Normal Distribution for IGCSE AQA Maths: Key Points | IGCSE AQA 数学：正态分布考点精讲

📚 Normal Distribution for IGCSE AQA Maths: Key Points | IGCSE AQA 数学：正态分布考点精讲

This article covers everything you need to know about the normal distribution for the IGCSE AQA Mathematics (Higher Tier) exam. We will explore the curve’s properties, standardisation, using z-tables, and solving typical exam problems. By mastering these concepts, you can confidently tackle any related question.

本文涵盖 IGCSE AQA 数学（高阶）考试中关于正态分布的所有考点。我们将探讨曲线性质、标准化、使用 z 表以及解决典型考试问题。掌握这些概念后，你将能自信地应对任何相关问题。

1. Understanding the Normal Distribution | 了解正态分布

The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. It is widely used to model real-world data such as heights, weights, and test scores.

正态分布是一种对称的钟形连续概率分布。它广泛用于模拟现实世界数据，如身高、体重和考试成绩。

In IGCSE AQA, you will encounter normal distributions with a given mean μ and standard deviation σ, as well as the standard normal distribution with mean 0 and standard deviation 1.

在 IGCSE AQA 中，你会遇到给定均值 μ 和标准差 σ 的正态分布，以及均值为 0、标准差为 1 的标准正态分布。

A continuous distribution means the variable can take any value within an interval. The total area under the probability density curve equals 1.

连续分布意味着变量可以取区间内的任何值。概率密度曲线下的总面积为 1。

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

The normal curve is bell-shaped and symmetric about the mean μ. This means the left half is a mirror image of the right half.

正态曲线呈钟形，并关于均值 μ 对称。这意味着左半部分是右半部分的镜像。

The curve never touches the horizontal axis, extending infinitely in both directions. The area under the curve represents probability.

曲线从不接触横轴，向两侧无限延伸。曲线下的面积代表概率。

The location of the mean determines the centre of the curve, while the standard deviation σ controls its spread. A smaller σ gives a taller, narrower peak; a larger σ gives a flatter, wider shape.

均值的位置决定曲线的中心，标准差 σ 控制其分散程度。σ 越小，峰值越高越窄；σ 越大，曲线越扁平越宽。

3. Mean, Median, Mode and Standard Deviation | 均值、中位数、众数和标准差

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

对于完全正态分布，均值、中位数和众数全部相等，且都位于曲线的峰值处。

The standard deviation measures the average distance of data points from the mean. Approximately 68% of data lies within one standard deviation of the mean.

标准差衡量数据点与均值的平均距离。大约 68% 的数据落在均值的一个标准差范围内。

In exam problems, you are usually given μ and σ. Use these parameters to describe the distribution fully.

在考试题目中，通常会给定 μ 和 σ。使用这些参数来完整描述分布。

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

The Empirical Rule states: about 68% of data falls within μ ± σ, about 95% within μ ± 2σ, and about 99.7% within μ ± 3σ.

经验法则指出：大约 68% 的数据落在 μ ± σ 内，大约 95% 落在 μ ± 2σ 内，大约 99.7% 落在 μ ± 3σ 内。

μ ± σ → 68% | μ ± 2σ → 95% | μ ± 3σ → 99.7%

This rule helps quickly estimate probabilities without a z-table. For example, if heights are normal with μ = 170 cm, σ = 10 cm, then around 95% of people are between 150 cm and 190 cm.

该法则有助于在没有 z 表时快速估算概率。例如，若身高服从正态分布 μ = 170 cm，σ = 10 cm，那么大约 95% 的人身高在 150 cm 到 190 cm 之间。

5. Standardising: The z-score Formula | 标准化：z 分数公式

To use standard normal tables, convert an x-value to a z-score. The z-score represents how many standard deviations x is above or below the mean.

要使用标准正态表，需将 x 值转换为 z 分数。z 分数表示 x 与均值相差几个标准差。

z = (x − μ) / σ

For the standard normal distribution, μ = 0 and σ = 1, so z is simply the value itself.

对于标准正态分布，μ = 0，σ = 1，因此 z 分数就是数值本身。

Always include the sign: positive z means x is above the mean, negative z means x is below the mean.

始终注意符号：正 z 意味着 x 高于均值，负 z 意味着 x 低于均值。

Example: If μ = 50, σ = 5, and x = 62, then z = (62 – 50)/5 = 12/5 = 2.4.

示例：若 μ = 50，σ = 5，x = 62，则 z = (62 – 50)/5 = 12/5 = 2.4。

6. Using the Standard Normal Distribution Table | 使用标准正态分布表

The standard normal table gives the cumulative probability P(Z ≤ z) for z ≥ 0. For instance, P(Z ≤ 1.00) = 0.8413.

标准正态表给出 z ≥ 0 时的累积概率 P(Z ≤ z)。例如，P(Z ≤ 1.00) = 0.8413。

Below is a small extract from a typical cumulative table:

以下是一个典型累积分布表的摘录：

z	P(Z ≤ z)
0.00	0.5000
0.50	0.6915
1.00	0.8413
1.50	0.9332
2.00	0.9772

Because the curve is symmetric, probabilities for negative z-values are derived from the positive side: P(Z ≤ –a) = 1 – P(Z ≤ a).

由于曲线是对称的，负 z 值的概率可从正侧推导：P(Z ≤ –a) = 1 – P(Z ≤ a)。

In the AQA exam, you may be provided with a full table or a percentage points table for inverse lookups. Always check which table is given before starting calculations.

在 AQA 考试中，你可能会得到完整的表格或用于逆查找的百分点表。开始计算前务必确认所提供的表格类型。

7. Calculating Probabilities: P(Z < z) | 计算概率：P(Z < z)

To find the probability that a standard normal variable is less than a given value, simply look up the z-score in the cumulative table. For P(Z < 1.25), find the row for 1.2 and column for 0.05, giving 0.8944.

要求标准正态变量小于给定值的概率，只需在累积表中查找 z 分数。求 P(Z < 1.25) 时，找到 1.2 所在行和 0.05 所在列，得到 0.8944。

If the z-score is not exactly in the table, use linear interpolation as a rough estimate, though in IGCSE the required values are usually directly given or can be found exactly.

如果 z 分数不完全在表中，可使用线性插值粗略估计，但 IGCSE 通常要求的值可直接或精确找到。

For any continuous distribution, P(Z < z) = P(Z ≤ z) because the probability of exactly one value is zero.

对于任何连续分布，P(Z < z) = P(Z ≤ z)，因为恰好取某一值的概率为零。

8. Calculating Probabilities for Intervals and Tails | 计算区间和尾部概率

To find the probability between two z-values a and b: P(a < Z < b) = P(Z < b) – P(Z < a).

要求两个 z 值 a 和 b 之间的概率：P(a < Z < b) = P(Z < b) – P(Z < a)。

For example, P(0.5 < Z < 1.5) = P(Z < 1.5) – P(Z < 0.5) = 0.9332 – 0.6915 = 0.2417.

例如，P(0.5 < Z < 1.5) = P(Z < 1.5) – P(Z < 0.5) = 0.9332 – 0.6915 = 0.2417。

To find a right-tail probability P(Z > a), use the complement rule: P(Z > a) = 1 – P(Z < a). So P(Z > 1.8) = 1 – 0.9641 = 0.0359.

要求右尾概率 P(Z > a)，使用补集法则：P(Z > a) = 1 – P(Z < a)。因此 P(Z > 1.8) = 1 – 0.9641 = 0.0359。

If the z-value is negative, use symmetry: P(Z < –a) = P(Z > a) = 1 – P(Z < a).

若 z 值为负，利用对称性：P(Z < –a) = P(Z > a) = 1 – P(Z < a)。

9. Inverse Normal: Finding z from a Given Probability | 逆正态：由给定概率求 z

Sometimes the exam asks you to find the z-score corresponding to a given cumulative probability p. Look inside the standard normal table for the probability closest to p and read off the corresponding z.

有时考试会要求找出与给定累积概率 p 相对应的 z 分数。在标准正态表内查找最接近 p 的概率值，并读出对应的 z 值。

For example, if P(Z < z) = 0.975, find 0.9750 inside the table – it corresponds to z = 1.96.

例如，如果 P(Z < z) = 0.975，在表中找到 0.9750 – 对应 z = 1.96。

If the percentage points table is provided, you may directly read the z-value for a right-tail probability α, such as z = 1.645 for α = 0.05.

如果提供了百分点表，你可以直接读出右尾概率 α 对应的 z 值，例如 α = 0.05 时 z = 1.645。

Once you have z, convert back to x using the formula: x = μ + zσ.

一旦求得 z，便可用公式换回 x：x = μ + zσ。

10. Solving Real-Life Problems | 解决实际问题

Real-world problems require you to identify μ and σ, calculate the relevant z-score, use the table to find probabilities, and then interpret the result in context.

解决实际问题时，你需要确定 μ 和 σ，计算相应的 z 分数，查表求概率，然后结合情境解释结果。

Example: The weights of apples are normally distributed with μ = 150 g and σ = 20 g. Find the proportion of apples weighing more than 175 g.

示例：苹果重量服从正态分布，μ = 150 g，σ = 20 g。求重量超过 175 g 的苹果比例。

z = (175 – 150)/20 = 1.25. P(Z > 1.25) = 1 – P(Z < 1.25) = 1 – 0.8944 = 0.1056. So about 10.6% of apples exceed 175 g.

z = (175 – 150)/20 = 1.25。P(Z > 1.25) = 1 – P(Z < 1.25) = 1 – 0.8944 = 0.1056。因此大约 10.6% 的苹果超过 175 g。

Always check whether the question asks for a proportion, a percentage, or a specific number of items out of a total.

务必检查题目要求的是比例、百分比还是特定数量的个体。

11. Exam Tips and Common Pitfalls | 考试技巧与常见陷阱

Draw a sketch of the normal curve and shade the required area. This helps you decide whether to subtract from 1 or from another probability.

画出正态曲线草图并涂色所求区域。这有助于决定是从 1 还是从另一概率中减去。

Remember that the table typically gives P(Z < z). Adjust your calculations if the question involves 'greater than' or 'between' probabilities.

记住表格通常给出的是 P(Z < z)。如果题目涉及‘大于’或‘之间’的概率，需要调整计算。

Do not mix up μ and σ with the standard normal parameters. Always standardise before using the z-table unless already given standard normal data.

不要混淆 μ、σ 和标准正态参数。除非数据已经是标准正态，否则在使用 z 表前必须先标准化。

Watch out for negative z-values. Use P(Z < –a) = 1 – P(Z < a) carefully, and confirm with a sketch.

注意负 z 值。谨慎使用 P(Z < –a) = 1 – P(Z < a)，并通过草图确认。

When doing inverse normal, be precise with the probability value; use the closest table entry. If interpolation is needed, show your working clearly.

进行逆正态计算时，需精确匹配概率值；使用最接近的表值。若需插值，清楚地写出解答过程。

Practise past paper questions to get familiar with the wording and typical mark allocations in AQA exams.

练习历年真题，熟悉 AQA 考试中的措辞和典型分值分配。

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Normal Distribution for IGCSE AQA Maths: Key Points | IGCSE AQA 数学：正态分布 考点精讲