📚 The Standard Normal Distribution (Z-Distribution) | 标准正态分布（Z分布）
The standard normal distribution, often called the Z-distribution, is a special normal distribution with a mean of 0 and a standard deviation of 1. It serves as a universal reference that allows us to calculate probabilities for any normal distribution through a process called standardisation. In IB Mathematics, mastering the Z-distribution is essential for hypothesis testing, confidence intervals, and solving a wide range of real-world probability problems.
标准正态分布，通常称为Z分布，是一种均值为0、标准差为1的特殊正态分布。它作为一个通用参考，通过标准化的过程，我们可以计算任何正态分布的概率。在IB数学中，掌握Z分布对于假设检验、置信区间以及解决各种现实概率问题至关重要。
1. Definition and Notation | 定义与符号
The standard normal distribution is denoted by N(0, 1²) or simply N(0, 1). A random variable that follows this distribution is usually written as Z, so we say Z ~ N(0, 1). The probability density function is symmetric about zero, and the total area under the curve equals 1.
标准正态分布记为 N(0, 1²) 或简写为 N(0, 1)。服从该分布的随机变量通常写作 Z，因此我们说 Z ~ N(0, 1)。其概率密度函数关于零对称，曲线下的总面积为1。
2. Key Properties | 关键特性
ution is symmetric and bell-shaped. Its mean, median, and mode are all 0. Approximately 68% of the data lies within ±1 standard deviation, 95% within ±1.96, and 99.7% that Z equals any single value is zero; we always calculate probabilities over intervals.
Z分布是对称的钟形曲线。其均值、中位数和众数均为0。约68%的数据落在±1个标准差内，95%落在±1.96内，99.7%落在±2.58内。由于是连续分布，Z取任何单一值的概率为零；我们总是计算区间上的概率。
3. Standardisation (Z-score) | 标准化（Z分数）
Any normal random variable X ~ N(μ, σ²) can be transformed into a standard normal variable Z using the formula:
任何正态随机变量 X ~ N(μ, σ²) 都可以通过以下公式转化为标准正态变量 Z：
Z = (X − μ) / σ
This process is called standardisation, and the resulting Z value tells us how many standard deviations X is above or below the mean. A positive Z indicates X is above the mean; a negative Z indicates it is below.
这个过程称为标准化，得到的Z值告诉我们 X 距离均值有多少个标准差。正的 Z 表示 X 高于均值；负的 Z 表示低于均值。
4. Understanding the Z-Table | 理解Z表
ecimal in the columns, and the body shows the cumulative probability from −∞ up to that z.
累积标准正态表，即Z表，给出了给定z值下的概率 Φ(z) = P(Z ≤ z)。大多数IB考试局都会提供这样的表格。表格通常以一位小数的z值作为行，第二位小数作为列，表内显示从−∞ 到该z值的累积概率。
| z | Φ(z) = P(Z ≤ z) |
|---|---|
| 0.00 | 0.5000 |
| 1.00 | 0.8413 |
| 1.645 | 0.9500 |
| 1.96 | 0.9750 |
| 2.58 | 0.9951 |
These key values are worth memorising because they are frequently used for confidence intervals and significance tests.
这些关键值值得记住，因为它们常用于置信区间和显著性检验。
5. Left-Tail Probabilities | 左尾概率
n the Z-table. For example, P(Z < 1.25) = Φ(1.25) ≈ 0.8944. This tells us that about 89.44% of the standard normal distribution lies below 1.25.
要计算 Z 小于给定值 a 的概率，我们只需在Z表中查找 Φ(a)。例如，P(Z < 1.25) = Φ(1.25) ≈ 0.8944。这告诉我们，标准正态分布中约89.44%的区域位于1.25以下。
roperty: Φ(−a) = 1 − Φ(a). For instance, P(Z < −0.75) = 1 − Φ(0.75) ≈ 1 − 0.7734 = 0.2266.
如果 a 是负数，我们利用对称性：Φ(−a) = 1 − Φ(a)。例如，P(Z < −0.75) = 1 − Φ(0.75) ≈ 1 − 0.7734 = 0.2266。
6. Right-Tail Probabilities | 右尾概率
to the right of a. For example, P(Z > 1.645) = 1 − 0.9500 = 0.0500, which is exactly the 5% significance level often used in one-tailed tests.
Z 大于某值 a 的概率由 P(Z > a) = 1 − Φ(a) 给出，表示曲线下 a 右侧的面积。例如，P(Z > 1.645) = 1 − 0.9500 = 0.0500，这正是单尾检验中常用的5%显著性水平。
Because of symmetry, P(Z > a) for positive a is equal to P(Z < −a). This relationship is extremely useful when your Z-table only provides positive z-values.
由于对称性，正数 a 的 P(Z > a) 等于 P(Z < −a)。当你的Z表只提供正值z时，这个关系非常有用。
7. Probabilities Between Two Values | 两值之间的概率
− 0.6915 = 0.2417. This gives the proportion of the distribution falling between a and b.
要计算 P(a < Z < b)，我们用累积概率相减：Φ(b) − Φ(a)。例如，P(0.5 < Z < 1.5) = Φ(1.5) − Φ(0.5) ≈ 0.9332 − 0.6915 = 0.2417。这给出了落在 a 和 b 之间的分布比例。
.2 < Z < 0.8) = Φ(0.8) − Φ(−1.2) = Φ(0.8) − [1 − Φ(1.2)] = 0.7881 − (1 − 0.8849) = 0.6730.
当上限或下限为负数时，要仔细处理符号，运用对称性。例如，1.2 < Z < 0.8) = Φ(0.8) − Φ(−1.2) = Φ(0.8) − [1 − Φ(1.2)] = 0.7881 − (1 − 0.8849) = 0.6730。
8. Inverse Normal: Finding Z from Probability | 逆正态：从概率求Z
o find z such that P(Z < z) = 0.975, we search the table for 0.9750 and discover z ≈ 1.96.
我们经常需要找到对应给定累积概率 p 的 Z 分数，这称为逆正态问题。我们在Z表内部查找最接近 p 的值，然后读出相应的 z 值。例如，要找 z 使得 P(Z < z) = 0.975，我们在表中寻找 0.9750，发现 z ≈ 1.96。
rst convert it to a left-tail probability: P(Z < z) = 1 − 0.01 = 0.99, then find z ≈ 2.33.
如果给定的是右尾概率，如 P(Z > z) = 0.01，我们先将其转化为左尾概率：P(Z < z) = 1 − 0.01 = 0.99，然后找到 z ≈ 2.33。
When using a calculator or GDC, the invNorm function can perform this directly; however, understanding the table method is crucial for paper-based exams.
使用计算器或GDC时，invNorm函数可以直接完成这一操作；然而，理解查表法对于笔试考试至关重要。
9. Application: Exam Scores | 应用：考试分数
In IB exams, a typical problem involves converting a normally distributed raw score into a Z-score to find percentiles or the probability of scoring above a certain mark. For instance, suppose exam scores are normally distributed with mean 62 and standard deviation 8. What proportion of students score above 74?
在IB考试中，一个典型的问题是将正态分布的原始分数转化为Z分数，以查找百分位数或得分高于某个分数的概率。例如，假设考试分数服从均值为62、标准差为8的正态分布，有多少比例的学生得分高于74？
8 = 1.5. Then P(X > 74) = P(Z > 1.5) = 1 − Φ(1.5) ≈ 1 − 0.9332 = 0.0668. Hence, about 6.68% of students score above 74.
首先计算 Z = (74 − 62) / 8 = 12 / 8 = 1.5。然后 P(X > 74) = P(Z > 1.5) = 1 − Φ(1.5) ≈ 1 − 0.9332 = 0.0668。因此，大约6.68%的学生得分高于74。
Z-scores are also used to compare performances across different assessments. A student with a higher Z-score has performed better relative to the respective group, regardless of the original scale.
Z分数也用于比较不同评估中的表现。Z分数更高的学生相对于各自群体表现得更好，无论原始尺度如何。
10. Common Mistakes and Tips | 常见错误与提示
One frequent error is confusing the standard deviation of the sampling distribution (standard error) with the population standard deviation when standardising sample means. Remember to use σ/√n for the distribution of sample means.
一个常见错误是在标准化样本均值时，将抽样分布的标准差（标准误）与总体标准差混淆。记住，样本均值的分布要用 σ/√n。
ulative probability from −∞. Always check what your table provides and adjust accordingly.
另一个错误是错误地解读Z表：有些表格给出的是从0到z的面积，而非从−∞ 开始的累积概率。务必检查你所用表格提供的内容并进行相应调整。
When using inverse normal, ensuring the correct tail probability is critical. Sketching the normal curve and shading the required area can help avoid many simple errors.
在使用逆正态时，确保正确的尾部概率至关重要。绘制正态曲线并涂鸦所需区域有助于避免许多简单错误。
11. Summary | 总结
The standard normal distribution is a cornerstone of inferential statistics. By transforming any normal variable into a Z-score, we can standardise the problem and use a single probability table. The key skills are calculating Z-scores, using the cumulative table for left, right, and between probabilities, and performing inverse lookups. With practice, these techniques become straightforward and open the door to confidence intervals and hypothesis testing in the IB syllabus.
标准正态分布是推断统计学的基石。通过将任何正态变量转化为Z分数，我们可以将问题标准化并使用统一的概率表。核心技能包括计算Z分数、使用累积表求解左尾、右尾和区间概率，以及执行逆向查找。经过练习，这些技巧将变得简单，并为IB课程中的置信区间和假设检验打开大门。
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