7.4 The Standard Normal Distribution

Introduction

The standard normal distribution is a special case of the normal distribution. It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one. The normal random variable of a standard normal distribution is called a standard score or a z-score.

Every normal random variable X can be transformed into a z-score via the following equation:

Z = (X - μ) / σ

where Z is the standard score, X is a normal random variable, μ is the mean of X, and σ is the standard deviation of X.

引言

标准正态分布是正态分布的一种特殊情况。当正态随机变量的均值为零且标准差为一时，就得到了标准正态分布。标准正态分布的随机变量被称为标准分数或Z分数。

任何正态随机变量X都可以通过以下公式转换为Z分数：

Z = (X - μ) / σ

其中Z是标准分数，X是正态随机变量，μ是X的均值，σ是X的标准差。

定义：标准正态分布

标准正态分布是一种均值为0，标准差为1的正态分布，通常记为：

Z ~ N(0, 1²)

其概率密度函数为：

f(z) = (1/√(2π)) * e^(-z²/2)

Why Standard Normal Distribution?

The standard normal distribution is important because any normal distribution can be transformed into the standard normal distribution. This transformation allows us to use a single table (the standard normal table) to find probabilities for any normal distribution.

Without standardization, we would need a different table for every possible combination of mean and standard deviation, which would be impractical.

标准正态分布之所以重要，是因为任何正态分布都可以转换为标准正态分布。这种转换使我们能够使用单一的表格（标准正态分布表）来查找任何正态分布的概率。

如果没有标准化，我们将需要为每一种可能的均值和标准差组合准备不同的表格，这是不切实际的。

The Standardization Process

标准化过程涉及将原始正态分布的值转换为Z分数。这个过程包括以下步骤：

确定原始正态分布的均值(μ)和标准差(σ)
对每个原始值(X)，使用公式Z = (X - μ)/σ计算Z分数
Z分数表示原始值距离均值有多少个标准差
使用标准正态分布表或计算器查找对应的概率

注意：

Z分数为正表示原始值大于均值，Z分数为负表示原始值小于均值，Z分数为零表示原始值等于均值。

Properties of the Standard Normal Distribution

The standard normal distribution has the following properties:

It is symmetric around the mean of 0
The total area under the curve is 1
About 68% of the area is within 1 standard deviation of the mean
About 95% of the area is within 2 standard deviations of the mean
About 99.7% of the area is within 3 standard deviations of the mean

标准正态分布具有以下性质：

关于均值0对称
曲线下的总面积为1
约68%的面积位于均值的1个标准差范围内
约95%的面积位于均值的2个标准差范围内
约99.7%的面积位于均值的3个标准差范围内

图1：标准正态分布曲线，显示了±1σ范围内的68%区域

Using the Standard Normal Table

The standard normal table (also known as the z-table) provides the cumulative probability P(Z ≤ z) for various values of z. To use the table:

Locate the first two digits of z in the left column
Locate the third digit of z in the top row
The intersection of the row and column gives the cumulative probability

For example, to find P(Z ≤ 1.23):

Find 1.2 in the left column
Find 0.03 in the top row
The intersection gives 0.8907, so P(Z ≤ 1.23) = 0.8907

标准正态分布表（也称为Z表）提供了不同z值对应的累积概率P(Z ≤ z)。使用该表的步骤如下：

在左侧列找到z的前两位数字
在顶部行找到z的第三位数字
行和列的交点即为累积概率

例如，要查找P(Z ≤ 1.23)：

在左侧列找到1.2
在顶部行找到0.03
交点处的值为0.8907，因此P(Z ≤ 1.23) = 0.8907

Example 1: Using the Standard Normal Table

Find the following probabilities using the standard normal table:

a) P(Z ≤ 1.5)

b) P(Z > 0.8)

c) P(-1.2 < Z < 0.9)

Solution:

a) P(Z ≤ 1.5) = 0.9332

b) P(Z > 0.8) = 1 - P(Z ≤ 0.8) = 1 - 0.7881 = 0.2119

c) P(-1.2 < Z < 0.9) = P(Z ≤ 0.9) - P(Z ≤ -1.2) = 0.8159 - 0.1151 = 0.7008

Standardization in Practice

在实际应用中，标准化是解决正态分布问题的关键步骤。让我们通过以下步骤来解决正态分布问题：

步骤1：识别问题中的正态分布参数（均值μ和标准差σ）

步骤2：使用标准化公式Z = (X - μ)/σ将问题转换为标准正态分布

步骤3：使用标准正态分布表或计算器查找相应的概率值

步骤4：如果需要，将结果转换回原始问题的情境

Example 2: Application of Standardization

The heights of adult males in a certain country are normally distributed with mean 175 cm and standard deviation 7 cm.

a) What is the probability that a randomly selected male is taller than 185 cm?

b) What percentage of males are between 165 cm and 180 cm tall?

Solution:

Given: X ~ N(175, 7²)

a) P(X > 185)

First, standardize:

Z = (185 - 175) / 7 ≈ 1.4286

Then, P(X > 185) = P(Z > 1.43) = 1 - P(Z ≤ 1.43) = 1 - 0.9236 = 0.0764

b) P(165 < X < 180)

Standardize both values:

Z₁ = (165 - 175) / 7 ≈ -1.4286

Z₂ = (180 - 175) / 7 ≈ 0.7143

P(165 < X < 180) = P(-1.43 < Z < 0.71) = P(Z ≤ 0.71) - P(Z ≤ -1.43) = 0.7611 - 0.0764 = 0.6847

So approximately 68.47% of males are between 165 cm and 180 cm tall.

Inverse Normal Calculations

Sometimes we need to find the z-score corresponding to a given probability. This is called an inverse normal calculation. To do this:

Identify the cumulative probability p = P(Z ≤ z)
Look up this probability in the standard normal table to find the corresponding z-score
Alternatively, use the inverse normal function on a calculator

Once we have the z-score, we can convert it back to the original distribution using:

X = μ + Zσ

有时我们需要找到对应于给定概率的z分数。这被称为逆正态计算。执行此操作的步骤如下：

确定累积概率p = P(Z ≤ z)
在标准正态分布表中查找此概率，找到对应的z分数
或者，使用计算器上的逆正态函数

一旦我们有了z分数，我们可以使用以下公式将其转换回原始分布：

X = μ + Zσ

Example 3: Inverse Normal Calculation

The weights of packages sent through a certain shipping company are normally distributed with mean 5 kg and standard deviation 0.8 kg.

a) Find the weight that corresponds to the 90th percentile.

b) Find the weights that define the middle 50% of all packages.

Solution:

Given: X ~ N(5, 0.8²)

a) The 90th percentile corresponds to P(Z ≤ z) = 0.90

From the standard normal table, z ≈ 1.28

Convert back to original distribution:

X = 5 + 1.28 × 0.8 = 5 + 1.024 = 6.024 kg

b) The middle 50% means 25% below and 25% above, so we need the 25th and 75th percentiles.

For 25th percentile: P(Z ≤ z₁) = 0.25, so z₁ ≈ -0.67

For 75th percentile: P(Z ≤ z₂) = 0.75, so z₂ ≈ 0.67

Convert back to original distribution:

X₁ = 5 + (-0.67) × 0.8 = 5 - 0.536 = 4.464 kg

X₂ = 5 + 0.67 × 0.8 = 5 + 0.536 = 5.536 kg

So the middle 50% of packages weigh between approximately 4.46 kg and 5.54 kg.

Summary

标准正态分布要点总结：

1. 标准正态分布是均值为0，标准差为1的正态分布，记为Z ~ N(0, 1²)

2. 任何正态分布都可以通过标准化公式Z = (X - μ)/σ转换为标准正态分布

3. 标准化使我们能够使用单一的标准正态分布表计算任何正态分布的概率

4. 对于逆问题，可以使用逆正态函数找到对应于特定概率的Z值

5. 标准化是解决正态分布问题的关键工具，在统计分析中有广泛应用