📚 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 ∼ 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 ∼ 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 ∼ N(μ, σ2) , 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 ∼ N(μ, σ2) ，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。
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 ≤ Z ≤ 1) ≈ 0.6827
约68%的面积位于1个标准差之内： P(-1 ≤ Z ≤ 1) ≈ 0.6827
About 95% lies within 2 standard deviations: P(-2 ≤ Z ≤ 2) ≈ 0.9545
约95%的面积位于2个标准差之内： P(-2 ≤ Z ≤ 2) ≈ 0.9545
About 99.7% lies within 3 standard deviations: P(-3 ≤ Z ≤ 3) ≈ 0.9973
约99.7%的面积位于3个标准差之内： P(-3 ≤ Z ≤ 3) ≈ 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 ∼ N(μ, σ2) , then the standardised variable is defined as:
若 X ∼ N(μ, σ2) ，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 μ and the standard deviation σ of X .
写出 X 的均值 μ 与标准差 σ 。
Compute the z-score using Z = (X - μ) / σ .
使用 Z = (X - μ) / σ 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 = μ + Zσ .
当题目给出概率时，将公式反解以求得 X ： X = μ + Zσ 。
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 Φ(z) = P(Z ≤ 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 Φ(z) = P(Z ≤ 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 ≤ a) , a > 0 | Read Φ(a) directly 直接读取 Φ(a) |
| P(Z ≥ a) , a > 0 | Compute 1 - Φ(a) 计算 1 - Φ(a) |
| P(Z ≤ -a) , a > 0 | Compute 1 - Φ(a) 计算 1 - Φ(a) |
| P(a ≤ Z ≤ b) , 0 < a < b | Compute Φ(b) - Φ(a) 计算 Φ(b) - Φ(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 ∼ N(50, 16) . Find P(X > 56) .
设 X ∼ N(50, 16) ，求 P(X > 56) 。
Step 1: Identify parameters. The mean μ = 50 and the standard deviation σ = √16 = 4 .
第一步：确定参数。均值 μ = 50 ，标准差 σ = √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 - Φ(1.5) . From the table, Φ(1.5) = 0.9332 .
第三步：利用对称性与查表： P(Z > 1.5) = 1 - Φ(1.5) 。查表得 Φ(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 ∼ N(100, 25) . Find the value k such that P(X ≤ k) = 0.90 .
设 X ∼ N(100, 25) ，求满足 P(X ≤ k) = 0.90 的 k 值。
First, find the z-value such that Φ(z) = 0.90 . From the table, the closest value is z ≈ 1.2816 . Then invert the standardisation formula:
首先，找到使 Φ(z) = 0.90 的z值。查表可得最近的值为 z ≈ 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 σ2 instead of σ in the denominator of the z-score formula. Remember that variance is σ2 , 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 σ2 而非 σ 。请记住方差是 σ2 ，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。
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