正态分布近似，即 X ≈ N(np, np(1-p))。AQA 通常使用的经验准则是 np > 5 且 n(1-p) > 5。这种近似可以简化二项分布概率的计算，特别是结合标准正态分布表时。

9. Continuity Correction | 连续性校正

Because the normal distribution is continuous while the binomial is discrete, we apply a continuity correction when approximating binomial probabilities. For P(X ⩽ a) we use P(X < a + 0.5) under the approximating normal; for P(X ⩾ a) we use P(X > a – 0.5); for P(X = a) we use P(a – 0.5 < X < a + 0.5). Failing to apply the continuity correction is a common mistake and will lose marks.

由于正态分布是连续型而二项分布是离散型，在近似二项概率时必须进行连续性校正。对于 P(X ⩽ a)，应在近似正态下计算 P(X < a + 0.5)；对于 P(X ⩾ a)，计算 P(X > a – 0.5)；对于 P(X = a)，计算 P(a – 0.5 < X < a + 0.5)。忘记进行连续性校正是常见错误，将导致失分。

10. Choosing Between Exact Binomial and Normal Approximation | 精确二项与正态近似的选择

In the exam, you must first check the validity conditions (np > 5, n(1-p) > 5). If satisfied, proceed with normal approximation plus continuity correction. Otherwise, use exact binomial calculations, possibly with the formula P(X = r) = nCr p^r (1-p)^(n-r) or cumulative tables provided. Always state clearly which method you are using and show the continuity correction step.

在考试中，你首先必须检查有效性条件（np > 5, n(1-p) > 5）。若满足，则使用带连续性校正的正态近似。否则，应使用精确二项计算，可能需要用到公式 P(X = r) = nCr p^r (1-p)^(n-r) 或所附的累积表格。务必清楚说明所用的方法，并展示连续性校正的步骤。

11. Applying the Normal Distribution in Context | 正态分布的实际应用

Real-world contexts often require careful identification of μ and σ. For instance, ‘the weight of apples is normally distributed with mean 150 g and standard deviation 20 g. Find the probability that a randomly chosen apple weighs less than 130 g.’ Always define the variable, write X ~ N(150, 20²), standardise and sketch. Questions may also involve sums and differences of independent normal variables, where you use the rules: E(aX + bY) = aμₓ + bμᵧ and Var(aX + bY) = a²σₓ² + b²σᵧ² if X and Y are independent.

实际背景题往往需要仔细识别 μ 和 σ。例如，“苹果的重量服从正态分布，均值为 150 g，标准差为 20 g。求随机选取一个苹果重量小于 130 g 的概率。”始终定义变量，写出 X ~ N(150, 20²)，做标准化并画草图。题目还可能涉及独立正态变量的和与差，此时使用规则：如果 X 与 Y 独立，则 E(aX + bY) = aμₓ + bμᵧ，Var(aX + bY) = a²σₓ² + b²σᵧ²。

12. Common Mistakes and Exam Tips | 常见错误与考试技巧

Avoid these frequent errors: forgetting to square σ when writing variance; misreading σ² as σ; misapplying the symmetry Φ(-z) = 1 – Φ(z); using the wrong tail when performing inverse lookups; ignoring the continuity correction in binomial approximations; confusing ‘greater than’ with ‘at least’. Exam tip: always draw a quick sketch of the bell curve, label the area required, and write down the standardisation step before picking up your calculator or table. This will drastically reduce careless mistakes.

避免以下常见错误：写下方差时忘记对 σ 平方；将 σ² 误读为 σ；错误应用对称性 Φ(-z) = 1 – Φ(z)；反向查表时弄错左右尾；二项近似时忽略连续性校正；混淆“大于”和“至少”。考试技巧：画一个快速的正态曲线草图，标出所需面积，并在拿起计算器或查表之前写下标准化步骤。这将极大减少粗心失分。

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Normal Distribution | 正态分布 考点精讲