📚 IGCSE Maths Normal Distribution: Key Exam Points | IGCSE 数学：正态分布 考点精讲
The normal distribution is one of the most important continuous probability distributions in IGCSE Mathematics. It models many natural phenomena and is essential for statistical inference. In this revision guide, we'll cover the core concepts, the empirical rule, z-scores, using standard normal tables, and exam strategies.
正态分布是IGCSE数学中最重要的连续概率分布之一，它模拟了许多自然现象，对统计推断至关重要。在本复习指南中，我们将涵盖核心概念、经验法则、Z分数、标准正态表的使用以及考试策略。
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. It is defined by its mean (μ, mu) and standard deviation (σ, sigma). The total area under the curve equals 1, representing total probability. In IGCSE exams, you may see questions about heights, weights, test scores, or measurement errors.
正态分布是一种对称的钟形连续概率分布。它由其均值（μ）和标准差（σ）定义。曲线下的总面积为1，代表总概率。在IGCSE考试中，你可能会遇到关于身高、体重、考试成绩或测量误差的题目。
2. Key Properties of the Normal Curve | 正态曲线的主要性质
The normal curve is symmetric about its mean, meaning the left half is a mirror image of the right half. It is unimodal (has a single peak at the mean) and the tails extend infinitely in both directions, getting closer and closer to the horizontal axis but never touching it (asymptotic). The curve is bell-shaped, with most data clustered around the centre and less frequent extreme values.
正态曲线关于均值对称，即左半部分是右半部分的镜像。它是单峰的（在均值处有一个峰值），且尾部向两侧无限延伸，越来越接近水平轴但永不相交（渐近线）。曲线呈钟形，大部分数据集中在中心附近，极端值较少出现。
3. Mean, Median and Mode | 均值、中位数和众数
In a perfectly normal distribution, the mean, median, and mode are all equal and lie at the centre of the distribution. This is a quick check for normality: if a dataset is approximately symmetric and bell-shaped, these three measures will be very close. IGCSE questions may ask you to identify the mean from a given normal curve.
在完美的正态分布中，均值、中位数和众数完全相等，且都位于分布的中心。这是检验正态性的一种快速方法：如果数据集大致对称且呈钟形，这三个指标会非常接近。IGCSE题目可能会要求你根据给定的正态曲线找出均值。
4. Standard Deviation and Variance | 标准差和方差
a taller, narrower curve. A larger σ gives a flatter, wider curve. The variance is σ². Remember, 68% of observations fall within ±1σ of the mean, as described by the empirical rule.
标准差（σ）衡量分布的离散程度。σ越小，数据点越紧密地聚集在均值周围，曲线更高更窄；σ越大，曲线越平坦宽阔。方差是σ²。请记住，根据经验法则，68%的观测值落在均值±1σ范围内。
5. The Empirical Rule (68–95–99.7) | 经验法则 (68–95–99.7)
for IGCSE normal distribution problems. It states that for a normal distribution: about 68% of data lie within 1 standard deviation of the mean (μ ± σ), about 95% within 2 standard deviations (μ ± 2σ), and about 99.7% iations (μ ± 3σ). You can use these percentages to estimate probabilities without a table.
经验法则对IGCSE正态分布问题至关重要。它指出：约68%的数据落在均值±1个标准差内（μ ± σ），约95%落在±2个标准差内（μ ± 2σ），约99.7%落在±3个标准差内（μ ± 3σ）。你可以用这些百分比在没有表格的情况下估算概率。
| Interval | Probability |
|---|---|
| μ ± σ | 68% |
| μ ± 2σ | 95% |
| μ ± 3σ | 99.7% |
6. Standard Normal Distribution and Z-scores | 标准正态分布与Z分数
The standard normal distribution has mean 0 and standard deviation 1. To convert any normal value x into a z-score, use the formula:
标准正态分布的均值为0，标准差为1。要将任意正态值x转换为z分数，请使用公式：
z = (x − μ) / σ
The z-score tells you how many standard deviations x is from the mean. Positive z means above the mean, negative means below. Once you have z, you can look up the corresponding probability in standard normal tables.
z分数告诉你x距离均值有多少个标准差。正z值表示高于均值，负值表示低于均值。得到z后，你就可以在标准正态表中查找相应的概率。
7. Using Standard Normal Tables | 使用标准正态分布表
he curve to the left of z. For example, if z = 1.00, Φ(1.00) ≈ 0.8413, meaning about 84.13% ove the mean. To find the probability of being above a certain z, use P(Z > z) = 1 − Φ(z).
标准正态表给出累积概率Φ(z) = P(Z ≤ z)，即z左边曲线下的面积。例如，若z = 1.00，Φ(1.00) ≈ 0.8413，表示约84.13%的值低于均值以上1个标准差。要求大于某个z值的概率，使用P(Z > z) = 1 − Φ(z)。
Below is a small section of a typical table:
以下是一个典型表格的一小部分：
| z | 0.00 | 0.01 | 0.02 |
|---|---|---|---|
| 0.0 | 0.5000 | 0.5040 | 0.5080 |
| 1.0 | 0.8413 | 0.8438 | 0.8461 |
| 2.0 | 0.9772 | 0.9778 | 0.9783 |
hether your exam table gives the probability from -∞ to z or uses a different arrangement.
对于z = 1.02（行1.0，列0.02），Φ = 0.8461。务必检查你的考试表格是给出从 -∞ 到z的概率，还是采用其他排列方式。
8. Inverse Normal: Finding the Value from Probability | 反向正态：根据概率求值
Sometimes you are given a probability or percentile and need to find the corresponding x-value. First, look up the z-score that gives that cumulative probability in the table. Then convert back to the original distribution using:
有时题目会给出一个概率或百分位数，需要求出对应的x值。首先，在表中查找给出该累积概率的z分数。然后用以下公式转换回原始分布：
x = μ + z σ
the 90th percentile (P=0.90), find z ≈ 1.28 (from table), then x = 50 + 1.28 × 10 = 62.8.
例如，假设μ = 50，σ = 10，需要求第90百分位数的值（P=0.90），查表得z ≈ 1.28，则x = 50 + 1.28 × 10 = 62.8。
9. Real-life Applications | 实际应用
IGCSE normal distribution questions often involve contexts like: scores on an exam (e.g., the top 5% of students get an A*), quality control (diameters of bolts within tolerance), or natural measurements (heights of a population). Understand how to identify μ and σ from the problem and set up the correct area/probability.
IGCSE正态分布题目常涉及以下情境：考试成绩（如成绩前5%的学生获得A*）、质量控制（螺栓直径在公差范围内）或自然测量（人口身高）。要懂得如何从题目中识别μ和σ，并正确建立区域/概率的对应关系。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
A frequent error is confusing the probability for an interval with a one-tailed probability. For instance, when asked for the proportion between μ−σ and μ+σ, use the empirical 68%, not half of it. Another mistake is reading the table incorrectly – always check whether your z-table gives the area to the left or from 0 to z. Also, never mix up σ² (variance) with σ (standard deviation).
一个常见错误是把区间概率与单尾概率混淆。例如，当题目要求μ−σ到μ+σ之间的比例时，应使用经验法则68%，而非一半。另一个错误是读表不当——务必检查你的z表给出的是左侧面积还是从0到z的面积。此外，切勿混淆σ²（方差）和σ（标准差）。
11. Exam Tips and Tricks | 考试技巧与诀窍
lities. If your z-values are negative, use symmetry: P(Z < −a) = P(Z > a) = 1 − Φ(a). Memorise the empirical rule percentages – estions. Always round your final answer to an appropriate degree of accuracy as specified.
在使用表格前，务必快速画出正态曲线并涂出所需区域。这有助于避免混淆左尾和右尾概率。如果z值为负，利用对称性：P(Z < −a) = P(Z > a) = 1 − Φ(a)。记住经验法则的百分比——这可以帮你在许多IGCSE题目上节省时间。始终按题目要求将最终答案四舍五入到合适的精度。
12. Example Exam-style Questions | 考试形式例题
Question: The weights of apples from an orchard are normally distributed with mean 150 g and standard deviation 12 g. Find the probability that a randomly chosen apple weighs less than 138 g.
题目：某果园苹果的重量服从均值为150 g、标准差为12 g的正态分布。求随机选取的一个苹果重量小于138 g的概率。
Solution: −1.00) = P(Z > 1.00) = 1 − Φ(1.00) = 1 − 0.8413 = 0.1587. So the probability is about 15.9%.
解答：首先计算 z = (138 − 150) / 12 = −1.00。利用对称性，P(Z < −1.00) = P(Z > 1.00) = 1 − Φ(1.00) = 1 − 0.8413 = 0.1587。因此概率约为15.9%。
Question: In a test, scores are normally distributed with mean 65 and standard deviation 8. What score marks the top 10% of students?
题目：某测试成绩服从均值65、标准差8的正态分布。前10%学生的分数线是多少？
Solution: Top 10% means 90% Then x = μ + zσ = 65 + 1.28 × 8 = 75.24. Therefore a score of about 75.2 marks the top 10%.
解答：前10%意味着90%的学生低于分数线。查找Φ(z) = 0.90 → z ≈ 1.28。然后 x = 65 + 1.28 × 8 = 75.24。因此大约75.2分是前10%的界线。
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