The Normal Distribution in IB Mathematics | IB数学：正态分布

📚 The Normal Distribution in IB Mathematics | IB数学：正态分布

The normal distribution is one of the most important continuous probability distributions in IB Mathematics. It is used to model data that clusters around a central mean, such as heights, test scores, and measurement errors. This article provides a comprehensive, exam-focused review of the normal distribution for both Analysis and Approaches (AA) and Applications and Interpretation (AI) students.

正态分布是IB数学中最重要的连续概率分布之一。它用于模拟围绕中心均值聚集的数据，例如身高、考试成绩和测量误差。本文为数学分析与方法（AA）和应用与解释（AI）的学生提供全面、紧扣考点的复习。

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

The normal distribution is a continuous probability distribution defined by two parameters: the mean μ and the standard deviation σ. Its graph is the familiar symmetric bell-shaped curve. The total area under the curve is exactly 1, representing the total probability.

正态分布是一种连续概率分布，由两个参数决定：均值 μ 和标准差 σ。它的图形是熟悉的对称钟形曲线。曲线下的总面积为1，表示总概率。

A normal random variable X is often written as X ~ N(μ, σ²). Here μ determines the centre of the distribution, while σ measures the spread. A smaller σ produces a taller, narrower curve; a larger σ produces a flatter, wider curve.

正态随机变量 X 常记作 X ~ N(μ, σ²)。其中 μ 决定分布的中心，σ 衡量分布的离散程度。σ 越小，曲线越高而窄；σ 越大，曲线越低而宽。

2. The Probability Density Function | 概率密度函数

The probability density function (PDF) of a normal distribution is given by the formula:

f(x) = (1 / (σ√(2π))) · e^(−(x−μ)² / (2σ²))

This formula is provided in the IB formula booklet, so you usually do not need to memorise it. However, you must understand how the parameters μ and σ affect the shape and position of the curve.

该公式在IB公式手册中提供，通常不需要记忆。但你必须理解 μ 和 σ 如何影响曲线的形状和位置。

Because X is continuous, the probability that X equals any single exact value is zero. Instead, probabilities are always calculated over intervals, such as P(a ≤ X ≤ b), which is the area under the curve between a and b.

由于 X 是连续型随机变量，X 恰好等于某个具体值的概率为零。因此，概率总是通过区间来计算，例如 P(a ≤ X ≤ b)，即曲线下 a 到 b 之间的面积。

3. Properties of the Normal Curve | 正态曲线的性质

The normal curve has several important properties that are frequently tested in IB exams:

正态曲线有几个在IB考试中经常考查的重要性质：

Symmetry: The curve is perfectly symmetric about the mean μ. This means P(X ≤ μ) = P(X ≥ μ) = 0.5.

对称性：曲线关于均值 μ 完全对称。这意味着 P(X ≤ μ) = P(X ≥ μ) = 0.5。
Unimodal: The curve has a single peak at x = μ, where the mode and median are equal to the mean.

单峰性：曲线在 x = μ 处有唯一的峰值，且众数和中位数均等于均值。
Inflection points: The curve changes from concave down to concave up at x = μ − σ and x = μ + σ.

拐点：曲线在 x = μ − σ 和 x = μ + σ 处由凹向下变为凹向上。
Asymptotic tails: The tails of the curve approach the x-axis but never actually touch it.

渐进尾部：曲线的尾部无限接近 x 轴，但永远不会触及 x 轴。

4. The Standard Normal Distribution and Z-Scores | 标准正态分布与Z分数

The standard normal distribution is a special case with mean 0 and standard deviation 1, denoted Z ~ N(0, 1). Any normal variable X can be transformed into a standard normal variable Z using the z-score formula:

标准正态分布是均值为0、标准差为1的特殊情况，记作 Z ~ N(0, 1)。任何正态变量 X 都可以通过 z 分数公式转换为标准正态变量 Z：

z = (x − μ) / σ

The z-score measures how many standard deviations a value x is above or below the mean. A positive z-score means x is above the mean; a negative z-score means x is below the mean.

z 分数衡量一个值 x 高于或低于均值多少个标准差。z 分数为正表示 x 高于均值；z 分数为负表示 x 低于均值。

In the IB syllabus, you are expected to use your GDC (graphical display calculator) to find probabilities and inverse values directly. However, understanding z-scores is still essential for interpreting results and for hypothesis testing.

在IB课程中，你被期望使用图形计算器（GDC）直接计算概率和反查值。然而，理解 z 分数对于解释结果和进行假设检验仍然至关重要。

5. Calculating Probabilities with Your GDC | 使用图形计算器计算概率

Most IB exam questions require you to calculate probabilities such as P(X < a), P(X > b), or P(a < X < b). You should use the normal CDF function on your GDC, typically found under the "DISTR" or "Distribution" menu.

大多数IB考题要求你计算诸如 P(X < a)、P(X > b) 或 P(a < X < b) 的概率。你应该使用计算器上的正态累积分布函数（normal CDF），通常位于“DISTR”或“Distribution”菜单中。

On a TI-84, the command is normalcdf(lower, upper, μ, σ). For example, if X ~ N(100, 15²), then P(90 < X < 110) = normalcdf(90, 110, 100, 15). On many calculators you can also enter the z-scores directly.

在TI-84上，命令是 normalcdf(下限, 上限, μ, σ)。例如，若 X ~ N(100, 15²)，则 P(90 < X < 110) = normalcdf(90, 110, 100, 15)。在许多计算器上，你也可以直接输入 z 分数。

Always sketch a rough graph of the normal curve and shade the required area. This helps you choose the correct lower and upper bounds and avoids careless mistakes.

始终画一个正态曲线的草图，并标出所需区域。这有助于你选择正确的上下限，避免粗心错误。

6. Inverse Normal Calculations | 正态分布的反向计算

In inverse normal problems, you are given a probability or percentile and asked to find the corresponding x value. For example, “Find the value of a such that P(X < a) = 0.85." This is equivalent to finding the 85th percentile of the distribution.

在反向正态问题中，你已知概率或百分位数，需要求对应的 x 值。例如，“求 a 使得 P(X < a) = 0.85。”这相当于求分布的第85个百分位数。

On a TI-84, use the invNorm command: invNorm(area, μ, σ). For a left-tail probability, enter the area directly. For a right-tail probability, use invNorm(1 − area, μ, σ) or a symmetric property.

在TI-84上，使用 invNorm 命令：invNorm(面积, μ, σ)。对于左尾概率，直接输入面积。对于右尾概率，使用 invNorm(1 − 面积, μ, σ) 或利用对称性。

Be careful with “more than” language. If you want P(X > a) = 0.10, then a is at the 90th percentile, so use invNorm(0.90, μ, σ).

注意“大于”这类表述。如果你想 P(X > a) = 0.10，则 a 是在第90个百分位数，应使用 invNorm(0.90, μ, σ)。

7. Normal Approximation to the Binomial Distribution | 正态近似二项分布

When the number of trials n is large and the probability of success p is not too close to 0 or 1, the binomial distribution B(n, p) can be approximated by a normal distribution with mean μ = np and variance σ² = np(1 − p).

当试验次数 n 很大且成功概率 p 不太接近0或1时，二项分布 B(n, p) 可以用正态分布近似，其均值 μ = np，方差 σ² = np(1 − p)。

A common rule of thumb is that both np and n(1 − p) should be at least 5 (or 10, depending on the textbook) for the approximation to be reliable.

一个常用的经验法则是 np 和 n(1 − p) 都应至少为5（或10，取决于教材），近似才可靠。

For example, if X ~ B(200, 0.4), then X can be approximated by N(80, 48). You would then use the normal distribution to estimate probabilities such as P(X ≤ 90).

例如，若 X ~ B(200, 0.4)，则 X 可近似为 N(80, 48)。然后使用正态分布来估计诸如 P(X ≤ 90) 的概率。

8. Continuity Correction | 连续性修正

Because the binomial distribution is discrete and the normal distribution is continuous, a continuity correction is needed when approximating. This adjusts the interval boundaries by 0.5.

由于二项分布是离散的而正态分布是连续的，近似时需要连续性修正。这通过将区间边界调整0.5来实现。

The table below summarises the corrections for common inequalities:

下表总结了常见不等式的修正方法：

Binomial statement	Normal approximation
P(X ≤ k)	P(X ≤ k + 0.5)
P(X < k)	P(X ≤ k − 0.5)
P(X ≥ k)	P(X ≥ k − 0.5)
P(X > k)	P(X ≥ k + 0.5)

For example, P(X ≤ 20) becomes P(X ≤ 20.5) in the normal approximation, while P(X < 20) becomes P(X ≤ 19.5).

例如，P(X ≤ 20) 在正态近似中变成 P(X ≤ 20.5)，而 P(X < 20) 变成 P(X ≤ 19.5)。

9. Hypothesis Testing with the Normal Distribution | 使用正态分布进行假设检验

In IB Mathematics, hypothesis testing is a key skill in both AA and AI. For a population mean μ with known variance, we use the z-test based on the normal distribution. The test statistic is:

在IB数学中，假设检验是AA和AI的关键技能。对于方差已知的总体均值 μ，我们使用基于正态分布的 z 检验。检验统计量为：

z = (x̄ − μ₀) / (σ / √n)

Here x̄ is the sample mean, μ₀ is the hypothesized population mean, σ is the population standard deviation, and n is the sample size.

其中 x̄ 是样本均值，μ₀ 是假设的总体均值，σ 是总体标准差，n 是样本量。

You compare the calculated z-value with the critical value from a standard normal table (or using your GDC). If |z| is greater than the critical value, you reject the null hypothesis at the chosen significance level.

你将计算出的 z 值与标准正态表（或使用计算器）的临界值比较。如果 |z| 大于临界值，则在选定的显著性水平下拒绝原假设。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Students often make the following mistakes in normal distribution questions:

学生在正态分布题目中常犯以下错误：

Forgetting to use the correct lower or upper bound when using normalcdf. For a left tail, use −∞ as the lower bound, which can be entered as −10⁹⁹.

使用 normalcdf 时忘记正确的上下限。对于左尾，用 −∞ 作为下限，可以输入 −10⁹⁹。
Confusing σ and σ². If a problem gives the variance as 25, then σ = 5.

混淆 σ 和 σ²。如果题目给出方差为25，则 σ = 5。
Applying continuity correction incorrectly or forgetting it entirely in binomial approximation.

在二项近似中错误地应用或完全忘记连续性修正。
Using the z-score formula when asked for the actual x value; remember to rearrange and solve for x.

在题目要求求 x 的值时仍使用 z 分数公式；记住要变形解出 x。

Always define your random variable clearly, write down the distribution, and sketch the curve. Award-winning exam responses show clear method and final answer to 3 significant figures.

始终明确写出随机变量，标明分布，并画草图。获得高分的答题步骤应展示清晰的方法，最终答案保留3位有效数字。

11. Real-World Applications | 实际应用

Normal distributions arise naturally in many real-world contexts. Examples include IQ scores (mean 100, standard deviation 15), heights and weights of populations, blood pressure measurements, and measurement errors in scientific experiments.

正态分布广泛应用于现实世界。例如智商分数（均值100，标准差15）、人群的身高和体重、血压测量值以及科学实验中的测量误差。

The empirical rule (68-95-99.7 rule) states that about 68% of data lies within one standard deviation of the mean, 95% within two, and 99.7% within three. This rule is a quick way to estimate probabilities without a calculator.

经验法则（68-95-99.7法则）指出，约68%的数据落在均值的一个标准差内，95%落在两个标准差内，99.7%落在三个标准差内。该法则是快速估算概率的方法。

In IB exam problems, you may be asked to interpret real data in terms of the normal model, such as finding the percentage of products that fall outside a given specification limit. Remember to contextualise your answer.

在IB考试题中，你可能会被要求用正态模型解释真实数据，例如找出超出规定规格限制的产品百分比。记住要结合上下文给出答案。

12. Summary | 总结

The normal distribution is a central topic in IB Mathematics. You should be comfortable with its properties, able to compute probabilities and inverse values using your GDC, and understand how to apply the normal approximation to the binomial distribution with continuity correction.

正态分布是IB数学的核心内容。你应该熟练掌握其性质，能够使用计算器计算概率和反查值，并理解如何应用正态近似二项分布及连续性修正。

Practice past-paper questions, familiarise yourself with calculator shortcuts, and always verify that your answers are plausible in context. With systematic revision and careful exam technique, you can master the normal distribution and earn full marks on these questions.

多做真题，熟悉计算器快捷操作，并始终检查答案在上下文中是否合理。通过系统复习和细致的答题技巧，你可以掌握正态分布并在相关题目中获得满分。

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The Normal Distribution in IB Mathematics | IB数学：正态分布