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。

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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。

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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。

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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。

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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。

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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。

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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。

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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。

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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 + Zsigma ).

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

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。

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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。

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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%。

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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。

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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。

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