A-Level Mathematics: Normal Distribution Basics | A-Level数学：正态分布基础讲解

📚 A-Level Mathematics: Normal Distribution Basics | A-Level数学：正态分布基础讲解

The normal distribution is one of the most important probability models in A-Level Mathematics. This article explains its key ideas in a clear, step-by-step way, from the bell curve to standardisation and probability tables.

正态分布是A-Level数学中最重要的概率模型之一。本文从钟形曲线到标准化和概率表，循序渐进地讲解其核心概念。

1. What Is a Normal Distribution? | 什么是正态分布？

A normal distribution is a continuous probability distribution that is symmetrical and bell-shaped. Most values cluster around the centre, and extreme values are rare.

正态分布是一种连续型概率分布，图像对称且呈钟形。大部分数据集中在中心附近，极端值很少出现。

We write X ~ N(μ, σ²) to show that the random variable X follows a normal distribution with mean μ and variance σ².

我们用 X ~ N(μ, σ²) 表示随机变量 X 服从均值为 μ、方差为 σ² 的正态分布。

The normal distribution is widely used in statistics, science and business to describe data that vary naturally around an average.

正态分布广泛用于统计、科学和商业领域，用来描述围绕平均值自然波动的数据。

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2. Key Properties of the Normal Curve | 正态曲线的主要性质

The curve is symmetric about the mean μ, so the left half is a mirror image of the right half.

曲线关于均值 μ 对称，因此左半部分是右半部分的镜像。

For a normal distribution, the mean, median and mode are all equal and lie at the centre of the distribution.

对于正态分布，均值、中位数和众数都相等，并且位于分布的中心。

The total area under the curve is exactly 1, which represents the total probability of all possible outcomes.

曲线下的总面积为 1，这表示所有可能结果的总概率。

The tails of the curve never touch the horizontal axis; they approach it asymptotically as x moves further away from μ.

曲线的尾部永远不会接触横轴；当 x 距离 μ 越来越远时，尾部会渐近地接近横轴。

onvex at the points x = μ − σ and x = μ + σ. These points are called points of inflection.

曲线在 x = μ − σ 和 x = μ + σ 处由凹变凸，这两点称为拐点。

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3. The Parameters μ and σ | 参数 μ 和 σ

The mean μ controls the position of the curve. If μ increases, the whole curve shifts to the right; if μ decreases, it shifts to the left.

均值 μ 控制曲线的位置。如果 μ 增大，整条曲线向右移动；如果 μ 减小，曲线向左移动。

The standard deviation σ controls the spread of the curve. A larger σ produces a shorter, wider curve; a smaller σ produces a taller, narrower curve.

标准差 σ 控制曲线的离散程度。σ 越大，曲线越矮越宽；σ 越小，曲线越高越窄。

The variance is written as σ², and the standard deviation is the positive square root of the variance.

方差写作 σ²，标准差是方差的正平方根。

The probability density function of a normal distribution is given by:

正态分布的概率密度函数为：

f(x) = 1 / (σ√2π) × e^( −(x − μ)² / (2σ²) )

This formula looks complicated, but in your exam you rarely need to use it directly. Instead, you use standard tables or a calculator.

这个公式看起来复杂，但在考试中你很少需要直接使用它。相反，你会使用标准正态分布表或计算器。

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4. The Empirical Rule | 经验法则

The empirical rule gives a quick estimate of how much data lies within one, two and three standard deviations of the mean.

经验法则可以快速估计有多少数据落在平均值的一个、两个和三个标准差范围内。

For any normal distribution:

对于任何正态分布：

Interval	Percentage within this interval
μ ± 1σ	≈ 68%
μ ± 2σ	≈ 95%
μ ± 3σ	≈ 99.7%

This means that nearly all values lie within three standard deviations of the mean.

这意味着几乎所有数值都落在平均值的三倍标准差范围内。

The empirical rule is useful for checking whether your computed probability is reasonable.

经验法则可用于检查你计算出的概率是否合理。

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5. The Standard Normal Distribution and Z-Scores | 标准正态分布与Z分数

Every normal distribution can be transformed into the standard normal distribution, which has mean 0 and standard deviation 1.

任何正态分布都可以转换为标准正态分布，它的均值为 0，标准差为 1。

The standard normal distribution is written as Z ~ N(0, 1).

标准正态分布写作 Z ~ N(0, 1)。

To transform a value x from X ~ N(μ, σ²) into a z-score, use this formula:

要把来自 X ~ N(μ, σ²) 的数值 x 转换为 z 分数，使用以下公式：

z = (x − μ) / σ

The z-score tells us how many standard deviations x is above or below the mean.

z 分数告诉我们 x 比均值高出或低出多少个标准差。

For example, if z = 2, then x is 2 standard deviations above the mean.

例如，如果 z = 2，那么 x 比均值高出 2 个标准差。

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6. Finding Probabilities Using the Normal Distribution Table | 使用正态分布表计算概率

The standard normal table gives the area to the left of a given z-value. This area is written as Φ(z), which means P(Z < z).

标准正态分布表给出某个 z 值左侧的面积。这个面积记作 Φ(z)，表示 P(Z < z)。

To find P(Z > z), we use the complement rule:

要求 P(Z > z)，我们使用补集规则：

P(Z > z) = 1 − Φ(z)

To find the probability that Z lies between two values a and b, we subtract:

要计算 Z 在 a 和 b 之间的概率，我们需要相减：

P(a < Z < b) = Φ(b) − Φ(a)

The table below shows some commonly used cumulative probabilities.

下表列出了一些常用的累积概率。

z	Φ(z) = P(Z < z)
0.00	0.5000
1.00	0.8413
1.64	0.9495
1.96	0.9750
2.00	0.9772
2.58	0.9950

Always draw a sketch of the bell curve and shade the region you need before looking up the table.

在查表之前，一定要画出钟形曲线并标出你需要的区域。

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7. Inverse Normal Calculations | 正态分布的逆运算

Sometimes you are given a probability and asked to find the corresponding z-score. This is called an inverse normal calculation.

有时题目会给你一个概率，然后要求你找出对应的 z 分数。这称为正态分布的逆运算。

For example, suppose we want the value of z such that P(Z < z) = 0.95.

例如，假设我们想找到这样的 z，使得 P(Z < z) = 0.95。

Using statistical tables or a calculator, we find:

使用统计表或计算器，我们可以找到：

z ≈ 1.6449

This means that 95% of the data in a standard normal distribution lies below 1.6449.

这意味着在标准正态分布中，95% 的数据位于 1.6449 以下。

On most calculators, this function is called invNorm or Inverse Normal. You enter the probability, mean 0 and standard deviation 1.

在大多数计算器上，这个功能称为 invNorm 或 Inverse Normal。你输入概率、均值 0 和标准差 1 即可。

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8. Working with Any Normal Distribution | 处理任意正态分布

ral normal distribution, we first standardise the value or boundary using z = (x − μ) / σ.

要解决涉及一般正态分布的问题，我们首先使用 z = (x − μ) / σ 对数值或边界进行标准化。

Example: Let X ~ N(100, 16²). Find P(X < 120).

例：设 X ~ N(100, 16²)。求 P(X < 120)。

Step 1: Calculate the z-score.

第一步：计算 z 分数。

z = (120 − 100) / 16 = 20 / 16 = 1.25

Step 2: Look up the table value.

第二步：查表获取数值。

P(X < 120) = P(Z < 1.25) ≈ 0.8944

So the probability is approximately 0.8944.

因此概率约为 0.8944。

We can also find the probability between two values. For example, find P(90 < X < 120).

我们还可以求两个值之间的概率。例如，求 P(90 < X < 120)。

The z-scores are:

对应的 z 分数为：

z₁ = (90 − 100) / 16 = −0.625

z₂ = (120 − 100) / 16 = 1.25

Therefore:

因此：

P(−0.625 < Z < 1.25) = Φ(1.25) − Φ(−0.625) ≈ 0.8944 − 0.2660 = 0.6284

This gives a probability of approximately 0.6284.

得到的概率约为 0.6284。

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9. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Mistake 1: Using σ² as the standard deviation. Remember that in X ~ N(μ, σ²), σ² is the variance and σ is the standard deviation.

错误一：把 σ² 当作标准差。记住在 X ~ N(μ, σ²) 中，σ² 是方差，σ 是标准差。

Mistake 2: Forgetting to use the complement rule. For P(X > a), write 1 − P(X < a).

错误二：忘记使用补集规则。对于 P(X > a)，要写成 1 − P(X < a)。

Mistake 3: Misreading the direction of the inequality. Always check whether you need the area to the left or to the right.

错误三：误读不等号方向。始终确认你需要的是左侧面积还是右侧面积。

us distribution, P(X = a) = 0, so strict and non-strict inequalities give the same result.

错误四：忘记 X 是连续型变量。在连续分布中，P(X = a) = 0，因此严格不等号和非严格不等号得到相同的结果。

Mistake 5: Not drawing a diagram. A quick sketch of the bell curve with the shaded region can prevent many sign errors.

错误五：不画图。快速画一条钟形曲线并标出阴影区域，可以避免许多符号错误。

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10. Final Exam Tips | 考试要点总结

Always state the distribution clearly, for example X ~ N(20, 5²), and identify the mean and standard deviation before calculating.

始终清楚写出分布，例如 X ~ N(20, 5²)，并在计算前确认均值和标准差。

When using a calculator, check whether your calculator asks for the standard deviation or the variance. This is a common source of wrong answers.

使用计算器时，检查计算器要求输入的是标准差还是方差。这是常见错误答案的来源。

After finding any probability, check that it lies between 0 and 1. If it is negative or greater than 1, something has gone wrong.

算出任何概率后，检查它是否在 0 到 1 之间。如果为负数或大于 1，说明出了问题。

Finally, remember that 95% of the data lies within 1.96 standard deviations of the mean. This is a useful benchmark in many statistical questions.

最后，记住 95% 的数据位于平均值左右 1.96 个标准差之内。这是许多统计问题中有用的基准值。

With careful standardisation, clear diagrams and consistent use of tables or calculators, you can solve normal distribution questions confidently.

只要仔细进行标准化、清楚画图，并规范使用表格或计算器，你就能自信地解决正态分布问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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