Exercise 13G: Normal Distribution | 练习13G：正态分布

📚 Exercise 13G: Normal Distribution | 练习13G：正态分布

In Exercise 13G, we tackle a core topic in IB Mathematics: the normal distribution. This powerful continuous probability model appears everywhere, from natural measurements like heights and test scores to errors in scientific experiments. Mastering it means you can calculate probabilities, find critical values, and solve real-world problems with confidence — exactly the skills this exercise builds.

在练习13G中，我们处理IB数学的一个核心主题：正态分布。这个强大的连续概率模型无处不在，从身高、考试成绩等自然测量值，到科学实验中的误差。掌握它，你就能自信地计算概率、寻找临界值并解决实际问题——这正是本练习所要培养的技能。

1. What is a Normal Distribution? | 什么是正态分布？

A normal distribution is a continuous probability distribution that is symmetric and bell‑shaped. It is fully defined by two parameters: the mean μ (mu), which locates the centre, and the standard deviation σ (sigma), which controls the spread. We write X ~ N(μ, σ²) to say that a random variable X follows a normal distribution with mean μ and variance σ².

正态分布是一种对称的钟形连续概率分布。它完全由两个参数定义：均值μ（mu）决定中心位置，标准差σ（sigma）决定离散程度。我们记作 X ~ N(μ, σ²)，表示随机变量X服从均值为μ、方差为σ²的正态分布。

The curve's highest point sits at x = μ, and it approaches — but never touches — merges from many independent, additive effects, as explained by the Central Limit Theorem.

曲线最高点位于 x = μ 处，随着 x 远离均值，曲线无限接近但永不触及横轴。这正是中心极限定理所描述的理想化形状，由众多独立、可加效应共同作用而产生。

2. Key Properties of the Normal Curve | 正态曲线的重要性质

The normal probability density function has several properties that you must know for IB examinations: symmetry, asymptotic behaviour, and the empirical (68–95–99.7) rule.

正态概率密度函数有几条性质必须在IB考试中掌握：对称性、渐近行为以及经验法则（68–95–99.7规则）。

Symmetry: abilities on opposite sides of the centre are equal as long as the distances are the same.

对称性：曲线关于均值完全对称。这意味着 P(X < μ) = P(X > μ) = 0.5，且只要距离相等，中心两侧的概率也相等。

Asymptotic tail behaviour: The tails extend infinitely in both directions, so the function is never zero, but probabilities for extreme values become vanishingly small.

渐近尾部行为：两侧尾部无限延伸，因此函数值永不为零，但极端值出现的概率会变得极小。

Empirical rule: About 68% of the data lies within 1σ of μ, about 95% within 2σ, and about 99.7% within 3σ. This rule provides quick estimates without a calculator.

经验法则：大约68%的数据落在 μ ± 1σ 内，约95%落在 μ ± 2σ 内，约99.7%落在 μ ± 3σ 内。这一法则可在不用计算器时进行快速估算。

3. The Standard Normal Distribution | 标准正态分布

To compare different normal distributions, we use the standard normal distribution Z ~ N(0, 1), where the mean is 0 and the standard deviation is 1. Any normal variable X can be transformed into a standard score, called a z‑score.

为了比较不同的正态分布，我们使用标准正态分布 Z ~ N(0, 1)，其均值为0、标准差为1。任何正态变量X 都可以转化为标准分数，称为z值。

The z‑score tells us how many standard deviations an observation x is from the mean. It is calculated by subtracting μ and dividing by σ:

z值告诉我们某个观测值 x 距离均值有多少个标准差。计算方法是用 x 减去 μ 再除以 σ：

z = (x – μ) / σ

With standardisation, the whole family of normal curves collapses into a single curve, making it possible to use a standard table or built‑in calculator functions to find probabilities.

通过标准化处理，所有正态曲线族都归结为一条曲线，从而可以使用标准表格或计算器内置函数来求概率。

4. Using the GDC for Normal Probabilities | 使用图形计算器求正态概率

IB exams assume you are proficient with your graphical display calculator (GDC). To find the probability P(a < X < b) when X ~ N(μ, σ²), you typically use the normal cumulative distribution function (normalcdf).

IB考试默认你能熟练使用图形计算器（GDC）。若X ~ N(μ, σ²)，要计算概率 P(a < X < b)，通常使用正态累积分布函数（normalcdf）。

For Texas Instruments calculators the syntax is normalcdf(lower bound, upper bound, μ, σ) rs the menu sequence is similar. If the lower bound is -∞, use a very small number such as –10⁹⁹; for the upper bound of ∞, use 10⁹⁹.

对于德州仪器计算器，语法为 normalcdf(下限, 上限, μ, σ)。卡西欧计算器的菜单顺序类似。若下限为 –∞，可输入一个极小的数如 –10⁹⁹；若上限为 ∞，则输入 10⁹⁹。

Always sketch a quick bell‑curve and shade the relevant area. This visual check helps avoid common errors, such as giving the complement probability or confusing left‑tail with right‑tail areas.

始终快速画出钟形曲线并涂上相应区域。这种视觉检查有助于避免常见错误，例如给出互补概率或混淆左尾与右尾面积。

5. Inverse Normal: From Probability to Value | 逆正态分布：从概率反推取值

Often a problem states a probability and asks you to find the corresponding x‑value. For instance, "Find the height that 90% of students are shorter than." This requires the inverse normal function, usually labelled invNorm.

题目经常给出一个概率，要求你找出对应的 x 值。例如，“求90%学生身高低于的那个高度。”这时就要用到逆正态函数，通常标记为 invNorm。

On a TI calculator, the syntax is invNorm(area to the left, μ, σ). The function returns the x‑value such that the area under the normal curve to the left of x equals the given probability. Always confirm whether the question asks for a left‑tail, right‑tail, or central area.

在TI计算器上，语法为 invNorm(左侧面积, μ, σ)。该函数返回的 x 值使得正态曲线下左侧面积等于给定的概率。做题时务必确认题目要求的是左尾、右尾还是中心区域。

For a right‑tail area of p, the left‑tail area is 1 – p. For a symmetric central area, use the appropriate split, remembering that if the central area is c, the left bound has area (1 – c)/2 to its left.

对于右尾面积 p，左尾面积为 1 – p。对于对称的中心区域，需进行合理的拆分，记住若中心面积为 c，则左侧边界左侧的面积为 (1 – c)/2。

6. Reading the Standard Normal Table | 读取标准正态表

While the GDC is the primary tool, understanding a standard normal table remains valuable — especially for concept reinforcement and proof‑based questions. The table gives the cumulative probability P(Z < z) for positive z‑values.

虽然GDC是主要工具，但理解标准正态表仍然很有价值——尤其在巩固概念和解答证明类问题时。该表格给出了正z值对应的累积概率 P(Z < z)。

Because of symmetry, you can deduce probabilities for negative z. For example, P(Z < –z) = 1 – P(Z < z). A small extract of such a table looks like this:

利用对称性，可以推导出负z值的概率。例如，P(Z < –z) = 1 – P(Z < z)。标准正态表的一个小片段如下：

广告 / Sponsored

z	0.00	0.01	0.02	0.03
0.0	0.5000	0.5040	0.5080	0.5120
1.0	0.8413	0.8438	0.8461	0.8485

In an exam, you will likely be told when to use tables, but being able to cross‑check your GDC result with a rough table value is a powerful accuracy skill.

考试中很可能会指定何时使用表格，但能够用粗略的表值交叉验证GDC结果，是一项强大的准确性技能。

7. Finding Unknown Parameters μ and σ | 求解未知参数 μ 和 σ

Some IB questions give two probability conditions and ask you to find the mean μ or standard deviation σ. This involves setting up two equations using z‑scores and solving simultaneously.

有些IB题目会给出两个概率条件，要求你求出均值 μ 或标准差 σ。这需要利用z值建立两个方程并联立求解。

For instance, if P(X < 50) = 0.3 and P(X > 80) = 0.1, standardising yields (50 – μ)/σ = z₁ and (80 – normal values of 0.3 and 0.9 respectively. Solving the two linear equations gives μ and σ.

例如，若 P(X < 50) = 0.3 且 P(X > 80) = 0.1，标准化后可得到 (50 – μ)/σ = z₁ 和 (80 – μ)/σ = z₂，其中 z₁ 和 z₂ 分别是 0.3 和 0.9 对应的逆正态取值。求解这两个线性方程即可得到 μ 和 σ。

Always use the left‑tail probability for inverse normal. So for a right‑tail condition, convert it before substituting. This technique is tested frequently in HL papers and appears in Exercise 13G as well.

始终使用左尾概率来做逆正态运算。因此对于右尾条件，应先进行转换再代入。这一技巧在HL试卷中经常考查，练习13G同样会涉及。

8. Applying the Normal Distribution in Context | 正态分布的实际应用

IB places strong emphasis on modelling real‑world scenarios. Typical contexts include: heights and weights of populations, lifetimes of components, measurement errors, and examination scores. You are expected to interpret the results in context.

IB非常强调对现实情景的建模。典型情景包括：人群的身高和体重、零件寿命、测量误差以及考试成绩。你需要结合具体情境解释结果。

For example, "The masses of apples from an orchard follow N(150 g, 20² g). Find the percentage of apples weighing less than 130 g." A correct answer might be "about 15.9%," followed by a sentence linking this to the orchard's quality control.

例如，“某果园苹果的质量服从 N(150 g, 20² g)。求重量低于130 g的苹果所占的百分比。”正确答案可能是“约15.9%”，随后用一句话将这一结果与果园的质量控制联系起来。

Always round your final answer to a sensible precision — usually three significant figures unless the problem specifies otherwise — and include units where appropriate.

最终答案总要取合理的精度——除非题目另有规定，一般取三位有效数字——并在适当处标明单位。

9. The Normal Approximation to the Binomial | 二项分布的正态近似

If X ~ n the textbook), the binomial distribution can be approximated by a normal distribution: X ～ N(np, npq), where q = 1 – p.

如果 X ~ B(n, p) 且 np 和 nq 均大于5（或10，视教材而定），二项分布便可用正态分布来近似：X ～ N(np, npq)，其中 q = 1 – p。

This is especially useful for large n, where binomial calculations become tedious. Applying a continuity correction is essential for accuracy: the binomial integer k corresponds to the normal interval from k – 0.5 to k + 0.5.

这在 n 很大时尤其有用，因为手工进行二项计算会非常繁琐。为保证准确，必须使用连续性校正：二项分布中的整数 k 对应正态分布中从 k – 0.5 到 k + 0.5 的区间。

In AI HL, this approximation is tested explicitly; in AA, it can appear as an extension. Even if Exercise 13G focuses on the pure normal, this concept often follows closely.

在AI HL中，这一近似会明确考查；在AA中它可能作为拓展出现。即便练习13G聚焦于纯粹的正态分布，这一概念往往紧随其后。

10. Common Mistakes and How to Avoid Them | 常见错误与避坑指南

Even strong students make avoidable errors when working with normal distributions. The most frequent mistakes include: using variance instead of standard deviation, forgetting to square σ when writing the distribution, and misreading a right‑tail probability as a left‑tail one.

即使优秀的学生在处理正态分布时也会犯一些本可避免的错误。最常见的错误包括：误用方差代替标准差、书写分布时忘记将 σ 平方，以及将右尾概率误当作左尾概率。

Another classic pitfall is directly entering a z‑score when the calculator expects the original x‑value, or vice versa. Always double‑check the function you are using: normalcdf works with the raw distribution, while the standard normal table works with Z.

另一个经典陷阱是，当计算器需要原始x值时却直接输入了z分数，反之亦然。务必仔细核对正在使用的功能：normalcdf 处理的是原始分布，而标准正态表处理的是Z。

Keep a labelled sketch next to your working. Write down μ, σ, the boundaries, and the shaded region. This habit reduces guesswork and makes your reasoning clear to examiners.

在草稿纸上画一幅带标注的草图。写下 μ、σ、边界并涂上阴影区域。这个习惯能减少猜测，并使你的推理过程对考官清晰可见。

11. Strategy for Solving Exercise 13G Problems | 练习13G题目的解题策略

tically. First, identify the random variable and write down its distribution in the form X ~ nd, translate the words into a mathematical probability statement, such as P(X > a) = 0.2.

每道题都要系统作答。第一步，明确随机变量，并将其分布写成 X ~ N(μ, σ²) 的形式。第二步，将文字转化为数学概率语句，例如 P(X > a) = 0.2。

Third, draw and shade a quick curve. Fourth, decide whether the problem requires a forward calculation (normalcdf) or an inverse calculation (invNorm). Finally, carry out the computation and write your answer in the context of the problem.

第三步，快速画出曲线并涂上阴影。第四步，判断题目需要正向计算（normalcdf）还是逆向计算（invNorm）。最后，完成计算并将答案放回题目情境中表述。

If the question involves two unknowns, set up simultaneous z‑score equations as described earlier. If it is an approximation problem, verify that the np > 5 condition holds and apply continuity correction before using the normal distribution.

如果题目涉及两个未知数，则按照前面描述的方法建立z值联立方程。如果是近似问题，先验证 np > 5 条件是否满足，并在使用正态分布前进行连续性校正。

12. Summary and Key Takeaways | 总结与要点回顾

erstone of IB statistics. You are now equipped to standardise any observation using z = (x – μ)/σ, to find probabilities with GDC or tables, to invert probabilities to find x‑values, and to handle real‑world modelling questions confidently.

正态分布是IB统计学的基石。你现在已经能够用 z = (x – μ)/σ 对任意观测值进行标准化，用GDC或表格求概率，逆用概率反推x值，并自信地处理现实建模问题。

Remember the three key rules: sketch the curve, check whether you need left‑tail or right‑tail, and never confuse σ with σ². With consistent practice, Exercise 13G will become a straightforward step on your path to mastering statistical inference.

请牢记三条关键规则：画出曲线、分清左尾还是右尾，以及绝不要混淆 σ 和 σ²。随之持续练习，练习13G将成为你掌握统计推断道路上的轻松一步。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）