Normal Distribution in GCSE CIE Maths | GCSE CIE 数学：正态分布考点精讲

📚 Normal Distribution in GCSE CIE Maths | GCSE 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 GCSE CIE Mathematics, you are expected to understand its shape, properties, and how to use it to calculate probabilities for real-world data. This revision guide covers the key concepts, formulas, and problem-solving techniques you need to master for your exam.

正态分布是统计学中最重要的连续概率分布之一。在 GCSE CIE 数学中，你需要理解它的形状、性质，并学会如何利用它对现实数据进行概率计算。这份复习指南涵盖了考试中必须掌握的核心概念、公式和解题技巧。

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

The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. It is often used to model variables that cluster around a mean, such as heights, weights, and exam scores. The curve is defined by two parameters: the population mean (μ) and the population standard deviation (σ).

正态分布是一种连续型概率分布，图形对称且呈钟形。它常用来模拟围绕均值聚集的变量，如身高、体重和考试成绩。该曲线由两个参数决定：总体均值（μ）和总体标准差（σ）。

The total area under the normal curve equals 1, representing total probability. The curve extends infinitely in both directions along the horizontal axis, never touching it. The peak of the curve is located at the mean, and the spread is determined by the standard deviation.

正态曲线下的总面积等于 1，代表总概率。曲线沿水平轴向两侧无限延伸，但永不触轴。曲线的峰值位于均值处，分布的离散程度由标准差决定。

2. Key Properties of the Normal Curve | 正态曲线的关键性质

The normal curve has several important properties that are always tested. It is symmetrical about the mean μ. This means that values equidistant from the mean have the same probability density. The mean, median, and mode are all equal and located at the centre of the distribution.

正态曲线有几个常考的重要性质。它关于均值 μ 对称，这意味着距离均值等距离的数值具有相同的概率密度。平均数、中位数和众数三者相等，均位于分布的中心。

The curve changes concavity at points one standard deviation away from the mean; these are called the points of inflection. About 68% of the data lies within 1σ of μ, about 95% within 2σ, and about 99.7% within 3σ. This is known as the empirical rule or the 68–95–99.7 rule.

曲线在距离均值一个标准差处改变凹凸性，这些点称为拐点。大约 68% 的数据落在 μ ± 1σ 范围内，约 95% 落在 μ ± 2σ，约 99.7% 落在 μ ± 3σ。这被称为经验法则或 68–95–99.7 规则。

3. The Standard Normal Distribution | 标准正态分布

The standard normal distribution is a special normal distribution with a mean of 0 and a standard deviation of 1. It is denoted by the letter Z. Any normal variable X with mean μ and standard deviation σ can be transformed into a Z-score using the formula:

标准正态分布是一个特殊的正态分布，其均值为 0，标准差为 1，用字母 Z 表示。任何具有均值 μ 和标准差 σ 的正态变量 X 都可以通过以下公式转化为 Z 分数：

Z = (X – μ) ÷ σ

The Z-score tells you how many standard deviations an observation is above or below the mean. A positive Z-score indicates the value is above the mean, while a negative Z-score indicates it is below the mean.

Z 分数表示一个观测值距离均值多少个标准差。正 Z 分数表示该值高于均值，负 Z 分数表示低于均值。

Standardisation allows us to use a single normal distribution table (the standard normal table) to find probabilities and critical values for any normal distribution. This is essential because we cannot have a separate table for every possible μ and σ.

标准化过程让我们能够使用单一的分布表（标准正态表）来查找任意正态分布的概率和临界值。这一点至关重要，因为我们不可能为每一种 μ 和 σ 都单独制表。

4. Using the Standard Normal Table | 使用标准正态表

The standard normal table gives the cumulative probability P(Z < z) for a given z-value. That is, the area under the curve to the left of z. Most CIE exam tables provide these left-tail probabilities for positive z-values. For negative z-values, you use the symmetry property: P(Z < –z) = 1 – P(Z < z).

标准正态表给出了给定 z 值的累积概率 P(Z < z)，即曲线下 z 左侧的面积。大多数 CIE 考试表格为正值 z 提供左尾概率。对于负 z 值，利用对称性：P(Z < –z) = 1 – P(Z < z)。

When asked to find P(Z > z), subtract the table value from 1: P(Z > z) = 1 – P(Z < z). For the probability between two values, P(a < Z < b) = P(Z < b) – P(Z < a). Always sketch a quick bell curve and shade the required area to avoid mistakes.

当要求求 P(Z > z) 时，用 1 减去表值：P(Z > z) = 1 – P(Z < z)。对于两值之间的概率，P(a < Z < b) = P(Z < b) – P(Z < a)。务必快速画出钟形曲线并涂出所需区域，以避免错误。

5. Calculating Probabilities for Any Normal Variable | 计算任意正态变量的概率

To find probabilities for a normal variable X ∼ N(μ, σ²), first standardise the boundary value(s) using Z = (X – μ) ÷ σ. Then use the standard normal table to find the required probability. Remember that σ² is the variance, so σ is the square root of the given variance.

要求正态变量 X ∼ N(μ, σ²) 的概率，首先用 Z = (X – μ) ÷ σ 对边界值进行标准化。然后使用标准正态表求所需的概率。记住 σ² 是方差，因此 σ 是给定方差的平方根。

Example: If X ∼ N(100, 15²), find P(X > 115). First, Z = (115 – 100) ÷ 15 = 1.00. From the table, P(Z < 1.00) = 0.8413, so P(X > 115) = 1 – 0.8413 = 0.1587.

示例：若 X ∼ N(100, 15²)，求 P(X > 115)。首先，Z = (115 – 100) ÷ 15 = 1.00。查表得 P(Z < 1.00) = 0.8413，因此 P(X > 115) = 1 – 0.8413 = 0.1587。

Always check whether the question asks for ‘less than’, ‘greater than’, or ‘between’ two values. Write down the standardisation step clearly; marks are often awarded for correct substitution even if the final table reading has a minor slip.

务必检查题目问的是“小于”、“大于”还是“介于”两个值之间。清楚地写出标准化步骤；即使最后的查表出现了小差错，正确的代入往往也能得到分数。

6. Inverse Normal: Finding the Value Given a Probability | 逆向正态：已知概率求数值

Sometimes you are given a probability (or percentage) and asked to find the corresponding value of X. This is called an inverse normal problem. Start by using the standard normal table in reverse to find the z-value corresponding to the given cumulative probability.

有时题目会给出一个概率（或百分比），让你求对应的 X 值。这称为逆向正态问题。首先反向使用标准正态表，找出与给定累积概率相对应的 z 值。

For example, if P(Z < z) = 0.975, the table gives z = 1.96. Then substitute into the standardisation formula rearranged to make X the subject:

例如，若 P(Z < z) = 0.975，查表得 z = 1.96。然后代入标准公式并变形为求 X 的式子：

X = μ + z × σ

If the table only gives positive z-values and you need a negative z, remember that if the cumulative probability is less than 0.5, the z-value will be negative. For instance, if P(Z < z) = 0.025, then by symmetry, z = –1.96 because P(Z < –1.96) = 0.025.

如果表格只给出正 z 值而你需要负 z 值，记住若累积概率小于 0.5，z 值将为负。例如，若 P(Z < z) = 0.025，则由对称性，z = –1.96，因为 P(Z < –1.96) = 0.025。

7. The 68–95–99.7 Rule in Context | 情境中的 68–95–99.7 规则

The empirical rule is a quick way to estimate probabilities without a table if the boundary values happen to be exactly 1, 2, or 3 standard deviations from the mean. For example, if exam scores are normally distributed with μ = 60 and σ = 10, then about 68% of students scored between 50 and 70, and about 95% scored between 40 and 80.

如果边界值恰好在距离均值 1、2 或 3 个标准差处，经验法则是一种无需查表即可快速估算概率的方法。例如，若考试成绩服从正态分布，μ = 60，σ = 10，则约 68% 的学生得分在 50 到 70 之间，约 95% 在 40 到 80 之间。

Remember that the percentages refer to the proportion of data falling within those intervals. The rule also helps to check the reasonableness of answers obtained from a table. If you calculate P(40 < X < 80) and get 0.83, you know it should be around 0.95, so a mistake was made.

记住这些百分比指的是落入这些区间的数据比例。该规则也有助于核查从表中所得答案的合理性。如果你计算 P(40 < X < 80) 得到 0.83，就知道它大约应为 0.95，说明出错了。

In the exam, you may be asked to sketch a normal curve and label the percentages for μ ± σ, μ ± 2σ, and μ ± 3σ. Ensure your sketch is symmetric and bell-shaped, with the mean clearly marked at the centre.

考试中可能会让你画出正态曲线并标出 μ ± σ、μ ± 2σ 和 μ ± 3σ 的百分比。确保你的草图对称、呈钟形，并在中心清晰地标出均值。

8. Finding Unknown Mean or Standard Deviation | 求未知均值或标准差

Some problems give you a probability and a corresponding X value, along with either μ or σ, and ask you to find the unknown parameter. You set up the standardisation equation using the given information and then solve for the missing value.

有些题目会给出一个概率和一个对应的 X 值，以及 μ 或 σ 中的一个，要求你找出未知参数。你需要利用所给信息建立标准化方程，然后解出缺失的值。

For instance, if X ∼ N(μ, 4²) and P(X < 20) = 0.8413, you know from the table that P(Z < 1.0) = 0.8413, so (20 – μ) ÷ 4 = 1.0, giving μ = 16. If both μ and σ are unknown, you will need two equations from two different given probabilities.

例如，若 X ∼ N(μ, 4²) 且 P(X < 20) = 0.8413，从表中可知 P(Z < 1.0) = 0.8413，因此 (20 – μ) ÷ 4 = 1.0，得 μ = 16。若 μ 和 σ 均未知，则需要从两个不同的给定概率得到两个方程。

Always show the step where you equate the Z expression to the z-value obtained from the table. This demonstrates your understanding of the inverse process and secures valuable method marks.

务必展示你将 Z 表达式与表得 z 值等同起来的步骤。这能体现你对逆向过程的理解，并确保获得宝贵的方法分。

9. Approximating Binomial with Normal | 用正态分布近似二项分布

When a binomial distribution has a large number of trials n, and p is not too close to 0 or 1, we can approximate it using a normal distribution. The conditions usually required are np > 5 and nq > 5 (where q = 1 – p).

当二项分布的试验次数 n 很大，且 p 不太接近 0 或 1 时，我们可以用正态分布来近似。通常要求的条件是 np > 5 且 nq > 5（其中 q = 1 – p）。

The approximating normal distribution has mean μ = np and variance σ² = npq. Because we are approximating a discrete distribution with a continuous one, we apply a continuity correction. For example, P(X ≥ 10) becomes P(X > 9.5) in the normal approximation.

近似的正态分布具有均值 μ = np 和方差 σ² = npq。由于我们用连续分布近似离散分布，需要应用连续性校正。例如，P(X ≥ 10) 在正态近似中变为 P(X > 9.5)。

This topic appears in some CIE GCSE extended or additional maths papers. Always check the syllabus; if included, practice applying the half-unit correction and then standardise to find the probability.

此主题出现在某些 CIE GCSE 扩展或附加数学试卷中。务必核对考纲；若包含在内，请练习应用半单位校正，然后标准化求概率。

10. Common Mistakes and Exam Tips | 常见错误与应试技巧

One frequent error is using the variance instead of the standard deviation in the Z formula. Remember, σ is the square root of the variance. Another mistake is reading the table incorrectly: always check whether the table gives the cumulative probability to the left or to the right.

一个常见错误是在 Z 公式中用了方差而非标准差。记住，σ 是方差的平方根。另一个错误是读表有误：务必检查表格给出的是左尾还是右尾的累积概率。

For inverse problems, many students forget to convert the given percentage to a decimal probability. If a question states “the top 5%”, you need to find the z-value for an area of 0.95 to the left. Drawing a diagram is the single most effective way to avoid confusion.

在逆向问题中，许多学生忘记将给定的百分比转换为小数概率。若题目说“最高的 5%”，你需要找到左侧面积为 0.95 所对应的 z 值。画图是避免混淆的最有效方法。

Always label your axes when sketching, indicate the mean, and shade the required region. State clearly that you assume the data is normally distributed if the question doesn’t explicitly say so; this justifies your use of the normal model.

画草图时务必标注坐标轴，标出均值，并涂出所需区域。如果题目未明确说明，要清晰声明你假设数据服从正态分布；这为使用正态模型提供了理由。

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Normal Distribution in GCSE CIE Maths | GCSE CIE 数学：正态分布 考点精讲