📚 9 The normal distribution | 正态分布
The normal distribution is one of the most important continuous probability distributions in statistics. It models a vast range of natural phenomena, from heights and weights to measurement errors and test scores. Understanding its properties and applications is essential for any IB Mathematics student, as it appears in both Analysis & Approaches and Applications & Interpretation courses. In this chapter, we explore the characteristics of the normal curve, learn to calculate probabilities using standardization and technology, and extend the model to approximate binomial distributions.
正态分布是统计学中最重要的连续概率分布之一。它模拟了大量自然现象，从身高、体重到测量误差和考试分数。理解其性质和应用对每位 IB 数学学生都至关重要，因为它在分析与方法和应用与解释课程中均有出现。本章我们将探索正态曲线的特征，学习使用标准化和技术计算概率，并将该模型扩展到二项分布的近似。
1. What is the Normal Distribution? | 什么是正态分布？
A normal distribution is a continuous probability distribution defined by its bell-shaped probability density function. The random variable X can take any real value, and the total area under the curve equals 1. The distribution is determined entirely by two parameters: the mean μ, which locates the centre, and the standard deviation σ, which controls the spread. We write X ~ N(μ, σ²) to indicate that X follows a normal distribution with mean μ and variance σ².
正态分布是一种连续概率分布，由其钟形概率密度函数定义。随机变量 X 可取任何实数值，曲线下的总面积为 1。该分布完全由两个参数决定：均值 μ 规定了中心位置，标准差 σ 控制了分散程度。我们记作 X ~ N(μ, σ²)，表示 X 服从均值为 μ、方差为 σ² 的正态分布。
Many real-world variables are approximately normally distributed. For example, the heights of adult men, the blood pressure readings in a population, or the errors produced by a machine often follow a normal pattern. The central limit theorem further explains why the normal distribution appears so frequently in statistical sampling.
许多现实世界变量都近似服从正态分布。例如，成年男性的身高、人群中的血压读数或机器产生的误差常常遵循正态模式。中心极限定理进一步解释了为什么正态分布在统计抽样中如此频繁地出现。
2. The Probability Density Function | 概率密度函数
The probability density function (PDF) of a normal random variable X ~ N(μ, σ²) is given by the formula:
正态随机变量 X ~ N(μ, σ²) 的概率密度函数由以下公式给出：
f(x) = 1/(σ√2π) e−(x−μ)²/(2σ²)
In this expression, e is Euler' The exponent contains the squared deviation of x from the mean, divided by twice the variance. This formula ensures the curve is symmetric about μ and that the total area under the curve equals exactly 1.
在此表达式中，e 是欧拉数（≈ 2.71828），π ≈ 3.14159。指数部分包含 x 与均值偏差的平方除以两倍方差。该公式确保了曲线关于 μ 对称，并且曲线下的总面积恰好等于 1。
The PDF itself does not give probabilities directly; instead, probabilities are found by integrating f(x) over an interval. Because the integral has no elementary closed form, we rely on numerical methods, standard tables, or, in the IB course, your graphical display calculator (GDC).
概率密度函数本身并不直接给出概率；相反，概率是通过对 f(x) 在区间上积分得到的。由于该积分没有初等的封闭形式，我们依赖数值方法、标准表格，或在 IB 课程中使用图形计算器（GDC）。
3. Properties of the Normal Curve | 正态曲线的性质
The normal curve has several defining features that are essential for problem-solving. First, it is perfectly symmetric about the mean μ, so the area to the left of μ is exactly 0.5, and the area to the right is also 0.5. The mean, median, and mode all coincide at the centre.
正态曲线有几个定义性特征，对解题至关重要。首先，它关于均值 μ 完全对称，因此 μ 左侧的面积恰好为 0.5，右侧的面积也为 0.5。均值、中位数和众数都在中心处重合。
Second, the curve is asymptotic to the horizontal axis; it extends infinitely in both directions but never touches the x-axis. Third, the total area under the curve equals 1, representing a 100% probability that the random variable falls somewhere on the real line. Fourth, the spread is governed by σ: a smaller σ produces a taller, narrower peak, while a larger σ flattens and widens the curve.
其次，曲线渐近于水平轴；它向两端无限延伸但永不触及 x 轴。第三，曲线下的总面积为 1，意味着随机变量落在实数轴某处的概率为 100%。第四，分散程度由 σ 决定：较小的 σ 产生更高更窄的峰，而较大的 σ 使曲线变得扁平宽阔。
Inflection points occur exactly one standard deviation away from the mean, at μ − σ and μ + σ. At these points, the curvature changes direction. This geometric property helps in sketching the curve accurately.
拐点恰好位于距离均值一个标准差的位置，即 μ − σ 和 μ + σ 处。在这些点上，曲率改变方向。这一几何性质有助于精确绘制曲线。
4. The Standard Normal Distribution | 标准正态分布
To work with normal probabilities without integrating every time, we transform any normal random variable to a standard normal variable Z ~ N(0, 1). The transformation is called standardization:
为避免每次都对正态分布进行积分，我们将任何正态随机变量转换为标准正态变量 Z ~ N(0, 1)。这种转换称为标准化：
Z = (X − μ) / σ
graphic calculators, statisticians used printed tables of Φ(z). In the IB exam, you may still be asked to read such a table or, more commonly, to use the normCdf and invNorm functions on your GDC.
标准正态分布的均值为 0，标准差为 1。其概率密度常记为 φ(z)，累积分布函数为 Φ(z) = P(Z ≤ z)。在图形计算器出现之前，统计学家使用印刷的 Φ(z) 表。在 IB 考试中，你仍可能被要求查阅此类表格，但更常见的是使用 GDC 上的 normCdf 和 invNorm 功能。
< Z < (b − μ)/σ). This means once you know how to find probabilities for Z, you can solve problems for any normal distribution.
该转换保留了面积性质：P(a < X < b) = P((a − μ)/σ < Z < (b − μ)/σ)。这意味着一旦你懂得如何求 Z 的概率，就能解决任何正态分布的问题。
5. Calculating Probabilities with Technology | 使用技术计算概率
In IB Mathematics, you are expected to use your GDC to compute normal probabilities efficiently. The key functions are:
在 IB 数学中，你需要使用 GDC 高效地计算正态概率。关键函数如下：
normCdf(lower, upper, μ, σ): returns the probability P(lower < X < upper) for X ~ N(μ, σ²).
normCdf(下限, 上限, μ, σ)：返回 X ~ N(μ, σ²) 中 P(下限 < X < 上限) 的概率。
invNorm(area, μ, σ): returns the value x such that P(X < x) = area (the left-tail probability).
invNorm(面积, μ, σ)：返回使得 P(X < x) = 面积 的值 x（左尾概率）。
a < Z < a) = 2P(Z < a) − 1. Always sketch a diagram to avoid calculator mistakes.
在处理标准正态分布时，可省略 μ 和 σ（默认值为 0 和 1）。对于单尾问题，要正确指定尾部。如果需要右尾概率，使用补集规则：P(X > k) = 1 − P(X < k)。对于对称区间，P(−a < Z < a) = 2P(Z < a) − 1。务必画出示意图，以避免计算器错误。
6. Inverse Normal Calculations | 逆正态计算
obability and ask for the corresponding x-value. For example, find the height such that 90% 70, 6) directly yields the 90th percentile. If the question asks for a symmetric central interval containing 95% of the data, remember that the tails each contain 2.5% e left boundary corresponds to invNorm(0.025, μ, σ) and the right to invNorm(0.975, μ, σ).
逆正态问题给出概率，要求找出对应的 x 值。例如，给定 μ = 170 厘米、σ = 6 厘米，求满足 90% 人群身高低于该值的高度。直接使用 invNorm(0.9, 170, 6) 即可得到第 90 个百分位。若问题要求求出包含 95% 数据的对称中心区间，请记住两尾各含 2.5%，因此左边界对应 invNorm(0.025, μ, σ)，右边界对应 invNorm(0.975, μ, σ)。
IB exam questions often present a “find the mean” or “find the standard deviation” > 40) = 0.1, you can translate these to Z-scores and solve for μ and σ simultaneously.
IB 考试题目常通过给出两个概率来设置“求均值”或“求标准差”的情境。你需要利用 invNorm 或手动标准化建立两个方程，并求解一个线性方程组。例如，若 P(X < 20) = 0.3 且 P(X > 40) = 0.1，可将其转换为 Z 分数，并同时解出 μ 和 σ。
7. The Empirical Rule | 经验法则
For any normal distribution, roughly 68% of observations lie within one standard deviation of the mean (μ ± σ), about 95% lie within two standard deviations (μ ± 2σ), and approximately 99.7% fall within three standard deviations (μ ± 3σ). This is known as the 68–95–99.7 rule.
对于任何正态分布，大约 68% 的观测值落在均值的一个标准差范围内（μ ± σ），约 95% 落在两个标准差范围内（μ ± 2σ），约 99.7% 落在三个标准差范围内（μ ± 3σ）。这被称为 68–95–99.7 法则。
pproximations, but they provide a quick sanity check for answers. If you calculate that 80% o reinforces the idea that extreme values are rare in a normal distribution, with only 0.3% of data beyond ±3σ.
这些百分比是近似值，但它们提供了快速查验答案的合理性依据。若你计算出 80% 的数据落在 μ ± σ 内，就知道出错了。经验法则还强化了一个概念：在正态分布中极端值是罕见的，仅 0.3% 的数据超出 ±3σ 范围。
8. Normal Approximation to the Binomial | 二项分布的正态近似
approximating normal distribution has mean μ = np and variance σ² = np(1 − p), so we use Y ~ N(np, np(1 − p)).
当二项随机变量 B(n, p) 的样本量足够大时，其形状非常接近正态曲线。这使得我们可以用正态分布来近似二项概率，当 n 很大时计算更快。近似的正态分布的均值为 μ = np，方差为 σ² = np(1 − p)，因此我们使用 Y ~ N(np, np(1 − p))。
r a more accurate approximation. Always check these conditions before applying the normal approximation.
该近似仅在满足一定条件时才被认为有效。IB 中最常引用的准则是 np ≥ 5 且 n(1 − p) ≥ 5。有些教科书推荐 np ≥ 10 且 n(1 − p) ≥ 10，以获得更精确的近似。在应用正态近似之前，务必检查这些条件。
9. Continuity Correction | 连续性校正
lt; k + 0.5) using the normal variable Y. For P(X ≤ k) we use P(Y < k + 0.5), and for P(X ≥ k) we use P(Y > k − 0.5).
由于二项分布是离散的而正态分布是连续的，必须进行校正以提高精度。连续性校正涉及将离散的 x 值调整 0.5。例如，要近似二项分布的 P(X = k)，我们使用正态变量 Y 计算 P(k − 0.5 < Y < k + 0.5)。对于 P(X ≤ k)，使用 P(Y < k + 0.5)；对于 P(X ≥ k)，使用 P(Y > k − 0.5)。
Forgetting the continuity correction is a common IB examination error. The mark scheme often explicitly awards marks for using it, especially when approximating a single probability. When n increases, the effect of the correction becomes smaller, but it is still required for precise work unless stated otherwise.
忘记连续性校正是 IB 考试中常见的错误。评分方案通常会明确对使用校正给予分数，尤其是在近似单个概率时。随着 n 的增加，校正的效果变小，但除非另有说明，仍须使用它以保证精确性。
10. Choosing the Right Method | 选择正确的方法
In an IB question, you must decide whether to use exact binomial calculations, a normal approximation, or direct normal distribution. If the variable is inherently continuous and you are given μ and σ, use normCdf or invNorm. If the variable counts successes in n trials, first check if the distribution is exactly binomial. If n is small or the success probability is extreme, use the exact binomial formula or binomCdf on the GDC.
在 IB 考题中，你必须决定是使用精确二项计算、正态近似还是直接使用正态分布。如果变量本身是连续的并给出了 μ 和 σ，使用 normCdf 或 invNorm。如果变量计算的是 n 次试验中的成功次数，首先检查分布是否严格为二项分布。如果 n 较小或成功概率极端，应使用精确二项公式或 GDC 上的 binomCdf。
ion with continuity correction is appropriate and often saves time. However, always verify whether the question specifically instructs a method. In the IB “Applications and Interpretation” course, emphasis is placed on using technology, while “Analysis and Approaches” may require more algebraic manipulation.
如果 n 很大且满足 np ≥ 5 和 n(1 − p) ≥ 5 的条件，使用带连续性校正的正态近似是合适的，并常能节省时间。但务必核对题目是否明确指定了方法。在 IB “应用与解释”课程中，强调使用技术，而“分析与方法”则可能要求更多的代数操作。
11. Typical IB Exam Questions and Strategies | 典型 IB 考题与策略
Typical questions include: “Given X ~ N(24, 9), find P(X > 26)”, “Find the interquartile range of a normal distribution”, “The weights of apples are normally distributed with mean 150 g and standard deviation 20 g. Find the probability that a randomly chosen apple weighs between 140 g and 165 g.” For each, draw a normal curve, shade the required area, and set up the correct calculator syntax.
典型题目包括：“已知 X ~ N(24, 9)，求 P(X > 26)”、“求正态分布的四分位距”、“苹果的重量服从均值为 150 克、标准差为 20 克的正态分布。求随机选取的一个苹果重量在 140 克到 165 克之间的概率。”对于每道题，画出正态曲线，标出所需区域，并设置正确的计算器语法。
utions, such as comparing scores from different tests, use Z-scores to make a fair comparison. The object with the higher Z-score performed better relative to its distribution. Another common IB task is to determine unknown μ or σ from probability statements— , often using the invNorm function to get the Z-value corresponding to the given left-tail probability.
当处理两个正态分布时，例如比较不同测试的分数，使用 Z 分数进行公平比较。Z 分数较高的对象相对于其分布表现更好。另一个常见的 IB 任务是，根据概率语句确定未知的 μ 或 σ——建立 Z = (x − μ)/σ 并仔细求解，通常使用 invNorm 函数获取与给定左尾概率对应的 Z 值。
12. Summary and Key Takeaways | 总结与要点
The normal distribution underpins much of inferential statistics and modeling. Remember that the curve is symmetric, the total area is 1, and probabilities correspond to areas under the curve. Standardization translates any normal problem into a standard normal context, simplifying the computation. Always make a sketch to confirm the region you need. The empirical rule gives quick estimates, while technology provides precise answers.
正态分布是推断统计和建模的基石。请记住，曲线是对称的，总面积为 1，概率对应于曲线下的面积。标准化将任何正态问题转换为标准正态情境，简化了计算。务必画出草图以确认所需的区域。经验法则提供快速估算，而技术则提供精确答案。
(1 − p)), and apply the continuity correction to bridge the discrete-continuous gap. This topic connects core probability concepts and equips you with tools to tackle real-world data analysis. Practice plenty of past paper questions to build confidence in both calculator skills and theoretical reasoning.
对于二项近似，检查 np ≥ 5 和 n(1 − p) ≥ 5，设 μ = np、σ = √(np(1 − p))，并应用连续性校正来弥合离散与连续的差距。该主题将核心概率概念联系起来，并为你提供了处理现实世界数据分析的工具。练习大量历年真题，以增强对计算器技能和理论推理的信心。
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