📚 Normal Distribution in IGCSE WJEC Maths | IGCSE WJEC 数学：正态分布 考点精讲
The normal distribution is one of the most important concepts in IGCSE WJEC Mathematics. It models continuous data that cluster around a central mean, such as heights, weights, or exam scores. This article covers all the key points you need for the exam: properties of the normal curve, the empirical rule, z-scores, using the standard normal table, and solving both forward and inverse problems. Let’s dive in with clear explanations and paired English–Chinese guidance.
正态分布是 IGCSE WJEC 数学中最重要的概念之一。它用来描述围绕一个中心均值聚集的连续数据，比如身高、体重或者考试成绩。本文覆盖了考试需要的所有要点：正态曲线的性质、经验法则、z 分数、标准正态分布表的使用，以及正向求解和反向求解问题。我们通过清晰的中英对照讲解一起深入理解。
1. What is a Normal Distribution? | 什么是正态分布？
A normal distribution is a continuous probability distribution that is symmetric and bell-shaped. Many natural phenomena follow this pattern, so it is sometimes called the ‘bell curve’. The curve shows how data values are spread: most values cluster near the mean, and fewer appear as you move away from the centre.
正态分布是一种连续型概率分布，具有对称的钟形形状。许多自然现象都遵循这种模式，因此它常被称为“钟形曲线”。曲线显示了数据的分布情况：大多数值聚集在均值附近，离中心越远，出现的频率越低。
In the WJEC specification, you are expected to recognise the normal shape, understand that the total area under the curve equals 1, and that the curve never touches the horizontal axis – it extends infinitely in both directions.
在 WJEC 考纲中，你需要能识别正态形状，理解曲线下的总面积等于 1，并且曲线永远不与横轴相交——它向两个方向无限延伸。
2. Properties of the Normal Curve | 正态曲线的性质
The normal curve is symmetric about the mean (μ). This means if you fold the curve at the mean, the two halves match perfectly. The mean, median and mode are all equal and lie at the centre.
正态曲线关于均值（μ）对称。也就是说，如果你在均值处将曲线对折，两半会完全重合。均值、中位数和众数都相等，且位于中心位置。
The spread of the curve is determined by the standard deviation (σ). A larger σ makes the curve flatter and wider; a smaller σ makes it taller and narrower. However, the total area underneath always stays at 1, representing total probability.
曲线的分散程度由标准差（σ）决定。σ 越大，曲线越扁平、越宽；σ 越小，曲线越高、越窄。但不管形状如何，曲线下的总面积始终保持为 1，代表总概率。
Key properties to remember: the curve is symmetrical; mean=μ, sd=σ; area = 1; asymptotic to the x-axis.
需要记住的关键性质：曲线对称；均值为 μ，标准差为 σ；总面积为 1；以 x 轴为渐近线。
3. The Role of Mean (μ) and Standard Deviation (σ) | 均值 (μ) 和标准差 (σ) 的作用
The mean μ determines the location of the centre of the curve. Changing μ shifts the whole curve left or right without changing its shape. The standard deviation σ controls the spread. Two normal distributions can have the same mean but different standard deviations.
均值 μ 决定了曲线中心的位置。改变 μ 会使整个曲线左右平移，但形状不变。标准差 σ 控制分散程度。两个正态分布可以有相同的均值，但标准差不同。
For example, heights of adult women and men might both follow normal distributions but with different means and slightly different standard deviations. WJEC questions often ask you to compare curves and identify which has a larger mean or standard deviation.
例如，成年女性和男性的身高可能都服从正态分布，但均值和标准差略有不同。WJEC 的题目经常会要求你比较曲线，并判断哪一个均值更大或者标准差更大。
4. The Empirical Rule: 68-95-99.7 | 经验法则：68-95-99.7
The Empirical Rule states that for a normal distribution: about 68% of data falls within 1 standard deviation of the mean (μ ± 1σ), about 95% within 2 standard deviations (μ ± 2σ), and about 99.7% within 3 standard deviations (μ ± 3σ).
经验法则指出，对于正态分布：大约 68% 的数据落在均值 ±1 个标准差范围内（μ ± 1σ），大约 95% 落在均值 ±2 个标准差范围内（μ ± 2σ），大约 99.7% 落在均值 ±3 个标准差范围内（μ ± 3σ）。
You can use this rule to quickly estimate probabilities without a table. For instance, if the mean test score is 50 and σ=10, then roughly 95% of students score between 30 and 70. Remember these percentages represent the area under the curve between those limits.
你可以利用这个法则在不查表的情况下快速估计概率。例如，如果考试平均分是 50，σ = 10，那么大约 95% 的学生分数在 30 到 70 之间。记住这些百分比表示的是这些界限之间的曲线下面积。
5. 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 Z ~ N(0, 1). Any normal distribution X ~ N(μ, σ²) can be converted to the standard normal using the z-score formula.
标准正态分布是一种特殊的正态分布，均值为 0，标准差为 1，记作 Z ~ N(0, 1)。任何正态分布 X ~ N(μ, σ²) 都可以通过 z 分数公式转换成标准正态分布。
Standardisation removes the units and allows us to use a single probability table for all normal distributions. The table gives the cumulative probability Φ(z) = P(Z ≤ z). This is the foundation for all normal distribution calculations in the exam.
标准化去除了单位，让我们能够用同一张概率表处理所有正态分布。这张表给出了累积概率 Φ(z) = P(Z ≤ z)。这是考试中所有正态分布计算的基础。
6. Calculating z-Scores | 计算 z 分数
The z-score tells you how many standard deviations a value x is from the mean. The formula is:
z 分数表示某个值 x 距离均值有多少个标准差。公式为：
z = (x − μ) / σ
For example, if μ=60, σ=8, and x=72, then z = (72−60)/8 = 1.5. This means x is 1.5 standard deviations above the mean. If x is less than the mean, z will be negative.
例如，如果 μ=60，σ=8，x=72，那么 z = (72−60)/8 = 1.5。这意味着 x 比均值高 1.5 个标准差。如果 x 小于均值，z 将为负数。
Always remember: a positive z-score indicates the value is above the mean; a negative z-score indicates it is below. The sign is crucial when using the standard normal table.
始终记住：正 z 分数表示数值高于均值；负 z 分数表示数值低于均值。在使用标准正态分布表时，符号非常关键。
7. Using the Standard Normal Distribution Table | 使用标准正态分布表
The standard normal table gives Φ(z), the probability that Z is less than or equal to a given z-value. In WJEC exams, you will be given an extract of this table. It typically provides probabilities for positive z values from 0 to about 3.49.
标准正态分布表给出了 Φ(z)，即 Z 小于或等于某个给定 z 值的概率。在 WJEC 考试中，你会得到这份表格的节选。它通常提供从 0 到约 3.49 的正 z 值对应的概率。
To read the table: find the row for the first decimal of z and the column for the second decimal. For example, for z=1.25, locate the row for 1.2 and the column for 0.05, giving a probability of about 0.8944. That means P(Z < 1.25) = 0.8944.
查表方法：找到 z 的第一位小数所在的行，和第二位小数所在的列。例如，对于 z=1.25，定位到 1.2 行和 0.05 列，得到的概率约为 0.8944。这表示 P(Z < 1.25) = 0.8944。
For negative z values, use symmetry: P(Z < −a) = 1 − P(Z < a). Thus you can handle any z-score with the same table.
对于负 z 值，利用对称性：P(Z < −a) = 1 − P(Z < a)。这样你就可以用同一张表处理任何 z 分数。
8. Finding Probabilities: Less Than a Value | 求概率：小于某个值
To find P(X < x) for a normal distribution X ~ N(μ, σ²), first convert x to z using z = (x − μ) / σ, then look up Φ(z) in the table. That gives the answer directly.
要求正态分布 X ~ N(μ, σ²) 下 P(X < x)，首先用公式 z = (x − μ) / σ 将 x 转换为 z，然后在表中查找 Φ(z)。这个值就直接给出了答案。
Example: The weight of cereal boxes is normal with μ=500 g and σ=10 g. Find the probability a box weighs less than 495 g. z = (495−500)/10 = −0.5. P(Z < −0.5) = 1 − P(Z < 0.5) = 1 − 0.6915 = 0.3085. So about 30.85% of boxes weigh less than 495 g.
例子：谷物盒的重量服从正态分布，μ=500 g，σ=10 g。求一盒重量小于 495 g 的概率。z = (495−500)/10 = −0.5。P(Z < −0.5) = 1 − P(Z < 0.5) = 1 − 0.6915 = 0.3085。所以大约 30.85% 的盒子重量低于 495 g。
9. Finding Probabilities: Greater Than a Value | 求概率：大于某个值
To find P(X > x), use the complement rule: P(X > x) = 1 − P(X < x). So after finding the z-score, look up Φ(z) and subtract it from 1.
要求 P(X > x)，使用补集法则：P(X > x) = 1 − P(X < x)。因此求出 z 分数后，查表得到 Φ(z)，然后用 1 减去它。
Continuing the cereal example: probability a box weighs more than 510 g. z = (510−500)/10 = 1. P(Z < 1) = 0.8413, so P(X > 510) = 1 − 0.8413 = 0.1587.
继续谷物盒的例子：一盒重量大于 510 g 的概率。z = (510−500)/10 = 1。P(Z < 1) = 0.8413，所以 P(X > 510) = 1 − 0.8413 = 0.1587。
Always remember that the total area is 1, so ‘greater than’ is the right-tail probability.
始终记住总面积为 1，所以“大于”对应的是右尾概率。
10. Finding Probabilities: Between Two Values | 求概率：介于两个值之间
For P(a < X < b), find the z-scores for both a and b, then look up their cumulative probabilities and subtract: Φ(z_b) − Φ(z_a).
对于 P(a < X < b)，分别求出 a 和 b 的 z 分数，然后查找它们的累积概率并相减：Φ(z_b) − Φ(z_a)。
Example: For the cereal boxes, probability a box weighs between 495 g and 505 g. z₁ = −0.5, z₂ = 0.5. P(Z<0.5)=0.6915, P(Z<−0.5)=0.3085. Difference = 0.3830. So about 38.3% of boxes are within this range.
例子：对于谷物盒，一盒重量在 495 g 到 505 g 之间的概率。z₁ = −0.5，z₂ = 0.5。P(Z<0.5)=0.6915，P(Z<−0.5)=0.3085。差值 = 0.3830。所以大约 38.3% 的盒子在这个范围内。
This method works for any interval. Just make sure you subtract the lower bound probability from the upper bound probability.
这种方法适用于任何区间。只要确保用上限概率减去下限概率即可。
11. Inverse Normal: Finding Values from Probabilities | 反向查找：已知概率求值
Sometimes a question gives a probability and asks you to find the corresponding x-value, e.g. ‘Find the weight exceeded by only 10% of boxes.’ This is the inverse normal problem.
有时题目会给出一个概率，要求你找出对应的 x 值，例如“找出只有 10% 的盒子超过的重量”。这就是反向正态问题。
Steps: First, identify the tail probability. If it’s ‘exceeded by 10%’, then P(X > x) = 0.10, so P(X < x) = 0.90. Lookup the z-score that gives Φ(z)=0.90. From the table, z ≈ 1.2816 (or exact from the provided extract). Then un-standardise: x = μ + zσ.
步骤：首先，确定尾概率。如果是“超过 10%”，那么 P(X > x) = 0.10，因此 P(X < x) = 0.90。查找使得 Φ(z) = 0.90 的 z 分数。从表中得到 z ≈ 1.2816（或使用提供的节选值）。然后去标准化：x = μ + zσ。
For the cereal boxes with μ=500, σ=10, and upper 10% tail, z=1.2816 gives x=500+1.2816×10=512.816 ≈ 512.8 g. Thus 10% of boxes weigh more than 512.8 g.
对于谷物盒，μ=500，σ=10，上侧 10% 尾部，z=1.2816 得到 x=500+1.2816×10=512.816 ≈ 512.8 g。所以 10% 的盒子重于 512.8 g。
12. Real-World Application and Exam Tips | 实际应用与考试技巧
Normal distribution questions often involve real contexts: heights, weights, test scores, machine settings, or volumes. Always sketch a quick bell curve, label the mean, shade the area of interest, and note down the z-score formula. This reduces errors.
正态分布题目经常涉及真实情境：身高、体重、考试分数、机器设定或体积。始终快速画一个钟形曲线草图，标出均值，涂出感兴趣的区域，并写下 z 分数公式。这能减少错误。
In the exam, check whether the question requires a probability or a data value. Use the empirical rule for quick estimates when values are 1, 2, or 3 standard deviations from the mean. For other values, use the standard normal table precisely. Remember that tables vary; always use the one provided in the exam.
考试时，要辨别题目是需要求概率还是求数据值。当数值恰好距离均值 1、2 或 3 个标准差时，可以用经验法则快速估算。对于其他值，精确使用标准正态分布表。记住不同的表可能稍有差异，始终使用考试提供的表。
Finally, communicate clearly: state the distribution, show the standardisation, and give your final answer with appropriate units or as a percentage. Now you are ready to tackle any normal distribution question in your IGCSE WJEC Maths exam!
最后一点，表达要清晰：写出分布，展示标准化过程，最终答案要带上合适的单位或者用百分比表示。现在你已经准备好应对 IGCSE WJEC 数学考试中任何正态分布题目了！
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