📚 Exercise 24I: Mastering Normal Distribution for IB Math | 练习24I：掌握IB数学正态分布
Exercise 24I in the IB Mathematics SL/HL course brings together all the key skills of the normal distribution. Whether you are calculating probabilities, finding unknown means or standard deviations, or applying the normal model to real‑world data, this mixed revision set challenges your understanding and your fluency with the GDC. This article revisits every essential concept, from the bell curve to inverse normal calculations, so you can tackle Exercise 24I with confidence.
IB数学SL/HL课程中的练习24I汇集了正态分布的全部核心技能。无论你需要计算概率、反求未知的均值或标准差，还是将正态模型应用到实际数据，这套综合性复习题都在检验你的理解和图形计算器熟练度。本文从钟形曲线到逆正态计算，重新梳理每一个关键概念，帮助你有把握地完成练习24I。
1. The Shape and Parameters of a Normal Curve | 正态曲线的形状与参数
A normal random variable X ~ N(μ, σ²) has a bell‑shaped probability density function that is symmetric about the mean μ. The spread of the curve is determined by the standard deviation σ — a larger σ flattens and widens the curve, while a smaller σ makes it taller and narrower.
正态随机变量 X ~ N(μ, σ²) 的概率密度函数呈钟形，关于均值 μ 对称。曲线的“宽窄”由标准差 σ 决定——σ 越大曲线越扁宽，σ 越小曲线越高窄。
The total area under any normal density curve equals 1. Because the distribution is continuous, probabilities are found as areas under the curve, not as heights at a single point.
任意正态密度曲线下的总面积恒为 1。由于分布是连续的，概率表示为曲线下的面积，而非某一点的高度。
These two parameters, μ and σ, are the starting point for every problem in Exercise 24I. Always identify them before you reach for your calculator.
参数 μ 和 σ 是练习24I中每一题的出发点。动用计算器之前，请务必先找出这两个值。
2. Standardisation and Z‑scores | 标准化与Z分数
To compare values from different normal distributions, we convert them to the standard normal variable Z ~ N(0, 1) using the formula:
为了比较来自不同正态分布的数据，我们使用下列标准分公式将其转换为标准正态变量 Z ~ N(0, 1)：
z = (x − μ) / σ
This z‑score tells you how many standard deviations a value x lies above or below the mean. A positive z indicates a value larger than μ; a negative z indicates a value smaller than μ.
z分数告诉你某一个数值 x 位于均值之上或之下几个标准差。z 为正表示数值大于 μ，为负则表示数值小于 μ。
Exercise 24I frequently asks you to find probabilities like P(X < a) by standardising to P(Z < z). While a GDC can do this directly, understanding z‑scores helps you spot symmetrical shortcuts and avoid sign errors.
练习24I经常要求你通过标准化求 P(X < a) 之类的概率。尽管GDC可以一步到位，但理解z分数能帮你发现对称捷径，避免正负号错误。
3. Using the Normal Cumulative Distribution Function | 使用正态累积分布函数
With a graphic display calculator, you can find P(a < X < b) by using the normal CDF (often called normCdf). Enter the lower bound a, upper bound b, mean μ, and standard deviation σ.
借助图形计算器，你可以通过正态累积分布函数（通常称为 normCdf）求出 P(a < X < b)。依次输入下界 a、上界 b、均值 μ 和标准差 σ。
−∞ (e.g. −10⁹⁹) and a very large number for +∞ (e.g. 10⁹⁹), or use the calculator’s built‑in −E99 and E99 options. Always sketch a quick curve to visualise the required area — this is the best guard against ‘upper tail’ vs ‘lower tail’ confusion.
对于无界尾端，用极小的数代替 −∞（如 −10⁹⁹），用极大的数代替 +∞（如 10⁹⁹），或直接使用计算器内建选项 −E99 与 E99。动笔之前快速画一幅曲线草图，明确所求区域——这是防止“上尾”与“下尾”混淆的最佳办法。
Many marks in Exercise 24I are lost through simple input mistakes. Double‑check that you are using σ, not σ², in the CDF menu.
练习24I中很多失分都源于简单的输入错误。请再三确认在CDF菜单中使用的是 σ 而非 σ²。
4. The Inverse Normal: Finding Values from Probabilities | 逆正态分布：从概率反求数值
When the question gives a probability and asks you to find an unknown x‑value, you need the inverse normal function (invNorm). You supply the left‑tail area, the mean, and the standard deviation.
当题目给出概率并要求你找出未知的 x 值时，就需要使用逆正态函数（invNorm）。你需要提供左尾面积、均值和标准差。
For example, to find the score that separates the bottom 10% from the top 90% = 50, σ = 4, calculate invNorm(0.10, 50, 4). Be careful: if the question refers to a right‑tail area of 15%, you must first convert it to a left‑tail area of 0.85.
例如，在 μ = 50，σ = 4 的分布中，要找出分割后10%与前90%的分数，只需计算 invNorm(0.10, 50, 4)。务必留意：如果题目给了右尾面积 15%，你必须先将其转换为左尾面积 0.85。
Some parts of Exercise 24I link inverse normal with simultaneous equations — you may be asked to find μ and σ given two probability statements. Set up a pair of z‑score equations and solve them algebraically or with a solver.
练习24I的某些部分将逆正态与方程组联系起来——可能会给出两个概率条件，要求反求 μ 和 σ。此时需建立一对标准分方程，并通过代数方法或求解器解决。
5. Working with the Standard Normal Table | 使用标准正态分布表
ble to read the standard normal table is a valuable skill. The table gives cumulative left‑tail probabilities Φ(z) = P(Z < z) for z ≥ 0.
尽管IB考试允许使用计算器，但阅读标准正态分布表仍是一项宝贵技能。该表提供的是 z ≥ 0 时的累计左尾概率 Φ(z) = P(Z < z)。
For negative z, use Φ(−z) = 1 − Φ(z). And to find probabilities between two z‑ quickly check calculator output and solve problems where the table is explicitly required.
对负 z 值，利用 Φ(−z) = 1 − Φ(z)。若要求两个 z 值之间的概率，则计算 Φ(z₂) − Φ(z₁)。熟练使用表格能够帮你快速验证计算器输出，也能应对明确要求查表的题目。
6. Symmetry and the Empirical Rule | 对称性与经验法则
The symmetry of the normal curve leads to several useful facts:
正态曲线的对称性带来几条颇有用途的性质：
Roughly 68% of the data lie within 1σ of the mean; 95% within 2σ; 99.7% within 3σ. (This is the empirical rule.)
大约 68% 的数据落在距离均值 1σ 以内；95% 在 2σ 以内；99.7% 在 3σ 以内。（这就是经验法则。）
P(X > μ + kσ) = P(X < μ − kσ).
P(X > μ + kσ) = P(X < μ − kσ).
The median equals the mean: P(X < μ) = P(X > μ) = 0.5.
中位数等于均值：P(X < μ) = P(X > μ) = 0.5.
Exercise 24I often exploits these properties to reduce workload. If a question asks for P(μ − 2σ < X < μ + 2σ), you can immediately answer 0.95 without touching a calculator.
练习24I经常利用这些性质来简化步骤。若题目要求 P(μ − 2σ < X < μ + 2σ)，你不必动用计算器就能径直写下答案 0.95。
7. Real‑World Applications Mixed in Exercise 24I | 练习24I中的实际应用题
The mixed exercises often present data in context: the lengths of bolts, masses of apples, or examination scores. The steps are consistent:
混合练习常常将数据置于实际背景中：螺栓长度、苹果质量或考试分数。解决步骤是一致的：
Identify the random variable and its distribution X ~ N(μ, σ²).
确定随机变量及其分布 X ~ N(μ, σ²)。
Extract the given μ and σ from the question stem or an earlier part.
从题干或先前小问中提取已知的 μ 与 σ。
Translate words like ‘exceed’, ‘at most’, ‘between’ into probability statements.
将“超过”“至多”“介于”等字眼转译为概率表述。
Decide whether you need a CDF, inverse normal, or a combination.
判断你需要使用CDF、逆正态还是二者的组合。
Some of the trickier tasks in Exercise 24I ask, ‘Find the symmetric interval about the mean that contains a certain percentage of the data.’ For a coverage of p%, set the lower bound at μ − k and the upper bound at μ + k, then use P(−k < X − μ < k) or the z‑score version to solve for k.
练习24I中更有挑战的一类题目是：“求一个关于均值对称、包含一定百分比数据的区间。” 若覆盖率为 p%，则可设下界为 μ − k、上界为 μ + k，再利用 P(−k < X − μ < k) 或其标准分形式解出 k。
8. Linking Normal Distribution to Other Topics | 正态分布与其他知识点的联系
Exercise 24I can include cross‑topic problems. You may need to combine the normal distribution with:
练习24I有时会融合跨章节的内容。你可能需要将正态分布与以下知识点结合：
| Connection | Example in Exercise 24I |
|---|---|
| Binomial distribution & normal approximation | Using N(np, np(1−p)) when n is large and/or the continuity correction is mentioned. |
| Conditional probability | P(X > a | X > b) rewritten as P(X > a) / P(X > b). |
| Sum and difference of independent normal variables | If X ~ N(μ₁,σ₁²) and Y ~ N(μ₂,σ₂²) are independent, then X ± Y ~ N(μ₁ ± μ₂, σ₁² + σ₂²). |
Recognising these links is often the key to unlocking a high‑mark question in the IB exam.
识别出这些联系往往是解锁IB考试中高分题目的钥匙。
9. Calculator Techniques and Checks | 计算器技巧与核查
Get to know your GDC’s normal functions thoroughly. Many students fail to notice that their calculator can store intermediate results. After finding a z‑score, store it and use it in a second calculation to avoid rounding errors.
请彻底熟悉你的GDC中的正态函数。许多学生未意识到计算器可以存储中间结果。求出z分数后，将其存储并在后续运算中调用，以避免舍入误差。
Always perform a ‘ballpark check’: if the answer you obtain is a probability greater than 1 or a negative standard deviation, you have made a setup error. Sketching the bell curve and shading the area takes only a few seconds but can save you a significant number of marks.
永远做一个“数量级检查”：若你得到的概率大于1或标准差为负数，必定是设定出错了。随手画一条钟形曲线并涂上阴影仅需几秒，却能为你挽回不少分数。
10. Worked Example from a Typical Exercise 24I | 练习24I典型例题详解
Problem: The weights of bags of flour are normally distributed with mean 1.02 kg and standard deviation 0.03 kg. A bag is rejected if it weighs less than 0.98 kg. Find the probability a randomly chosen bag is rejected.
题目： 袋装面粉的重量服从正态分布，均值为 1.02 kg，标准差为 0.03 kg。若一袋重量低于 0.98 kg 则被拒收。求随机选取的一袋被拒收的概率。
Let X ~ 1.333. From the standard normal table, Φ(−1.333) ≈ 0.0912. The probability is about 0.091.
设 X ~ N(1.02, 0.03²)。需求 P(X < 0.98)。使用GDC：normCdf(−E99, 0.98, 1.02, 0.03) ≈ 0.0912。或标准化：z = (0.98 − 1.02) / 0.03 = −1.333。查标准正态表，Φ(−1.333) ≈ 0.0912。概率约为 0.091。
Extension: The factory wants to adjust the mean so that only 4% he new mean. We require P(X < 0.98) = 0.04. Using invNorm(0.04, μ_new, 0.03) = 0.98. The z‑score for left‑ . Then −1.751 = (0.98 − μ_new) / 0.03 → μ_new ≈ 1.0325 kg. This is exactly the type of two‑step reasoning Exercise 24I rewards.
延伸： 工厂希望调整均值，使得仅有 4% 的袋子被拒收，σ 保持不变。求新的均值。我们需要 P(X < 0.98) = 0.04。利用 invNorm(0.04, μ_new, 0.03) = 0.98。左尾面积 0.04 对应的 z 分为 invNorm(0.04, 0, 1) ≈ −1.751。于是 −1.751 = (0.98 − μ_new) / 0.03 → μ_new ≈ 1.0325 kg。这正是练习24I所青睐的两步推理模式。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图