Exercise 21G.1: Normal Distribution | 练习21G.1：正态分布

📚 Exercise 21G.1: Normal Distribution | 练习21G.1：正态分布

In IB Mathematics, Exercise 21G.1 typically marks the start of a formal study of the normal distribution, one of the most fundamental continuous probability distributions in statistics. This exercise introduces the bell-shaped curve, its parameters μ and σ, and the process of calculating probabilities for normally distributed variables. Whether you are following the Applications and Interpretation or the Analysis and Approaches syllabus, mastering this topic is essential for both Paper 2 and the internal assessment. The following article breaks down the key ideas, worked examples, and examiner tips linked to Exercise 21G.1.

在IB数学课程中，练习21G.1通常标志着对正态分布正式学习的开始，正态分布是统计学中最基本的连续概率分布之一。该练习介绍了钟形曲线、参数μ与σ，以及计算服从正态分布变量的概率的方法。无论你学习的是《应用与解释》还是《分析与方法》课程，掌握这一主题对试卷二和内部评估都至关重要。以下文章将拆解与练习21G.1相关的核心概念、实例讲解和考官建议。

1. What Is the Normal Distribution? | 什么是正态分布？

The normal distribution is a continuous probability distribution that is symmetric about the mean μ, with its shape determined by the standard deviation σ. It is often called the Gaussian distribution and appears naturally in countless real-world variables such as heights, test scores, and measurement errors. In Exercise 21G.1, you first encounter the notation X ~ N(μ, σ²), which means that the random variable X follows a normal distribution with mean μ and variance σ². The total area under the probability density curve equals 1, representing the certainty that the variable takes some value within its range.

正态分布是一种关于均值μ对称的连续概率分布，其形状由标准差σ决定。它常被称为高斯分布，广泛存在于身高、考试分数、测量误差等真实变量中。在练习21G.1中，你首先会遇到记号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。概率密度曲线下的总面积等于1，代表变量在其范围内取某个值的概率为1。

2. Key Properties of the Normal Curve | 正态曲线的关键性质

The normal curve is bell-shaped and unimodal, with the mean, median and mode all coinciding at the centre. The curve is perfectly symmetric, so the area to the left of μ is 0.5 and to the right is also 0.5. As σ increases, the curve becomes flatter and more spread out; as σ decreases, it becomes steeper and more concentrated around μ. Approximately 68% of the data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. These empirical rules are frequently tested in IB exam questions that build on Exercise 21G.1.

正态曲线呈钟形、单峰，均值、中位数和众数在中心重合。曲线完全对称，因此μ左侧的面积为0.5，右侧也为0.5。当σ增大时，曲线变平、更为分散；当σ减小时，曲线变陡、更集中于μ附近。大约68%的数据落入均值±1σ范围内，95%落入±2σ内，99.7%落入±3σ内。这些经验法则在基于练习21G.1的IB试题中经常考查。

3. Understanding the Parameters μ and σ | 理解参数μ和σ

The parameter μ determines the location of the centre of the distribution. Changing μ shifts the entire curve left or right along the horizontal axis without altering its shape. The parameter σ controls the spread; a larger σ means greater variability and a wider, lower peak. In Exercise 21G.1 you are often asked to sketch normal curves with different means and standard deviations, or to compare two normal distributions visually. Remember that the notation N(μ, σ²) uses the variance σ², so always be careful to extract the standard deviation correctly when computing probabilities.

参数μ决定了分布中心的位置。改变μ会使整条曲线沿水平轴左右平移，而形状不变。参数σ控制离散程度；σ越大意味着变异性越大，峰值更低且分布更宽。在练习21G.1中，你经常需要绘制不同均值和标准差的正态曲线，或者从视觉上比较两个正态分布。请牢记记号N(μ, σ²)使用的是方差σ²，因此在计算概率时务必正确提取标准差。

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

The standard normal distribution is a special case with mean 0 and standard deviation 1, denoted by Z ~ N(0, 1). Its probability density function is simpler, and its cumulative probabilities are tabulated in the IB formula booklet. Any normal variable can be transformed into a standard normal variable using the z-score. In Exercise 21G.1, you learn to perform this standardisation and to use the standard normal table, although the IB exam also expects you to use a graphical display calculator (GDC) efficiently.

标准正态分布是均值为0、标准差为1的特殊情形，记作Z ~ N(0, 1)。其概率密度函数更为简洁，且累积概率已在IB公式手册中列表给出。任何正态变量都可以通过z分数转化为标准正态变量。在练习21G.1中，你将学习如何进行这种标准化并使用标准正态表，不过IB考试还要求你高效使用图形计算器(GDC)。

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5. Standardising with Z-Scores | 用z分数进行标准化

The z-score tells you how many standard deviations a raw value x is away from the mean. The formula is:

z = (x − μ) / σ

A positive z-score means x is above μ, a negative z-score means x is below μ. In Exercise 21G.1, typical problems give μ, σ, and a specific x-value, then ask you to find the corresponding z-score and use it to determine P(X < x) or P(X > x). This step is fundamental because your GDC’s normalcdf function can directly handle non-standard distributions, but the conceptual understanding of standardisation remains vital for more complex problems.

z分数表示原始值x距离均值多少个标准差。公式为：z = (x − μ) / σ。若z为正，说明x高于μ；若z为负，说明x低于μ。在练习21G.1中，典型问题会给出μ、σ和某个特定x值，要求找出相应的z分数并计算P(X < x)或P(X > x)。这一步骤至关重要，因为尽管GDC的normalcdf函数可直接处理非标准分布，但深入理解标准化概念对解决更复杂的问题仍不可或缺。

6. Calculating Probabilities with GDC | 使用GDC计算概率

IB exams strongly emphasise the use of the GDC to find normal probabilities. You will use the normalcdf (or Ncd) function, which requires a lower bound, an upper bound, μ, and σ. For example, to find P(X < 25) for X ~ N(30, 4²), you set lower bound to a very small number (e.g. −10⁹⁹), upper bound to 25, μ = 30, σ = 4. The GDC returns the cumulative probability. Exercise 21G.1 builds fluency in choosing bounds correctly: left-tail, right-tail, or between two values. Always sketch a diagram, shade the required area, and label the probability statement before using the calculator.

IB考试非常重视使用GDC计算正态概率。你将使用normalcdf（或Ncd）函数，该函数需要下界、上界、μ和σ。例如，对于X ~ N(30, 4²)，求P(X < 25)，可将下界设为极小值（如−10⁹⁹）、上界设为25、μ=30、σ=4，GDC即返回累积概率。练习21G.1培养正确选择边界的能力：左尾、右尾或介于两值之间。在使用计算器前，始终画出示意图、涂出所需区域并标出概率陈述。

7. Inverse Normal Calculations | 逆正态计算

Sometimes Exercise 21G.1 problems reverse the process: given a probability, find the corresponding x-value. Here you use the invNorm (inverse normal) function on your GDC. You must input the left-tail cumulative probability, μ, and σ. The calculator returns the value k such that P(X < k) equals the given probability. Be careful with right-tail probabilities: if you are asked for the value with 10% above it, first convert to 90% below it. This skill is tested frequently in questions about quartiles, percentiles, or warranty limits that appear shortly after Exercise 21G.1.

有时候练习21G.1的问题会反过来：给定一个概率，求相应的x值。此时需使用GDC上的invNorm（逆正态）函数。你需要输入左尾累积概率、μ和σ。计算器将返回满足P(X < k)等于给定概率的k值。处理右尾概率时要小心：若要求上方10%对应的值，应首先转换为下方90%。这一技能常见于与四分位数、百分位数或保修期限相关的题目中，这些题目紧随练习21G.1之后出现。

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8. Working with Non-Standard Normal Distributions | 处理非标准正态分布

Not all normal variables have μ = 0 and σ = 1; in fact, most real data do not. Exercise 21G.1 explicitly introduces general normal distributions and shows how the GDC eliminates the need to standardise manually each time. However, you should still be able to convert a probability statement into a z-score probability when required, for example to use the printed standard normal table. The key is to maintain consistency: if using the z-score route, transform the boundary as well, then use P(Z < z). Both methods must yield identical results.

并非所有正态变量的μ都等于0、σ都等于1；事实上，大部分真实数据并非如此。练习21G.1明确引入了一般正态分布，并展示了GDC如何免去每次手工标准化的步骤。然而，你仍应能够在需要时将概率陈述转化为z分数概率，例如使用打印出的标准正态表时。关键在于保持一致性：若采用z分数路线，需同时转换边界，然后使用P(Z < z)。两种方法必须给出完全相同的结果。

9. Common Student Mistakes | 学生常见错误

One frequent error in Exercise 21G.1 is confusing variance σ² with standard deviation σ. The notation X ~ N(μ, σ²) explicitly uses variance, so always take the square root before entering σ into your GDC. Another mistake is choosing the wrong tail: for P(X > a), students sometimes forget to subtract the lower tail from 1, or they set an incorrect upper bound. Additionally, rounding too early can lead to inaccurate final answers. In IB exams, you should store intermediate values in the GDC and round only the final answer to three significant figures unless otherwise stated.

练习21G.1中一个常见的错误是混淆方差σ²与标准差σ。记号X ~ N(μ, σ²)明确使用方差，因此在将σ输入GDC之前，务必先开平方根。另一个错误是选错尾部：对于P(X > a)，学生有时忘记用1减去下尾概率，或者设置了错误的上界。此外，过早四舍五入也可能导致最终答案不准确。在IB考试中，你应在GDC中存储中间值，并仅将最终答案四舍五入至三位有效数字，除非题目另有说明。

10. Real-World Applications | 实际应用

The normal distribution is used extensively in quality control, finance, and natural sciences. In the context of Exercise 21G.1, you will encounter problems about battery lifetimes, heights of students, and exam scores. For instance, a question may state that the lifetime of a mobile phone battery is normally distributed with a mean of 20 hours and a standard deviation of 2.5 hours, and then ask for the probability that a battery lasts more than 24 hours. By modelling real situations, you see why the normal distribution is such a powerful statistical tool.

正态分布被广泛用于质量控制、金融和自然科学领域。在练习21G.1的背景中，你会遇到关于电池寿命、学生身高和考试成绩的问题。例如，题目可能描述手机电池的寿命服从均值为20小时、标准差为2.5小时的正态分布，然后要求计算电池续航超过24小时的概率。通过对真实情境建模，你将理解为何正态分布是一种如此强大的统计工具。

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11. Connection to the IB Syllabus | 与IB课程大纲的联系

Exercise 21G.1 aligns with several syllabus statements for both AI and AA routes. For AI SL/HL, it falls under Topic 4: Statistics and Probability, specifically the normal distribution and calculations with GDC. For AA SL/HL, normal distribution appears under Topic 4 as well, though AA students may also encounter it in the context of the central limit theorem. In both courses, the conceptual understanding gained from Exercise 21G.1 lays the groundwork for hypothesis testing, confidence intervals, and bivariate analysis that follow in the curriculum.

练习21G.1与AI和AA路径的多项大纲要求相对应。对于AI SL/HL，它属于主题4：统计与概率，特别是正态分布及GDC计算。AA SL/HL的正态分布也出现在主题4中，不过AA学生还可能在中心极限定理中遇到它。在这两类课程中，通过练习21G.1建立的概念理解为后续的假设检验、置信区间和双变量分析奠定了基础。

12. Exam Tips and Study Strategies | 考试技巧与学习策略

When tackling Exercise 21G.1, always write down the distribution notation and the probability you are finding before reaching for your GDC. Sketch a quick normal curve and shade the region. This not only reduces errors but also earns method marks in the exam. Practise using both normalcdf and invNorm functions until the steps become automatic. Finally, make summary cards: one side with the empirical rule percentages, the other with GDC keystrokes for your specific calculator model. Consistent practice with past paper questions that mirror Exercise 21G.1 will build confidence and speed.

在处理练习21G.1时，先写下分布记号和要求解的概率，再使用GDC。快速画出正态曲线并涂上阴影区域。这不仅能减少错误，还能在考试中赢得方法分。反复练习normalcdf和invNorm函数，直到操作自动化。最后，制作复习卡片：一面写上经验法则的百分比，另一面写下你所用计算器型号的具体按键步骤。通过反复练习与练习21G.1类似的历年真题，你将建立信心并提高解题速度。

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