IGCSE CIE Maths: Normal Distribution Exam Focus | IGCSE CIE 数学：正态分布考点精讲

📚 IGCSE CIE Maths: Normal Distribution Exam Focus | IGCSE CIE 数学：正态分布考点精讲

The normal distribution is one of the most important continuous probability distributions in statistics. In the IGCSE CIE Maths syllabus, you are expected to understand its shape, apply the empirical rule, use standard normal distribution tables, and solve problems involving normally distributed data. This guide covers every key exam point to help you achieve top marks.

正态分布是统计学中最重要的连续概率分布之一。在 IGCSE CIE 数学考纲中，你需要理解它的形状、应用经验法则、使用标准正态分布表，并解决涉及正态分布数据的应用题。本文涵盖所有关键考点，助你夺得高分。

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

A normal distribution is a bell-shaped, symmetric probability distribution defined by its mean (μ) and standard deviation (σ). Many natural phenomena – such as heights, weights, and test scores – approximately follow a normal distribution. The area under the entire curve represents the total probability, which is always 1.

正态分布是一种钟形、对称的概率分布，由均值 (μ) 和标准差 (σ) 决定。许多自然现象（如身高、体重和考试成绩）都近似服从正态分布。整条曲线下方的总面积代表总概率，恒为 1。

2. Key Features of the Normal Curve | 正态曲线的关键特征

The curve is symmetric about the mean, so the mean, median, and mode are all equal. It has a single peak at the centre, and the tails extend infinitely in both directions without touching the horizontal axis. The spread of the curve is determined by the standard deviation σ: a larger σ makes the curve wider and flatter, while a smaller σ makes it narrower and taller.

曲线关于均值对称，因此均值、中位数和众数都相等。它在中心处有单一峰值，尾部向两边无限延伸但永远不会触及横轴。曲线的分散程度由标准差 σ 决定：σ 越大，曲线越宽、越平坦；σ 越小，曲线越窄、越高耸。

The notation X ~ N(μ, σ²) is used to denote that a random variable X follows a normal distribution with mean μ and variance σ². Note that the second parameter is the variance, not the standard deviation.

我们常用记号 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。请留意第二个参数是方差，而非标准差。

3. Mean and Standard Deviation – The Two Parameters | 两个参数：均值和标准差

A normal distribution is completely determined by μ and σ. μ locates the centre of the distribution, while σ controls its spread. Changing μ shifts the curve left or right; changing σ stretches or squeezes it. In the special case where μ = 0 and σ = 1, we obtain the standard normal distribution, denoted Z ~ N(0, 1).

正态分布完全由 μ 和 σ 确定。μ 决定了分布的中心位置，σ 控制其离散程度。改变 μ 会使曲线左右平移；改变 σ 则会拉伸或压缩曲线。在 μ = 0、σ = 1 的特殊情形下，我们得到标准正态分布，记为 Z ~ N(0, 1)。

X ~ N(μ, σ²)

4. The Empirical Rule (68–95–99.7% Rule) | 经验法则（68 – 95 – 99.7% 规则）

For any normal distribution, the approximate percentage of data that falls within 1, 2, and 3 standard deviations from the mean is given by the empirical rule. This is extremely useful for quick estimates without a table.

对于任意正态分布，数据落在均值附近 1、2、3 个标准差范围内的近似百分比由经验法则给出。这在无需查表时非常有用，可快速估算。

Interval	Approximate Probability
μ ± σ	68%
μ ± 2σ	95%
μ ± 3σ	99.7%

For example, if test scores are normally distributed with μ = 50 and σ = 10, then roughly 68% of students score between 40 and 60, and about 95% score between 30 and 70.

例如，若考试分数服从 μ = 50、σ = 10 的正态分布，则约 68% 的学生得分在 40 至 60 之间，约 95% 在 30 至 70 之间。

5. Standardising to the Standard Normal Distribution | 标准化到标准正态分布

To find probabilities for any normal distribution, we convert the raw score x into a standardised score Z, which tells us how many standard deviations x is away from the mean. The formula is:

要计算任意正态分布的概率，我们需要将原始分数 x 转换为标准分数 Z，它表示 x 距离均值有多少个标准差。公式如下：

z = (x – μ) / σ

After standardising, the variable Z always has a mean of 0 and a standard deviation of 1, so we can use the standard normal distribution table.

标准化之后，变量 Z 始终具有均值 0 和标准差 1，因此我们可使用标准正态分布表。

6. Using the Standard Normal Distribution Table (Finding Probabilities) | 使用标准正态分布表（求概率）

The standard normal table gives the cumulative probability Φ(z) = P(Z ≤ z) for a given z-value. In the IGCSE CIE exam, you will be provided with a table, but you must first check whether it gives the cumulative probability or the probability between 0 and z. Most commonly it is cumulative.

标准正态分布表给出了给定 z 值对应的累积概率 Φ(z) = P(Z ≤ z)。在 IGCSE CIE 考试中，你会得到一份表格，但务必先检查它给出的是累积概率还是 0 到 z 之间的概率。最常见的是累积概率表格。

Example: Find P(Z < 1.23). If the table is cumulative, look up 1.23 to read the probability directly. For P(Z > 1.23), use symmetry: P(Z > 1.23) = 1 – P(Z ≤ 1.23). For P(–1.5 < Z < 2), compute P(Z < 2) – P(Z < –1.5) = P(Z < 2) – [1 – P(Z < 1.5)].

示例：求 P(Z < 1.23)。若表格为累积表，直接查找 1.23 即可。求 P(Z > 1.23) 时，利用对称性：P(Z > 1.23) = 1 – P(Z ≤ 1.23)。求 P(–1.5 < Z < 2) 时，计算 P(Z < 2) – P(Z < –1.5) = P(Z < 2) – [1 – P(Z < 1.5)]。

Always draw a sketch and shade the required region to avoid sign errors.

始终画出草图并给目标区域涂上阴影，以避免符号错误。

7. Solving Problems Involving Any Normal Distribution | 求解任意正态分布问题

General steps for exam questions:

考试题的一般求解步骤：

1. Identify that the variable X is normally distributed: write X ~ N(μ, σ²).
2. Extract the values of μ, σ, and the x-value(s) from the question.
3. Calculate the z-score: z = (x – μ) / σ.
4. Use the standard normal table to find the required probability.
5. Interpret the result in the context of the problem.

1. 确认变量 X 服从正态分布：写出 X ~ N(μ, σ²)。
2. 从题目中提取 μ、σ 以及 x 的值。
3. 计算 z 分数：z = (x – μ) / σ。
4. 查标准正态表求得所需概率。
5. 结合问题背景解释结果。

Example: The masses of apples from an orchard are normally distributed with mean 150 g and standard deviation 20 g. Find the probability that a randomly chosen apple weighs less than 130 g.
Solution: X ~ N(150, 20²). z = (130 – 150) / 20 = –1.00. From the table, P(Z < –1.00) = 1 – P(Z < 1.00) ≈ 1 – 0.8413 = 0.1587.

示例：果园苹果的质量服从均值为 150 g、标准差为 20 g 的正态分布。求随机选取一个苹果质量小于 130 g 的概率。
解：X ~ N(150, 20²)。z = (130 – 150) / 20 = –1.00。查表，P(Z < –1.00) = 1 – P(Z < 1.00) ≈ 1 – 0.8413 = 0.1587。

8. Reverse Look-up: Given a Probability, Find the Cut-off Score | 反向查表：已知概率求临界分数

Sometimes you are given a probability or percentile and need to find the corresponding x-value. First, find the z-value from the table that satisfies the given probability, then apply the inverse formula:

有时题目给出一个概率或百分位数，让你求对应的 x 值。首先从表中找出满足该概率的 z 值，然后使用逆公式：

x = μ + zσ

For instance, if the top 10% of scores are to be given an A* grade, find the z-value such that P(Z > z) = 0.10, so P(Z < z) = 0.90. Look up 0.90 in the cumulative table to get z ≈ 1.2816, then plug into x = μ + zσ.

例如，若要给前 10% 的分数评为 A*，找到满足 P(Z > z) = 0.10 的 z 值，即 P(Z < z) = 0.90。在累积表中查找 0.90 得到 z ≈ 1.2816，然后代入 x = μ + zσ 计算临界分数。

Be careful with the left tail: if P(Z < z) = 0.05, z will be negative and equal to –1.6449 (by symmetry).

注意左侧尾部情形：若 P(Z < z) = 0.05，z 为负数，根据对称性得 z = –1.6449。

9. Common Exam Question Types | 常见考试题型

IGCSE CIE maths questions on normal distribution typically fall into these categories:

IGCSE CIE 数学正态分布考题通常分为以下几类：

Finding probabilities for a given x: Standardise and use the table to find P(X < a), P(X > b), or P(a < X < b).
给定 x 求概率：标准化后用表求 P(X < a)、P(X > b) 或 P(a < X < b)。
Reverse normal problems: Given a probability, find the x-value, often used to determine pass marks or warranty limits.
反向正态问题：已知概率求 x 值，常用于确定及格线或保修期限。
Comparing two distributions: Use z-scores to compare performances across subjects with different means and standard deviations.
比较两个分布：使用 z 分数比较来自不同均值和标准差的学科表现。
Finding unknown μ or σ: Set up equations using z = (x – μ)/σ and solve simultaneously when two probabilities are given.
求未知 μ 或 σ：利用 z = (x – μ)/σ 建立方程，当题目给出两个概率时可联立求解。

10. Common Mistakes to Avoid | 常见错误警示

Avoid these pitfalls to keep your solutions accurate:

避开以下陷阱，确保解答准确：

– Confusing variance σ² with standard deviation σ when writing the distribution or substituting into the z-formula.
– Forgetting to standardise before consulting the standard normal table.
– Misidentifying the table type – assuming it gives P(0 < Z < z) when it is actually cumulative, or vice versa.
– Misapplying symmetry: remember that P(Z < –a) = P(Z > a) = 1 – P(Z < a).
– Rounding too early; keep at least 4 decimal places during intermediate calculations.

– 在写出分布或代入 z 公式时，混淆方差 σ² 和标准差 σ。
– 忘记先标准化就直接查阅标准正态表。
– 误判表格类型——以为提供的是 P(0 < Z < z)，实际却是累积概率，反之亦然。
– 错误使用对称性：记住 P(Z < –a) = P(Z > a) = 1 – P(Z < a)。
– 过早舍入；中间计算步骤至少保留四位小数。

11. Tips for the IGCSE CIE Exam | IGCSE CIE 考试技巧

Always sketch a normal curve and shade the area of interest. Mark the mean and the x-value(s). This visual aid will guide your use of the table and help you decide whether to add or subtract probabilities. Use clear notation, such as X ~ N(μ, σ²) and Z = (x – μ)/σ, to show the examiner your logical flow. Check whether the question asks for a probability, a proportion, or a percentage, and present your final answer accordingly.

务必画出正态曲线草图，并给目标区域涂上阴影。标出均值和 x 值。这一视觉辅助会指引你正确查表，并帮你决定是加还是减概率。使用清晰的记法，如 X ~ N(μ, σ²) 和 Z = (x – μ)/σ，向阅卷老师展示你的逻辑流程。检查题目要求的是概率、比例还是百分比，并相应呈现最终答案。

When doing a reverse problem, double-check that your z-value makes sense relative to the shaded area – a z-value for a left-tail 5% should be negative, for example.

做反向题时，再次确认 z 值相对于阴影区域是否合理——例如左尾 5% 对应的 z 值应为负数。

12. Summary of Key Formulae and Concepts | 考点总结

These are the essential elements you must carry into the exam:

以下是你进入考场前必须掌握的核心要点：

Normal distribution notation: X ~ N(μ, σ²)
Standard normal distribution: Z ~ N(0, 1)
Z-score formula: z = (x – μ) / σ
Inverse formula: x = μ + zσ
Empirical rule: 68% – 95% – 99.7%
Cumulative probability lookup and reverse lookup using the provided table
Symmetry and complement rules: P(Z < –a) = 1 – P(Z < a)

Connect every problem to a sketch, standardise carefully, and always verify your answer is reasonable in context. With consistent practice on past paper questions, normal distribution problems become highly predictable marks.

将每个问题都与草图联系起来，仔细标准化，并始终验证答案在上下文中是否合理。通过不断练习历年真题，正态分布题将成为非常稳定易得的得分点。

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IGCSE CIE Maths: Normal Distribution Exam Focus | IGCSE CIE 数学：正态分布 考点精讲