Edexcel Maths: Normal Distribution Key Concepts | Edexcel 数学：正态分布考点精讲

📚 Edexcel Maths: Normal Distribution Key Concepts | Edexcel 数学：正态分布考点精讲

The normal distribution is one of the most important continuous probability distributions in A-level Mathematics. From natural phenomena like heights and weights to examination marks and measurement errors, this bell-shaped curve provides a powerful model for real-world data. In this article, we break down every key concept required for Edexcel Mathematics – including the standard normal distribution, standardisation, use of tables, inverse normal calculations, linear transformations of mean and variance, and the normal approximation to the binomial distribution with continuity correction. Whether you are studying for AS or A2, mastering these fundamentals will give you the confidence to tackle any normal distribution problem that appears on your paper.

正态分布是 A-level 数学中最重要的连续概率分布之一。从身高、体重等自然现象到考试分数和测量误差，这条钟形曲线为真实数据提供了强大的模型。本文详细拆解 Edexcel 数学要求掌握的每一个核心考点——包括标准正态分布、标准化、查表、逆正态计算、均值与方差的线性变换以及二项分布的正态近似（含连续性校正）。无论你正在备考 AS 还是 A2，掌握这些基础都将让你充满信心地应对试卷上出现的任何正态分布问题。

1. The Shape and Properties of the Normal Curve | 正态曲线的形状与性质

The normal distribution is a symmetric, bell-shaped continuous distribution defined for all real numbers. Its total area under the curve equals 1, representing the sum of all probabilities. The curve is unimodal – it has a single peak at the mean μ – and is asymptotic to the horizontal axis, meaning the tails extend indefinitely without ever touching zero. Two key parameters determine its exact shape: the population mean μ, which fixes the centre of the curve, and the population standard deviation σ (or variance σ²), which controls the spread. About 68% of observations lie within one standard deviation of the mean, roughly 95% within two standard deviations, and nearly 99.7% within three standard deviations.

正态分布是一种对称的钟形连续分布，定义域为全体实数。曲线下总面积等于 1，代表所有概率之和。该曲线是单峰的——在均值 μ 处有一个最高点——并且以水平轴为渐近线，意味着尾部无限延伸但永不触及零轴。两个关键参数决定了它的精确形状：总体均值 μ 确定曲线的中心位置，总体标准差 σ（或方差 σ²）控制离散程度。大约 68% 的观测值落在均值的一个标准差范围内，约 95% 落在两个标准差内，而几乎 99.7% 落在三个标准差内。

2. Probability Density Function (PDF) | 概率密度函数

The probability density function of a normal random variable X ~ N(μ, σ²) is given by f(x) = (1 / (σ√(2π))) e^(−(x−μ)²/(2σ²)). Although you will not be asked to integrate this function in the Edexcel exam, you must understand that the probability of X taking any exact value is zero for a continuous distribution; probabilities are found as areas under the curve between two limits. The notation X ~ N(μ, σ²) tells you that X follows a normal distribution with mean μ and variance σ². Remember that the second parameter in the bracket is the variance, not the standard deviation – a common source of error.

正态随机变量 X ~ N(μ, σ²) 的概率密度函数为 f(x) = (1 / (σ√(2π))) e^(−(x−μ)²/(2σ²))。虽然在 Edexcel 考试中不要求对这个函数进行积分，但你必须理解：在连续分布中，X 取任何精确值的概率都是零；概率是作为曲线下方两个界限之间的面积求得的。记号 X ~ N(μ, σ²) 表示 X 服从均值为 μ、方差为 σ² 的正态分布。请注意括号中的第二个参数是方差而非标准差——这是一个常见的错误来源。

3. The Standard Normal Distribution Z ~ N(0, 1) | 标准正态分布 Z ~ N(0, 1)

The standard normal distribution is the special case where the mean is 0 and the variance (and standard deviation) is 1. We denote this variable by Z. Its probability density is symmetric about zero, and statistical tables provide cumulative probabilities Φ(z) = P(Z < z) for z ≥ 0. These tables are essential tools in the exam, as all normal probability questions can be converted into problems about Z. For negative z-values, you must use symmetry: P(Z < −a) = 1 − P(Z < a), and P(Z > −a) = P(Z < a).

标准正态分布是均值为 0、方差（及标准差）为 1 的特殊情形。我们用 Z 来表示这个变量。其概率密度关于零对称，统计表提供 z ≥ 0 时的累积概率 Φ(z) = P(Z < z)。这些表格是考试中必不可少的工具，因为所有正态概率问题都可以转化为关于 Z 的问题。对于负的 z 值，必须利用对称性：P(Z < −a) = 1 − P(Z < a)，并且 P(Z > −a) = P(Z < a)。

4. Standardisation – Converting X to Z | 标准化——将 X 转化为 Z

Any normal variable X ~ N(μ, σ²) can be standardised by subtracting the mean and dividing by the standard deviation. The formula is Z = (X − μ) / σ. The resulting variable Z follows N(0, 1). This process lets you use the standard normal table to find probabilities for any X. For example, to find P(X < a), calculate z = (a − μ) / σ, then look up Φ(z). To find P(X > b), use 1 − Φ((b − μ)/σ). For an interval, P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ).

任何正态变量 X ~ N(μ, σ²) 都可以通过减去均值并除以标准差来实现标准化。公式为 Z = (X − μ) / σ。得到的变量 Z 服从 N(0, 1)。这一过程使你能够使用标准正态表求出任何 X 的概率。例如，求 P(X < a) 时，先计算 z = (a − μ) / σ，然后查表得 Φ(z)。求 P(X > b) 时，用 1 − Φ((b − μ)/σ)。对于区间概率，P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ)。

5. Using the Normal Distribution Table | 使用正态分布表

Edexcel provides the cumulative standard normal distribution table for positive z-values up to about 3.5. The table gives Φ(z) = P(Z < z). For a value like z = 1.25, locate the row for 1.2 and the column for 0.05; their intersection gives the probability, say 0.8944. If you need P(Z > 1.25), compute 1 − 0.8944 = 0.1056. To find probabilities for a z-value not directly tabulated, such as P(Z < 1.253), linear interpolation is expected. Check your specification to confirm whether interpolation is required or whether rounding to two decimal places is acceptable for your exam series.

Edexcel 提供了一张标准正态分布的累积分布表，列出了正 z 值直至约 3.5 的数据。该表给出 Φ(z) = P(Z < z)。例如，对于 z = 1.25，找到 1.2 所在行与 0.05 所在列的交叉点，即可读出概率值，如 0.8944。如果你需要求 P(Z > 1.25)，则计算 1 − 0.8944 = 0.1056。当需要求某个未直接列出的 z 值对应的概率时，例如 P(Z < 1.253)，可能会要求进行线性插值。请查阅考纲确认你的考试季是否要求插值，还是允许四舍五入保留两位小数。

6. Inverse Normal Calculations | 逆正态计算

Often the question provides a probability and asks you to find the corresponding value of X. First, apply the inverse normal (percentage point) table to Z. For a left‑tail probability p, find the z-value such that Φ(z) = p. Edexcel provides a separate percentage points table that gives z-values for common probabilities (e.g. 0.9, 0.95, 0.99) or you may use the main table backwards. Once you have the standardised value z, convert back to X using X = μ + σ z. If the probability is a right‑tail or a central area, adjust accordingly. For instance, if P(X > k) = 0.05, then P(X < k) = 0.95, so find the z-value for 0.95 and set k = μ + σ z.

考题通常会给出一个概率并要求你求出相应的 X 值。首先，对 Z 使用逆正态（百分点）表。对于左尾概率 p，找出使得 Φ(z) = p 的 z 值。Edexcel 提供了一张单独的百分点表，列出了常见概率（如 0.9, 0.95, 0.99）下的 z 值，你也可以反向使用主表格。一旦得到标准化值 z，就用 X = μ + σ z 换回 X。如果概率是右尾的或中心区域的，需相应做出调整。例如，若 P(X > k) = 0.05，则 P(X < k) = 0.95，因此找出对应于 0.95 的 z 值，并设 k = μ + σ z。

7. Linear Transformations of Mean and Variance | 均值与方差的线性变换

If X ~ N(μ, σ²) and Y = aX + b, where a and b are constants, then Y is also normally distributed. Its mean and variance follow the rules: E(Y) = aE(X) + b = aμ + b, Var(Y) = a² Var(X) = a²σ². Hence Y ~ N(aμ + b, a²σ²). This property is vital when combining and scaling normal variables. For the sum or difference of two independent normal variables, X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²), the sum S = X + Y has mean μ₁ + μ₂ and variance σ₁² + σ₂², while the difference D = X − Y has mean μ₁ − μ₂ and variance σ₁² + σ₂² (variances add in both cases).

若 X ~ N(μ, σ²) 且 Y = aX + b，其中 a 和 b 为常数，则 Y 也服从正态分布。其均值和方差遵循以下规则：E(Y) = aE(X) + b = aμ + b，Var(Y) = a² Var(X) = a²σ²。因此 Y ~ N(aμ + b, a²σ²)。这一性质在组合和缩放正态变量时至关重要。对于两个独立正态变量 X ~ N(μ₁, σ₁²) 与 Y ~ N(μ₂, σ₂²) 的和与差，和 S = X + Y 有均值 μ₁ + μ₂ 和方差 σ₁² + σ₂²，差 D = X − Y 有均值 μ₁ − μ₂ 和方差 σ₁² + σ₂²（两种情况下方差均相加）。

8. Finding Unknown μ or σ | 求未知的 μ 或 σ

A common exam question gives two probability statements involving an unknown normal distribution and asks you to find μ and/or σ. You form two simultaneous equations by standardising both pieces of information. For example, given P(X < 20) = 0.3085 and P(X > 50) = 0.0228, find the z-values (−0.5 and 2.0 respectively from tables), then write (20 − μ)/σ = −0.5 and (50 − μ)/σ = 2.0. Solve the pair of linear equations to obtain μ ≈ 26.7 and σ ≈ 13.3. Accuracy with the standard normal table and algebraic manipulation is essential.

一种常见的考题是给出与一个未知正态分布相关的两条概率信息，要求你求出 μ 和/或 σ。你需要将两条信息标准化，形成两个联立方程。例如，已知 P(X < 20) = 0.3085 和 P(X > 50) = 0.0228，从表中查出 z 值（分别为 −0.5 和 2.0），然后列出 (20 − μ)/σ = −0.5 和 (50 − μ)/σ = 2.0。解这对线性方程可得 μ ≈ 26.7 和 σ ≈ 13.3。正确使用标准正态表并进行准确的代数运算至关重要。

9. Normal Approximation to the Binomial | 二项分布的正态近似

When the number of trials n is large and probability of success p is not too close to 0 or 1, the binomial distribution B(n, p) can be approximated by a normal distribution with mean μ = np and variance σ² = np(1 − p). As a rule of thumb, the approximation is suitable when np > 5 and n(1 − p) > 5. To improve accuracy, a continuity correction is applied: a discrete binomial count x is treated as covering the continuous interval [x − 0.5, x + 0.5]. For example, the binomial probability P(X ≥ 15) becomes P(Y > 14.5) under the normal approximation Y ~ N(np, np(1−p)).

当试验次数 n 很大且成功概率 p 不太接近 0 或 1 时，二项分布 B(n, p) 可以用均值为 μ = np、方差为 σ² = np(1 − p) 的正态分布来近似。作为经验法则，当 np > 5 且 n(1 − p) > 5 时该近似是合适的。为提高精度，需要进行连续性校正：将离散的二项计数值 x 视作覆盖连续区间 [x − 0.5, x + 0.5]。例如，二项概率 P(X ≥ 15) 在正态近似 Y ~ N(np, np(1−p)) 下转化为 P(Y > 14.5)。

10. Continuity Correction in Detail | 连续性校正详解

Applying the continuity correction correctly is often the difference between full marks and losing marks. The adjustment always moves the boundary by 0.5 in the direction that includes or excludes the discrete value. For P(X = k), use P(k − 0.5 < Y < k + 0.5). For P(X ≤ k), use P(Y < k + 0.5). For P(X < k), use P(Y < k − 0.5). For P(X ≥ k), use P(Y > k − 0.5). For P(X > k), use P(Y > k + 0.5). Always sketch the normal curve and shade the region to confirm the corrected boundary.

正确进行连续性校正往往是获得满分与失分的区别。这一调整总是将边界沿包含或排除离散值的方向移动 0.5。对于 P(X = k)，使用 P(k − 0.5 < Y < k + 0.5)。对于 P(X ≤ k)，使用 P(Y < k + 0.5)。对于 P(X < k)，使用 P(Y < k − 0.5)。对于 P(X ≥ k)，使用 P(Y > k − 0.5)。对于 P(X > k)，使用 P(Y > k + 0.5)。请务必画出正态曲线并给相关区域涂上阴影，以确认校正后的边界。

11. Choosing Between Normal and Binomial | 在正态与二项分布之间做出选择

Reading the question carefully is essential. If the situation involves a fixed number of independent trials with two outcomes and the parameter n is given, use the binomial distribution unless the conditions for a normal approximation are satisfied and the question asks for an approximate probability. If the data is continuous or a sample mean is being modelled, the normal distribution is usually the correct choice. Pay attention to phrases like ‘normally distributed’, ‘approximated by a normal’, or ‘assume a normal model’.

仔细阅读题目至关重要。如果所涉情境包含固定次数的独立试验、具有两种可能结果且给定了参数 n，则应使用二项分布，除非满足正态近似的条件且题目要求的是近似概率。如果数据是连续的，或模型涉及样本均值，通常应选用正态分布。注意诸如“服从正态分布”、“用正态近似”或“假定一个正态模型”等说法。

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

Do not confuse variance and standard deviation; when a variance is given, take the square root to get σ before standardising. For negative z-values, always use symmetry rather than trying to look up a negative table. In inverse normal questions, sketch a diagram and label the known area and the unknown boundary. When using the normal approximation, remember the continuity correction – forgetting it is one of the most frequent errors. Finally, always round your final answer to a sensible degree of accuracy (often 3 significant figures) and show your working clearly to gain method marks even if arithmetic slips occur.

不要混淆方差与标准差；当给定方差时，先开平方根得到 σ 再进行标准化。对于负 z 值，务必利用对称性，而不是试图去查找负值表格。在逆正态问题中，绘制示意图并标出已知面积和未知边界。使用正态近似时，务必记得连续性校正——忘记校正是最常见的错误之一。最后，始终将最终答案四舍五入到合理的精度（通常为 3 位有效数字），并清晰地展示解题步骤，这样即使出现计算错误也能获得方法分。

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Edexcel Maths: Normal Distribution Key Concepts | Edexcel 数学：正态分布 考点精讲