📚 IB WJEC Mathematics: Normal Distribution Key Points | IB WJEC 数学：正态分布 考点精讲
The normal distribution is one of the most important continuous probability distributions in statistics, underpinning many concepts in IB and WJEC Mathematics. A solid understanding of its properties, the use of standard normal tables, and the ability to apply it in real-world contexts are essential for exam success. This article covers all the key points you need to master, from basic definitions to the normal approximation of the binomial distribution.
正态分布是统计学中最重要的连续概率分布之一，也是 IB 和 WJEC 数学的基石。扎实掌握其性质、标准正态分布表的使用以及在现实情境中的应用能力，是考试成功的关键。本文涵盖了从基本定义到二项分布的正态近似等所有必考要点。
1. Definition and Key Properties | 定义与核心性质
The normal distribution is a continuous probability distribution defined by its mean μ and standard deviation σ. Its probability density function forms the classic 'bell curve', symmetric about the mean. The total area under the curve equals 1, representing the total probability.
正态分布是由均值 μ 和标准差 σ 定义的连续概率分布。其概率密度函数形成经典的“钟形曲线”，关于均值对称。曲线下的总面积为 1，代表总概率。
Key properties include: the mean, median and mode are all equal; the curve is asymptotically approaching the horizontal axis but never touching it; about 68% of data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ (the empirical rule). For a random variable X ~ N(μ, σ²), the distribution is fully specified.
核心性质包括：均值、中位数和众数均相等；曲线渐近地趋近于横轴但永不接触；大约 68% 的数据落在均值 ±1σ 范围内，95% 落在 ±2σ 内，99.7% 落在 ±3σ 内（经验法则）。对于随机变量 X ~ N(μ, σ²)，分布由这两个参数完全确定。
2. Standard Normal Distribution and Z-Scores | 标准正态分布与 Z 分数
The standard normal distribution has a mean of 0 and a variance of 1, denoted Z ~ viding by the standard deviation: Z = (X − μ) / σ. This process is called standardisation.
标准正态分布的均值为 0，方差为 1，记作 Z ~ N(0, 1)。任何一个正态变量 X 都可以通过减去均值后除以标准差转化为标准正态变量 Z：Z = (X − μ) / σ。这一过程称为标准化。
The Z-score represents how many standard deviations a data point is from the mean. It allows us to compare values from different normal distributions and to use a single set of probability tables. Standardisation is the key step in virtually all normal distribution problems in WJEC exams.
Z 分数表示一个数据点距离均值有多少个标准差。它使我们能够比较来自不同正态分布的值，并使用同一套概率表。标准化几乎是所有 WJEC 考题中解决正态分布问题的关键步骤。
3. Reading the Normal Distribution Table | 查阅正态分布表
e Φ(z) = P(Z ≤ z) for positive z-values. For example, Φ(1.00) = 0.8413, meaning that 84.13% of values lie below 1 standard deviation above the mean.
大多数考试局（包括 WJEC）提供标准正态分布的累积分布表。这些表格给出了正面 z 值的 Φ(z) = P(Z ≤ z)。例如 Φ(1.00) = 0.8413，意味着 84.13% 的数据落在均值上方 1 个标准差以下。
a). The table typically provides values to 4 decimal places, and you must be able to look up z-scores quickly and accurately, as well as perform inverse look-ups.
要查找负 z 值的概率，应利用对称性：P(Z ≤ −a) = 1 − P(Z ≤ a)。同时，P(a < Z < b) = P(Z < b) − P(Z < a)。正态分布表通常精确到四位小数，你必须能够快速准确地查找 z 分数，并能进行反向查找。
4. Finding Probabilities for Given X-values | 由 X 值求概率
Given X ~ a) = 1 − P(X ≤ a) because the normal distribution is continuous (equality has zero probability).
已知 X ~ N(μ, σ²)，要求 P(X > a) 或 P(a < X < b)，首先对边界值进行标准化得到 z 值，然后使用标准正态分布表。对于 P(X > a)，注意 P(X > a) = 1 − P(X ≤ a)，因为正态分布是连续的（等于某点的概率为零）。
mly chosen apple weighs more than 165 g. You would standardise: z = (165 − 150)/12 = 1.25, then find P(Z > 1.25) = 1 − 0.8944 = 0.1056. Always sketch the bell curve to check the area you are looking for.
典型考题：苹果的质量服从均值为 150 g、标准差为 12 g 的正态分布。求随机选取的一个苹果质量大于 165 g 的概率。标准化：z = (165 − 150)/12 = 1.25，然后求得 P(Z > 1.25) = 1 − 0.8944 = 0.1056。始终画出钟形曲线来确认你要求的是哪部分面积。
5. Inverse Normal: Finding X for a Given Probability | 正态分布反查：由概率求 X
o find the Z-score that satisfies the cumulative probability. Then convert back to the original scale using X = μ + σ·Z.
当给定概率时，你可能需要求出对应的 X 值。首先，使用标准正态分布反查表（或倒查表格）找到满足累积概率的 Z 分数。然后通过 X = μ + σ·Z 转换回原始刻度。
For example, if the top 10% ) = 0.90. From tables, z ≈ 1.2816. Then X = 58 + 12×1.2816 ≈ 73.4. So a score of 74 (or 73.4 depending on context) is the cut-off.
例如，如果考试成绩前 10% 将被授予 A 等，成绩服从 N(58, 144)，求获得 A 等所需的最低分数。首先，找到 z 使得 P(Z > z) = 0.10，即 P(Z ≤ z) = 0.90。查表得 z ≈ 1.2816。然后 X = 58 + 12×1.2816 ≈ 73.4。所以 74 分（或 73.4 分，视上下文而定）是分界线。
6. Normal Approximation to the Binomial Distribution | 二项分布的正态近似
For a binomial random variable X ~ th mean μ = np and variance σ² = np(1−p). This is particularly useful when n is large and p is not too close to 0 or 1.
对于大 n 值的二项随机变量 X ~ B(n, p)，精确概率计算可能非常繁琐。当条件满足时，我们可以用均值为 μ = np、方差为 σ² = np(1−p) 的正态分布进行近似。这在 n 很大且 p 不过于接近 0 或 1 时特别有用。
The WJEC specification requires you to state the approximating distribution as X ~ N(np, np(1−p)). The approximation improves as n increases and as p gets closer to 0.5. Always check the conditions before applying the approximation.
WJEC 考纲要求你将近似分布表述为 X ~ N(np, np(1−p))。n 越大、p 越接近 0.5，近似效果越好。在应用近似前务必检查条件。
7. Conditions for the Normal Approximation | 正态近似的条件
ficiently symmetric and the approximation is valid. For example, if n=50 and p=0.3, np=15 and n(1−p)=35, both >5, so approximation is appropriate.
通常使用的经验法则是 np 和 n(1−p) 均需大于 5（有些教材使用 10）。这能保证分布足够对称，近似有效。例如，若 n=50, p=0.3，np=15, n(1−p)=35，均大于 5，因此可以近似。
th large n the binomial distribution is skewed, and a Poisson approximation might be more suitable. However, WJEC mostly focuses on cases where the normal approximation is valid.
如果 p 非常小，比如 p=0.01，即使 n 很大，二项分布也是偏态的，此时泊松近似可能更合适。但 WJEC 主要关注正态近似有效的情况。
8. Continuity Correction | 连续性校正
nomial probabilities. For P(X ≥ a), use P(X > a − 0.5); for P(X ≤ a), use P(X < a + 0.5); and for P(X = a), use P(a − 0.5 < X < a + 0.5).
由于二项分布是离散的，而正态分布是连续的，在近似二项概率时我们需要进行连续性校正。对于 P(X ≥ a)，使用 P(X > a − 0.5)；对于 P(X ≤ a)，使用 P(X < a + 0.5)；对于 P(X = a)，使用 P(a − 0.5 < X < a + 0.5)。
This adjustment accounts for the fact that the discrete probability bar at a is replaced by the area under the normal curve from a−0.5 to a+0.5. Omitting the continuity correction is a common mistake that leads to inaccurate answers and loss of marks in WJEC exams.
这一调整考虑到离散概率中在 a 处的条形被替换为从 a−0.5 到 a+0.5 的正态曲线下方面积。遗漏连续性校正是 WJEC 考试中常见的错误，会导致答案不准确乃至失分。
9. Solving Problems Involving Sample Means | 样本均值相关问题
The Central Limit Theorem states that for a random sample of size n from any population with mean μ and variance σ², the sample mean X̄ follows approximately a normal distribution with mean μ and variance σ²/n, provided n is large enough. This is crucial when dealing with sums or averages of observations.
中心极限定理指出，对于来自任意总体（均值为 μ，方差为 σ²）的容量为 n 的随机样本，只要 n 足够大，样本均值 X̄ 近似服从均值为 μ、方差为 σ²/n 的正态分布。这在处理观测值的总和或均值时至关重要。
t of 16 bars is N(50, 4/16) = N(50, 0.25). The standard deviation of the sample mean is σ/√n = 2/√16 Always adjust the variance correctly when dealing with the mean of multiple observations.
例如，如果一块巧克力的重量服从 N(50, 4)，那么 16 块巧克力重量的均值服从 N(50, 4/16) = N(50, 0.25)。样本均值的标准差为 σ/√n = 2/√16 = 0.5。处理多次观测的均值时，务必正确调整方差。
10. Exam Strategies and Common Errors | 应试策略与常见错误
Always draw a diagram of the normal curve to visualise the required probability or region. State the distribution clearly (e.g., X ~ N(100, 225)), and write out the standardisation step fully. Keep track of whether you are working with a greater-than or less-than probability, and adjust accordingly.
始终画出正态曲线图以直观显示所求的概率或区域。清楚写出分布（例如 X ~ N(100, 225)），并完整写出标准化步骤。注意区分是大于还是小于的概率，并相应调整。
Common pitfalls include: forgetting to take the square root of variance to get σ; using the wrong tail of the distribution; misreading the table; failing to apply continuity correction in binomial approximations; and confusing P(Z < a) with P(Z > a). Practising a wide range of past paper questions will help you internalise these techniques.
常见陷阱包括：忘记对方差开平方以得到 σ；使用了分布的错误的尾部；查表读错；在二项近似中未应用连续性校正；混淆 P(Z < a) 和 P(Z > a)。练习大量的历年真题将帮助你内化这些技巧。
11. Real-World Applications and Modelling | 实际应用与建模
Normal distributions model many natural phenomena—heights, weights, measurement errors, and test scores. In WJEC problems, you might be asked to determine warranty periods, quality control thresholds, or the probability of a component lasting beyond a certain time. Recognising when a normal model is appropriate is a key skill.
正态分布可用于建模许多自然现象——身高、体重、测量误差和考试成绩。在 WJEC 考题中，你可能被要求确定保修期限、质量控制阈值，或某个部件超过一定使用寿命的概率。能够识别何时使用正态模型是一项关键能力。
When using a sample to estimate population parameters, the normal distribution provides the theoretical foundation for confidence intervals and hypothesis tests, which are further developed in A-Level and IB Mathematics. A thorough understanding of the normal distribution now will pay dividends in more advanced topics.
在使用样本估计总体参数时，正态分布为置信区间和假设检验提供了理论基础，这些内容在 A-Level 和 IB 数学中会进一步展开。现在透彻理解正态分布将为更高级的主题打下坚实基础。
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