📚 Exercise 24B: Mastering Normal Distribution Calculations | 练习24B：掌握正态分布计算
In IB Mathematics, few topics carry as much weight in both the exams and real-world applications as the normal distribution. Exercise 24B typically builds on the basic properties introduced earlier, challenging students to calculate probabilities, work backward to find means or standard deviations, and apply the bell curve to genuine scenarios. This revision guide walks you through the key techniques you need to solve every problem in Exercise 24B with confidence, explaining each step in both English and Chinese so that language is never a barrier to understanding.
在 IB 数学中，正态分布无论是在考试还是实际应用中都具有举足轻重的地位。练习 24B 通常会在此前介绍的基本性质基础上进一步深入，要求学生计算概率、逆向求解均值或标准差，并将钟形曲线应用于真实场景。本篇复习指南将带你掌握解决练习 24B 中每一道题所需的关键技巧，每一步都用中英双语讲解，让语言不再成为理解的障碍。
1. Understanding the Normal Distribution | 理解正态分布
A normal distribution is a continuous probability distribution that is symmetric around its mean, often written as X ~ N(μ, σ²). The total area under the bell-shaped curve equals 1, and the probability that a randomly chosen value falls within a given interval corresponds to the area under the curve over that interval. In Exercise 24B you will see problems that require you to interpret this geometric meaning: the vertical axis is not the probability itself, but the probability density.
正态分布是一种关于其均值对称的连续概率分布，通常记作 X ~ N(μ, σ²)。钟形曲线下的总面积等于 1，随机选取的值落在某个区间内的概率就是该区间内曲线下的面积。在练习 24B 中，你会遇到要求你理解这一几何意义的题目：纵轴本身不是概率，而是概率密度。
k about probabilities over intervals. Exercise 24B often tests this distinction by asking for 'less than' or 'greater than' probabilities, where you need to include the whole left or right tail.
很多学生会混淆密度高度与实际概率。请始终记住：对于连续分布，P(X = a) = 0；你只能讨论区间上的概率。练习 24B 常通过询问“小于”或“大于”的概率来考察这一区别，此时你需要包含整个左尾或右尾。
2. Standardising Variables: Z-Scores | 变量标准化：Z分数
To use standard normal tables, you first convert any normal variable X into the standard normal Z ~ N(0, 1²) using the formula:
Z = (X - μ) / σ
This transformation tells you how many standard deviations a particular value lies from the mean. A positive Z means the value is above the mean; a negative Z means it is below. Exercise 24B is built around your ability to perform this standardisation quickly and accurately.
要使用标准正态表，你首先需要利用公式将任意正态变量 X 转换为标准正态变量 Z ~ N(0, 1²)：Z = (X - μ) / σ。这一变换告诉你某个特定值距离均值多少个标准差。Z 为正表示该值高于均值，为负表示低于均值。练习 24B 正是围绕你能否快速准确地完成这一标准化过程而设计的。
Even if you have a powerful graphing display calculator (GDC), you are expected to show the standardisation step in written solutions. Examiners want to see the Z-expression before you look up or compute probabilities. This not only earns method marks but also helps you catch errors when μ or σ is incorrectly substituted.
即使你拥有功能强大的图形计算器，在书面解答中仍须展示标准化步骤。阅卷人希望看到你在查表或计算概率之前写出 Z 表达式。这不仅帮助你获得方法分，还能让你在错误代入 μ 或 σ 时及时发现错误。
3. Using the Z-Table for Probability | 使用Z表求概率
0.05, giving approximately 0.8944. For negative Z-values, use symmetry: Φ(−z) = 1 − Φ(z).
标准正态表给出的是累积概率 Φ(z) = P(Z ≤ z)，即给定 Z 值左侧的面积。当练习 24B 要求计算 P(Z < 1.25) 时，你找到 1.2 行和 0.05 列，得到约 0.8944。对于负 Z 值，利用对称性：Φ(−z) = 1 − Φ(z)。
Some tables provide the probability from 0 to z; make sure you know which type is provided in your formula booklet. If the table shows the area between 0 and z, then for P(Z < z) with z > 0 you add 0.5. Mixing up table types is a classic mistake that Exercise 24B problems are designed to expose.
有些表格提供的是 0 到 z 之间的概率；你务必弄清公式手册中提供的是哪种类型的表格。如果表格显示的是 0 到 z 之间的面积，那么当 z > 0 时，P(Z < z) 需要加上 0.5。混淆表格类型是练习 24B 专门针对的一个典型错误。
4. Inverse Normal Calculations | 逆正态计算
given a probability and asked to find the corresponding X-value or Z-value. For instance, 'Find k such that P(X < k) = 0.95.' ind the Z-value that leaves 0.95 to the left, which is about 1.645. Then solve k = μ + Zσ.
练习 24B 经常将问题反过来：给定概率，要求你找出相应的 X 值或 Z 值。例如，“求 k 使得 P(X < k) = 0.95。”你首先使用计算器的逆正态功能，或者反过来查 Z 表，找出左侧面积 0.95 对应的 Z 值，约为 1.645。然后求解 k = μ + Zσ。
A common trap is confusing 'to the right' d the Z-value to look up is the one for 0.90. Always draw a quick sketch of the bell curve and shade the area of interest before performing any inverse calculation.
一个常见陷阱是混淆“右侧”概率。如果题目给出 P(X > c) = 0.10，那么左侧面积为 0.90，需要查找的 Z 值是 0.90 对应的值。进行任何逆运算之前，务必快速画出钟形曲线草图并涂黑感兴趣的区域。
5. Working with Non-Standard Normal Distributions | 处理非标准正态分布
every problem in Exercise 24B hands you a standard normal distribution; you must handle X ~ und by standardising: Z = (180 − 170) / 8 = 1.25, then P(Z > 1.25) = 1 − Φ(1.25) ≈ 0.1056.
练习 24B 中的题目并非都直接给出标准正态分布；你必须处理具有特定 μ 和 σ 的 X ~ N(μ, σ²)。例如，若学生身高服从正态分布，均值 170 厘米，标准差 8 厘米，则身高高于 180 厘米的学生比例可通过标准化求得：Z = (180 − 170) / 8 = 1.25，然后 P(Z > 1.25) = 1 − Φ(1.25) ≈ 0.1056。
ercise 24B deliberately includes variances like σ² = 36 to see if you remember that σ = 6.
当给出的是方差 σ² 而非标准差时，务必在标准化前开平方。写成 Z = (X − μ) / σ² 是一个代价高昂的符号错误，会让你的整个概率计算无效。练习 24B 故意给出如 σ² = 36 的方差，看你是否记得 σ = 6。
6. Solving Real-World Problems | 解决现实问题
e 24B asks you to translate a written description into a mathematical statement. For instance, 'The top 5% . If marks are normally distributed with μ = 62 and σ = 9, find the minimum mark for an A.' ≥ m) = 0.05, convert to P(Z ≥ z) = 0.05, find z ≈ 1.645, and then m = 62 + 1.645×9 ≈ 76.8.
IB 考试题目会将正态分布融入电池寿命、考试分数或包装重量等情景中。练习 24B 要求你将文字描述转化为数学表达式。例如，“成绩排名前 5% 的考生获得 A 等级。若分数服从正态分布，均值 62，标准差 9，求获得 A 的最低分数。”此时你需要设 P(X ≥ m) = 0.05，转化为 P(Z ≥ z) = 0.05，求出 z ≈ 1.645，然后 m = 62 + 1.645×9 ≈ 76.8。
Always define your variable clearly at the start of the solution: 'Let X be the mark of a randomly selected student, X ~ N(62, 9²).' This simple sentence shows the examiner your modelling process and frequently earns the first mark even if the subsequent arithmetic contains a mistake.
在解答开头一定要清晰地定义变量：“设 X 为一名随机选取的学生的分数，X ~ N(62, 9²)。”这个简单的句子向阅卷人展示了你的建模过程，即使后续的算术计算有误，也常常能获得第 1 分。
7. Symmetry Properties | 对称性质
(a) − 1. Exercise 24B often tests these to see if you can avoid redundant calculator work.
正态曲线关于均值对称，这为你提供了三个基本关系：P(Z ≤ −a) = P(Z ≥ a)；P(μ − c ≤ X ≤ μ + c) = 2 P(Z ≤ c/σ) − 1；以及很重要的 P(−a ≤ Z ≤ a) = 2 Φ(a) − 1。练习 24B 常会考察这些关系，看你能否避免重复的计算器操作。
ity that a value lies more than 1.5 standard deviations from the mean, you can simply recognise it as P(|Z| own demonstrates a deep understanding and saves time. Avoid the temptation to compute each tail separately.
当题目要求计算某个值距离均值超过 1.5 个标准差的概率时，你只需认出来这是 P(|Z| > 1.5) = 2 × [1 − Φ(1.5)]。写下这一步能展示出深刻的理解，还能节省时间。尽量避免分别计算每条尾巴的概率。
8. Between Two Values | 两个值之间的概率
Exercise 24B contains a significant number of 'between' zₐ). Beginners sometimes mistakenly add the probabilities or use the larger Z as the whole left area; drawing the normal curve and shading the strip between a and b prevents this.
练习 24B 包含大量“介于”问题：P(a < X < b)。正确的方法是同时标准化两个边界：P( (a − μ)/σ < Z < (b − μ)/σ ) = Φ(zᵦ) − Φ(zₐ)。初学者有时会错误地把概率相加，或者把较大的 Z 当成整个左侧面积；画出正态曲线并涂黑 a 与 b 之间的长条区域可以避免这类错误。
For symmetrical intervals like 'within one standard deviation of the mean' ects you to obtain a more precise answer, typically P(−1 < Z < 1) ≈ 0.6827, not just 0.68.
对于如“均值加减一个标准差范围内”这样的对称区间，计算是直截了当的，因为你可以用 68-95-99.7 法则进行粗略验算。不过，练习 24B 要求你得出更精确的答案，通常是 P(−1 < Z < 1) ≈ 0.6827，而不仅仅是 0.68。
9. Finding Unknown Mean or Standard Deviation | 求未知均值或标准差
gives two probability statements and asks you to find μ and σ simultaneously. For example, 'P(X < 50) = 0.15 and P(X > 80) = 0.08.' ng to the given left/right areas. Solving these two simultaneous equations yields μ and σ.
练习 24B 中有一类特别有挑战性的题目，会给出两个概率陈述，要求你同时求出 μ 和 σ。例如，“P(X < 50) = 0.15 且 P(X > 80) = 0.08。”你需要建立两个 Z 方程：(50 − μ)/σ = z₁ 和 (80 − μ)/σ = z₂，其中 z₁ 与 z₂ 是对应给定左/右面积的 Z 值。求解这组联立方程即可得到 μ 和 σ。
d 80 − μ = 1.405σ. Subtracting the first from the second eliminates μ and gives σ, then μ.
注意符号的精确性。若 P(X < 50) = 0.15，左侧面积为 0.15，因此 z₁ 为负值，因为 0.15 < 0.5。Z 表给出 Φ(−1.036) ≈ 0.15，故 z₁ ≈ −1.036。对于 P(X > 80) = 0.08，左侧面积为 0.92，得出 z₂ ≈ +1.405。接着求解：50 − μ = −1.036σ，80 − μ = 1.405σ。第二式减第一式消去 μ，求出 σ，再代回求 μ。
10. Common Mistakes to Avoid | 常见错误避免
One frequent slip is forgetting to square the standard deviation when writing the distribution. Writing X ~ N(170, 8) instead of X ~ N(170, 8²) implies a variance of 8, which completely distorts the spread. The IB data booklet expects the second parameter to be the variance, so be explicit.
一个常见疏忽是，在写分布时忘记将标准差平方。写成 X ~ N(170, 8) 而不是 X ~ N(170, 8²)，意味着方差为 8，这会彻底扭曲离散程度。IB 数据手册要求第二个参数是方差，因此请明确写出。
Another error is using the Z-table forward when an inverse calculation is needed, or mixing up the 'greater than' probability with the table's left-tail entry. A quick sketch eliminates this. Finally, avoid rounding Z-values too early; keep at least three decimal places during intermediate steps to prevent final answer drift.
另一个错误是，在需要逆运算时却正向使用 Z 表，或者把“大于”概率与表格的左尾条目弄混。快速画个草图就能消除这种错误。最后，避免过早对 Z 值进行四舍五入；中间步骤至少保留三位小数，以防最终答案出现偏差。
11. Exam-Style Questions Walkthrough | 考试风格题解析
Consider this typical Exercise 24B problem: ' components that last more than 550 hours. (b) The manufacturer wishes that no more than 2% of components fail before a guaranteed lifetime L. Find L.' For (a), Z = (550 − 500)/40 = 1.25, P(Z > 1.25) = 1 − 0.8944 = 0.1056.
思考这道典型的练习 24B 题目：“某组件的寿命服从正态分布，均值 500 小时，标准差 40 小时。(a) 求寿命超过 550 小时的组件比例。(b) 制造商希望保证寿命 L 之前失效的组件不超过 2%。求 L。”对于 (a)，Z = (550 − 500)/40 = 1.25，P(Z > 1.25) = 1 − 0.8944 = 0.1056。
.84 hours. The guarantee can be set at about 418 hours. Clearly state the answer in context and include units.
对于 (b) 部分，我们需要第 2 百分位数（左尾面积 0.02）。面积 0.02 对应的 Z 值约为 −2.054（利用逆正态求得）。于是 L = μ + Zσ = 500 + (−2.054 × 40) = 500 − 82.16 = 417.84 小时。保证寿命可以定为约 418 小时。答案要置于具体情境中，并带上单位。
12. Summary and Key Takeaways | 总结与要点
Exercise 24B is designed to solidify your fluency with the normal distribution toolbox. The core cycle — standardise, use the Z-table or inverse normal, and interpret the result — repeats in virtually every question. Master this cycle, and you turn a seemingly complex problem into a mechanical sequence of steps.
练习 24B 旨在巩固你熟练运用正态分布工具的能力。“标准化——查 Z 表或用逆正态——解读结果”这一核心循环几乎贯穿每一道题。掌握这个循环，你就把一个看似复杂的问题变成了机械的步骤序列。
Remember to always check whether the given parameter is variance or standard deviation, draw a diagram to confirm the direction of the inequality, and carry out inverse calculations with care. By working through all the variations in Exercise 24B, you build the precision and speed needed for the IB examination.
请记住，务必检查给出的参数是方差还是标准差，画图确认不等式的方向，并谨慎进行逆运算。通过做完练习 24B 中的各种变体，你将培养出 IB 考试所需的精准度与速度。
| Key Z-values | Area to left |
| −1.645 | 0.05 |
| −1.036 | 0.15 |
| 0.0 | 0.5 |
| 1.036 | 0.85 |
| 1.645 | 0.95 |
| 2.326 | 0.99 |
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