📚 The Normal Distribution | 正态分布
The normal distribution is arguably the most important probability model in A‑Level Mathematics. It underpins inferential statistics, models natural variation in countless real‑world measurements, and appears as a limiting case for other distributions. This article covers the Edexcel specification for the normal distribution, including its probability density function, standardisation, use of statistical tables, inverse calculations, and the normal approximation to the binomial distribution.
正态分布可以说是 A‑Level 数学中最重要的概率模型。它为推断统计奠定基础，能够刻画无数真实测量中的自然变异，并且作为其他分布的极限形式出现。本文涵盖 Edexcel 考试大纲中关于正态分布的全部内容，包括概率密度函数、标准化过程、统计表的使用、反向计算以及二项分布的正态近似。
1. Continuous Distributions and the Bell Curve | 连续型分布与钟形曲线
In statistics, a continuous random variable can take any value within an interval. The normal distribution models such a variable with a smooth, bell‑shaped curve that is defined for all real numbers. The total area under the curve equals 1, representing total probability.
在统计学中，连续型随机变量可以取区间内的任意值。正态分布用一条光滑的钟形曲线来刻画这种变量，该曲线定义在整个实数轴上。曲线下的总面积等于 1，代表总概率。
It is sometimes called the Gaussian distribution after Carl Friedrich Gauss, who used it extensively in astronomical work. Many natural phenomena — from heights and body mass to exam scores and measurement errors — tend to cluster around a central value with symmetrically thinning tails.
有时它也因卡尔·弗里德里希·高斯的广泛应用而被称为高斯分布。许多自然现象——从身高、体重到考试成绩和测量误差——往往围绕一个中心值聚集，两侧尾部对称地衰减。
2. The Probability Density Function | 概率密度函数
The probability density function (pdf) of a normal random variable X with mean μ and standard deviation σ is given by the formula
均值为 μ、标准差为 σ 的正态随机变量 X 的概率密度函数由下式给出
f(x) = 1 / (σ √2π) · e‑½[(x − μ) / σ]²
arithms. The parameters μ and σ completely determine the shape and location of the curve. Note that f(x) itself is not a probability; the probability that X lies in an interval [a, b] is the area under f(x) from a to b.
其中 π 是常数 3.14159...，e ≈ 2.71828 是自然对数的底。参数 μ 和 σ 完全决定了曲线的形状和位置。注意 f(x) 本身并不是概率；X 落在区间 [a, b] 内的概率是 f(x) 从 a 到 b 与 x 轴所围的面积。
points of inflection occur at distances σ away from the mean, at x = μ − σ and x = μ + σ.
f(x) 的图像关于直线 x = μ 对称，这意味着正态分布的均值、中位数和众数三者合一。曲线的拐点出现在离均值为 ±σ 处，即 x = μ − σ 和 x = μ + σ 的位置。
3. Mean and Variance | 均值与方差
For a random variable X that follows a normal distribution with mean μ and variance σ², we write X ~ N(μ, σ²). The variance, σ², measures the spread of the distribution; a larger σ flattens the curve and widens the tail region.
对于一个服从均值为 μ、方差为 σ² 的正态分布的随机变量 X，我们记作 X ~ N(μ, σ²)。方差 σ² 度量分布的离散程度；σ 增大时曲线变扁平，尾部区域加宽。
Three important empirical rules hold approximately for any normal distribution:
- About 68% of observations lie within 1 standard deviation of the mean (μ ± 1σ).
- About 95% lie within 2 standard deviations (μ ± 2σ).
- About 99.7% lie within 3 standard deviations (μ ± 3σ).
在任何一个正态分布中，大体上有以下经验规律：
- 约 68% 的观测值落在均值 ±1 个标准差 (μ ± σ) 范围内。
- 约 95% 落在均值 ±2 个标准差 (μ ± 2σ) 范围内。
- 约 99.7% 落在均值 ±3 个标准差 (μ ± 3σ) 范围内。
These percentages are derived from the standard normal table and are invaluable for quick estimates.
这些百分比可以从标准正态表推出，对于快速估算非常有用。
4. The Standard Normal Distribution | 标准正态分布
The simplest case is the standard normal distribution, Z ~ N(0, 1), where μ = 0 and σ = 1. The probability density function of Z reduces to
最简单的情形是标准正态分布 Z ~ N(0, 1)，其中 μ = 0、σ = 1。此时 Z 的概率密度函数简化为
ϕ(z) = 1 / √2π · e‑z²/2
on. The table typically gives the cumulative distribution function Φ(z) = P(Z ≤ z) for non‑negative values of z.
由于任何正态变量都可以转化为 Z，因此仅对标准正态分布出版表格。表通常给出对非负 z 值的累积分布函数 Φ(z) = P(Z ≤ z)。
The shape of Φ(z) is a sigmoid curve that increases from 0 to 1. Because of symmetry, Φ(‑z) = 1 − Φ(z), which allows us to deal with negative z‑values without an explicit table for negative arguments.
Φ(z) 的图像是一条从 0 增加到 1 的 S 形曲线。由于对称性，Φ(‑z) = 1 − Φ(z)，这让我们无需专门为负数列表也能处理负的 z 值。
5. The Z‑Score and Standardisation | Z 分数与标准化
To work with a general normal variable X ~ N(μ, σ²), we convert it to the standard normal variable Z using the transformation
为了处理一般的正态变量 X ~ N(μ, σ²)，我们通过以下变换将其转化为标准正态变量 Z
Z = (X − μ) / σ
The quantity (x − μ) / σ is called the z‑score of x. It measures how many standard deviations x is above or below the mean. A positive z‑score indicates a value above the mean; a negative z‑score indicates a value below the mean.
量 (x − μ) / σ 称为 x 的 z 分数。它度量 x 位于均值上方或下方多少个标准差。正值表示高于均值，负值表示低于均值。
This standardisation is the bridge between real‑world data and the standard normal table. Once we find a z‑score, we can look up the corresponding probability or, inversely, find the x‑value corresponding to a given probability.
标准化过程搭建了现实数据与标准正态表之间的桥梁。一旦求得 z 分数，就能查阅对应的概率，或者反过来，由给定的概率求 x 值。
6. Using Standard Normal Tables | 使用标准正态分布表
f Φ(z) for 0 ≤ z ≤ 4.0, with increments of 0.01. A small extract looks like the one below.
Edexcel 提供了一张关于 0 ≤ z ≤ 4.0（步长 0.01）的 Φ(z) 表。下面是一小段摘录。
| z | Φ(z) |
|---|---|
| 0.00 | 0.5000 |
| 0.50 | 0.6915 |
| 1.00 | 0.8413 |
| 1.50 | 0.9332 |
| 1.96 | 0.9750 |
| 2.00 | 0.9772 |
| 2.58 | 0.9951 |
To use the table, first write the z‑ n the top row. The intersection gives 0.9495, meaning P(Z ≤ 1.64) = 0.9495. For negative z‑values, apply symmetry: Φ(‑1.64) = 1 − 0.9495 = 0.0505.
使用表格时，先将 z 分数写成两位小数。将其拆分为行部分（整数位加第一位小数）和列部分（第二位小数）。例如，要查 Φ(1.64)：在最左列找到 1.6，在最上行找到 0.04，交叉处即为 0.9495，因此 P(Z ≤ 1.64) = 0.9495。对于负的 z 值，利用对称性：Φ(‑1.64) = 1 − 0.9495 = 0.0505。
7. Calculating Probabilities | 计算概率
All probability problems involving a normal variable X ~ N(μ, σ²) are tackled by a three‑step procedure:
所有涉及正态变量 X ~ N(μ, σ²) 的概率问题都可以按三步走：
(1) Restate the question as an interval for X, e.g. P(a < X < b).
(2) Convert the boundaries to z‑scores: z₁ = (a − μ) / σ, z₂ = (b − μ) / σ.
(3) Express the probability in terms of Z using the cumulative table and symmetry. Sketching a bell curve with the relevant region shaded is strongly recommended.
(1) 将问题重新表述为关于 X 的区间，如 P(a < X < b)。
(2) 将上下界转化为 z 分数：z₁ = (a − μ) / σ，z₂ = (b − μ) / σ。
(3) 用累积表和对称性将概率表达成关于 Z 的式子。强烈建议画出钟形曲线并给相关区域涂上阴影。
Common patterns include:
P(Z > z) = 1 − Φ(z).
P(z₁ < Z < z₂) = Φ(z₂) − Φ(z₁).
For intervals symmetric about the mean, use the fact P(‑z < Z < z) = 2Φ(z) − 1.
常见的模式有：
P(Z > z) = 1 − Φ(z)。
P(z₁ < Z < z₂) = Φ(z₂) − Φ(z₁)。
对于关于均值对称的区间，利用 P(‑z < Z < z) = 2Φ(z) − 1 即可。
8. Inverse Normal Calculations | 反向正态计算
In many examination questions you are given a probability and asked to find the corresponding value of x. The technique is the reverse of standardisation.
在许多考题中，会给定一个概率，要求找出对应的 x 值。方法是标准化的逆过程。
First, identify the area to the left of the unknown z‑value. If a right‑tail probability p is given, then the left‑tail area is 1 − p. Search the body of the table for the area closest to the desired value, then read off the corresponding z to two decimal places. If the required critical value is a standard one (like 1.6449 for 95% or 1.9600 for 97.5%), you may use those exact constants.
首先，确定未知 z 值左侧的面积。如果给出的是右尾概率 p，那么左尾面积是 1 − p。在表体中寻找最接近所需面积的数值，然后读出对应的 z 值（保留两位小数）。如果所需的临界值是标准值（如 95% 对应 1.6449，97.5% 对应 1.9600），可以直接使用这些精确常数。
Finally, apply the inverse transformation: x = μ + σz. This equation turns the z‑score back into the original measurement scale.
最后，应用逆变换：x = μ + σz。这个等式将 z 分数还原到原始的测量尺度。
9. Finding Unknown Mean or Standard Deviation | 求未知均值或标准差
p₁ and P(X > b) = p₂. By standardising both boundaries, you obtain two equations linking z‑scores obtained from the tables.
有些题目会提供两条独立的概率信息，从而可以建立关于 μ 和 σ 的联立方程。例如，已知 P(X < a) = p₁ 且 P(X > b) = p₂。将两个边界标准化后，可以得到两个由表查得的 z 分数所构成的方程。
Solving these equations simultaneously yields the unknown parameters. Careful rounding at the final step is essential to maintain accuracy; Edexcel usually expects answers to be given to 3 significant figures or an appropriate degree of accuracy.
解这个方程组就能得到未知参数。最后一步的精确取整非常重要；Edexcel 通常要求答案保留三位有效数字或适当的精度。
10. Normal Approximation to the Binomial | 二项分布的正态近似
When the number of trials n is large, the binomial distribution X ~ B(n, p) can be approximated by a normal distribution Y ~ , where q = 1 − p. The approximation is considered reasonable when both np ≥ 5 and nq ≥ 5.
当试验次数 n 很大时，二项分布 X ~ B(n, p) 可以用正态分布 Y ~ N(np, npq) 来近似，其中 q = 1 − p。通常认为当 np ≥ 5 且 nq ≥ 5 时，近似是合理的。
The rationale comes from the Central Limit Theorem, which states that the sum of independent identically distributed random variables tends to a normal distribution as the sample size grows. A binomial variable is precisely a sum of Bernoulli trials, so the normal approximation is a direct consequence.
其依据来自中心极限定理：独立同分布随机变量之和，随着样本容量增大，趋向于正态分布。二项变量恰是伯努利试验之和，因此正态近似是其直接推论。
a binomial probability such as P(X ≤ r), we calculate z = (r − μ) / σ with μ = np and σ = √npq, but a correction for continuity is required to improve accuracy.
要近似二项概率如 P(X ≤ r)，我们用 μ = np，σ = √npq 计算 z = (r − μ) / σ，但必须进行连续性校正以提高精度。
11. Continuity Correction | 连续性校正
continuity correction adjusts the discrete boundary by ±0.5 to better match the histogram.
由于二项分布是离散的，而正态分布是连续的，用光滑曲线下的面积代替离散概率会引入误差。连续性校正将离散边界加上或减去 0.5，以更好地匹配直方图。
Typical adjustments are:
- P(X ≤ r) → use an upper boundary of r + 0.5.
- P(X ≥ r) → use a lower boundary of r − 0.5.
- P(X = r) → use the interval (r − 0.5) to (r + 0.5).
典型的校正方式为：
- P(X ≤ r) → 上界取 r + 0.5。
- P(X ≥ r) → 下界取 r − 0.5。
- P(X = r) → 用区间 (r − 0.5) 到 (r + 0.5)。
After applying the continuity correction, standardise the adjusted boundary and use the normal table as usual. This small tweak reduces the maximum error and is expected in all Edexcel binomial‑normal approximation answers.
应用连续性校正后，将调整过的边界标准化，并照常使用正态表。这个微调可以减小最大误差，所有 Edexcel 有关二项-正态近似的答案都应包含这一校正。
12. Real‑World Applications and Exam Tips | 实际应用与应考提示
The normal distribution models a vast range of real‑world situations: heights of adults, measurement errors in scientific experiments, quality control on production lines, and even stock returns in finance. Whenever you see a problem about 'the middle 90% of values' or 'the top 5% of performers', you are dealing with normal distribution concepts.
正态分布可对大量真实情境建模：成年人的身高、科学实验中的测量误差、生产线上的质量控制，甚至金融中的股票收益率。每当遇到“中间的 90%”或“前 5% 的表现者”等问题时，就是在处理正态分布的概念。
formula explicitly before substituting numbers. Check whether final answers are reasonable — for example, a probability should lie between 0 and 1, and a z‑score beyond ±4 is extremely rare.
面对 Edexcel 考试，始终要画草图。清楚标出均值和边界，给所求区域涂上阴影。在代入数值之前写出标准化公式。检查最终答案是否合理——例如概率应介于 0 和 1 之间，z 分数超出 ±4 的情况极为罕见。
When approximating a binomial, never forget the continuity correction unless the question explicitly states otherwise. Practising past‑paper questions will build speed in reading tables and recognising symmetric patterns.
在近似二项分布时，除非题目明确说明否则千万别忘记连续性校正。多做历年真题可以提高查表速度和识别对称模式的能力。
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