📚 4: The Standard Normal Distribution | 4: 标准正态分布
The normal distribution is the most important continuous probability distribution in statistics. In A-Level Mathematics, a special case called the standard normal distribution simplifies calculations and forms the basis for working with any normal variable. This article explains key concepts, how to standardise, use tables, and apply the distribution to exam-style problems.
正态分布是统计学中最重要的连续概率分布。在A-Level数学中，一个称为标准正态分布的特殊情形能够简化计算，并构成处理任意正态变量的基础。本文将解释关键概念、如何进行标准化、使用分布表，以及如何将分布应用到考试题型中。
1. Introduction to the Normal Distribution | 正态分布简介
A continuous random variable X follows a normal distribution with mean μ and variance σ² if its probability density function is the familiar bell-shaped curve. The notation is X ~ N(μ, σ²). The curve is symmetric about μ, and its spread is determined by σ.
若连续随机变量 X 的概率密度函数呈现为熟悉的钟形曲线，则称 X 服从均值为 μ、方差为 σ² 的正态分布，记作 X ~ N(μ, σ²)。该曲线关于 μ 对称，其散布程度由 σ 决定。
The total area under the curve is 1, representing total probability. Practical examples include heights, weights, and measurement errors, which often follow an approximately normal distribution.
曲线下总面积为 1，代表总概率。常见的实际例子包括身高、体重和测量误差，它们通常近似服从正态分布。
2. The Standard Normal Distribution Z ~ N(0, 1) | 标准正态分布 Z ~ N(0, 1)
The standard normal distribution is a normal distribution with mean 0 and variance 1. It is denoted by Z ~ N(0, 1), where Z is the standard normal random variable. This distribution is used as a reference to find probabilities for any normal distribution after standardisation.
标准正态分布是均值为 0、方差为 1 的正态分布，记作 Z ~ N(0, 1)，其中 Z 为标准正态随机变量。该分布可作为参考，在对任意正态分布进行标准化之后求算概率。
The probability density function of Z is φ(z) = (1/√2π) e−z²/2 l distributions, tables of cumulative probabilities Φ(z) = P(Z ≤ z) are readily available.
Z 的概率密度函数为 φ(z) = (1/√2π) e−z²/2。与一般正态分布不同，累积概率表 Φ(z) = P(Z ≤ z) 可供直接查阅。
3. Properties of the Standard Normal Curve | 标准正态曲线的性质
The curve extends indefinitely in both directions, approaching but never touching the horizontal axis.
标准正态曲线关于 z = 0 对称，这意味着 Φ(0) = 0.5。曲线向两侧无限延伸，逐渐趋近但永不触及横轴。
Approximately 68% of the area lies within ±1 standard deviation (between z = −1 and z = 1), about 95% within ±2, and 99.7% within ±3. This is the empirical rule.
大约 68% 的面积落在 ±1 个标准差之内（z = −1 到 z = 1 之间），约 95% 落在 ±2 之内，99.7% 落在 ±3 之内。这就是经验法则。
symmetry, P(Z ≤ −z) = 1 − Φ(z). Also, P(−z ≤ Z ≤ z) = 2Φ(z) − 1, a commonly used identity.
由于对称性，P(Z ≤ −z) = 1 − Φ(z)。同时有 P(−z ≤ Z ≤ z) = 2Φ(z) − 1，这是一个常用恒等式。
4. The Standard Normal Table (Cumulative Distribution Function Φ(z)) | 标准正态分布表（累积分布函数 Φ(z)）
le, Φ(1.25) = 0.8944. Most tables give cumulative probabilities from the far left up to z.
标准正态分布表给出了正 z 值对应的 Φ(z) = P(Z ≤ z) 值。例如，Φ(1.25) = 0.8944。多数表格提供从最左端到 z 的累积概率。
w 1.2 and column 0.05 intersect at 0.8944. For negative z, use symmetry: Φ(−z) = 1 − Φ(z).
使用表格时，将 z 拆分为行（前两位数字）和列（第二位小数）。对于 z = 1.25，在行 1.2 与列 0.05 相交处得到 0.8944。对于负 z，利用对称性：Φ(−z) = 1 − Φ(z)。
A small excerpt of the table is shown below:
| z | Φ(z) |
|---|---|
| 0.00 | 0.5000 |
| 0.50 | 0.6915 |
| 1.00 | 0.8413 |
| 1.50 | 0.9332 |
| 2.00 | 0.9772 |
This table confirms that roughly 84% of observations fall below 1 standard deviation above the mean in a standard normal distribution.
该表证实，在标准正态分布中约 84% 的观测值落在高于均值 1 个标准差以内。
5. Standardising a Normal Variable: z = (X − μ) / σ | 正态变量的标准化：z = (X − μ) / σ
Any normal random variable X ~ N(μ, σ²) can be transformed into a standard normal Z by subtracting the mean and dividing by the standard deviation:
z = (x − μ) / σ
任意正态随机变量 X ~ N(μ, σ²) 都可以通过减去均值并除以标准差转换为标准正态变量 Z：
z = (x − μ) / σ
x) = P(Z ≤ z). The z-value tells us how many standard deviations x is away from the mean.
这个变换保持概率不变：P(X ≤ x) = P(Z ≤ z)。z 值告诉我们 x 距离均值有多少个标准差。
- If x > μ, z is positive.
- 若 x > μ，则 z 为正。
- If x < μ, z is negative.
- 若 x < μ，则 z 为负。
Standardisation is essential because it allows us to use the standard normal table for any normal distribution.
标准化至关重要，因为它使我们能够将标准正态分布表用于任意正态分布。
6. Finding Probabilities Using the Standard Normal Table | 使用标准正态分布表求概率
To find P(X < a) for X ~ (μ, σ²), first compute zₐ = (a − μ) / σ, then read Φ(zₐ) from the table. For example, if X ~ N(50, 4²) and we need P(X < 55), z = (55 − 50)/4 = 1.25, giving 0.8944.
要求 X ~ N(μ, σ²) 的 P(X < a)，先计算 zₐ = (a − μ) / σ，再从表中读取 Φ(zₐ)。例如，若 X ~ N(50, 4²) 且需计算 P(X < 55)，则 z = (55 − 50)/4 = 1.25，得到 0.8944。
nt rule: P(X > a) = 1 − Φ(zₐ). For the same distribution, P(X > 55) = 1 − 0.8944 = 0.1056.
若要求 P(X > a)，使用补集法则：P(X > a) = 1 − Φ(zₐ)。对于同一个分布，P(X > 55) = 1 − 0.8944 = 0.1056。
7. Working with Upper-Tail Probabilities | 处理上尾概率
Edexcel tables sometimes provide upper-tail probabilities directly, but it is more common to use the cumulative table. Always sketch a bell curve and shade the required region to avoid confusion.
Edexcel 的表格有时直接给出上尾概率，但更常见的是使用累积概率表。始终绘制钟形曲线并涂出所需区域，以避免混淆。
For P(Z > z), we find 1 − Φ(z). Because of symmetry, P(Z < −z) gives the same value as P(Z > z). This is helpful when dealing with negative values or two-tailed problems.
对于 P(Z > z)，我们计算 1 − Φ(z)。由对称性可知，P(Z < −z) 与 P(Z > z) 相等。这在处理负值或双尾问题时很有帮助。
8. Finding Probabilities for a Range: P(a < X < b) | 求区间概率：P(a < X < b)
nd b, standardise both: z₁ = (a − μ)/σ, z₂ = (b − μ)/σ. Then P(a < X < b) = Φ(z₂) − Φ(z₁).
要求 X 介于两个值 a 与 b 之间的概率，先对两者进行标准化：z₁ = (a − μ)/σ, z₂ = (b − μ)/σ。则 P(a < X < b) = Φ(z₂) − Φ(z₁)。
For instance, with X ~ 13, and Φ(−0.6667) ≈ 0.2525 (using symmetry). The probability is 0.8413 − 0.2525 = 0.5888.
例如，假设 X ~ N(100, 15²)，求 P(90 < X < 115)。z₁ = (90 − 100)/15 = −0.6667，z₂ = (115 − 100)/15 = 1.00。Φ(1.00) = 0.8413，Φ(−0.6667) ≈ 0.2525（利用对称性）。概率为 0.8413 − 0.2525 = 0.5888。
9. Inverse Normal: Finding z-values from Probabilities | 逆正态：由概率求 z 值
that Φ(z) = p by looking for p inside the table and reading the z-value from the margins.
有时会给出概率，需要求出对应的 z 值。这就是逆正态运算。若 P(Z < z) = p，则在表内寻找 p 并从表格边缘读出 z 值，使得 Φ(z) = p。
between 0.9495 (z = 1.64) and 0.9505 (z = 1.65), so linear interpolation gives z ≈ 1.645.
例如，要找使得 P(Z < z) = 0.95 的 z，先在表中找到 0.95（或最接近的值）。它位于 0.9495 (z = 1.64) 和 0.9505 (z = 1.65) 之间，因此线性插值给出 z ≈ 1.645。
ve z. Use the symmetry: if P(Z < z) = 0.1, find z* with Φ(z*) = 0.9, then z = −z* ≈ −1.28.
对于小于 0.5 的下尾概率 p，可能对应负 z 值。利用对称性：若 P(Z < z) = 0.1，找出 Φ(z*) = 0.9 的 z*，则 z = −z* ≈ −1.28。
10. Finding Unknown Mean or Standard Deviation | 求未知的均值或标准差
A typical A-Level question gives two pieces of probability information about a normal distribution and asks for μ or σ. Set up equations using the standardising formula and use inverse normal values.
常见的A-Level题目会给出两条关于正态分布的概率信息，要求求出 μ 或 σ。利用标准化公式建立方程，并使用逆正态值。
Suppose X ~ = −0.674. From P(X > 40) = 0.1, P(X < 40) = 0.9, giving z ≈ 1.282, so (40 − μ)/σ = 1.282.
假设 X ~ N(μ, σ²)。已知 P(X < 20) = 0.25 且 P(X > 40) = 0.1。由 P(X < 20) = 0.25，0.25 对应的 z 值约为 −0.674（因 Φ(−0.674) ≈ 0.25）。因此 (20 − μ)/σ = −0.674。由 P(X > 40) = 0.1，得 P(X < 40) = 0.9，对应 z ≈ 1.282，于是 (40 − μ)/σ = 1.282。
Solve the two equations simultaneously: subtract the first from the second to eliminate μ, find σ, then back-substitute to obtain μ. This is a standard exam technique.
联立两式求解：用第二式减去第一式以消去 μ，求出 σ，再回代求得 μ。这是标准的考试技巧。
11. Approximating Binomial and Other Distributions (Brief) | 二项分布等的正态近似（简述）
When a binomial distribution X ~ ontinuity correction of ±0.5 is applied when converting a discrete variable to continuous.
当二项分布 X ~ B(n, p) 的 n 很大且 p 接近 0.5 时，可以用均值为 μ = np、方差为 σ² = np(1 − p) 的正态分布来近似。从离散变量转换为连续变量时需进行 ±0.5 的连续性校正。
For example, P(X ≤ 10) for binomial becomes approximately P(Y < 10.5) using Y ~ tandard normal distribution is then used to compute the probability after standardising Y.
例如，二项分布的 P(X ≤ 10) 可近似为使用 Y ~ N(np, npq) 的 P(Y < 10.5)。对 Y 进行标准化后，即可用标准正态分布计算概率。
This topic appears in the Edexcel specification, but standard normal skills remain the central focus.
该主题出现在 Edexcel 考纲中，但标准正态的技能仍是重中之重。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
- Forgetting to standardise: Always convert your X-value to a z-value before using the table.
- 忘记标准化：在使用表格前，务必将 X 值转换为 z 值。
- Mixing up σ and σ²: The standardising formula uses σ, not the variance. Take the square root if given variance.
- 混淆 σ 与 σ²：标准化公式中使用的是标准差 σ，而不是方差。若已知方差，需先开方。
- Symmetry errors: When looking up negative z or reverse probabilities, draw a sketch to confirm your method.
- 对称性错误：在查找负 z 值或逆概率时，绘制草图以确认方法正确。
- Continuity correction: Needed for binomial approximations, but not for normal distributions themselves.
- 连续性校正：仅在二项分布近似时需要，本身的正态分布不需要。
- Always state the distribution: Begin with 'Let X ~ N(μ, σ²)' and show standardisation steps clearly.
- 始终写明分布：以“设 X ~ N(μ, σ²)”开头，并清晰展示标准化步骤。
The standard normal distribution is a powerful tool; mastery comes from practice with varied problems and careful use of the table.
标准正态分布是一种强有力的工具；熟练掌握需要通过练习不同类型的题目并仔细使用分布表。
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