📚 Normal Distribution | 正态分布
The normal distribution is one of the most important continuous probability distributions in A-Level Mathematics. It is used to model a wide range of real-world variables, from heights and weights to measurement errors. Understanding its shape, parameters, and how to calculate probabilities is essential for success in the Edexcel statistics component.
正态分布是A-Level数学中最重要的连续概率分布之一。它广泛应用于许多现实变量的建模，从身高体重到测量误差。理解它的形状、参数以及概率计算方法，对Edexcel统计部分取得好成绩至关重要。
1. Definition and Notation | 定义与记号
A continuous random variable X follows a normal distribution with mean μ and variance σ². We write X ~ N(μ, σ²). The mean determines the centre of the distribution, while the standard deviation σ controls the spread. If the variance is given directly as σ², the standard deviation is its square root.
连续随机变量X服从均值为μ、方差为σ²的正态分布，记作X ~ N(μ, σ²)。均值决定了分布的中心，而标准差σ控制离散程度。如果直接给出方差σ²，标准差则是它的算术平方根。
2. Shape and Properties of the Normal Curve | 正态曲线形状与性质
The probability density function of a normal distribution is a symmetric, bell-shaped curve. Key properties include: the curve is symmetrical about the mean μ, the mode and median are also equal to μ, and the total area under the curve equals 1. Approximately 68% of data lies within 1σ of μ, 95% within 2σ, and 99.7% within 3σ.
正态分布的概率密度函数是一条对称的钟形曲线。主要性质包括：曲线关于均值μ对称，众数和中位数也等于μ，曲线下的总面积为1。大约68%的数据落在μ±σ内，95%在μ±2σ内，99.7%在μ±3σ内。
3. Standard Normal Distribution – Z-distribution | 标准正态分布——Z分布
To easily calculate probabilities, any normal variable X ~ ransformed into the standard normal variable Z. The transformation uses the formula Z = (X – mean 0 and variance 1. Standardising allows us to use a single set of probability tables.
为方便计算概率，任何正态变量X ~ N(μ, σ²)都可以转化为标准正态变量Z。转化公式为Z = (X – μ) / σ。此时Z服从N(0, 1)分布，均值为0，方差为1。标准化后便可统一使用概率表。
Z = (X – μ) / σ
4. Using the Standard Normal Table | 使用标准正态表
≤ z), i.e. the area to the left of z. For negative Z values, use the symmetry property Φ(–z) = 1 – Φ(z). Always draw a diagram to visualise the required area before reading the table.
Edexcel考试提供正Z值的累积分布函数表，通常记作Φ(z)。Φ(z)表示P(Z ≤ z)，即z左侧的面积。对于负Z值，利用对称性Φ(–z) = 1 – Φ(z)。查表前务必画出示意图，明确所需面积对应的区域。
5. Calculating Probabilities for X ~ N(μ, σ²) | 正态分布概率计算
To find P(X < a), first standardise: Z = (a – μ) / σ, then look up Φ(z). For P(X > a), use 1 – Φ(z). For an interval P(a < X < b), compute Φ(z₂) – e corresponding standardised values. Always subtract the smaller area from the larger one.
求P(X < a)时，先标准化得Z值，再查表Φ(z)。若求P(X > a)，用1 – Φ(z)。若计算区间概率P(a < X < b)，则求Φ(z₂) – Φ(z₁)，其中z₁与z₂分别对应a和b的标准化值。总是用较大面积减去较小面积。
6. Finding Values Given Probabilities – Inverse Normal | 已知概率反求值
responds to a left-hand tail, use z directly; for right-hand tails, take care to find the correct z by reversing the area.
若题目给出概率，需求对应的x值，可使用逆正态方法。从标准正态表内查找最接近的概率以得到z值，再利用x = μ + zσ求解。当概率对应左尾部时直接使用z；若为右尾部，需注意将面积取反以确定正确的z。
x = μ + zσ
7. Mean and Variance of the Normal Distribution | 正态分布的均值与方差
The parameters μ and σ² are the population mean and variance. In many A-Level problems, μ and σ are unknown and must be estimated from sample data. Linear combinations of independent normal variables also follow a normal distribution: if X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²), then aX + bY ~ N(aμ₁ + bμ₂, a²σ₁² + b²σ₂²).
参数μ和σ²分别表示总体均值和方差。在许多A-Level题目中，μ和σ未知，需通过样本数据进行估计。独立正态变量的线性组合仍服从正态分布：若X ~ N(μ₁, σ₁²)且Y ~ N(μ₂, σ₂²)，则aX + bY ~ N(aμ₁ + bμ₂, a²σ₁² + b²σ₂²)。
8. Normal Approximation to the Binomial | 二项分布的正态近似
istribution B(n, p) can be approximated by a normal distribution with μ = np and σ² = np(1 – e. This improves accuracy because the binomial is discrete while the normal is continuous.
当n较大且p接近0.5时，二项分布B(n, p)可用正态分布近似，此时μ = np，σ² = np(1 – p)。必须进行连续性校正：将P(X ≤ k)替换为P(X < k + 0.5)的正态概率。这能提高精确度，因为二项分布是离散的，而正态分布是连续的。
X ~ B(n, p) ≈ N(np, np(1 – p))
9. Real-world Applications and Modelling | 实际应用与建模
Exam questions often describe a real context, such as the weight of cereal boxes, the IQ of a population, or the diameter of machine parts. You must identify μ and σ from the context, define the random variable, and correctly interpret probabilities in words. Always state your final answer in the context of the problem, e.g. “the probability that a randomly selected box weighs less than 500 g is 0.0228”.
考题常借助真实场景，例如谷物包装的重量、人群智商或机器零件的直径。你需要从上下文中识别μ和σ，定义随机变量，并正确用文字解释概率。最终答案务必放回问题情境中，如“随机选取一盒重量少于500 克的概率为0.0228”。
10. Common Exam Mistakes | 考试常见错误
Many students forget to draw the bell curve, which leads to errors in subtracting areas. Others confuse variance σ² with standard deviation σ when standardising. Another common mistake is omitting the continuity correction in normal approximation. Also, remember that the table gives cumulative probabilities Φ(z), not P(Z > z) directly. Always double-check whether you need the area to the left or to the right.
许多学生容易忘记画钟形曲线，导致面积相减时出错。另一些人标准化时分不清方差σ²和标准差σ。另一个常见错误是在正态近似时省略连续性校正。还要记住，查表给出的是累积概率Φ(z)，而非直接给出P(Z > z)。始终检查你需要的究竟是左侧还是右侧的面积。
11. Using Technology: Calculator Skills | 使用计算器技能
In the Edexcel A-Level exam, certain statistical calculators can find normal probabilities directly without standardising. You can input μ, σ, and the x-values to get P(X < a) or P(a < X < b). However, exam command words often require showing the standardising step and using tables, so only use calculator results to verify your work unless the question explicitly allows it.
在Edexcel A-Level考试中，部分统计计算器可直接计算正态概率而无需标准化。你可以输入μ、σ和x值得到P(X < a)或P(a < X < b)。但考题的指令词通常要求写出标准化步骤并使用表格，因此除非题目明确允许，否则计算器仅用于验证答案。
12. Practice and Revision Tips | 练习与复习建议
ed diagrams with shaded areas for every exercise. Flashcards with key formulae like Z = (X – andardisation and inverse methods. Consistent practice builds confidence for exam success.
通过大量练习历年真题来掌握正态分布。重点训练辨别题目是要求正向求概率还是反向求x值。做题时务必画出带阴影区域的示意图。记忆卡片上有Z = (X – μ)/σ和x = μ + zσ等关键公式，有助于巩固标准化与逆运算方法。持续练习能建立信心，在考试中游刃有余。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图