📚 The Standard Normal Distribution (Z-Distribution) Explained | 标准正态分布（Z分布）讲解
The normal distribution is one of the most important continuous probability distributions in statistics. When a normal distribution has a mean of 0 and a standard deviation of 1, it is called the standard normal distribution, often denoted as Z. Understanding the Z-distribution allows IB students to calculate probabilities, compare data sets, and solve real-world problems involving normally distributed variables.
正态分布是统计学中最重要的连续概率分布之一。当一个正态分布的均值为 0、标准差为 1 时，我们称之为标准正态分布，通常用 Z 表示。理解 Z 分布可以帮助 IB 学生计算概率、比较不同数据集，并解决涉及正态分布变量的实际问题。
1. The Normal Distribution and Its Parameters | 正态分布及其参数
A normal distribution is a symmetric, bell-shaped distribution completely described by two parameters: the mean μ and the standard deviation σ. The mean determines the center of the curve, while the standard deviation determines the spread. The total area under the curve equals 1, representing the total probability.
正态分布是一种对称的钟形分布，由两个参数完全决定：均值 μ 和标准差 σ。均值决定曲线的中心位置，标准差决定曲线的分散程度。曲线下的总面积为 1，代表总概率。
The probability density function of a normal distribution is given by
正态分布的概率密度函数为
f(x) = (1 / (σ √(2π))) · e^(−(x−μ)² / (2σ²))
This formula is rarely used directly in IB exams. Instead, we use the property that any normal distribution can be transformed into the standard normal distribution.
这个公式在 IB 考试中很少直接使用。相反，我们利用任何正态分布都可以转化为标准正态分布这一性质。
2. Standardization: Introducing the Z-Score | 标准化：引入 Z 分数
The Z-score measures how many standard deviations a data point x is away from the mean. The transformation formula is:
Z 分数衡量数据点 x 距离均值有多少个标准差。转换公式为：
Z = (x − μ) / σ
Here, μ is the mean and σ is the standard deviation of the original normal distribution. After this transformation, Z follows the standard normal distribution with mean 0 and standard deviation 1.
其中 μ 是原正态分布的均值，σ 是标准差。经过这一转换后，Z 服从均值为 0、标准差为 1 的标准正态分布。
A positive Z-score indicates the value is above the mean, while a negative Z-score indicates it is below the mean. For example, if μ = 70 and σ = 5, then x = 80 gives Z = (80 − 70) / 5 = 2, meaning 80 is 2 standard deviations above the mean.
正的 Z 分数表示该值高于均值，负的 Z 分数表示低于均值。例如，若 μ = 70，σ = 5，则 x = 80 对应的 Z = (80 − 70) / 5 = 2，说明 80 比均值高 2 个标准差。
3. Properties of the Standard Normal Distribution | 标准正态分布的性质
1. The standard normal distribution has mean 0 and standard deviation 1.
1. 标准正态分布的均值为 0，标准差为 1。
2. It is perfectly symmetric about the vertical line Z = 0.
2. 它关于 Z = 0 这条竖线完全对称。
3. The total area under the curve is 1.
3. 曲线下的总面积为 1。
4. The curve approaches the horizontal axis but never touches it, forming asymptotes at both ends.
4. 曲线向横轴无限延伸但永不相交，两端形成渐近线。
5. Approximately 68% of the data lies within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This is known as the Empirical Rule.
5. 约 68% 的数据位于均值左右 1 个标准差内，95% 位于 2 个标准差内，99.7% 位于 3 个标准差内。这称为经验法则。
These properties are fundamental when interpreting probabilities in IB questions.
这些性质是解读 IB 题目中概率的基础。
4. Probabilities and Area Under the Curve | 概率与曲线下面积
For a continuous distribution, the probability that X takes any single exact value is 0. Instead, we calculate the probability that X lies in an interval, which corresponds to the area under the curve over that interval.
对于连续分布，X 恰好取某个具体值的概率为 0。我们通常计算 X 落在某个区间内的概率，这对应于该区间上曲线下的面积。
Using the standard normal variable Z, we write probabilities as:
使用标准正态变量 Z，我们将概率表示为：
P(Z < a), P(Z > a), P(a < Z < b)
By symmetry, P(Z < −a) = P(Z > a) and P(Z > −a) = P(Z < a). These identities are extremely useful for finding probabilities from standard normal tables or calculators.
根据对称性，P(Z < −a) = P(Z > a)，且 P(Z > −a) = P(Z < a)。这些恒等式在通过标准正态表或计算器查找概率时非常有用。
For example, P(Z < 1.5) is the area to the left of Z = 1.5. In many IB exams, you will use a GDC (graphical display calculator) or a provided z-table to find such values.
例如，P(Z < 1.5) 是 Z = 1.5 左侧的面积。在许多 IB 考试中，你可以使用图形计算器（GDC）或提供的 z 表来查找这些值。
5. Using the Z-Table and GDC | 使用 Z 表和计算器
A standard normal table (z-table) usually gives the area to the left of a given z-score, i.e., P(Z < z). For example, to find P(Z < 1.25), locate 1.2 in the row and 0.05 in the column; the table entry gives the probability.
标准正态表（z 表）通常给出给定 z 分数左侧的面积，即 P(Z < z)。例如，要求 P(Z < 1.25)，在行中找到 1.2，在列中找到 0.05，交叉处的表值即为概率。
On a GDC (such as a TI-84 or Casio), you can use the normalcdf function:
在图形计算器（如 TI-84 或 Casio）上，可以使用 normalcdf 函数：
normalcdf(lower, upper, μ, σ)
For the standard normal distribution, set μ = 0 and σ = 1. To find P(Z < 1.5), enter normalcdf(−∞, 1.5, 0, 1). In practice, use a very large negative number like −10⁹ for negative infinity.
对于标准正态分布，设 μ = 0，σ = 1。要求 P(Z < 1.5)，输入 normalcdf(−∞, 1.5, 0, 1)。实际操作中，可以用一个很大的负数如 −10⁹ 来表示负无穷。
For P(Z > a), use normalcdf(a, ∞, 0, 1). This directly gives the right-tail probability.
对于 P(Z > a)，使用 normalcdf(a, ∞, 0, 1)。这会直接给出右尾概率。
6. Inverse Problems: Finding Z from a Probability | 反向问题：由概率求 Z
Sometimes the probability is given and we need to find the corresponding z-score. This is called an inverse normal problem. On a GDC, use the invNorm function:
有时概率已知，需要求出对应的 z 分数。这称为正态分布的反向问题。在图形计算器上，使用 invNorm 函数：
invNorm(area, μ, σ)
Here, “area” represents the probability to the left of the unknown z-score. For example, if P(Z < z) = 0.8, then z = invNorm(0.8, 0, 1) ≈ 0.8416.
这里的“area”表示未知 z 分数左侧的概率。例如，若 P(Z < z) = 0.8，则 z = invNorm(0.8, 0, 1) ≈ 0.8416。
For right-tail problems, remember to use 1 − probability as the left area. If P(Z > z) = 0.05, then P(Z < z) = 0.95, so z = invNorm(0.95) ≈ 1.645.
对于右尾问题，记得使用 1 − 概率作为左侧面积。若 P(Z > z) = 0.05，则 P(Z < z) = 0.95，因此 z = invNorm(0.95) ≈ 1.645。
7. Converting Real Data to Z-Scores | 将实际数据转换为 Z 分数
Given a normal distribution with mean μ and standard deviation σ, the probability that X is less than a value x can be found by first computing the z-score and then using the standard normal table:
给定均值为 μ、标准差为 σ 的正态分布，要求 X 小于某个值 x 的概率，可以先计算 z 分数，再使用标准正态表：
P(X < x) = P(Z < (x − μ) / σ)
This is the most common procedure in IB exam questions. For example, if test scores are normally distributed with μ = 60 and σ = 8, then P(X < 70) can be calculated by Z = (70 − 60) / 8 = 1.25, so P(Z < 1.25) ≈ 0.8944.
这是 IB 考试中最常见的步骤。例如，考试成绩服从正态分布，μ = 60，σ = 8，则 P(X < 70) 可通过 Z = (70 − 60) / 8 = 1.25 计算，因此 P(Z < 1.25) ≈ 0.8944。
Similarly, for an interval a < X < b, convert both endpoints to z-scores and subtract the cumulative probabilities:
类似地，对于区间 a < X < b，将两个端点转换为 z 分数，然后相减累积概率：
P(a < X < b) = P(Z < z_b) − P(Z < z_a)
This method works for any normal distribution, making the z-distribution the bridge between real data and probability.
这种方法适用于任何正态分布，因此 z 分布成为连接实际数据与概率之间的桥梁。
8. Common Exam Mistakes | 常见考试错误
1. Using the wrong mean or standard deviation when standardizing. Always check whether the question gives μ and σ or asks for the z-score directly.
1. 标准化时用错均值或标准差。做题时务必检查题目给出的是 μ 和 σ，还是直接求 z 分数。
2. Forgetting to use 1 − P for right-tail probabilities. Many students compute P(Z < a) when asked for P(Z > a).
2. 求右尾概率时忘记使用 1 − P。许多学生在题目要求 P(Z > a) 时却计算了 P(Z < a)。
3. Confusing inverse normal with normalcdf. If the probability is given and you need the boundary value, use invNorm, not normalcdf.
3. 混淆反向正态与 normalcdf。如果已知概率需要求边界值，应使用 invNorm，而不是 normalcdf。
4. Ignoring symmetry. For negative z-values, use P(Z < −a) = P(Z > a) to simplify calculations.
4. 忽略对称性。对于负的 z 值，利用 P(Z < −a) = P(Z > a) 来简化计算。
5. Rounding too early. Keep at least 4 decimal places during intermediate steps to maintain accuracy in the final answer.
5. 过早四舍五入。中间步骤至少保留 4 位小数，以保证最终答案的准确性。
9. Worked Example (IB Style) | 例题（IB 风格）
Example: The heights of a group of students are normally distributed with mean 170 cm and standard deviation 6 cm. (a) Find the probability that a randomly chosen student is taller than 178 cm. (b) Find the height value below which 90% of students lie.
例题：某组学生的身高服从正态分布，均值为 170 cm，标准差为 6 cm。(a) 求随机选择一名学生身高超过 178 cm 的概率。(b) 求 90% 的学生身高低于哪个值。
Solution (a): Standardize: Z = (178 − 170) / 6 = 1.3333. We need P(Z > 1.3333). Using a GDC, P(Z > 1.3333) = 1 − P(Z < 1.3333) ≈ 1 − 0.9088 = 0.0912.
解答 (a)：标准化：Z = (178 − 170) / 6 = 1.3333。我们需要 P(Z > 1.3333)。使用计算器，P(Z > 1.3333) = 1 − P(Z < 1.3333) ≈ 1 − 0.9088 = 0.0912。
Solution (b): We need the 90th percentile. Find z such that P(Z < z) = 0.9. From inverse normal, z ≈ 1.2816. Then convert back: x = μ + zσ = 170 + 1.2816 × 6 ≈ 177.69 cm.
解答 (b)：我们需要第 90 百分位数。求 z 使得 P(Z < z) = 0.9。由反向正态得 z ≈ 1.2816。再转换回原值：x = μ + zσ = 170 + 1.2816 × 6 ≈ 177.69 cm。
Therefore, about 9.12% of students are taller than 178 cm, and 90% of students are shorter than approximately 177.7 cm.
因此，约 9.12% 的学生身高超过 178 cm，而 90% 的学生身高低于约 177.7 cm。
10. Summary: The Power of Z | 总结：Z 的力量
The standard normal distribution is a elegant tool that allows us to standardize any normal distribution and compare values across different scales. Mastery of z-scores, cumulative probabilities, and inverse normal calculations is essential for IB Mathematics Applications & Interpretation and Analysis & Approaches; it appears in Paper 2 and the internal assessment as well.
标准正态分布是一个优雅的工具，它让我们能够标准化任何正态分布，并在不同尺度之间比较数值。熟练掌握 z 分数、累积概率和反向正态计算对于 IB 数学应用与解释以及分析与方法课程至关重要；它在试卷二和内部评估中都会出现。
Remember the key steps: standardize using Z = (x − μ) / σ, use symmetry to handle negative values, and use invNorm for reverse questions. With practice, these problems become routine and highly scoring.
记住关键步骤：使用 Z = (x − μ) / σ 标准化，利用对称性处理负值，并在反向问题中使用 invNorm。通过练习，这些问题会变得常规且容易得分。
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