📚 Normal Distribution | 正态分布
The normal distribution is one of the most important continuous probability distributions in statistics. Its bell-shaped curve appears naturally in many real-world situations, from measurement errors to heights of people. In IB Mathematics, understanding the normal distribution is essential for both the Analysis and Approaches (AA) and Applications and Interpretation (AI) courses. This article covers the key concepts, calculations, and applications to help you master the topic.
正态分布是统计学中最重要的连续概率分布之一。它的钟形曲线广泛存在于许多现实场景中，从测量误差到人们的身高。在IB数学中，理解正态分布对分析与方法（AA）以及应用与解释（AI）课程都至关重要。本文将涵盖关键概念、计算方法和应用，帮助你掌握这一内容。
1. What is a Normal Distribution? | 什么是正态分布？
A normal distribution is a symmetric, bell-shaped distribution defined by two parameters: the mean (μ) and the standard deviation (σ). The probability density function is given by:
正态分布是一个对称的钟形分布，由两个参数定义：均值（μ）和标准差（σ）。其概率密度函数为：
f(x) = (1/(σ√(2π))) · e^[ –(x – μ)² / (2σ²) ]
The notation X ~ N(μ, σ²) means that the random variable X is normally distributed with mean μ and variance σ². Note that the second parameter is the variance, not the standard deviation.
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。注意第二个参数是方差，而不是标准差。
The graph of the normal distribution is symmetric about the mean, and the total area under the curve is 1. This property allows us to use area to represent probability.
正态分布曲线关于均值对称，曲线下的总面积为1。这一性质使我们能用面积表示概率。
2. Properties of the Normal Curve | 正态曲线的性质
The normal curve is bell-shaped and symmetric. The mean, median, and mode are all equal. The curve has inflection points at μ ± σ, where the curvature changes from concave down to concave up.
正态曲线呈钟形且对称。平均值、中位数和众数相等。曲线在 μ ± σ 处有拐点，在此曲率由凹向下变为凹向上。
The tails of the distribution approach but never touch the horizontal axis. This means the random variable can take any real value, but extreme values are very unlikely.
分布的尾部趋近但永不触及水平轴。这意味着随机变量可以取任何实数值，但极端值非常不可能。
The total area under the normal curve is exactly 1. Therefore, the probability of the variable falling within a certain interval equals the area under the curve over that interval.
正态曲线下的总面积为1。因此，变量落在某一区间内的概率等于该区间上曲线下的面积。
3. Parameters: Mean and Standard Deviation | 参数：均值与标准差
The mean μ determines the location of the centre of the distribution. Changing μ shifts the curve left or right without changing its shape.
均值 μ 决定了分布中心的位置。改变 μ 会使曲线左右平移，而不改变其形状。
The standard deviation σ controls the spread of the distribution. A larger σ results in a flatter, wider curve; a smaller σ gives a taller, narrower curve. Two normal distributions with the same mean but different standard deviations are shown below conceptually:
标准差 σ 控制分布的离散程度。σ 越大，曲线越平缓宽大；σ 越小，曲线越陡峭窄小。下图概念性地展示了均值相同但标准差不同的两个正态分布：
Curve A (σ small): tall and concentrated; Curve B (σ large): short and spread out.
曲线A（σ小）：高而集中；曲线B（σ大）：矮而分散。
Because the area must remain 1, the peak height adjusts accordingly.
因为总面积必须保持为1，峰高会相应调整。
4. The Empirical Rule (68-95-99.7) | 经验法则（68-95-99.7）
For any normal distribution, approximately 68% of the data lie within 1 standard deviation of the mean (μ ± σ), about 95% within 2 standard deviations (μ ± 2σ), and about 99.7% within 3 standard deviations (μ ± 3σ).
对于任何正态分布，大约68%的数据落在均值左右一个标准差内（μ ± σ），约95%在两个标准差内（μ ± 2σ），约99.7%在三个标准差内（μ ± 3σ）。
This rule provides quick probability estimates without using tables. For example, if the heights of adult men are normally distributed with μ = 178 cm and σ = 7 cm, then 95% of men have heights between 164 cm and 192 cm. Only 2.5% are taller than 192 cm.
该法则无需查表即可快速估计概率。例如，如果成年男性身高服从正态分布，μ = 178 cm，σ = 7 cm，那么95%的男性身高在164 cm到192 cm之间。只有2.5%的男性身高超过192 cm。
Similarly, about 99.7% of observations fall within 3σ of the mean, which is helpful in quality control to define ‘almost all’ items.
类似地，约99.7%的观测值落在均值左右3σ范围内，这在质量控制中用于定义“几乎所有”产品都符合的范围。
5. Standard Normal Distribution and Z-Score | 标准正态分布与Z分数
The standard normal distribution is a special normal distribution with μ = 0 and σ = 1. It is denoted by Z ~ N(0, 1).
标准正态分布是一个特殊的正态分布，其均值 μ = 0、标准差 σ = 1。记为 Z ~ N(0, 1)。
To convert a value from a general normal distribution to the standard normal, we calculate the z-score:
要将一般正态分布的值转换为标准正态分布，我们计算Z分数：
Z = (X – μ) / σ
The z-score tells us how many standard deviations X is above or below the mean. If X = 130 from N(100, 15²), then Z = (130 – 100) / 15 = 2. This means 130 is 2 standard deviations above the mean.
Z分数告诉我们 X 在均值之上或之下多少个标准差。若 X = 130 且 X ~ N(100, 15²)，则 Z = (130 – 100) / 15 = 2，表示130高于均值2个标准差。
A negative z-score indicates the value is below the mean. For X = 85, Z = (85 – 100) / 15 = –1, meaning one standard deviation below the mean.
负的Z分数表示该值低于均值。例如 X = 85，Z = (85 – 100) / 15 = –1，表示低于均值1个标准差。
6. Using the Standard Normal Table | 使用标准正态表
The standard normal table (z-table) gives cumulative probabilities P(Z ≤ z) for the standard normal distribution. In IB examinations, you are provided with a formula booklet that includes such a table. It is essential to know how to read it correctly.
标准正态表（Z表）给出了标准正态分布的累积概率 P(Z ≤ z)。在IB考试中，你将得到包含此表的公式手册。正确阅读此表至关重要。
To find P(Z ≤ z), locate the row for the integer and first decimal, and the column for the second decimal. For example, to find P(Z ≤ 1.25), read row 1.2 and column 0.05; the table gives 0.8944. This means about 89.44% of observations fall below 1.25 standard deviations above the mean.
要查找 P(Z ≤ z)，找到整数和第一位小数的行，以及第二位小数的列。例如，要查找 P(Z ≤ 1.25)，在行1.2和列0.05的交叉处，表格给出0.8944。这意味着大约89.44%的观测值落在均值以上1.25个标准差之内。
Below is a small extract of a standard normal table for illustration. The body gives P(Z ≤ z) for selected z-values.
下面是标准正态表的一小部分摘录，用于说明。表格主体给出了选定z值的累积概率 P(Z ≤ z)。
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