Exercise 17E: Mastering the Normal Distribution | 练习17E：掌握正态分布

📚 Exercise 17E: Mastering the Normal Distribution | 练习17E：掌握正态分布

Exercise 17E in the IB Mathematics course, whether you are following the Analysis and Approaches or Applications and Interpretation syllabus, is dedicated to deepening your understanding of the normal distribution. This continuous probability distribution is fundamental to statistics and appears in a huge range of IB exam questions, from calculating probabilities for a single observation to finding unknown means or standard deviations. The goal of this article is to walk you through every essential concept, technique, and common pitfall so that you can tackle Exercise 17E with confidence and precision.

IB数学课程中的练习17E，无论是分析和方法（AA）还是应用和解释（AI）方向，都致力于加深你对正态分布的理解。这一连续型概率分布是统计学的基础，出现在大量IB考试题目中，从计算单个观测值的概率，到求解未知的均值或标准差。本文的目标是带你梳理每一个核心概念、解题技巧和常见误区，让你能够自信而精准地攻克练习17E。

1. The Normal Distribution and Its Properties | 正态分布及其性质

A normal distribution is a symmetric, bell-shaped continuous probability distribution defined by two parameters: the population mean μ and the population standard deviation σ. The total area under the probability density function is exactly 1, and the curve approaches but never touches the horizontal axis. Because the distribution is symmetric, the mean, median, and mode all coincide at the centre of the curve. Approximately 68% of the data lies within one standard deviation of the mean, about 95% within two standard deviations, and roughly 99.7% within three standard deviations.

正态分布是一种对称的钟形连续概率分布，由两个参数定义：总体均值μ和总体标准差σ。概率密度函数下的总面积正好为1，曲线向两端无限延伸但永不触及横轴。由于分布是对称的，均值、中位数和众数在曲线中心重合。大约68%的数据落在均值的一个标准差范围内，约95%落在两个标准差内，约99.7%落在三个标准差内。

When working with a normal random variable X, we write X ~ N(μ, σ²), where σ² is the variance. The smooth, unimodal shape of the normal curve makes it an excellent model for many natural phenomena, such as heights, weights, exam scores, and measurement errors. IB questions often provide the parameters explicitly, but you must be ready to extract them from a word problem or a real-world context.

当处理正态随机变量X时，我们记作X ~ N(μ, σ²)，其中σ²为方差。正态曲线平滑、单峰的形态使其成为许多自然现象的良好模型，比如身高、体重、考试分数和测量误差。IB题目通常会明确给出参数，但你必须能够从文字题或实际背景中提取它们。

---

2. The Standard Normal Distribution Z ~ N(0, 1) | 标准正态分布 Z ~ N(0, 1)

To standardise any normal observation, we convert the value x into a z-score using the formula:

z = (x − μ) / σ

This transformation shifts the distribution so that it has a mean of 0 and a standard deviation of 1. The resulting standard normal distribution, Z ~ N(0, 1), allows us to use a single set of probability tables or pre-programmed calculator functions. The z-score tells us how many standard deviations an observation is above or below the mean. A negative z-score indicates the value is below the mean, while a positive z-score shows it lies above.

为了对任意正态观测值进行标准化，我们使用以下公式将数值x转换为z分数：

z = (x − μ) / σ

这种变换将分布平移并缩放，使其均值为0、标准差为1。得到的标准正态分布Z ~ N(0, 1)使我们能够使用同一套概率表或计算器内置函数。z分数告诉我们一个观测值距离均值有多少个标准差。负的z分数表明该值低于均值，正的z分数则表明高于均值。

In the era of graphical display calculators (GDC), you can often compute normal probabilities directly without manual standardisation. However, the standardisation concept remains crucial for understanding inverse normal problems and for checking the reasonableness of your answers. You should be comfortable moving back and forth between an x value and its z-score.

在图形计算器（GDC）时代，你通常可以直接计算正态概率而无需手动标准化。然而，标准化概念对于理解逆正态问题以及检查答案的合理性仍然至关重要。你应该能够熟练地在x值及其z分数之间来回转换。

---

3. Calculating Probabilities Using Z-scores | 利用Z分数计算概率

Once you have a z-score, the probability that Z is less than a given value is found using the standard normal cumulative distribution function, denoted Φ(z). For instance, P(Z < 1.25) can be looked up in a table or obtained directly from a GDC using the normalcdf function with lower bound –∞ (or a very small number like –10⁹) and upper bound 1.25. Most IB questions will ask for probabilities such as P(X < a), P(X > a), or P(a < X < b).

一旦得到z分数，Z小于某一给定值的概率便可以通过标准正态累积分布函数Φ(z)求得。例如，P(Z < 1.25)可以在表格中查取，或通过GDC的normalcdf函数直接获得，下限为–∞（或一个非常小的数如–10⁹），上限为1.25。大多数IB题目会要求计算P(X < a)、P(X > a)或P(a < X < b)这样的概率。

When using a GDC for a non-standard normal variable, you can enter the distribution parameters directly: normalcdf(lower, upper, μ, σ). If no lower bound is needed, use a very small number; if no upper bound is needed, use a very large number. Always sketch a quick bell curve and shade the region of interest. This visual check will drastically reduce sign errors and help you decide whether to subtract from 1 or to use symmetry rules.

使用GDC处理非标准正态变量时，你可以直接输入分布参数：normalcdf(下限, 上限, μ, σ)。若不需要下限，用一个极小的数；若不需要上限，用一个极大的数。始终快速画一条钟形曲线并涂出目标区域。这种视觉检查将大幅减少符号错误，并帮助判断是否需要从1中减去，或者使用对称规则。

---

4. Using the Inverse Normal Function | 使用逆正态函数

The inverse normal function allows you to find the observation or z-score that corresponds to a given cumulative probability. Symbolically, if P(Z ≤ z₀) = p, then z₀ = Φ⁻¹(p). On a GDC, this is usually the invNorm function, where you supply the area to the left, the mean, and the standard deviation. IB questions frequently ask for the value that cuts off the top 5% or the middle 90% of the distribution.

逆正态函数可以帮助你找到与给定累积概率相对应的观测值或z分数。符号上，若P(Z ≤ z₀) = p，则z₀ = Φ⁻¹(p)。在GDC上，通常使用invNorm函数，需要提供左侧面积、均值和标准差。IB题目经常要求找出截断顶部5%或中间90%的数值。

For a standard normal distribution, invNorm(p) returns the z-score with cumulative probability p to its left. To find an x-value for X ~ N(μ, σ²), you can either standardise afterwards or simply use the GDC’s invNorm with the given μ and σ. Many students lose marks by confusing the tail area: invNorm always works with the left-tail probability, so if you are given a right-tail area of 0.05, you must input 0.95 as the area.

对于标准正态分布，invNorm(p)返回左侧累积概率为p的z分数。要找到X ~ N(μ, σ²)的x值，你可以事后标准化，也可以直接使用带给定μ和σ的GDC invNorm功能。很多学生因混淆尾部面积而失分：invNorm始终使用左侧尾部概率，因此如果给出右侧尾部面积为0.05，你必须输入0.95作为面积。

---

5. Symmetry and Complement Rules | 对称性与互补规则

The symmetry of the normal curve gives rise to several handy relationships. Since the total area under the curve is 1, we have P(Z > a) = 1 − P(Z < a). Similarly, P(Z < −a) = P(Z > a). These complement and symmetry rules allow you to compute probabilities for negative z-scores when your table only provides positive values, or to handle questions that ask for the probability outside an interval.

正态曲线的对称性衍生出几个便捷的关系式。由于曲线下总面积为1，我们有P(Z > a) = 1 − P(Z < a)。类似地，P(Z < −a) = P(Z > a)。借助这些互补与对称规则，当你的表格只提供正值时，你可以计算出负z分数的概率，或者处理那些询问区间外概率的题目。

For example, to find P(Z > −1.5), note that the area to the right of −1.5 is the same as the area to the left of +1.5, so P(Z > −1.5) = P(Z < 1.5). This trick is extremely useful in inverse normal problems where the given probability is less than 0.5, forcing you to consider a negative z-score.

例如，要求P(Z > −1.5)，注意到−1.5右侧的面积与+1.5左侧的面积相等，因此P(Z > −1.5) = P(Z < 1.5)。这个技巧在逆正态问题中极为有用，此时给定的概率可能小于0.5，从而迫使你考虑一个负的z分数。

---

6. Working with Non-standard Normal Distributions | 应对非标准正态分布

Most real data will not have μ = 0 and σ = 1, so you must become fluent in handling general normal distributions. The key is to either transform the boundary values to z-scores and then use the standard normal distribution, or to let your GDC handle the original parameters. If you standardise manually, remember that the inequality direction remains unchanged, and you are really performing a linear transformation of the random variable.

大多数实际数据的均值和标准差都不会是0和1，因此你必须熟练处理一般正态分布。关键在于要么将边界值变换为z分数后使用标准正态分布，要么让GDC直接处理原始参数。如果手动标准化，请记住不等号方向保持不变，实质上是对随机变量进行了一次线性变换。

A typical IB question might state: X ~ N(62, 15²). Find P(55 < X < 70). You can compute the z-scores: z₁ = (55 − 62) / 15 ≈ −0.467, z₂ = (70 − 62) / 15 ≈ 0.533. Then P(−0.467 < Z < 0.533) = P(Z < 0.533) − P(Z < −0.467). Apply symmetry to the negative part if using a table.

一道典型的IB题目可能会说：X ~ N(62, 15²)。求P(55 < X < 70)。你可以计算z分数：z₁ = (55 − 62) / 15 ≈ −0.467，z₂ = (70 − 62) / 15 ≈ 0.533。然后P(−0.467 < Z < 0.533) = P(Z < 0.533) − P(Z < −0.467)。若使用表格，对负值部分应用对称性。

---

7. Real-world Applications: IB Exam-style Problems | 实际应用：IB考试风格问题

IB exam questions often embed the normal distribution in practical contexts. You might be asked about the lifespan of batteries, the masses of apples, the times for a chemical reaction, or the marks in a large examination. In each case, you need to identify μ and σ from the wording. Phrases like “normally distributed with mean … and standard deviation …” are direct, but sometimes you must infer the standard deviation from a variance or from a statement like “95% of the data lies within 10 units of the mean.”

IB考试题目常常将正态分布嵌入实际情境中。你可能会遇到关于电池寿命、苹果质量、化学反应时间或大型考试的分数等问题。在每种情况下，你都需要从措辞中识别出μ和σ。像“服从均值为…、标准差…的正态分布”这样的表述是直接的，但有时你必须从方差或类似“95%的数据在均值10个单位以内”的陈述中推断出标准差。

When interpreting real-world results, remember that a normally distributed continuous variable theoretically assigns a probability to any interval, no matter how small. In an exam, you can treat strict and non-strict inequalities identically because the probability of an exact single point is zero in a continuous distribution. Thus P(X < 70) equals P(X ≤ 70).

在解释实际结果时，记住一个正态分布的连续变量理论上会给任意区间分配概率，无论区间多小。在考试中，你可以将严格不等式和非严格不等式同等对待，因为在连续分布中，恰好等于某一点的概率为零。因此P(X < 70)等于P(X ≤ 70)。

---

8. Step-by-step Example: Finding Probability Between Two Values | 逐步示例：求两值之间的概率

Problem: The volumes of juice in a pack are normally distributed with mean 250 ml and standard deviation 3 ml. Find the probability that a randomly selected pack contains between 245 ml and 253 ml.

题目：某果汁包装的体积服从均值为250 ml、标准差为3 ml的正态分布。求随机抽取一包果汁体积在245 ml到253 ml之间的概率。

Solution: Use your GDC with normalcdf(245, 253, 250, 3). The screen displays approximately 0.7936. Alternatively, calculate z-scores: z₁ = (245 − 250)/3 ≈ −1.667, z₂ = (253 − 250)/3 = 1.000. Then P(−1.667 < Z < 1) = Φ(1) − Φ(−1.667). Using symmetry, Φ(−1.667) = 1 − Φ(1.667) ≈ 1 − 0.9522 = 0.0478, and Φ(1) ≈ 0.8413. So the probability is 0.8413 − 0.0478 = 0.7935. The slight difference is due to rounding. Always draw a bell curve with the mean and the boundaries labelled to confirm your result is plausible.

解答：使用GDC，输入normalcdf(245, 253, 250, 3)。屏幕显示约0.7936。或者计算z分数：z₁ = (245 − 250)/3 ≈ −1.667，z₂ = (253 − 250)/3 = 1.000。那么P(−1.667 < Z < 1) = Φ(1) − Φ(−1.667)。利用对称性，Φ(−1.667) = 1 − Φ(1.667) ≈ 1 − 0.9522 = 0.0478，Φ(1) ≈ 0.8413。因此概率为0.8413 − 0.0478 = 0.7935。微小的差异源于四舍五入。始终画出标有均值和边界的钟形曲线，以核实结果合理。

---

9. Example: Inverse Normal to Find Unknown Mean | 示例：逆正态求未知均值

Problem: The scores in a test are normally distributed with standard deviation 12. If 10% of the candidates score more than 80 marks, find the mean score. This is a classic inverse normal problem where the right-tail area is known.

题目：某测试成绩服从正态分布，标准差为12。若有10%的考生得分超过80分，求平均分。这是一道典型的已知右侧尾部面积的逆正态问题。

Solution: Since 10% is above 80, the area to the left of 80 is 0.90. For the standard normal distribution, find z such that P(Z < z) = 0.90. Using invNorm(0.90, 0, 1) we get z ≈ 1.2816. Now use the standardising formula: z = (x − μ)/σ → 1.2816 = (80 − μ)/12. Solve for μ: μ = 80 − 12 × 1.2816 ≈ 80 − 15.38 = 64.62. Therefore the mean score is approximately 64.6. Always check: if the mean were lower than 80, the 10% tail makes sense.

解答：由于10%在80分以上，80分左侧的面积为0.90。对于标准正态分布，找到满足P(Z < z) = 0.90的z。使用invNorm(0.90, 0, 1)得到z ≈ 1.2816。现在利用标准化公式：z = (x − μ)/σ → 1.2816 = (80 − μ)/12。解出μ：μ = 80 − 12 × 1.2816 ≈ 80 − 15.38 = 64.62。因此平均分约为64.6。务必检验：均值低于80分时，10%的右侧尾部是合理的。

---

10. Common Mistakes to Avoid | 常见错误避免

One frequent error is using the standard deviation σ when the variance σ² is given. Always check the notation: if a question says “variance 25”, then σ = 5. Another mistake is forgetting to check whether the question asks for a left-tail or right-tail probability when using invNorm. If you need a right-tail percentage, subtract it from 1 before entering it into the inverse function.

一个常见错误是，当给出方差σ²时，却误用为标准差σ。务必检查符号：如果题目说“方差为25”，则σ = 5。另一个错误是在使用invNorm时忘记检查题目要求的是左侧尾部还是右侧尾部概率。如果需要右侧尾部百分比，先将其从1中减去再输入逆函数。

Students also sometimes mix up the inequality signs when standardising negative values or when applying symmetry. It is helpful to write down the probability statement in words first, then translate it step by step. Finally, be careful with calculator syntax: some models require the lower and upper bounds in a specific order, and entering them backwards can produce a negative probability or an error.

学生们有时在标准化负值或应用对称性时会弄混不等号方向。一个有效的办法是先用文字写出概率陈述，再逐步转化为数学表达式。最后，注意计算器语法：某些型号要求按特定顺序输入下限和上限，顺序颠倒可能导致负概率或出错。

---

11. Practice Tips for Exercise 17E | 练习17E的练习技巧

Begin by classifying each question: is it a straightforward “find the probability” type, an inverse normal problem, or a “find the unknown mean/standard deviation” challenge? Mark up the given information clearly and always draw a sketch. Use two colours – one for the given parameters and another for the target area. This habit speeds up your work and reduces careless errors.

开始时先对每道题分类：是简单的“求概率”型，还是逆正态问题，又或是“求未知均值/标准差”的挑战？清晰标注已知信息，并务必画出示意图。使用两种颜色——一种标出已知参数，另一种标出目标区域。这个习惯能加快解题速度并减少粗心错误。

Practise without a GDC for a few problems using the standard normal table, as this builds deep intuition about the meaning of z-scores. Then solve the same problems with your GDC to verify the answers. When you encounter word problems, re-read the final sentence to confirm what the question is really asking — is it the probability of being greater than a value, less than a value, or between two values? Misinterpreting the phrase “at least” or “exceeds” is a common source of lost marks.

找一些题目不用GDC、改用标准正态表练习，这能加深你对z分数含义的直觉。然后再用GDC解答相同题目以验证答案。遇到文字题时，重读最后一句以确认问题到底在问什么——是大于某个值的概率、小于某个值的概率还是介于两个值之间的概率？误解“至少”或“超过”等措辞是常见的失分原因。

---

12. Summary and Key Takeaways | 总结与关键要点

The normal distribution is a powerful tool for modelling continuous data, and Exercise 17E is designed to solidify your ability to move flexibly between raw data, z-scores, and probabilities. Master the standardising formula z = (x − μ)/σ and know how to use your GDC’s normalcdf and invNorm functions efficiently. Always identify whether a problem is a forward calculation (finding probability from boundaries) or an inverse problem (finding boundaries from probabilities).

正态分布是建模连续数据的强大工具，而练习17E旨在巩固你在原始数据、z分数和概率之间灵活转换的能力。掌握标准化公式z = (x − μ)/σ，并知道如何高效地使用GDC的normalcdf和invNorm功能。始终判断一道题是正向计算（由边界求概率）还是逆向问题（由概率求边界）。

Use symmetry and complement rules to simplify calculations, and never skip the step of sketching the normal curve. With consistent practice, recognising the appropriate technique will become second nature. By the end of Exercise 17E, you should feel fully equipped to handle any normal distribution question on your IB exam, whether it appears in Paper 1, Paper 2, or your Internal Assessment.

利用对称性和互补规则简化计算，绝不要跳过绘制正态曲线这一步骤。通过持续练习，识别相应技巧将成为你的第二本能。完成练习17E后，你应该感到充分具备应对IB考试中任何正态分布题目的能力，无论是在试卷一、试卷二还是内部评估中出现。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）