📚 Normal Distribution and Approximations: Edexcel A-Level Applied Maths | 正态分布与近似：Edexcel A-Level 应用数学
The normal distribution is one of the most important continuous probability distributions in the Edexcel A-Level Applied Mathematics syllabus. From modelling real-world variables such as heights and masses to serving as the foundation for statistical inference, a solid grasp of the normal curve, standardisation, and the normal approximation to the binomial is essential for top marks in both the AS and A-Level examinations. This revision guide walks you through the key concepts, calculations, and common pitfalls, using clear unicode notation and concise examples, so you can approach any normal distribution question with confidence.
正态分布是 Edexcel A-Level 应用数学大纲中最重要的连续型概率分布之一。从建模身高、质量等现实变量，到作为统计推断的基础，牢固掌握正态曲线、标准化以及二项分布的正态近似，对于在 AS 和 A-Level 考试中取得高分至关重要。本复习指南将带您梳理关键概念、计算方法和常见误区，采用清晰的 Unicode 符号和简洁示例，让您能够自信应对任何正态分布试题。
1. Introduction to Normal Distribution | 正态分布简介
The normal distribution, often called the Gaussian distribution, is a symmetric, bell-shaped continuous distribution defined by two parameters: the mean μ and the variance σ². A random variable X that follows a normal distribution is written as X ~ N(μ, σ²). Its probability density function reaches a maximum at the mean, and the curve approaches the horizontal axis asymptotically as |x - μ| increases.
正态分布，常被称为高斯分布，是一种由两个参数定义的对称钟形连续分布：均值 μ 和方差 σ²。服从正态分布的随机变量 X 记作 X ~ N(μ, σ²)。其概率密度函数在均值处达到最大值，随着 |x - μ| 增大，曲线渐近地趋向水平轴。
2. Probability Density Function and Parameters | 概率密度函数与参数
The function for the normal distribution is f(x) = (1 / (σ√2π)) e-(x - μ)² / (2σ²). In Edexcel exams you do not need to evaluate this function directly; instead you will work with cumulative probabilities from tables or a calculator. The parameter μ shifts the curve horizontally, while σ controls its spread: a larger σ produces a flatter, wider curve, and a smaller σ yields a taller, narrower peak.
正态分布的函数为 f(x) = (1 / (σ√2π)) e-(x - μ)² / (2σ²)。在 Edexcel 考试中，您无需直接计算该函数，而是使用表格或计算器处理累积概率。参数 μ 控制曲线的水平位置，而 σ 控制其离散程度：σ 越大曲线越扁平宽阔，σ 越小则峰越高且越窄。
3. Standard Normal Distribution | 标准正态分布
To make calculations universal, we transform any normal variable to the standard normal variable Z, where Z ~ N(0, 1). The transformation uses the formula:
Z = (X - μ) / σ
This makes the mean 0 and the standard deviation 1. Once standardised, the cumulative probability P(Z < z) can be read directly from the standard normal table, and the total area under the curve equals 1.
为使计算具有普适性，我们将任意正态变量转换为标准正态变量 Z，其中 Z ~ N(0, 1)。转换公式为：
Z = (X - μ) / σ
这样均值变为 0，标准差变为 1。标准化后，累积概率 P(Z < z) 可直接从标准正态表中查得，曲线下总面积为 1。
4. Using Standard Normal Tables | 使用标准正态分布表
egative z, use the symmetry Φ(-z) = 1 - Φ(z). Always sketch a quick diagram to decide whether you need a left-tail, right-tail, or between probability. For instance, P(Z > 1.5) = 1 - Φ(1.5), and P(-1.2 < Z < 2.0) = Φ(2.0) - Φ(-1.2) = Φ(2.0) - [1 - Φ(1.2)].
Edexcel 公式手册包含标准正态分布表，给出正 z 值下 Φ(z) = P(Z < z) 的数值，最高约至 z = 4。对于负 z，利用对称性 Φ(-z) = 1 - Φ(z)。务必快速画一个草图，判断需要左尾、右尾还是区间概率。例如 P(Z > 1.5) = 1 - Φ(1.5)，而 P(-1.2 < Z < 2.0) = Φ(2.0) - Φ(-1.2) = Φ(2.0) - [1 - Φ(1.2)]。
5. Calculating Probabilities for General Normal | 一般正态分布的概率计算
For X ~ ). Remember that when the standard deviation is given as, say, σ = 5, the variance is σ² = 25; many students mistakenly use σ instead of σ² in the denominator.
对于 X ~ N(μ, σ²)，要计算 P(X < a)，首先计算 z = (a - μ)/σ，然后查表 Φ(z)。要求 P(a < X < b)，则计算 z₁ = (a - μ)/σ 与 z₂ = (b - μ)/σ，然后 P = Φ(z₂) - Φ(z₁)。请注意，若已知标准差 σ = 5，则方差为 σ² = 25；许多学生错误地在分母中使用 σ 而非 σ²。
6. Inverse Normal Calculations | 逆正态计算
ing Φ(zₚ) = p using the percentage points table or calculator inverse function. Then back-transform: a = μ + σ × zₚ. If the question gives P(X > a) = q, convert to P(X < a) = 1 - q before finding z. Always check whether the problem supplies the variance or standard deviation, and draw a diagram to confirm the direction of the inequality.
当已知左尾概率 p，求对应的值 a 使得 P(X < a) = p 时，首先利用百分位点表或计算器逆函数求出满足 Φ(zₚ) = p 的标准正态分位数 zₚ。然后反变换：a = μ + σ × zₚ。若题目给出 P(X > a) = q，先转换为 P(X < a) = 1 - q 再求 z。务必确认题目给出的是方差还是标准差，并画图确认不等式的方向。
7. Normal Approximation to Binomial | 二项分布的正态近似
mean μ = np and variance σ² = np(1 - p). This is useful when n exceeds the range of binomial tables or when a calculator is not available for exact binomial probabilities. The approximation improves as n increases and is usually considered valid when both np > 5 and n(1 - p) > 5.
对于二项分布 B(n, p)，当 n 较大且 p 不接近 0 或 1 时，可用均值为 μ = np、方差为 σ² = np(1 - p) 的正态分布来近似。当 n 超出二项分布表的范围，或无法使用计算器求精确二项概率时，这种方法非常有用。随着 n 增大，近似效果更好，通常认为当 np > 5 且 n(1 - p) > 5 时近似有效。
8. Continuity Correction | 连续性校正
applied. For a binomial probability P(X = 15), approximate by P(14.5 < Y < 15.5), where Y ~ etting the continuity correction is a very common exam mistake and can cost several marks.
由于我们用连续分布来模拟离散分布，必须进行连续性校正。对于二项概率 P(X = 15)，近似为 P(14.5 < Y < 15.5)，其中 Y ~ N(np, np(1-p))。对于 P(X ≤ 12)，使用 P(Y < 12.5)；对于 P(X ≥ 20)，使用 P(Y > 19.5)。忘记连续性校正是考试中非常常见的错误，可能导致失去数分。
9. Conditions for Approximation | 近似的条件
e np and nq (where q = 1 - p) both exceed 5. If p is very small or very large, the binomial becomes skewed and the normal approximation is poor; in such cases a Poisson approximation might be more appropriate. Always justify the approximation in one sentence before applying it.
Edexcel 评分标准通常要求说明近似合适，因为 n 很大且 p 接近 0.5，或者更具体地，因为 np 与 nq（q = 1 - p）均大于 5。若 p 非常小或非常大，二项分布呈偏态，正态近似效果差；此时泊松近似可能更合适。在应用近似前，务必用一句话说明理由。
10. Worked Example | 实例解析
he approximate probability of obtaining between 70 and 85 heads inclusive. Solution: Let X ~ 80, variance σ² = 200 × 0.4 × 0.6 = 48, so σ ≈ 6.9282. Applying the normal approximation Y ~ 39. Then P = Φ(0.7939) - Φ(-1.5155) = Φ(0.7939) - [1 - Φ(1.5155)]. From tables, Φ(0.79) ≈ 0.7852, Φ(1.52) ≈ 0.9357, so P ≈ 0.7852 - (1 - 0.9357) = 0.7209.
一枚偏倚的硬币投掷 200 次，正面概率 p = 0.4。求正面次数介于 70 至 85（含）的近似概率。解：设 X ~ B(200, 0.4)。均值 μ = 200 × 0.4 = 80，方差 σ² = 200 × 0.4 × 0.6 = 48，故 σ ≈ 6.9282。应用正态近似 Y ~ N(80, 48)。经连续性校正，P(70 ≤ X ≤ 85) ≈ P(69.5 < Y < 85.5)。标准化：z1 = (69.5 - 80) / 6.9282 ≈ -1.5155，z2 = (85.5 - 80) / 6.9282 ≈ 0.7939。则概率 = Φ(0.7939) - Φ(-1.5155) = Φ(0.7939) - [1 - Φ(1.5155)]。查表得 Φ(0.79) ≈ 0.7852，Φ(1.52) ≈ 0.9357，故概率 ≈ 0.7852 - (1 - 0.9357) = 0.7209。
11. Common Exam Tips | 常见考试技巧
- Always specify when you use the normal approximation and state the values of μ and σ².
- Draw a bell curve and shade the area you are finding; this reduces inequality-direction errors.
- If using a calculator's normal CDF function, double-check you have entered the standard deviation, not the variance, and that you apply continuity correction when approximating a binomial.
- Round probabilities to at least 4 decimal places unless instructed otherwise, but avoid premature rounding of z-scores.
- 使用正态近似时必须说明，并写出 μ 与 σ² 的值。
- 画出钟形曲线并涂出所求区域，可减少不等式方向错误。
- 若使用计算器正态累积函数，请仔细检查输入的是标准差而非方差，并且在二项近似时应用连续性校正。
- 除非另有要求，概率保留至少 4 位小数，但避免过早舍入 z 值。
12. Summary | 总结
The normal distribution is a powerful tool that appears throughout the Edexcel A-Level syllabus, from basic probability to hypothesis testing. Mastery requires fluency in standardisation, accurate table reading, confidence with inverse normal processes, and careful application of the normal approximation to the binomial with continuity correction. Regular practice with past paper questions and on-screen calculator diagnostics will build the speed and accuracy needed for the exam.
正态分布是整个 Edexcel A-Level 大纲中一个强大的工具，从基础概率到假设检验都有应用。掌握它需要熟练进行标准化、精确查表、自信地进行逆正态求解，并谨慎地使用连续性校正对二项分布进行正态近似。通过反复练习历年试题和屏幕计算器自检，您将培养出考试所需的速度与精确度。
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