📚 Exercise 14H: The Normal Distribution | 练习 14H：正态分布
In the IB Mathematics: Applications and Interpretation or Analysis and Approaches courses, Exercise 14H typically deals with the normal distribution – a cornerstone of continuous probability. This article unpacks every essential skill, from reading bell curves to using a GDC for inverse calculations, helping you tackle the exercise with confidence.
在 IB 数学课程中，练习 14H 通常涉及正态分布——连续概率的基石。本文将解析每一项必备技能，从解读钟形曲线到使用图形计算器进行逆运算，帮助你自信地完成练习。
1. Understanding the Normal Distribution | 理解正态分布
The normal distribution is a continuous probability distribution that describes data clustering around a mean. Its graph is the famous bell-shaped curve, symmetric about the mean μ and with spread determined by the standard deviation σ. The total area under the curve equals 1, representing a 100% probability.
正态分布是一种连续概率分布，描述数据围绕均值聚集的情况。其图像就是著名的钟形曲线，关于均值 μ 对称，其分散程度由标准差 σ 决定。曲线下的总面积等于 1，代表 100% 的概率。
A normal random variable X is written as X ~ N(μ, σ²), where the second parameter is the variance σ², not the standard deviation. It is essential to distinguish between σ and σ² when inputting values into your GDC.
正态随机变量 X 记作 X ~ N(μ, σ²)，其中第二个参数是方差 σ²，而非标准差。在向图形计算器输入数值时，区分 σ 与 σ² 至关重要。
), it never touches the horizontal axis (asymptotic behaviour), and its shape depends entirely on μ and σ. A smaller σ gives a taller, narrower curve; a larger σ gives a flatter, wider curve.
正态曲线的性质包括：单峰（x = μ 处有一个峰）、永不触及横轴（渐近行为），并且其形状完全取决于 μ 和 σ。σ 越小，曲线越高越窄；σ 越大，曲线越扁越宽。
2. The Standard Normal Distribution and Z‑scores | 标准正态分布与 Z 值
distribution is the specific normal distribution with mean 0 and variance 1, denoted by Z ~ (X − μ) / σ. This transformation shifts the mean to 0 and scales the spread to a standard deviation of 1.
标准正态分布是均值为 0、方差为 1 的特殊正态分布，记作 Z ~ N(0, 1)。任何正态变量 X 都可以通过公式 Z = (X − μ) / σ 进行标准化。这种变换将均值移至 0，并将离散度缩放至标准差为 1。
Standardising is useful when solving problems that involve unknowns or when comparing different normal populations. In IB exams, however, you will usually use your GDC to work directly with the original distribution. Still, the concept of the z‑score is fundamental to inverse normal problems and proof-based questions.
标准化在涉及未知参数或比较不同正态总体时非常有用。不过在 IB 考试中，你通常会直接使用图形计算器处理原始分布。尽管如此，z 值的概念对于逆正态问题和证明类题目仍是基础。
3. Using the GDC to Find Probabilities | 使用图形计算器计算概率
For a given normal distribution X ~ N(μ, σ²), the probability that X falls in an interval [a, b] is calculated on a GDC using the normal cumulative density function (normalcdf). The syntax is typically normalcdf(lower bound, upper bound, μ, σ). Note that σ is the standard deviation, not the variance.
对于给定的正态分布 X ~ N(μ, σ²)，X 落在区间 [a, b] 内的概率可以在图形计算器上使用正态累积密度函数 (normalcdf) 计算。语法通常为 normalcdf(下限, 上限, μ, σ)。请注意，这里 σ 是标准差，而不是方差。
Example: Let X ~ N(100, 15²). To find P(X < 115), use normalcdf(−1E99, 115, 100, 15). The lower bound −1E99 represents negative infinity. The calculator returns approximately 0.841. This means there is an 84.1% chance of X being less than 115.
示例：设 X ~ N(100, 15²)。要计算 P(X < 115)，可使用 normalcdf(−1E99, 115, 100, 15)。下限 −1E99 表示负无穷大。计算器返回的结果约为 0.841。这意味着 X 小于 115 的概率为 84.1%。
To find a right‑tail probability such as P(X > 85), use normalcdf(85, 1E99, 100, 15). The result will be around 0.841 as well, due to symmetry. For the probability between two values, say P(90 < X < 110), input normalcdf(90, 110, 100, 15). The answer is roughly 0.497.
若要计算右尾概率，例如 P(X > 85)，可使用 normalcdf(85, 1E99, 100, 15)。由于对称性，结果也约为 0.841。若要求两个值之间的概率，比如 P(90 < X < 110)，则可输入 normalcdf(90, 110, 100, 15)。结果大约为 0.497。
4. Inverse Normal Calculations | 逆正态计算
The inverse normal function answers the question: given a left‑ vNorm(p, μ, σ). The function always returns the value corresponding to the given area on the left side of the distribution.
逆正态函数回答以下问题：给定左尾概率 p，求使 P(X < x) = p 成立的 x 值。在图形计算器上，使用 invNorm(p, μ, σ)。该函数始终返回与分布左侧给定面积对应的数值。
Example: If X ~ 0, 8²), find the 90th percentile. InvNorm(0.90, 60, 8) gives approximately 70.25. Thus, 90% of the data lies below 70.25. If you need a right‑tail value such that P(X > k) = 0.05, first convert to a left‑tail probability: P(X < k) = 1 − 0.05 = 0.95, then use invNorm(0.95, μ, σ).
示例：若 X ~ N(60, 8²)，求第 90 百分位数。invNorm(0.90, 60, 8) 得出的结果约为 70.25。因此，90% 的数据低于 70.25。如果你需要使 P(X > k) = 0.05 的右尾值，首先要转换为左尾概率：P(X < k) = 1 − 0.05 = 0.95，然后使用 invNorm(0.95, μ, σ)。
Always draw a quick sketch and shade the area corresponding to the probability before using invNorm. This prevents the common mistake of using the wrong tail probability.
在使用 invNorm 之前，始终画一个简图并标出与概率对应的面积。这可以避免因使用错误尾概率而导致的常见错误。
5. Finding the Mean or Standard Deviation | 求均值或标准差
IB exam questions frequently ask you to determine μ or σ when given two probability statements. The strategy is to standardise both conditions to the z‑score and solve the resulting simultaneous equations.
IB 考试题目经常要求你在已知两个概率语句的情况下确定 μ 或 σ。策略是将两个条件标准化为 z 分数，然后求解由此产生的联立方程。
Suppose X ~ N(μ, σ²). If P(X < 25) = 0.10 and P(X > 35) = 0.05, first find the z‑ = 1.6449. Solve these two linear equations simultaneously to find μ ≈ 28.90 and σ ≈ 3.71.
假设 X ~ N(μ, σ²)。若 P(X < 25) = 0.10 且 P(X > 35) = 0.05，首先使用 invNorm 在标准正态上求出 z 分数：面积 0.10，z₁ ≈ −1.2816；面积 0.95（因 P(X < 35) = 1 − 0.05 = 0.95），z₂ ≈ 1.6449。然后建立方程 (25 − μ) / σ = −1.2816 和 (35 − μ) / σ = 1.6449。同时求解这两个线性方程可得 μ ≈ 28.90 和 σ ≈ 3.71。
Always remember that you are solving for σ, not σ². Some students mistakenly square the standard deviation too early. Solve for σ first, then square it if the variance is required.
始终记住，你正在求解的是 σ，而不是 σ²。有些学生过早地将标准差平方。应先解出 σ，如果需要方差再将其平方。
6. The Normal Curve as a Probability Density Function | 作为概率密度函数的正态曲线
Mathematically, the normal probability density function (PDF) is given by f(x) = 1/(σ√2π is rarely used directly in calculations, but understanding it deepens your appreciation of the curve's shape.
从数学上讲，正态概率密度函数 (PDF) 由 f(x) = 1/(σ√2π) · e^(-(x−μ)²/(2σ²)) 给出。虽然在计算中很少直接使用此公式，但理解它可以加深你对曲线形状的认识。
The PDF confirms that the peak height is f(μ) = 1/(σ√2π), meaning a smaller σ produces a higher peak. The squared term in the exponent gives the curve its symmetry, and the two parameters μ and σ completely determine the distribution.
该 PDF 证实了峰值高度为 f(μ) = 1/(σ√2π)，这意味着 σ 越小，峰值越高。指数中的平方项赋予了曲线对称性，而 μ 和 σ 这两个参数完全决定了分布。
< a), which is why we can use inclusive or exclusive bounds interchangeably in continuous distributions.
由于正态分布是连续的，X 取任何单一精确值的概率均为零，即 P(X = a) = 0。因此，P(X ≤ a) = P(X < a)，这就是为什么在连续分布中可以互换使用包含或不包含边界的原因。
7. Symmetry and Key Percentiles | 对称性与关键百分位数
The empirical rule, also known as the 68–95–99.7 rule, states that for any normal distribution roughly 68% of values lie within μ ± σ, 95% within μ ± 2σ, and 99.7% This rule is a powerful tool for quick estimates and checking the plausibility of answers.
经验法则，也称为 68–95–99.7 规则，指出对于任何正态分布，大约 68% 的值位于 μ ± σ 范围内，95% 位于 μ ± 2σ 范围内，99.7% 位于 μ ± 3σ 范围内。该规则是快速估算和检查答案合理性的有力工具。
| Interval | 区间 | Approximate Probability | 近似概率 |
|---|---|
| μ ± σ | 0.6827 (≈ 68%) |
| μ ± 2σ | 0.9545 (≈ 95%) |
| μ ± 3σ | 0.9973 (≈ 99.7%) |
These interval probabilities are symmetric: about 16% of data falls above μ + σ and 16% below μ − σ. Similarly, 2.5% lies above μ + 2σ and 2.5% below μ − 2σ. Understanding these tail probabilities is frequently tested in quality‑control and prediction problems.
这些区间概率是对称的：约 16% 的数据高于 μ + σ，16% 低于 μ − σ。同样，2.5% 高于 μ + 2σ，2.5% 低于 μ − 2σ。理解这些尾部概率经常在质量控制与预测问题中考查。
8. Working with Combined Events | 组合事件的处理
You can extend normal probability to 'or' and 'and' scenarios when data values are independent. For instance, if the heights of two randomly selected students are independent normal variables, the probability that both exceed a certain value is the product of individual probabilities.
当数据值相互独立时，可以将正态概率扩展到“或”和“且”的场景。例如，如果随机选择的两名学生身高是独立的正态变量，则两人都超过某一值的概率就是各个概率的乘积。
Be cautious: 'or' SL) + P(X > USL). If the bounds are symmetric around the mean, this equals 2 × P(X < LSL).
注意：互斥区间的“或”概率可以直接相加。例如，在质量检验中，一件产品重量过轻 (X < LSL) 或过重 (X > USL) 的概率为 P(X < LSL) + P(X > USL)。如果上下限关于均值对称，这等于 2 × P(X < LSL)。
Another typical problem: given that 95% 1.96σ. Such reasoning shows that any normal interval probability can be expressed using z‑multipliers.
另一个典型题目：已知 95% 的包裹重量在 μ − d 和 μ + d 之间，求 d 关于 σ 的表达式。利用 invNorm(0.975, 0, 1) ≈ 1.96，可得 d = 1.96σ。这样的推理表明，任何正态区间概率都可以用 z 乘数表示。
9. Real‑World Applications | 实际应用
ften scaled to μ = 100, σ = 15), birth weights, examination marks, and measurement errors.
当变量受到许多微小的独立效应影响时，正态分布便自然出现。经典示例包括智商分数（通常标定为 μ = 100，σ = 15）、出生体重、考试成绩以及测量误差。
When applying the normal model, you should always check the context. For a dataset to be modelled by a normal distribution, it should be roughly symmetric, unimodal, and continuous. Outliers might indicate that the normal assumption is violated.
在应用正态模型时，应始终检查背景。要用正态分布模拟数据集，该数据集应大致对称、单峰且连续。异常值可能表明违反了正态性假设。
In IB exam questions, you often see wording like 'assume the masses of apples are normally distributed' or 'the lengths of metal rods follow a normal distribution'. Then you are asked to find the probability of selecting an item within a specific specification range, or to calculate the expected number of items in a sample that meet a criterion.
在 IB 考题中，常会看到诸如“假设苹果的重量服从正态分布”或“金属棒的长度服从正态分布”的表述。然后要求你求出所选产品落在特定规格范围内的概率，或计算样本中满足某条件的产品预期数量。
10. Common Pitfalls and Exam Tips | 常见错误与考试技巧
One frequent mistake is confusing variance with standard deviation when entering parameters into the GDC. Always check whether the question gives σ² or σ, and ensure you input σ. Another common error is using invNorm with a right‑tail probability directly — you must subtract it from 1 first.
一个常见错误是向图形计算器输入参数时混淆方差与标准差。务必检查题目给出的是 σ² 还是 σ，并确保你输入的是 σ。另一个常见错误是直接用右尾概率使用 invNorm——你必须先将其从 1 中减去。
For infinite bounds, use a very large number like 1E99 or –1E99. Avoid using the mean as a bound unless the question explicitly asks for a one‑sided probability around the mean. Always sketch the normal curve, label μ, σ, and shade the area of interest. This visual aid dramatically reduces careless mistakes.
对于无穷界限，请使用一个很大的数，如 1E99 或 –1E99。除非题目明确要求围绕均值的单边概率，否则不要使用均值作为边界。始终绘制正态曲线草图，标出 μ、σ 并涂黑感兴趣的区域。这种可视化辅助手段可大幅减少粗心错误。
When solving simultaneous equations for μ and σ, keep the z‑values to sufficient decimal places (at least four) to maintain accuracy. Rounding too early can lead to a final answer outside the accepted range. And remember: in IB questions, unless stated otherwise, give probabilities to three significant figures.
在求解关于 μ 和 σ 的联立方程时，要将 z 值保留足够的小数位（至少四位）以保持准确性。过早舍入可能导致最终答案超出可接受范围。同时记住：在 IB 题目中，除非另有说明，概率应给出三位有效数字。
Finally, practise reading the question carefully — p, then multiply by 500. The normal distribution gives p; the expected frequency is n × p.
最后，练习仔细审题——如果题目问在 500 件物品中，预计有多少件重量超过 5 kg，首先计算概率 p，然后乘以 500。正态分布给出 p；期望频数为 n × p。
Mastering Exercise 14H is not just about button-pressing on the GDC; it is about understanding the model deeply enough to interpret results and communicate conclusions accurately, a skill the IB values highly.
掌握练习 14H 不仅仅是会按图形计算器的按钮；而是要深入理解模型，以便准确解读结果并传达结论，这正是 IB 高度重视的一项技能。
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