📚 C - The standard normal distribution (Z-distribution) | C - 标准正态分布（Z分布）
The normal distribution is one of the most fundamental concepts in statistics and appears throughout the IB Mathematics curriculum. It models a vast range of natural and social phenomena, from heights and weights to measurement errors and test scores. Within this family of bell-shaped curves, the standard normal distribution — also called the Z‑distribution — plays a special role as the benchmark against which all other normal distributions are compared.
正态分布是统计学中最基本的概念之一，贯穿 IB 数学课程始终。它为从身高、体重到测量误差和考试成绩等各种自然和社会现象建模。在这族钟形曲线中，标准正态分布（也称 Z 分布）扮演着特殊角色，它是所有其他正态分布与之比较的基准。
1. The shape and parameters of a normal curve | 正态曲线的形状与参数
A normal distribution is a continuous probability distribution whose graph is a symmetric, bell‑shaped curve centred at the population mean μ. The spread of the curve is determined by the population standard deviation σ: a larger σ produces a wider, flatter bell, while a smaller σ makes the curve tall and narrow.
正态分布是一种连续概率分布，其图形是一条以总体均值 μ 为中心的对称钟形曲线。曲线的扩散程度由总体标准差 σ 决定：σ 越大，钟形越宽越扁；σ 越小，曲线越高越窄。
The probability density function (pdf) of a general normal variable X ~ N(μ, σ²) is given by:
一般正态变量 X ~ N(μ, σ²) 的概率密度函数为：
f(x) = (1/(σ√2π)) e−½[(x−μ)/σ]²
rve never touches the horizontal axis, meaning extreme values are possible but increasingly unlikely.
曲线下总面积等于 1，X 落在两个值 a 和 b 之间的概率就是 x = a 和 x = b 之间曲线下的面积。曲线永远不会触及横轴，这意味着极端值可能出现，但概率越来越小。
2. Why standardise? The need for a common scale | 为什么要标准化？——统一尺度的需要
There are infinitely many normal distributions, each with its own μ and σ. To make meaningful comparisons or to use statistical tables, we transform any normal variable into a standard measure that removes the units and centres the data at zero. This transformation is called standardisation.
正态分布有无穷多种，每一种都有自己的 μ 和 σ。为了进行有意义的比较或使用统计表，我们将任意正态变量转换为一种去除单位并以零为中心的标准度量，这一变换称为标准化。
The result of standardisation is a new random variable, conventionally denoted by Z, that tells us how many standard deviations an observation lies above or below its mean. Because the transformation is linear, Z retains the normality of the original data, but with a mean of 0 and a variance of 1.
标准化的结果是一个新随机变量，习惯上记作 Z，它告诉我们一个观测值距离其均值多少个标准差。由于变换是线性的，Z 保留了原始数据的正态性，但其均值为 0，方差为 1。
3. Defining the standard normal distribution | 标准正态分布的定义
The standard normal distribution, written Z ~ s mean μ = 0 and standard deviation σ = 1. Its probability density function simplifies to:
标准正态分布记作 Z ~ N(0, 1)，均值为 0，标准差为 1。其概率密度函数简化为：
φ(z) = (1/√2π) e−z²/2
ormal curve to the left of z. Because the total area is 1, we have Φ(∞) = 1 and Φ(−∞) = 0.
Z 的累积分布函数记为 Φ(z)。对任意实数 z，Φ(z) = P(Z ≤ z)，该值代表标准正态曲线下 z 左侧的面积。由于总面积为 1，有 Φ(∞) = 1，Φ(−∞) = 0。
Φ(z). This means the probability of obtaining a value less than −z equals the probability of exceeding +z, which often simplifies calculations.
对称性是一个关键性质：Φ(−z) = 1 − Φ(z)。这意味着取值小于 −z 的概率等于取值大于 +z 的概率，这常常能简化计算。
4. The Z‑score: measuring distance in standard deviations | Z 分数：用标准差度量距离
For any normally distributed variable X ~ N(μ, σ²), the Z‑score of an observation x is defined as:
对于任意正态变量 X ~ N(μ, σ²)，观测值 x 的 Z 分数定义为：
z = (x − μ) / σ
z = (75 − 60)/10 = 1.5, showing the student is 1.5 standard deviations above the average.
正的 z 表示 x 高于均值，负的 z 表示 x 低于均值。例如，若某次考试成绩 μ = 60，σ = 10，则 75 分的 Z 分数为 z = (75 − 60)/10 = 1.5，表明该生高出平均分 1.5 个标准差。
The Z‑score is dimensionless, allowing us to compare observations from completely different normal populations on one common scale. This is the heart of the standard normal distribution’s practical power.
Z 分数没有量纲，使我们能够在统一尺度上比较来自完全不同的正态总体的观测值，这正是标准正态分布在实践中强大威力的核心。
5. Properties of the Z‑distribution | Z 分布的性质
Because Z ~ tric about 0, and the points of inflection occur at z = ±1. Furthermore, the empirical (68–95–99.7) rule applies directly:
由于 Z ~ N(0, 1)，其均值、中位数和众数均为 0。分布关于 0 完全对称，拐点出现在 z = ±1 处。此外，经验法则（68–95–99.7 法则）直接适用：
- P(−1 < Z < 1) ≈ 0.6827 (about 68% of data lies within 1 standard deviation of the mean). / P(−1 < Z < 1) ≈ 0.6827（约 68% 的数据落在均值 1 个标准差内）。
- P(−2 < Z < 2) ≈ 0.9545 (about 95%). / P(−2 < Z < 2) ≈ 0.9545（约 95%）。
- < Z < 3) ≈ 0.9973 (nearly all data within 3 standard deviations). / P(−3 < Z < 3) ≈ 0.9973（几乎所有数据在 3 个标准差内）。
These facts are helpful for quick checks and understanding how rare extreme Z‑values are.
这些事实有助于快速核查并理解极端 Z 值有多罕见。
6. Reading the standard normal table | 查阅标准正态分布表
The standard normal table (Z‑ mal place across the columns. For example, to find Φ(1.25), locate row 1.2 and column 0.05, yielding 0.8944 — meaning 89.44% of the area lies to the left of z = 1.25.
标准正态分布表（Z 表）提供了 z ≥ 0 时的累积概率 Φ(z) = P(Z ≤ z)。典型的表格行给出 z 的整数与第一位小数，列给出第二位小数。例如，查找 Φ(1.25)，先找到行 1.2 和列 0.05，得到 0.8944——意味着 z = 1.25 左侧面积为 89.44%。
| z | Φ(z) |
|---|---|
| 0.00 | 0.5000 |
| 0.50 | 0.6915 |
| 1.00 | 0.8413 |
| 1.50 | 0.9332 |
| 2.00 | 0.9772 |
| 2.50 | 0.9938 |
| 3.00 | 0.9987 |
les give probabilities for the right tail, so always check which probability is displayed.
对于负的 z 值，利用对称性：Φ(−z) = 1 − Φ(z)。因此 Φ(−1.25) = 1 − 0.8944 = 0.1056。有些表给出右尾概率，因此务必先确认表格显示的是哪一类概率。
7. Calculating less than, greater than and between probabilities | 计算“小于”、“大于”和“介于”概率
Once you have Z‑scores, all probability questions can be answered by combining Φ‑values with the concepts of complement and symmetry.
一旦有了 Z 分数，所有概率问题都可以通过结合 Φ 值与补集和对称性概念来回答。
P(Z < a) = Φ(a) — read directly from the table if a ≥ 0, or use Φ(a) = 1 − Φ(|a|) if a < 0.
P(Z < a) = Φ(a)，若 a ≥ 0 可直接查表，若 a < 0 则利用 Φ(a) = 1 − Φ(|a|)。
P(Z > a) = 1 − Φ(a). For example, P(Z > 1.25) = 1 − 0.8944 = 0.1056.
P(Z > a) = 1 − Φ(a)。例如，P(Z > 1.25) = 1 − 0.8944 = 0.1056。
P(a < Z < b) ve, subtract the smaller Φ from the larger. If a is negative and b positive, use Φ(b) − Φ(a) where Φ(a) may be found via symmetry.
P(a < Z < b) = Φ(b) − Φ(a)。若 a 和 b 都为正，用较大的 Φ 减去较小的。若 a 为负、b 为正，可通过对称性求出 Φ(a) 再相减。
the original X scale only requires substituting z = (x − μ)/σ after finding the desired z‑values.
将这些概率转回原始 X 尺度，只需在求出所需的 z 值后代回 z = (x − μ)/σ 即可。
8. Inverse normal: from probability to Z‑value | 逆正态：从概率求 Z 值
Sometimes you are given a probability and asked to find the corresponding Z‑ mal problem. For instance, find z such that P(Z < z) = 0.95. Look inside the body of the Z‑ 500; you will find 0.9495 (z = 1.64) and 0.9505 (z = 1.65). Interpolation gives z ≈ 1.645.
有时题目会给出概率，要求找出对应的 Z 分数或数据值，这称为逆正态问题。例如，求满足 P(Z < z) = 0.95 的 z。在 Z 表主体内寻找最接近 0.9500 的概率，可找到 0.9495（z = 1.64）和 0.9505（z = 1.65），插值得到 z ≈ 1.645。
If you need the value x on the original scale, use the un‑standardising formula: x = μ + zσ. Thus, if μ = 50 and σ = 8, the 95th percentile is roughly 50 + 1.645 × 8 ≈ 63.16.
若需要原始尺度的值 x，使用去标准化公式：x = μ + zσ。因此，若 μ = 50，σ = 8，第 95 百分位数约为 50 + 1.645 × 8 ≈ 63.16。
Modern calculators and software offer inverse normal functions, but the table method is essential for building deep understanding and is often tested in exams.
现代计算器和软件提供逆正态函数，但表格方法对建立深刻理解至关重要，也常出现在考试中。
9. Verifying normality assumptions | 验证正态性假设
Applying the standard normal distribution assumes that the underlying data are normally distributed. In IB problems, you must sometimes check this condition, for example by looking at a histogram, a box plot or a normal probability plot. If the data are heavily skewed or have outliers, the normal model may be inappropriate.
使用标准正态分布的前提是原始数据服从正态分布。在 IB 考题中，有时需要检验这一条件，例如通过观察直方图、箱形图或正态概率图。如果数据严重偏斜或存在离群值，正态模型可能不适用。
Exam questions often state “assume the population is normally distributed” or provide summary statistics from a large, symmetric sample. When this is given, you can proceed with confidence that the z‑procedures are valid.
考题通常会说明“假设总体服从正态分布”，或提供来自大样本且对称的汇总统计量。当给出此类条件时，你就可以放心地使用 Z 方法。
10. Worked example: exam marks | 例题解析：考试成绩
Problem: A large class’s final marks are normally distributed with mean 68 and standard deviation 12. Find the proportion of students scoring (i) above 80, (ii) between 50 and 80, and (iii) find the mark that separates the top 10% of students.
题目：某大班期末成绩服从正态分布，均值为 68，标准差为 12。求 (i) 得分超过 80 的学生比例，(ii) 得分在 50 到 80 之间的比例，(iii) 划分出前 10% 学生的分数。
2 = −1.50. From the table, Φ(1.00) = 0.8413, Φ(−1.50) = 1 − Φ(1.50) = 1 − 0.9332 = 0.0668. (i) P(X > 80) = P(Z > 1.00) = 1 − 0.8413 = 0.1587 ≈ 15.9%. (ii) P(50 < X < 80) = P(−1.50 < Z < 1.00) = 0.8413 − 0.0668 = 0.7745 ≈ 77.5%. 82. Then x = μ + zσ = 68 + 1.282 × 12 ≈ 83.4. So a mark of about 83.4 separates the top 10%.
标准化：z₁ = (80−68)/12 = 1.00；z₂ = (50−68)/12 = −1.50。查表得 Φ(1.00)=0.8413，Φ(−1.50)=1−Φ(1.50)=1−0.9332=0.0668。
(i) P(X>80)=P(Z>1.00)=1−0.8413=0.1587≈15.9%。
(ii) P(50
11. Common mistakes and how to avoid them | 常见错误与避免方法
Mixing up Φ(z) and right‑tail probabilities. Always sketch a quick bell curve and shade the area you need. Label the z‑cutoff clearly. This visual check prevents you from using the complement incorrectly.
混淆 Φ(z) 和右尾概率。 始终快速画一条钟形曲线并给所需区域涂上阴影，清晰地标出 z 截断值。这个视觉检查能防止你错误地使用补集。
Forgetting to square σ when writing the variance. Remember, N(μ, σ²) means the second parameter is variance. If a problem gives standard deviation σ, you must square it when plugging into formulas that require variance.
忘记在写方差时将 σ 平方。 记住，N(μ, σ²) 表示第二个参数是方差。如果题目给出标准差 σ，在代入需要方差的公式时一定要平方。
Misreading the z‑table. e . Check whether your table gives cumulative area from -∞ (left tail) or something else. When in doubt, note that Φ(0) must equal 0.5 — this quickly confirms the table’s direction.
读错 Z 表。 确认你的表给出的是从 −∞ 开始的累积面积（左尾）还是其它。有疑问时，注意 Φ(0) 必然等于 0.5——这能快速确认表格的方向。
12. Summary and key connections | 总结与关键联系
The standard normal distribution transforms any normal problem into a universal, unit‑free framework. By mastering the Z‑score, the symmetry of the N(0,1) curve, and the use of the Z‑table (or calculator functions), you can solve a huge variety of probability questions efficiently. This topic also forms the foundation for hypothesis testing, confidence intervals, and many further statistical methods you will encounter in IB Mathematics and beyond.
标准正态分布将任意正态问题转化为一个统一、无单位的框架。通过掌握 Z 分数、N(0,1) 曲线的对称性以及 Z 表（或计算器功能）的使用，你可以高效地解决大量多样的概率问题。这一主题也为假设检验、置信区间以及你在 IB 数学及更远课程中会遇到的许多高级统计方法奠定了基础。
Always connect the geometry of the bell curve with numerical probabilities. If you can visualise the area, the arithmetic becomes straightforward. Practise with both table lookup and technology, and you will find the Z‑distribution a powerful and reliable tool.
始终将钟形曲线的几何形状与数值概率结合起来。如果你能将面积可视化，算术就会变得简单明了。通过表格查阅和技术工具并行练习，你会发现 Z 分布是一个强大且可靠的工具。
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