📚 Exercise 24G: Mastering Normal Distribution | 攻克正态分布
The normal distribution is the cornerstone of statistical analysis in IB Mathematics, and Exercise 24G is designed to solidify your ability to model continuous data, compute probabilities, and apply inverse normal techniques. This article breaks down every key concept, from z‑scores to GDC shortcuts, so you can approach exam questions with confidence and precision.
正态分布是IB数学中统计分析的基石，练习24G旨在巩固你建立连续数据模型、计算概率以及应用逆正态技巧的能力。本文将从z分数到GDC快捷操作逐一拆解核心概念，让你自信且精准地应对考试题目。
1. The Normal Curve and Its Parameters | 正态曲线及其参数
The normal distribution, often written as X ~ N(μ, σ²), is a symmetric bell‑shaped curve fully defined by its mean μ and variance σ². The total area under the curve equals 1, representing the total probability of all outcomes. The curve's spread is determined by the standard deviation σ, with approximately 68% of data within 1σ of the mean and 95% within 2σ.
正态分布通常记作 X ~ N(μ, σ²)，它是一条对称的钟形曲线，完全由其均值 μ 和方差 σ² 决定。曲线下的总面积等于1，代表所有结果的总概率。曲线的离散程度由标准差 σ 决定，约68%的数据落在均值±1σ范围内，95%落在±2σ范围内。
- μ shifts the curve left or right along the x‑axis; σ controls its narrowness or flatness.
- μ 决定曲线沿 x 轴左右平移；σ 控制其高矮胖瘦。
- The normal probability density function is symmetric around x = μ.
- 正态概率密度函数关于 x = μ 对称。
| Probability | Range |
| ~68.3% | μ − σ ≤ X ≤ μ + σ |
| ~95.4% | μ − 2σ ≤ X ≤ μ + 2σ |
| ~99.7% | μ − 3σ ≤ X ≤ μ + 3σ |
2. Z‑scores and Standardisation | Z分数与标准化
A z‑score tells you how many standard deviations a raw score x lies from the mean. Standardising with z = (x − μ) ÷ σ converts any normal distribution into the standard normal Z ~ N(0, 1). This allows us to use a single table or GDC function regardless of the original parameters.
z分数告诉你原始分数 x 距离均值多少个标准差。通过标准化公式 z = (x − μ) ÷ σ，任何正态分布都能转化为标准正态分布 Z ~ N(0, 1)。无论初始参数如何，我们都可以使用同一张表或GDC函数进行计算。
z = (x − μ) ÷ σ
If x = 85, μ = 75, σ = 6, then z = (85 − 75) ÷ 6 = 10 ÷ 6 ≈ 1.6667. This positive z‑score indicates x is 1.67 standard deviations above the mean.
若 x = 85，μ = 75，σ = 6，则 z = (85 − 75) ÷ 6 = 10 ÷ 6 ≈ 1.6667。这个正值 z 分数表明 x 比均值高出1.67个标准差。
Negative z‑ ow the mean, and the symmetry property P(Z < −a) = P(Z > a) often simplifies calculations.
负z分数对应低于均值的数值，利用对称性 P(Z < −a) = P(Z > a) 可以简化计算。
3. Using the GDC for Probability Calculations | 使用GDC计算概率
IB exam success depends heavily on fluent use of your graphics display calculator. To find P(a < X < b) for X ~ N(μ, σ²), use the normalcdf function. The typical syntax is normalcdf(lower bound, upper bound, μ, σ). For a left‑tail probability up to a value a, set the lower bound to a very small number like −10⁹⁹.
IB考试的成功很大程度上取决于能否熟练使用图形计算器。对于 X ~ N(μ, σ²)，求 P(a < X < b) 时使用 normalcdf 函数。典型语法为 normalcdf(下界, 上界, μ, σ)。对于左尾概率，可将下界设为一个极小值，例如 −10⁹⁹。
| Calculation | GDC Input |
| P(X < 120) with μ=100, σ=15 | normalcdf(−1E99, 120, 100, 15) |
| P(90 < X < 110) same distribution | normalcdf(90, 110, 100, 15) |
Always double‑check you are using σ, not σ², in the calculator arguments. Many errors come from accidentally entering the variance.
请务必检查计算器参数中输入的是 σ 而非 σ²。许多错误都源于误输了方差。
4. Inverse Normal: Finding x Given a Probability | 逆正态：已知概率求x
The inverse normal answers questions like 'What mark cuts off the top 10% of a distribution?' On the GDC, use invNorm(area to the left, μ, σ). If a problem gives the right‑tail area, subtract from 1 first. The output is the x‑value that satisfies P(X ≤ x) = specified left‑tail probability.
逆正态用于解答类似“整场考试前10%的学生分数需要达到多少？”的问题。在GDC上，使用 invNorm(左侧面积, μ, σ)。如果题目给的是右侧尾部面积，先将其从1中减去。输出的是满足 P(X ≤ x) = 指定左尾概率的 x 值。
invNorm( p, μ, σ ) → x
For example, to find the IQ score that separates the lowest 25% of a population with μ=100, σ=15, the left‑tail area is 0.25, so invNorm(0.25, 100, 15) ≈ 89.9.
例如，要找出将 IQ 分布的 25% 最低分分隔开来的分数，已知 μ=100，σ=15，左尾面积为 0.25，那么 invNorm(0.25, 100, 15) ≈ 89.9。
Remember that invNorm always requires a left‑tail cumulative probability. Draw a quick sketch to avoid mixing up left and right tails.
请记住 invNorm 始终需要输入左侧累积概率。快速画个草图，避免混淆左右尾部。
5. Finding Unknown μ or σ | 求解未知均值或标准差
determine μ and σ. This is typically solved by setting up two simultaneous equations in z‑ ite (45 − μ)/σ = invNorm(0.2) and (65 − μ)/σ = invNorm(0.8). Then solve the linear system.
有些IB题目会给出两个概率，要求你求解 μ 和 σ。这类问题通常需要通过建立两个 z 分数形式的方程来联立求解。例如，若 P(X < 45) = 0.2 且 P(X < 65) = 0.8，可列出 (45 − μ)/σ = invNorm(0.2) 和 (65 − μ)/σ = invNorm(0.8)，然后解这个线性方程组。
z₁ = (x₁ − μ)/σ, z₂ = (x₂ − μ)/σ
Since z₁ and z₂ are known from the inverse normal of the given areas, subtract the equations to isolate σ first, then back‑substitute for μ.
由于 z₁ 和 z₂ 可以从给定面积的逆正态求得，先将两个方程相减解出 σ，再回代求出 μ。
- Always store intermediate z‑values with full precision to avoid rounding errors.
- 始终以完整精度存储中间 z 值，避免舍入误差。
- Check your solutions by plugging μ and σ back into the normalcdf function.
- 将求出的 μ 和 σ 重新代入 normalcdf 函数进行验证。
6. Applications in Real‑World Contexts | 现实情境中的应用
Normal distributions model heights, weights, test scores, and measurement errors. In Exercise 24G, you will often meet problems about battery lifetimes, exam marks, or fill volumes. The key is translating textual conditions into mathematical bounds for normalcdf or invNorm.
正态分布可用于模拟身高、体重、考试分数和测量误差等。在练习 24G 中，你经常会遇到电池寿命、考试成绩或填充量的问题。关键在于将文字条件转化为 normalcdf 或 invNorm 的数学边界。
For a problem stating 'the middle 95% of lifetimes fall between 1200 and 1600 hours' fway between the bounds. Here μ = 1400, and you can find σ by solving P(L > 1600) = 0.025.
对于“中间 95% 的寿命落在 1200 至 1600 小时”这类题目，要能推断出 P(1200 < L < 1600) = 0.95，并且凭借对称性可知 μ 恰好位于两边界的中点。此例中 μ = 1400，然后通过解 P(L > 1600) = 0.025 求出 σ。
Always define the variable clearly: "Let X be the fill volume in ml, X ~ N(μ, σ²)" before writing any equation.
在写任何等式之前，务必明确定义变量：“设 X 为填充量（单位 ml），X ~ N(μ, σ²)”。
7. Normal Approximation to the Binomial | 二项分布的正态近似
arge n such that np ≥ 5 and n(1−p) ≥ 5, we can approximate it with a normal distribution Y ~ xercise 24G may include continuity correction, requiring boundaries to be shifted by ±0.5.
当二项随机变量 B(n, p) 的 n 足够大，满足 np ≥ 5 且 n(1−p) ≥ 5 时，我们可以用正态分布 Y ~ N(np, np(1−p)) 来近似。练习 24G 中可能包括连续性校正，需要将边界值平移 ±0.5。
P(a ≤ X ≤ b) ≈ P(a − 0.5 < Y < b + 0.5)
For example, to approximate P(X ≥ 210) for X ~ th continuity correction, P(X ≥ 210) → P(Y > 209.5) using normalcdf(209.5, 1E99, 200, 10).
例如，用正态分布近似计算 X ~ B(400, 0.5) 的 P(X ≥ 210) 时，先求 μ = 400×0.5 = 200，σ² = 400×0.5×0.5 = 100，σ = 10。再用连续性校正，P(X ≥ 210) → P(Y > 209.5)，即 normalcdf(209.5, 1E99, 200, 10)。
Without the continuity correction, the approximation is less accurate, and IB marks will often be dedicated to this step.
如果没有连续性校正，近似结果精度较差，而 IB 的分数往往就考察这关键一步。
8. Common Mistakes and How to Avoid Them | 常见错误与规避方法
Even strong students lose marks by confusing σ and σ², misreading 'greater than' questions as left‑tail, or forgetting to sketch. Another typical error is using invNorm to find a right‑tail percentile directly without subtracting from 1.
即使是优秀学生，也会因混淆 σ 和 σ²、将“大于”问题误读为左尾，或忘记画图而丢分。另一个典型错误是直接用 invNorm 求右侧百分位数而未先从 1 中减去对应面积。
- Draw a quick normal curve and shade the required area for every question.
- 每道题都快速画一条正态曲线，并标出所求面积。
- Write down the exact GDC command you plan to use before executing it.
- 在动手操作计算器之前，先写下你打算使用的 GDC 命令。
- If 'between' probabilities are asked, make sure lower < upper, otherwise the calculator returns an error.
- 如果求“介于”概率，务必确保下界小于上界，否则计算器会报错。
Rounding: give z‑scores to at least 3 decimal places and probabilities to 4 decimal places, unless the question states otherwise.
关于舍入：除非题目另有要求，z 分数至少保留三位小数，概率保留四位小数。
9. IB Exam‑Style Worked Examples | IB考试真题示例
Let's work through a typical Exercise 24G question: " = 20 g. Find the probability that a randomly chosen apple weighs between 140 g and 170 g." Using normalcdf(140, 170, 150, 20) yields approximately 0.5328.
我们来看一道典型的练习 24G 题目：“某果园苹果重量服从正态分布，μ = 150 g，σ = 20 g。随机选取一个苹果，求其重量介于 140 g 到 170 g 之间的概率。”使用 normalcdf(140, 170, 150, 20) 得到约 0.5328。
A second part might ask: "What weight is exceeded by only 8% of the apples?" Here we need the right‑tail x such that P(X > x) = 0.08. Left‑tail area = 0.92, so invNorm(0.92, 150, 20) ≈ 178.1 g.
第二部分可能问：“超过多大重量的苹果仅占总量的 8%？”此时需要求右侧 x 使得 P(X > x) = 0.08。左侧面积为 0.92，因此 invNorm(0.92, 150, 20) ≈ 178.1 g。
In exams, show the substitution into the z‑formula even if you use the GDC directly, as method marks may be awarded.
考试中，即使直接使用 GDC，也最好展示代入 z 分数的步骤，这样可能会获得方法分。
10. The Importance of the Standard Normal Table | 标准正态分布表的重要性
g the standard normal table deepens your conceptual grasp. The table gives cumulative left‑tail probabilities Φ(z) for Z ~ N(0, 1). For a negative z, use Φ(−z) = 1 − Φ(z).
虽然 GDC 是主要工具，但理解标准正态分布表能加深你的概念掌握。该表给出 Z ~ N(0, 1) 的累积左尾概率 Φ(z)。对于负 z 值，可利用 Φ(−z) = 1 − Φ(z)。
Interpolating between table values is sometimes required if the z‑score is not tabulated exactly. IB may test the ability to read a table and combine it with symmetry arguments.
如果 z 分数在表中没有精确对应，有时需要进行内插。IB 可能会考查学生查表并结合对称性论证的能力。
Φ(1.23) = 0.8907, Φ(−1.23) = 1 − 0.8907 = 0.1093
Practice matching your GDC results with table values in early exercises to build trust in your technology.
在初期练习中，尝试将 GDC 结果与查表值进行对比，以建立对计算工具的信任。
11. Linking Normal Distribution to Hypothesis Testing | 正态分布与假设检验的衔接
Exercise 24G is not an isolated topic; it prepares you for inferential statistics. In hypothesis testing, you will use normal distributions to compute p‑values and critical values. The invNorm function directly gives critical values for a given significance level α.
练习 24G 并非孤立主题，它为你学习推断统计做准备。在假设检验中，你将使用正态分布计算 p 值和临界值。invNorm 函数可直接给出给定显著性水平 α 下的临界值。
For a two‑tailed test at α = 0.05, the critical z‑values are ±1.96. This come from invNorm(0.025) and invNorm(0.975) for Z ~ N(0,1). The symmetry of the normal curve is fundamental to this logic.
对于 α = 0.05 的双尾检验，临界 z 值为 ±1.96。这源于对 Z ~ N(0,1) 使用 invNorm(0.025) 和 invNorm(0.975)。正态曲线的对称性是这一逻辑的基础。
Building a strong normal distribution foundation in Exercise 24G directly boosts your confidence in later chapters on t‑tests and confidence intervals.
在练习 24G 中打下坚实的正态分布基础，将直接提升你在后续关于 t 检验和置信区间章节中的信心。
12. Summary and Practice Strategy | 总结与练习策略
Mastering Exercise 24G requires moving beyond button‑pressing. Focus on interpreting the problem, drawing the curve, identifying the correct GDC function, and checking whether your answer makes sense in context. Re‑visit mixed exercises that combine finding μ/σ with inverse normal and binomial approximation.
攻克练习 24G 需要你超越机械按键的层面。要着重理解题目、画出曲线、确定正确的 GDC 函数，并检查答案在上下文中是否合理。反复练习那些需要同时用到求解 μ/σ、逆正态以及二项近似的综合题。
- Work through at least 10 varied problems from your textbook's mixed review.
- 从教材的综合复习中至少挑选 10 道不同类型的题目进行练习。
- Time yourself: a normal distribution exam question should take 3‑5 minutes.
- 计时练习：一道正态分布考题通常应在 3−5 分钟内完成。
- Keep a log of errors and re‑solve them after 48 hours.
- 建立错题日志，48 小时后重新作答这些题目。
With consistent practice and a clear method, you will transform Exercise 24G from a challenge into one of the most reliable sources of marks on your IB paper.
通过持续练习和清晰的方法，你将让练习 24G 从一道难关变成 IB 试卷上最稳妥的得分点之一。
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