Properties of the Normal Distribution and Probability Calculations | IB 数学：正态分布的性质与概率计算

📚 Properties of the Normal Distribution and Probability Calculations | IB 数学：正态分布的性质与概率计算

The normal distribution is one of the most important probability models in IB Mathematics. It appears throughout the Analysis and Approaches (AA) and Applications and Interpretation (AI) courses, especially in statistics and probability topics. Understanding its properties and mastering probability calculations is essential for exam success.

正态分布是 IB 数学中最核心的概率模型之一。无论是在分析与方法（AA）还是应用与解释（AI）课程中，它都会在统计与概率部分反复出现。掌握正态分布的性质和概率计算方法，对考试取得好成绩至关重要。

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

A normal distribution is a continuous probability distribution that is symmetric about its mean. It is often denoted as X ~ N(μ, σ²), where μ is the mean and σ is the standard deviation. The curve is bell-shaped, with most values clustering near the mean and fewer values appearing in the tails.

正态分布是一种关于均值对称的连续型概率分布，通常记作 X ~ N(μ, σ²)，其中 μ 为均值，σ 为标准差。其曲线呈钟形，大部分数据集中在均值附近，落在两端的数值越来越少。

The probability density function of a normal distribution is defined for every real number x and is given by the formula below.

正态分布的概率密度函数对每个实数 x 都有定义，其公式如下。

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

Here, e is Euler’s number, π is a mathematical constant, μ is the mean, and σ is the standard deviation. In practice, you rarely need to integrate this function by hand. Instead, you use a calculator, a statistical table, or standardised Z-scores.

其中 e 是自然常数，π 是数学常数，μ 是均值，σ 是标准差。实际解题时，通常不需要手动对这个函数求积分，而是借助计算器、统计表或标准化 Z 分数来完成。

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

The normal curve has several distinctive features that help you identify and interpret it correctly.

正态曲线具有若干显著特征，正确理解这些特征有助于你识别并解释正态分布。

The curve is symmetric about the mean μ. This means the left half is a mirror image of the right half.

曲线关于均值 μ 对称，即左半部分与右半部分互为镜像。
The total area under the curve is exactly 1, representing the total probability.

曲线下的总面积为 1，代表全部概率。
The mean, median, and mode are all equal and are located at the centre of the distribution.

均值、中位数与众数相等，且都位于分布的中心。
The curve is bell-shaped and has inflection points at x = μ – σ and x = μ + σ. These are the points where the curve changes from curving downward to curving upward.

曲线呈钟形，在 x = μ – σ 与 x = μ + σ 处存在拐点，即曲线由凹变凸或由凸变凹的点。
The tails of the curve approach but never touch the horizontal axis. Theoretically, values can extend to negative and positive infinity.

曲线的两端无限趋近于横轴但永不相交。理论上，取值可以延伸到正负无穷。

3. The Empirical Rule (68-95-99.7) | 经验法则（68-95-99.7）

The empirical rule is a quick way to estimate probabilities and percentages under a normal distribution. It states the following:

经验法则是一种快速估算正态分布概率和百分比的方法，其内容如下：

About 68% of the data lies within one standard deviation of the mean, i.e. between μ – σ and μ + σ.

约 68% 的数据落在均值的一个标准差之内，即在 μ – σ 与 μ + σ 之间。
About 95% of the data lies within two standard deviations of the mean, i.e. between μ – 2σ and μ + 2σ.

约 95% 的数据落在均值的两个标准差之内，即在 μ – 2σ 与 μ + 2σ 之间。
About 99.7% of the data lies within three standard deviations of the mean, i.e. between μ – 3σ and μ + 3σ.

约 99.7% 的数据落在均值的三个标准差之内，即在 μ – 3σ 与 μ + 3σ 之间。

For example, if the heights of a group of students are normally distributed with mean 170 cm and standard deviation 10 cm, then approximately 95% of students have heights between 150 cm and 190 cm.

例如，若一组学生的身高服从均值为 170 cm、标准差为 10 cm 的正态分布，则约有 95% 的学生身高在 150 cm 到 190 cm 之间。

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

Any normal distribution can be converted into the standard normal distribution Z ~ N(0, 1), which has mean 0 and standard deviation 1. This is done using the Z-score formula.

任何正态分布都可以转化为标准正态分布 Z ~ N(0, 1)，其均值为 0，标准差为 1。转化时使用 Z 分数公式。

Z = (X – μ) / σ

The Z-score measures how many standard deviations a value X lies above or below the mean. A positive Z-score means X is above the mean, while a negative Z-score means X is below the mean.

Z 分数衡量某个值 X 与均值相差多少个标准差。Z 分数为正说明 X 高于均值，为负说明 X 低于均值。

For example, if X ~ N(50, 4²), then the Z-score for X = 58 is (58 – 50) / 4 = 2. This means 58 is two standard deviations above the mean.

例如，若 X ~ N(50, 4²)，则 X = 58 的 Z 分数为 (58 – 50) / 4 = 2，表示 58 高于均值两个标准差。

5. Probability Calculations with the Normal Distribution | 正态分布的概率计算

In IB Mathematics, you need to find probabilities of the form P(X < a), P(X > a), or P(a < X < b). These correspond to areas under the normal curve.

在 IB 数学中，你需要计算形如 P(X < a)、P(X > a) 或 P(a < X < b) 的概率，这些概率对应正态曲线下的面积。

Because the total area under the curve is 1, the following relationships are always true for a continuous random variable:

由于曲线下总面积为 1，对连续型随机变量，以下关系恒成立：

P(X ≤ a) = P(X < a), because the probability of an exact single value is 0.

P(X ≤ a) = P(X < a)，因为连续型变量取某一精确值的概率为 0。
P(X > a) = 1 – P(X ≤ a).

P(X > a) = 1 – P(X ≤ a)。
P(a < X < b) = P(X < b) - P(X < a).

P(a < X < b) = P(X < b) - P(X < a)。

If the variable is standardised, then P(X < a) = Φ((a - μ)/σ), where Φ is the cumulative distribution function of the standard normal distribution.

若变量已标准化，则 P(X < a) = Φ((a - μ)/σ)，其中 Φ 是标准正态分布的累积分布函数。

6. Using the Standard Normal Table or GDC | 使用标准正态分布表或计算器

In the IB exam, you are expected to use a GDC (such as the TI-84 or Casio fx-9860G) for normal probability calculations. You can also use a standard normal distribution table in non-calculator contexts.

在 IB 考试中，你需要使用图形计算器（例如 TI-84 或 Casio fx-9860G）进行正态概率计算。在不允许使用计算器的题目中，也可以使用标准正态分布表。

When using the standard normal table, the table usually gives Φ(z) = P(Z < z). Follow these steps:

使用标准正态分布表时，表中所给的通常是 Φ(z) = P(Z < z)。具体步骤如下：

First, convert X to Z using the formula Z = (X – μ) / σ.

首先，利用公式 Z = (X – μ) / σ 将 X 转化为 Z。
Then, look up the Z-score in the table to find the cumulative probability P(Z < z).

然后，在表中查找 Z 分数对应的累积概率 P(Z < z)。
For P(Z > z), subtract the table value from 1.

若求 P(Z > z)，则用 1 减去表中的数值。

On a GDC, you can directly use the normal cumulative distribution function, often labelled normalCDF. You enter the lower bound, upper bound, mean, and standard deviation. For example, normalCDF(a, b, μ, σ) gives P(a < X < b).

使用图形计算器时，可以直接调用正态累积分布函数，通常标记为 normalCDF。你只需输入下限、上限、均值和标准差。例如 normalCDF(a, b, μ, σ) 输出 P(a < X < b)。

7. Inverse Normal Calculations | 逆正态计算

Sometimes you are given a probability and asked to find the corresponding value of X. This is called an inverse normal calculation, also known as finding the quantile or percentile.

有时题目会给出概率，要求你求出对应的 X 值。这类问题称为逆正态计算，也叫求分位数或百分位数。

The relation used is P(X < x) = p, and you must solve for x. On a GDC, the inverse normal function, often labelled invNorm, takes the probability p and the parameters μ and σ.

此时利用关系 P(X < x) = p 来求解 x。在图形计算器中，逆正态函数通常标记为 invNorm，输入概率 p、均值 μ 和标准差 σ 即可。

invNorm(p, μ, σ) = x

If you are working with the standard normal distribution, you can also use the table. Look inside the table for the probability closest to p, and then read the corresponding Z-score.

若使用标准正态分布，也可以查表。在表中找到最接近 p 的累积概率，再读出对应的 Z 分数。

For example, for Z ~ N(0, 1), the 90th percentile is approximately z = 1.2816. If X ~ N(100, 15²), then the 90th percentile is X = 100 + 1.2816 × 15 ≈ 119.2.

例如，对 Z ~ N(0, 1)，第 90 百分位数约为 z = 1.2816。若 X ~ N(100, 15²)，则其第 90 百分位数为 X = 100 + 1.2816 × 15 ≈ 119.2。

8. Normal Approximation to the Binomial Distribution | 二项分布的正态近似

In IB Mathematics, the normal distribution is also used to approximate a binomial distribution when n is large. The standard rule is that if n is large and both np and n(1 – p) are at least 5, the binomial distribution B(n, p) can be approximated by a normal distribution.

在 IB 数学中，当 n 很大时，正态分布也用于近似二项分布。常用规则是：若 n 足够大，且 np 与 n(1 – p) 均至少为 5，则二项分布 B(n, p) 可用正态分布近似。

B(n, p) ≈ N(np, np(1 – p))

Because the binomial distribution is discrete and the normal distribution is continuous, a continuity correction is required. For example, P(X = k) in the binomial model is approximated by P(k – 0.5 < X < k + 0.5) in the normal model.

由于二项分布是离散型分布，而正态分布是连续型分布，因此需要进行连续性修正。例如，二项分布中的 P(X = k) 可近似为正态分布中的 P(k – 0.5 < X < k + 0.5)。

Common continuity corrections:

常见的连续性修正如下：

P(X ≥ k) becomes P(X > k – 0.5).

P(X ≥ k) 改写为 P(X > k – 0.5)。
P(X ≤ k) becomes P(X < k + 0.5).

P(X ≤ k) 改写为 P(X < k + 0.5)。
P(X = k) becomes P(k – 0.5 < X < k + 0.5).

P(X = k) 改写为 P(k – 0.5 < X < k + 0.5)。

9. Exam-Style Worked Example | 考试题型示例

Let us work through a typical IB-style question. A machine fills bags of rice with mass X, measured in grams, where X is normally distributed with mean 500 g and standard deviation 12 g.

下面我们看一道典型的 IB 风格例题。某机器包装的大米质量 X（单位：克）服从正态分布，均值为 500 g，标准差为 12 g。

X ~ N(500, 12²)

(a) Find P(X > 520).

（a）求 P(X > 520)。

First, convert to a Z-score. Z = (520 – 500) / 12 = 1.6667. Then P(X > 520) = P(Z > 1.6667) = 1 – Φ(1.6667) ≈ 1 – 0.9522 = 0.0478. Using a GDC, normalCDF(520, ∞, 500, 12) gives the same result.

首先转换为 Z 分数：Z = (520 – 500) / 12 = 1.6667。因此 P(X > 520) = P(Z > 1.6667) = 1 – Φ(1.6667) ≈ 1 – 0.9522 = 0.0478。使用计算器 normalCDF(520, ∞, 500, 12) 可得相同结果。

(b) Find the mass m such that P(X < m) = 0.95.

（b）求质量 m，使得 P(X < m) = 0.95。

This is an inverse normal calculation. On a calculator, invNorm(0.95, 500, 12) ≈ 519.7 g. Thus, m ≈ 520 g. This means that 95% of bags weigh less than approximately 520 g.

这是逆正态计算。在计算器上输入 invNorm(0.95, 500, 12) ≈ 519.7 g。因此 m ≈ 520 g，也就是说约 95% 的袋装大米质量低于大约 520 g。

10. Common Pitfalls and Final Tips | 常见误区与最终建议

Many students lose marks on normal distribution questions due to avoidable errors. Here are the most common pitfalls and tips to overcome them.

许多学生在正态分布题目中因一些可以避免的错误而失分。以下是常见误区与应对建议。

Always check whether the problem gives σ or σ². If the variance is 144, the standard deviation is 12, not 144.

始终注意题目给的是 σ 还是 σ²。如果方差为 144，则标准差为 12，而不是 144。
Do not forget the continuity correction when approximating a binomial distribution.

使用正态分布近似二项分布时，不要忘记连续性修正。
For inverse normal questions, ensure the area you enter is the correct tail probability. If the problem asks for a lower percentile, use the cumulative probability directly.

在逆正态题目中，确保输入的面积是所要求的尾部概率。若题目求的是较低百分位数，应直接使用累积概率。
When using the normal distribution table, confirm whether the table gives the area to the left or to the right of the Z-score.

查标准正态分布表时，确认表中给出的是 Z 分数左侧还是右侧的面积。
Always draw a sketch of the normal curve with the mean labelled and the required region shaded. This reduces mistakes and helps the examiner follow your reasoning.

解题时先画出正态曲线草图，标出均值并涂阴影标出所求区域。这能减少错误，也便于考官理解你的思路。

Mastering the normal distribution requires practice. Work through past paper questions, memorise the key formulas, and learn to use your GDC efficiently before the exam.

掌握正态分布需要多加练习。建议认真刷历年真题，熟记关键公式，并在考前熟练掌握图形计算器的操作。

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Properties of the Normal Distribution and Probability Calculations | IB 数学：正态分布的性质与概率计算