📚 Normal Distribution for GCSE WJEC Maths | GCSE WJEC 数学：正态分布 考点精讲
The normal distribution is a fundamental concept in GCSE WJEC Maths, appearing in statistics and probability. Understanding its shape, properties and the 68–95–99.7 rule is essential for answering exam questions confidently. This guide breaks down all the key points you need to master the topic.
正态分布是 GCSE WJEC 数学统计与概率部分的核心概念。掌握它的形状、特性以及 68–95–99.7 规则，对于自信地解答考试题目至关重要。本文梳理了攻克该主题所需的所有考点。
1. Introduction to the Normal Distribution | 正态分布介绍
The normal distribution, often called the bell curve, is a continuous probability distribution that is symmetric around its mean. Many natural phenomena, such as heights, weights and test scores, follow a normal distribution.
正态分布通常被称作钟形曲线，是一种关于平均值对称的连续型概率分布。很多自然现象，如身高、体重和考试分数，都服从正态分布。
In probability theory, it models continuous data with most observations clustering around a central peak and probabilities tailing off symmetrically on both sides.
在概率论中，它用于对连续型数据建模，大多数观测值聚集在中心峰值周围，概率向两侧对称递减。
2. Key Characteristics of a Normal Distribution | 正态分布的主要特征
A normal distribution has several distinct features: it is bell-shaped and symmetric; the mean, median and mode are all equal; the total area under the curve equals 1; the curve never touches the horizontal axis (asymptotic); it is completely defined by its mean and standard deviation.
正态分布有以下几个显著特征：呈钟形且对称；平均数、中位数和众数全部相等；曲线下总面积为 1；曲线永不接触横轴（渐近线）；完全由其均值和标准差确定。
We often denote a normally distributed variable X with mean μ and variance σ² as:
我们通常将均值为 μ、方差为 σ² 的正态变量 X 表示为：
X ~ N(μ, σ²)
3. Mean, Median and Mode in a Normal Curve | 正态曲线中的平均数、中位数和众数
Because the curve is perfectly symmetric, the mean, median and mode coincide at the centre. This implies that 50% of the data falls below the mean and 50% above it. The centre of the distribution represents the most likely value.
由于曲线完全对称，平均数、中位数和众数在中心处重合。这意味着 50% 的数据位于均值以下，50% 位于均值以上。分布的中心代表了最可能出现的值。
4. The Effect of Standard Deviation on Shape | 标准差对曲线形状的影响
The standard deviation σ determines the spread or width of the bell curve. A smaller σ produces a taller, narrower curve, indicating that data points are tightly clustered around the mean. A larger σ results in a flatter, wider curve, reflecting greater variability. Changing the mean μ shifts the whole curve left or right without altering its shape.
标准差 σ 决定了钟形曲线的分散程度或宽度。较小的 σ 会产生更尖、更窄的曲线，表明数据点紧密聚集在均值周围。较大的 σ 则使曲线更扁平、更宽，反映出更大的变异性。改变均值 μ 会使整个曲线左右平移而不改变形状。
5. The 68–95–99.7 Rule (Empirical Rule) | 68-95-99.7 经验法则
The empirical rule describes the proportion of data within certain intervals for any normal distribution:
经验法则描述了任意正态分布中落在特定区间内的数据比例：
- 68% of observations fall within 1 standard deviation of the mean (μ ± σ).
- 中文：68% 的观测值落在均值 1 个标准差范围之内（μ ± σ）。
- 95% of observations fall within 2 standard deviations of the mean (μ ± 2σ).
- 中文：95% 的观测值落在 2 个标准差范围之内（μ ± 2σ）。
- 99.7% of observations fall within 3 standard deviations of the mean (μ ± 3σ).
- 中文：99.7% 的观测值落在 3 个标准差范围之内（μ ± 3σ）。
You can summarise the rule visually:
你可以用下图形象记忆：
68% within μ ± σ
95% within μ ± 2σ
99.7% within μ ± 3σ
6. Applying the Empirical Rule – Examples | 应用经验法则 – 实例
Example: Exam scores are normally distributed with mean μ = 60 and standard deviation σ = 10.
例：考试成绩服从均值为 60、标准差为 10 的正态分布。
Then we can immediately state:
我们可以马上得出：
| Interval 区间 | Percentage 百分比 |
|---|---|
| 50 to 70 (μ ± σ) | 68% |
| 40 to 80 (μ ± 2σ) | 95% |
| 30 to 90 (μ ± 3σ) | 99.7% |
What percentage of students scored above 80? Since 95% lie within 40–80, 5% lie outside this interval. Due to symmetry, 2.5% are below 40 and 2.5% are above 80. So the probability of scoring above 80 is 2.5%.
有多少百分比的学生分数高于 80？由于 95% 的分数在 40–80 范围内，因此 5% 落在这个区间之外。根据对称性，2.5% 低于 40，2.5% 高于 80。因此得分高于 80 的概率为 2.5%。
7. Using Symmetry to Find Probabilities | 利用对称性求概率
The normal curve's symmetry allows you to break down regions easily. The area within μ ± σ is 68%, so each half from μ to μ+σ (or μ to μ-σ) is 34%. The remaining area in each tail beyond μ ± σ is 16%. For μ ± 2σ, the middle 95% leaves 2.5% in each tail, and the regions between 1σ and 2σ from the mean hold 13.5% each side.
正态曲线的对称性让你可以轻松分解区域。μ ± σ 之内的面积为 68%，因此从 μ 到 μ+σ（或 μ 到 μ-σ）的每一半都是 34%。超出 μ ± σ 的每个尾部面积则为 16%。对于 μ ± 2σ，中间 95% 使每个尾部剩下 2.5%，而从均值 1σ 到 2σ 之间的区域每侧各占 13.5%。
These subdivisions are often tested:
这些细分区域经常被考查：
| Region 区域 | Approx. probability 近似概率 |
|---|---|
| μ to μ + σ | 34% |
| μ + σ to μ + 2σ | 13.5% |
| μ + 2σ to μ + 3σ | 2.35% |
| Beyond μ + 3σ (tail) 超出 μ + 3σ 的尾部 | 0.15% |
8. Interpreting Normal Distribution Problems | 解释正态分布问题
Typical WJEC GCSE questions provide a mean and standard deviation and ask for the percentage of data above or below a specific value, or between two values. The key is to determine how many standard deviations away that value is from the mean.
典型的 WJEC GCSE 题目会给出均值和标准差，要求计算数据高于或低于某个特定值的百分比，或介于两个值之间的百分比。关键在于判断该值距离均值有几个标准差。
3σ, apply the empirical rule directly. Read the question carefully to identify whether the desired region is a half, a tail or a combination. A quick sketch of the bell curve with the mean and cut-off points will help you avoid errors.
如果该值恰好是 μ ± σ、μ ± 2σ 或 μ ± 3σ，直接应用经验法则。仔细读题，判断所求区域是一半、一个尾部还是组合区域。快速画出一条带均值和分割点的钟形曲线将有助于避免错误。
9. Comparing Distributions | 比较分布
You may be asked to compare two or more normal distributions with different means and standard deviations. A distribution with a larger mean is shifted to the right on the same scale, indicating higher average values. A distribution with a smaller standard deviation is narrower and taller, signalling more consistent data.
考试可能会要求你比较两个或多个均值与标准差不同的正态分布。均值较大的分布在相同尺度上向右平移，表示平均值更高。标准差较小的分布更窄、更尖，表明数据更一致。
When describing distributions, always mention both the centre (mean) and the spread (standard deviation). For example: 'Class A has a higher mean score, but Class B shows a smaller standard deviation, meaning their scores are more clustered around the average.'
描述分布时，务必同时提及中心（均值）和离散程度（标准差）。例如：“A 班的平均分更高，但 B 班的标准差更小，说明他们的分数更集中在平均分附近。”
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Mistake 1: Confusing standard deviation with the range or variance. Remember, the empirical rule works strictly for normal distributions – do not apply it to skewed or unknown distributions.
错误 1：将标准差与极差或方差混淆。记住，经验法则严格适用于正态分布——不要把它应用到偏态或未知的分布上。
Mistake 2: Forgetting symmetry when calculating tail probabilities. For instance, 95% within ±2σ leaves 5% in total outside, not 5% in each tail. Each tail gets 2.5%.
错误 2：计算尾部概率时忘记对称性。例如，95% 在 ±2σ 内，意味着外部共 5%，而不是每个尾部 5%。每个尾部为 2.5%。
Mistake 3: Misapplying percentages when the given value is not an exact multiple of σ. At GCSE level, questions are designed so that the boundary values are always exact multiples; if not, you may need to use the fact that the total area is 1 and work with given information.
错误 3：当给定值不是 σ 的整数倍时错误应用百分比。在 GCSE 层次，题目设计通常会使边界值恰为精确倍数；若非如此，可能需要利用总面积为 1 并结合已知信息求解。
Exam Tip: ade the region of interest and label percentages. This visual approach significantly reduces careless mistakes.
考试技巧：始终画一个草图。标出均值，然后添加 μ ± σ、μ ± 2σ 和 μ ± 3σ 处的线条。把关心区域涂上阴影并标注百分比。这种视觉方法可以显著减少粗心错误。
11. Summary and Revision Checklist | 总结与复习清单
Make sure you can confidently explain and apply the following:
请确保你能自信地解释并应用以下内容：
- The normal curve is bell-shaped, symmetric and defined by μ and σ.
- 中文：正态曲线呈钟形、对称，由 μ 和 σ 确定。
- Mean = median = mode; total area under the curve = 1.
- 中文：平均数 = 中位数 = 众数；曲线下总面积为 1。
- 68% within μ ± σ, 95% within μ ± 2σ, 99.7% within μ ± 3σ.
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