Properties and Standardisation of the Standard Normal Distribution | 标准正态分布的性质与标准化方法

📚 Properties and Standardisation of the Standard Normal Distribution | 标准正态分布的性质与标准化方法

The standard normal distribution is one of the most important concepts in A Level statistics. It provides a universal reference for calculating probabilities under the normal model, and its standardisation formula transforms any normal distribution into a form that can be looked up in statistical tables. This article explores its key properties and the method of standardisation in detail.

标准正态分布是A Level统计学中最重要的概念之一。它为在正态模型下计算概率提供了通用参照，其标准化公式可以将任意正态分布转化为可查统计表的形式。本文将详细探讨其关键性质与标准化方法。

1. Definition and Notation | 定义与符号

A continuous random variable \( Z \) is said to follow a standard normal distribution if it is normally distributed with mean 0 and variance 1. This is written as \( Z \sim N(0, 1) \).

若一个连续随机变量 \( Z \) 服从均值为0、方差为1的正态分布，则称其服从标准正态分布，记作 \( Z \sim N(0, 1) \)。

The probability density function of \( Z \) is given by:

f(z) = (1 / √(2π)) × e^(−z² / 2)

Because the parameters are fixed at 0 and 1, this single curve serves as the reference for all normal distributions.

由于参数固定为0和1，这条唯一的曲线便成为所有正态分布的参照基准。

2. The Fundamental Role of the Standard Normal Distribution | 标准正态分布的基础地位

Not every normal distribution has convenient parameters. If \( X \sim N(\mu, \sigma^2) \), the probabilities depend on both the mean and the variance. Tables, however, are compiled for only one distribution: the standard normal.

并非每个正态分布都具有方便的参数。若 \( X \sim N(\mu, \sigma^2) \)，其概率同时取决于均值和方差。然而，统计表仅为一种分布编制：即标准正态分布。

By converting \( X \) into \( Z \), we reduce every normal problem to a single table lookup. This is why standardisation is not just a trick but a necessary step in solving real-world normal distribution problems.

通过将 \( X \) 转换为 \( Z \)，我们能把所有正态问题简化为一次查表。因此，标准化不仅是技巧，更是解决实际正态分布问题的必要步骤。

3. Symmetry About Zero | 关于零的对称性

The standard normal curve is perfectly symmetric about the vertical line \( z = 0 \). This means that the area to the left of \( -z \) is exactly equal to the area to the right of \( z \):

标准正态曲线关于垂直线 \( z = 0 \) 完全对称。这意味着 \( -z \) 左侧的面积恰好等于 \( z \) 右侧的面积：

P(Z ≤ −z) = P(Z ≥ z)

This symmetry greatly simplifies probability calculations. For example, if you need \( P(Z > 1.5) \), you do not need to integrate; you simply read \( P(Z < -1.5) \) from the table.

这一对称性极大简化了概率计算。例如，若需求 \( P(Z > 1.5) \)，无需积分，只需从表中读取 \( P(Z < -1.5) \) 即可。

4. Mean of 0 and Variance of 1 | 均值为0与方差为1

The mean of the standard normal distribution is exactly 0. This implies that the curve is centred at the origin and that the expected value of \( Z \) is zero. The median and mode also equal 0, because the distribution is unimodal and symmetric.

标准正态分布的均值恰好为0。这意味着曲线以原点为中心，且 \( Z \) 的期望值为0。由于该分布单峰且对称，其中位数和众数也均为0。

The variance of \( Z \) is exactly 1, so the standard deviation is also 1. A standard deviation of 1 defines the natural unit of measurement along the horizontal axis. Distances on the z-axis are therefore measured in standard deviation units.

\( Z \) 的方差恰好为1，因此标准差也为1。标准差为1定义了横轴上的自然度量单位。因此，z轴上的距离即以标准差为单位进行度量。

5. Inflection Points at ±1 | 拐点位于±1处

The standard normal density function has inflection points at \( z = -1 \) and \( z = +1 \). An inflection point is where the curve changes from being concave downward to concave upward, or vice versa.

标准正态密度函数在 \( z = -1 \) 与 \( z = +1 \) 处存在拐点。拐点是指曲线由凹向下转为凹向上（或反之）的位置。

These points divide the curve into three visually distinct regions. The central region between -1 and +1 is where the curve is convex, and the tails beyond ±1 are concave. This shape is the classic “bell” curve shape.

这两个点将曲线划分为三个视觉上不同的区域。-1至+1之间的中央区域曲线凸向下，而±1之外的两尾区域曲线凹向上。这一形状即为经典的”钟形”曲线。

6. The Empirical Rule and Standard Normal Deviations | 经验法则与标准正态偏差

The standard normal distribution obeys the empirical rule, which provides approximate probabilities for intervals around the mean:

标准正态分布遵循经验法则，该法则给出了均值附近区间概率的近似值：

About 68% of the area lies within 1 standard deviation: \( P(-1 \leq Z \leq 1) \approx 0.6827 \)

约68%的面积位于1个标准差之内：\( P(-1 \leq Z \leq 1) \approx 0.6827 \)
About 95% lies within 2 standard deviations: \( P(-2 \leq Z \leq 2) \approx 0.9545 \)

约95%的面积位于2个标准差之内：\( P(-2 \leq Z \leq 2) \approx 0.9545 \)
About 99.7% lies within 3 standard deviations: \( P(-3 \leq Z \leq 3) \approx 0.9973 \)

约99.7%的面积位于3个标准差之内：\( P(-3 \leq Z \leq 3) \approx 0.9973 \)

These figures are useful for quick estimation and for checking whether more precise table-based answers are plausible.

这些数值可用于快速估算，也可用来检验基于查表的精确答案是否合理。

7. The Standardisation Formula | 标准化公式

If \( X \sim N(\mu, \sigma^2) \), then the standardised variable is defined as:

若 \( X \sim N(\mu, \sigma^2) \)，则标准化变量定义为：

Z = (X − μ) / σ

This transformation subtracts the mean, centering the distribution at 0, and then divides by the standard deviation, scaling the spread to 1. As a result, \( Z \) follows a standard normal distribution.

该变换先减去均值，使分布的中心位于0；再除以标准差，将离散程度缩放为1。因此，\( Z \) 服从标准正态分布。

Every value of \( X \) maps to exactly one value of \( Z \). The z-score tells us how many standard deviations the original value lies above or below the mean.

每个 \( X \) 值恰好映射为一个 \( Z \) 值。z分数告诉我们原始值位于均值上方或下方多少个标准差。

8. Steps for Standardisation | 标准化的步骤

To standardise a normal random variable in practice, follow these steps:

在实际操作中，标准化一个正态随机变量需按以下步骤进行：

Write down the mean \( \mu \) and the standard deviation \( \sigma \) of \( X \).

写出 \( X \) 的均值 \( \mu \) 与标准差 \( \sigma \)。
Compute the z-score using \( Z = (X – \mu) / \sigma \).

使用 \( Z = (X – \mu) / \sigma \) 计算z分数。
Draw a sketch of the standard normal curve and shade the required region to avoid sign errors.

画出标准正态曲线草图并标出所需区域，以避免符号错误。
When the probability is given, invert the formula to solve for \( X \): \( X = \mu + Z\sigma \).

当题目给出概率时，将公式反解以求得 \( X \)：\( X = \mu + Z\sigma \)。

Mastering this order of operations prevents many common mistakes, particularly with inequalities that involve negative z-scores.

熟练这种操作顺序可以避免许多常见错误，尤其是涉及负z分数的不等式问题。

9. Using the Standard Normal Table | 标准正态分布表的使用

The standard normal table gives the cumulative probability \( \Phi(z) = P(Z \leq z) \) for non-negative values of z. To find probabilities involving negative z-values, symmetry is used.

标准正态分布表给出非负z值下的累积概率 \( \Phi(z) = P(Z \leq z) \)。对于涉及负z值的概率，需使用对称性。

Desired Probability 所需概率	Table Operation 查表操作
\( P(Z \leq a) \), \( a > 0 \)	Read \( \Phi(a) \) directly 直接读取 \( \Phi(a) \)
\( P(Z \geq a) \), \( a > 0 \)	Compute \( 1 – \Phi(a) \) 计算 \( 1 – \Phi(a) \)
\( P(Z \leq -a) \), \( a > 0 \)	Compute \( 1 – \Phi(a) \) 计算 \( 1 – \Phi(a) \)
\( P(a \leq Z \leq b) \), \( 0 < a < b \)	Compute \( \Phi(b) – \Phi(a) \) 计算 \( \Phi(b) – \Phi(a) \)

Always check whether the required area lies to the left, right, or between bounds before applying these operations.

在应用以上操作前，务必判断所求面积位于左侧、右侧还是区间内部。

10. Worked Example: Finding a Probability | 实例演示：求概率

Suppose \( X \sim N(50, 16) \). Find \( P(X > 56) \).

设 \( X \sim N(50, 16) \)，求 \( P(X > 56) \)。

Step 1: Identify parameters. The mean \( \mu = 50 \) and the standard deviation \( \sigma = \sqrt{16} = 4 \).

第一步：确定参数。均值 \( \mu = 50 \)，标准差 \( \sigma = \sqrt{16} = 4 \)。

Step 2: Standardise the value 56:

第二步：对56进行标准化：

z = (56 − 50) / 4 = 1.5

Step 3: Use symmetry and the table: \( P(Z > 1.5) = 1 – \Phi(1.5) \). From the table, \( \Phi(1.5) = 0.9332 \).

第三步：利用对称性与查表：\( P(Z > 1.5) = 1 – \Phi(1.5) \)。查表得 \( \Phi(1.5) = 0.9332 \)。

P(X > 56) = 1 − 0.9332 = 0.0668

Thus, approximately 6.68% of values exceed 56. This matches the empirical expectation that values more than 1 standard deviation above the mean occur with probability less than 16%.

因此，约有6.68%的值超过56。这与经验法则一致：超过均值1个标准差以上的值出现概率小于16%。

11. Worked Example: Finding a Boundary Value | 实例演示：求边界值

Suppose \( X \sim N(100, 25) \). Find the value \( k \) such that \( P(X \leq k) = 0.90 \).

设 \( X \sim N(100, 25) \)，求满足 \( P(X \leq k) = 0.90 \) 的 \( k \) 值。

First, find the z-value such that \( \Phi(z) = 0.90 \). From the table, the closest value is \( z \approx 1.2816 \). Then invert the standardisation formula:

首先，找到使 \( \Phi(z) = 0.90 \) 的z值。查表可得最近的值为 \( z \approx 1.2816 \)。随后反解标准化公式：

k = μ + zσ = 100 + (1.2816 × 5) = 106.408

Therefore, the 90th percentile of this distribution is approximately 106.4. This method applies to any percentile or boundary-value question.

因此，该分布的第90百分位数约为106.4。此方法适用于任何百分位数或边界值问题。

12. Common Pitfalls and Examination Tips | 常见易错点与应试提示

A frequent error is using \( \sigma^2 \) instead of \( \sigma \) in the denominator of the z-score formula. Remember that variance is \( \sigma^2 \), so the standard deviation is the square root of the variance.

常见错误之一是在z分数公式的分母中误用 \( \sigma^2 \) 而非 \( \sigma \)。请记住方差是 \( \sigma^2 \)，因此标准差是方差的平方根。

Another common mistake is forgetting to subtract the mean before dividing. The z-score always measures relative distance from the mean, not the distance from zero.

另一个常见错误是忘记在除以标准差之前先减去均值。z分数衡量的始终是相对均值的距离，而非相对零点的距离。

Avoid rounding z-scores too early. Keep at least four decimal places when looking up table values. In an exam, draw a small sketch before any calculation to verify whether the required probability is greater or less than 0.5.

应避免过早四舍五入z分数。查表时至少保留四位小数。考试中，在任何计算前先画一个简图，以验证所需概率是否大于或小于0.5。

Finally, always write down sufficient working, including the standardisation step. Many exam boards award method marks even when the final numerical answer is wrong.

最后，务必写出完整解题过程，包括标准化步骤。许多考试局即使最终数值有误，也会给方法分。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Properties and Standardisation of the Standard Normal Distribution | 标准正态分布的性质与标准化方法