IGCSE Edexcel Maths: Normal Distribution – Key Points Breakdown | IGCSE Edexcel 数学：正态分布 考点精讲

📚 IGCSE Edexcel Maths: Normal Distribution – Key Points Breakdown | IGCSE Edexcel 数学：正态分布 考点精讲

The normal distribution is one of the most important continuous probability distributions in IGCSE Edexcel Mathematics. It is used to model many real-world variables such as heights, weights, and exam scores. The distribution is symmetric, bell-shaped, and fully described by its mean (μ) and standard deviation (σ). In the IGCSE exam, you need to understand the properties of the normal curve, use the 68–95–99.7 rule to estimate probabilities, and calculate z‑scores to find proportions above or below a given value. This article covers all the key knowledge points, common question types, and essential formulas to help you tackle normal distribution problems with confidence.

正态分布是 IGCSE Edexcel 数学中最重要的连续型概率分布之一。它常常用来模拟真实世界中的许多变量，例如身高、体重和考试成绩。正态分布曲线呈对称的钟形，完全由均值(μ)和标准差(σ)描述。在 IGCSE 考试中，你需要理解正态曲线的性质，会用 68–95–99.7 法则估算概率，并会计算 z 分数来求大于或小于某个值的比例。本文涵盖所有重要的知识要点、常见题型和必备公式，帮助你自信地解答正态分布问题。

1. Understanding the Normal Distribution Curve | 理解正态分布曲线

The normal distribution curve is a continuous bell-shaped curve that is symmetric about the mean μ. The total area under the curve equals 1, representing total probability. The curve approaches the x‑axis but never touches it, meaning extreme values are possible but very unlikely. The peak of the curve is located at the mean, which is also the median and the mode in a perfect normal distribution.

正态分布曲线是一条连续的钟形曲线，关于均值 μ 对称。曲线下的总面积等于 1，代表总概率。曲线无限靠近 x 轴但永不相交，这意味着极端的取值可能出现但概率极低。曲线的最高点位于均值处，在完美的正态分布中，均值同时也是中位数和众数。

The shape of the curve is determined by the standard deviation σ. A smaller σ makes the curve taller and narrower, while a larger σ makes it flatter and wider. The position is set by μ, shifting left or right. In IGCSE questions, you are often given a sketch with μ and σ marked, and you must shade the required area.

曲线的形状由标准差 σ 决定。σ 越小，曲线越尖、越窄；σ 越大，曲线越扁、越宽。位置则由 μ 决定，曲线随之左右平移。在 IGCSE 试题中，通常会给出标有 μ 和 σ 的示意图，你需要根据题意在图上涂出相应的面积。

2. Properties of the Normal Distribution: Mean, Median and Mode | 正态分布的性质：均值、中位数和众数

In a perfectly symmetrical normal distribution, the mean, median and mode are all equal and lie at the centre. This property is often tested in multiple-choice questions or used to justify why a dataset might be normally distributed. If a histogram or frequency curve is roughly bell-shaped and symmetric, you can assume the mean ≈ median ≈ mode.

在完全对称的正态分布中，均值、中位数和众数三者相等且位于中心。这一性质经常出现在选择题中，或用于说明某个数据集为何可能服从正态分布。如果直方图或频率曲线大致呈钟形且对称，你可以认为均值 ≈ 中位数 ≈ 众数。

The symmetry also means that the probability of being a certain distance above the mean equals the probability of being the same distance below the mean. For example, P(X > μ + k) = P(X < μ – k). This symmetry is extremely helpful when solving problems without a full normal distribution table.

对称性还意味着：高于均值某一距离的概率，等于低于均值相同距离的概率。例如，P(X > μ + k) = P(X < μ – k)。在没有完整正态分布表可供查阅时，这种对称性在解题时非常有用。

3. The 68–95–99.7 Rule (Empirical Rule) | 68–95–99.7 法则（经验法则）

The empirical rule is a key tool for IGCSE normal distribution problems. It states that for a normal distribution:

经验法则是 IGCSE 正态分布问题中的核心工具。它指出，对于正态分布：

• About 68% of data lies within 1 standard deviation of the mean (μ ± σ).
  大约 68% 的数据落在均值 ± 1 个标准差的范围内 (μ ± σ)。

• About 95% of data lies within 2 standard deviations (μ ± 2σ).
  大约 95% 的数据落在 μ ± 2σ 内。

• About 99.7% of data lies within 3 standard deviations (μ ± 3σ).
  大约 99.7% 的数据落在 μ ± 3σ 内。

These percentages are approximate but widely accepted in IGCSE exam marking. You can use them to quickly estimate the proportion of values above or below a certain threshold without using z‑tables, if the boundary values are exactly 1, 2, or 3 standard deviations from the mean.

这些百分比是近似值，但在 IGCSE 考试评分中被广泛接受。如果边界值恰好是距离均值 1、2 或 3 个标准差，你可以用它们快速估算高于或低于某一阈值的数据比例，而无需查 z 表。

For example, if a test score is normally distributed with μ = 50 and σ = 10, then μ + σ = 60. The percentage of students scoring above 60 is roughly (100% – 68%) ÷ 2 = 16%. Similarly, the proportion between 40 and 60 is about 68%.

例如，某考试成绩服从正态分布，μ = 50，σ = 10，则 μ + σ = 60。分数高于 60 的学生比例约为 (100% – 68%) ÷ 2 = 16%。同样，40 到 60 分之间的比例约为 68%。

4. Understanding and Calculating the z‑Score | 理解并计算 z 分数

The z‑score measures how many standard deviations a data value x is away from the mean μ. It is given by the formula:

z 分数用于衡量某个数据值 x 距离均值 μ 有多少个标准差。其计算公式为：

z = (x – μ) / σ

A positive z‑score means x is above the mean; a negative z‑score means x is below the mean. In IGCSE, you will often need to find z first and then use a provided probability table or the 68–95–99.7 rule to determine the required proportion.

z 分数为正表示 x 高于均值；为负表示 x 低于均值。在 IGCSE 中，你通常需要先求出 z 值，然后使用给定的概率表或 68–95–99.7 法则来确定所需的比例。

Example: The lengths of nails are normally distributed with μ = 50 mm, σ = 2 mm. Find z for a nail of length 53 mm. z = (53 – 50) / 2 = 1.5. This nail is 1.5 standard deviations longer than the mean.

例题：钉子的长度服从正态分布，μ = 50 mm，σ = 2 mm。求一根长 53 mm 的钉子的 z 分数。z = (53 – 50) / 2 = 1.5。这根钉子的长度比均值多 1.5 个标准差。

IGCSE questions may provide a simplified z‑table or ask you to use the empirical rule only. When z is not 1, 2, or 3, you will normally be given the corresponding probability or a small section of the normal table. Always check the exam paper instructions.

IGCSE 试题可能会提供一个简化的 z 表，或者只要求你使用经验法则。当 z 值不是 1、2 或 3 时，通常都会给出相应的概率或一小段正态分布表。请务必仔细阅读试卷上的说明。

5. Using z‑Tables to Find Probabilities | 使用 z 表求概率

Standard normal tables give the cumulative probability P(Z < z) for z ≥ 0, i.e., the area to the left of a positive z‑score. Because the total area is 1, you can also find P(Z > z) = 1 – P(Z < z). Due to symmetry, P(Z < –z) = P(Z > z), and P(Z > –z) = P(Z < z). These relationships are essential when the z‑score is negative.

标准正态分布表给出 z ≥ 0 时的累积概率 P(Z < z)，即正 z 分数左侧的面积。因为总面积为 1，所以 P(Z > z) = 1 – P(Z < z)。利用对称性，P(Z < –z) = P(Z > z)，P(Z > –z) = P(Z < z)。当 z 分数为负值时，这些关系至关重要。

When solving problems:

解题步骤：

1. Draw a sketch of the normal curve, mark μ and the x value(s), and shade the required area.
   画出正态曲线草图，标出 μ 和 x 值，并涂出所求的面积。

2. Calculate the z‑score(s).
   计算 z 分数。

3. Use the table to find the probability for the z‑value(s) and apply symmetry where needed.
   查表得到对应 z 值的概率，并根据需要运用对称性。

4. Write the final answer as a decimal or percentage.
   最终答案写成小数或百分数。

IGCSE tables may give P(Z < z) for a range from 0 to 3.49 in small increments. Practice reading the table correctly: the row shows the first two digits, the column the second decimal place.

IGCSE 所用的表通常提供 0 到 3.49 之间各 z 值对应的 P(Z < z)。需要多加练习才能正确读表：行表示 z 值的前两位数字，列表示第二位小数。

6. Finding Probabilities for Ranges | 求某一区间内的概率

Exam questions frequently ask for the probability that a value lies between two limits, say a and b. The method is: P(a < X < b) = P(Z < z_b) – P(Z < z_a), where z_a and z_b are the z‑scores for a and b. If one limit is the mean, use symmetry and the fact that 50% of data lies below the mean.

考试中常要求求取值落在两个界限 a 与 b 之间的概率。方法是：P(a < X < b) = P(Z < z_b) – P(Z < z_a)，其中 z_a 和 z_b 分别是 a 与 b 的 z 分数。如果其中一个界限是均值，则利用对称性以及 50% 数据低于均值这一事实。

For example, test scores X ~ N(60, 8²). Find the proportion of scores between 52 and 68. z₁ = (52 – 60)/8 = –1, z₂ = (68 – 60)/8 = 1. P(–1 < Z < 1) = P(Z < 1) – P(Z < –1) = P(Z < 1) – [1 – P(Z < 1)] = 2P(Z < 1) – 1. If table gives P(Z < 1) = 0.8413, probability = 2(0.8413) – 1 = 0.6826, which is exactly the 68% from the empirical rule.

例如，考试成绩 X ~ N(60, 8²)。求分数在 52 与 68 之间的比例。z₁ = (52 – 60)/8 = –1，z₂ = (68 – 60)/8 = 1。P(–1 < Z < 1) = P(Z < 1) – P(Z < –1) = P(Z < 1) – [1 – P(Z < 1)] = 2P(Z < 1) – 1。若表给出 P(Z < 1) = 0.8413，则概率 = 2(0.8413) – 1 = 0.6826，这恰好就是经验法则中的 68%。

7. Reverse Normal Distribution – Finding x from a Given Probability | 逆向正态分布 – 已知概率求 x

Sometimes you are given a probability or a percentile and need to find the corresponding x-value. These “inverse normal” problems require you to first find the z‑score corresponding to the given cumulative probability from the table, then use x = μ + zσ.

有时题目会给出一个概率或百分位数，让你求对应的 x 值。这类“逆向正态分布”问题需要先根据给定的累积概率从表中查出对应的 z 分数，然后再用公式 x = μ + zσ。

For instance, a company’s delivery time X ~ N(30, 5²). What is the delivery time above which the slowest 10% deliveries lie? This means P(X > x) = 0.10, so P(X < x) = 0.90. Look up 0.9000 in the normal table (or the closest value) to find z ≈ 1.28. Then x = 30 + 1.28×5 = 36.4 minutes.

例如，某公司的配送时间 X ~ N(30, 5²)。问最慢的 10% 的配送时间高于多少分钟？即 P(X > x) = 0.10，因此 P(X < x) = 0.90。在正态表中查 0.9000（或最接近的值）得到 z ≈ 1.28。则 x = 30 + 1.28×5 = 36.4 分钟。

In IGCSE, the required z‑value may be given to you directly, or you may need to read it from a small table extract. Always check if the question expects you to use the 68–95–99.7 rule for simple percentiles (e.g., the top 2.5% corresponds to z = 1.96, because 95% is central).

在 IGCSE 中，所需的 z 值可能直接给出，也可能需要你从一小段表格中读取。请务必注意，题目是否期望你用 68–95–99.7 法则处理简单百分位数（例如，最高的 2.5% 对应 z = 1.96，因为中间有 95%）。

8. Standardising and Comparing Values from Different Distributions | 标准化及比较不同分布中的取值

The z‑score allows you to compare values from different normal distributions fairly. A higher z‑score indicates a better relative position. For example, student A scored 72 in a test with μ=60, σ=8; student B scored 68 in a test with μ=55, σ=6. Which performed better relative to their group? z_A = (72–60)/8 = 1.5; z_B = (68–55)/6 ≈ 2.17. Student B’s z‑score is higher, so B performed better relative to their own group.

z 分数使你能够公平地比较来自不同正态分布的数据。z 分数越高，说明相对位置越好。例如，学生 A 在一次考试中得 72 分，该考试 μ=60, σ=8；学生 B 在另一考试中得 68 分，该考试 μ=55, σ=6。谁在各自群体中表现更好？z_A = (72–60)/8 = 1.5；z_B = (68–55)/6 ≈ 2.17。学生 B 的 z 分数更高，因此 B 相对于自己所在的群体表现更好。

This type of question appears frequently in IGCSE modelling contexts. Remember that the raw scores cannot be directly compared; standardised scores must be used.

此类题目在 IGCSE 建模背景下经常出现。请记住，原始分数不能直接比较，必须使用标准化后的分数。

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

One typical mistake is confusing the standard deviation σ with the variance σ². In the normal distribution notation X ~ N(μ, σ²), the second parameter is the variance, not the standard deviation. Many students wrongly use σ² instead of σ when calculating z‑scores. Always check whether the question gives you the variance or the standard deviation.

一个典型错误是将标准差 σ 与方差 σ² 混淆。在正态分布的记法 X ~ N(μ, σ²) 中，第二个参数是方差，不是标准差。许多学生在计算 z 分数时错误地用 σ² 代替了 σ。请务必看清楚题目给的是方差还是标准差。

Another error is misapplying the symmetry. For P(X < 40) when μ=50 and σ=5, z = –2. Some students incorrectly look up z=2 and leave it as that, forgetting to do 1 – table value, or they quote a negative probability. Always sketch the curve and shade the area so you can visualise the required tail.

另一个错误是错误运用对称性。比如 μ=50、σ=5，求 P(X < 40) 时 z = –2。有些学生错误地查 z=2 就结束，忘了用 1 减去表值，或者写出了负的概率。一定要画出曲线图并涂色，把要求的尾部面积可视化。

Finally, when using the empirical rule, don’t forget that 68% inside μ±σ leaves 32% outside, which is split equally into 16% in each tail. Mixing up one-tail and two-tail probabilities is a frequent source of marks lost.

最后，使用经验法则时不要忘记：μ±σ 内有 68% 的数据，意味着外面有 32%，这 32% 被平均分成左右各 16%。将单尾概率与双尾概率混淆是常见的失分原因。

10. Setting Out Your Working Clearly | 清晰展示解题步骤

IGCSE examiners award method marks even if the final numerical answer is wrong. Always show: (1) the z‑score formula with substituted values, (2) a small sketch with μ and x positions shaded, (3) the probability statements, and (4) the final answer. If using the empirical rule, state the rule and the reasoning: e.g., “95% within 2σ therefore 2.5% above μ+2σ”. This structured approach will maximise your score.

IGCSE 考官会给方法分，即使最终数值答案有误。一定要展示：(1) 带人代数值的 z 分数公式；(2) 带有 μ 和 x 位置的涂色示意图；(3) 概率表达式；(4) 最终答案。如果用的是经验法则，要陈述该法则并给出推理，例如：“95% 在 2σ 以内，因此高于 μ+2σ 的比例为 2.5%”。这种结构化的方法会让你得到最高分。

Practise past paper questions and time yourself: drawing the curve need not be artistic – a simple symmetric bell shape with a vertical line for the mean and labelled values is sufficient. Always label the mean and the x-intercept you are interested in.

多练习历年真题并计时：画曲线无需画得很有艺术感——一个简单的对称钟形加上代表均值的竖线并标出数值就足够了。务必标出均值以及你关注的 x 值。

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