Standard Normal Distribution: Properties and Applications | 标准正态分布的特点与应用

📚 Standard Normal Distribution: Properties and Applications | 标准正态分布的特点与应用

The standard normal distribution is one of the most important concepts in A-Level Mathematics and Statistics. It provides a universal benchmark for calculating probabilities, finding percentiles, and conducting statistical inference.

标准正态分布是 A-Level 数学与统计学中最重要的概念之一。它提供了一个通用的基准，用于计算概率、查找分位数以及进行统计推断。

1. What Is the Standard Normal Distribution | 什么是标准正态分布

A normal distribution is a continuous probability distribution that is symmetric and bell-shaped. When the mean is 0 and the standard deviation is 1, it is called the standard normal distribution, often denoted by Z.

正态分布是一种连续型概率分布，具有对称性和钟形曲线。当均值为 0、标准差为 1 时，称之为标准正态分布，通常用 Z 表示。

Any normal random variable X with mean μ and standard deviation σ can be transformed into a standard normal variable using the Z-score formula:

任何均值为 μ、标准差为 σ 的正态随机变量 X，都可以通过 Z 分数公式转化为标准正态变量：

Z = (X − μ) / σ

This transformation measures how many standard deviations an observation is from the mean. It allows us to use a single standard table for all normal distributions.

该变换衡量一个观测值距离均值有多少个标准差，从而使我们能够用同一张标准正态分布表处理所有正态分布问题。

2. Probability Density Function | 概率密度函数

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

标准正态分布的概率密度函数（PDF）为：

φ(z) = (1 / √(2π)) · e^(−z²/2)

This function describes the shape of the bell curve. It is positive for all real values of z and the total area under the curve equals 1.

该函数描述了钟形曲线的形状。它对所有实数 z 都有正值，并且曲线下的总面积为 1。

The function reaches its maximum at z = 0, where the height is approximately 0.3989. It decreases symmetrically as z moves away from 0.

函数在 z = 0 处取得最大值，此时高度约为 0.3989。随着 z 远离 0，函数值对称地减小。

3. Key Properties of the Standard Normal Curve | 标准正态曲线的主要性质

Symmetry: The curve is perfectly symmetric about z = 0, so φ(−z) = φ(z).
对称性：曲线关于 z = 0 完全对称，因此 φ(−z) = φ(z)。
Mean, median and mode: All are equal to 0.
均值、中位数与众数：三者都等于 0。
Variance and standard deviation: Both equal 1.
方差与标准差：都等于 1。
Inflection points: The curve changes concavity at z = −1 and z = 1.
拐点：曲线在 z = −1 和 z = 1 处改变凹凸性。
Total area: The total probability under the curve is exactly 1.
总面积：曲线下的总概率恰好为 1。

4. The Empirical Rule | 经验法则

For a normal distribution, the empirical rule provides quick approximations for probabilities within certain numbers of standard deviations from the mean:

对于正态分布，经验法则提供了数据落在均值周围若干倍标准差范围内的概率近似值：

Interval \| 区间	Probability \| 概率
μ ± 1σ	≈ 0.68 (68%)
μ ± 2σ	≈ 0.95 (95%)
μ ± 3σ	≈ 0.997 (99.7%)

More precise values often used in exams are 1.96 for 95% and 2.576 for 99%.

考试中更常用的精确值是：95% 对应 1.96，99% 对应 2.576。

5. Using the Standard Normal Table | 使用标准正态分布表

The standard normal table gives the cumulative probability Φ(z) = P(Z ≤ z) for positive values of z. For negative z, we use symmetry: Φ(−z) = 1 − Φ(z).

标准正态分布表给出正 z 值的累积概率 Φ(z) = P(Z ≤ z)。对于负 z，利用对称性：Φ(−z) = 1 − Φ(z)。

The table usually provides Φ(z) for z values with two decimal places. The first digit and first decimal are found in the row, and the second decimal is found in the column.

分布表通常给出保留两位小数的 z 值所对应的 Φ(z)。z 的整数部分和第一位小数由行确定，第二位小数由列确定。

For example, to find Φ(1.33), locate row 1.3 and column 0.03 to obtain 0.9082.

例如，要查找 Φ(1.33)，先找到行 1.3，再找到列 0.03，得到 0.9082。

6. Standardisation and Z-Scores | 标准化与 Z 分数

The Z-score converts any normal random variable to the standard normal scale. It is defined as:

Z 分数将任意正态随机变量转换到标准正态尺度上，其定义为：

z = (x − μ) / σ

A positive z-score means the value is above the mean; a negative z-score means the value is below the mean.

正 Z 分数表示该值高于均值；负 Z 分数表示该值低于均值。

When working with a sample mean, the standard error σ/√n replaces σ:

当处理样本均值时，需要用标准误 σ/√n 代替 σ：

z = (x̄ − μ) / (σ / √n)

This is particularly important in hypothesis testing and confidence intervals.

这在假设检验和置信区间中尤为重要。

7. Calculating Normal Probabilities: Step-by-Step | 计算正态概率的步骤

Identify the mean μ and standard deviation σ of the normal distribution.
确定正态分布的均值 μ 和标准差 σ。
Convert the required value(s) into Z-scores using z = (x − μ)/σ.
使用 z = (x − μ)/σ 将所需值转换为 Z 分数。
Sketch the normal curve and shade the required region to avoid sign errors.
画出正态曲线并标出所需区域，以避免符号错误。
Use the standard normal table or a calculator to find the cumulative probability.
使用标准正态分布表或计算器查找累积概率。
Apply symmetry or complement rules if the region is not directly tabulated.
如果所需区域未在表中直接给出，则应用对称性或补集法则。

Example: Let X ~ N(100, 15²). Find P(X < 120).

示例：设 X ~ N(100, 15²)，求 P(X < 120)。

z = (120 − 100) / 15 = 1.3333

Using linear interpolation or a calculator, Φ(1.3333) ≈ 0.9088. Therefore P(X < 120) ≈ 0.9088.

通过线性插值或计算器，Φ(1.3333) ≈ 0.9088。因此 P(X < 120) ≈ 0.9088。

Another example: Find P(X > 85).

另一个示例：求 P(X > 85)。

z = (85 − 100) / 15 = −1

Φ(−1) = 1 − Φ(1) = 1 − 0.8413 = 0.1587. Therefore P(X > 85) = 1 − 0.1587 = 0.8413.

Φ(−1) = 1 − Φ(1) = 1 − 0.8413 = 0.1587。因此 P(X > 85) = 1 − 0.1587 = 0.8413。

8. Inverse Problems: Finding Percentiles | 逆问题：求分位数

Sometimes we are given a probability and need to find the corresponding value of X. This is the inverse of the cumulative probability problem.

有时我们已知概率，需要求对应的 X 值。这是累积概率问题的逆问题。

For example, find the value x such that P(X ≤ x) = 0.95 for X ~ N(100, 15²).

例如，设 X ~ N(100, 15²)，求满足 P(X ≤ x) = 0.95 的 x 值。

From the standard normal table, Φ(1.645) = 0.95. Thus z = 1.645.

由标准正态分布表可知，Φ(1.645) = 0.95。因此 z = 1.645。

x = μ + zσ = 100 + 1.645 × 15 = 124.675

This means 95% of the data lie below 124.675.

这意味着 95% 的数据都小于 124.675。

9. Applications in Inference | 在统计推断中的应用

The standard normal distribution plays a central role in confidence intervals and hypothesis testing when the population standard deviation is known.

在总体标准差已知的情况下，标准正态分布在置信区间和假设检验中扮演核心角色。

A 95% confidence interval for a population mean is:

总体均值的 95% 置信区间为：

x̄ ± 1.96 × (σ / √n)

For a hypothesis test, the test statistic is z = (x̄ − μ₀) / (σ / √n), where μ₀ is the hypothesised population mean.

对于假设检验，检验统计量为 z = (x̄ − μ₀) / (σ / √n)，其中 μ₀ 是假设的总体均值。

The critical values of z (e.g. 1.96 for a two-tailed test at 5% significance) come directly from the standard normal distribution.

z 的临界值（例如 5% 显著性水平下双侧检验的 1.96）直接来自标准正态分布。

10. Common Mistakes and Exam Tips | 常见错误与考试提示

Using the wrong standard deviation: For sample means, always use σ/√n instead of σ.
用错标准差：处理样本均值时，务必使用 σ/√n 而不是 σ。
Forgetting continuity correction: When approximating a binomial distribution with a normal distribution, apply the continuity correction.
忘记连续性修正：当用正态分布近似二项分布时，需要进行连续性修正。
Mixing up P(Z < z) and P(Z ≤ z): For a continuous distribution, these are equal because P(Z = z) = 0.
混淆 P(Z < z) 与 P(Z ≤ z)：对于连续分布，两者相等，因为 P(Z = z) = 0。
Sign errors with negative z: Use the symmetry property carefully.
负 z 的符号错误：要小心使用对称性。
Not sketching the curve: A quick sketch helps determine whether to subtract from 1 or add two probabilities.
不画曲线图：快速画图有助于判断是取补集还是将两个概率相加。

11. Summary | 总结

The standard normal distribution is the benchmark for all normal probability calculations. By standardising any normal variable with the Z-score formula, we can solve probability and percentile problems systematically.

标准正态分布是所有正态概率计算的基准。通过 Z 分数公式将任意正态变量标准化，我们可以系统地解决概率与分位数问题。

Remember the key properties: symmetry, mean 0, variance 1, and the empirical rule. Practise using the normal table and interpreting z-values correctly to avoid common errors in exams.

请记住关键性质：对称性、均值为 0、方差为 1，以及经验法则。多练习使用正态分布表，并正确解读 Z 值，以避免考试中的常见错误。

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Standard Normal Distribution: Properties and Applications | 标准正态分布的特点与应用