📚 Exercise 26C: Mastering Normal Distribution Problems | 练习26C：精通正态分布问题
In IB Mathematics, the normal distribution is one of the most widely applied continuous probability models. Exercise 26C challenges you to move beyond simple recognition and tackle a variety of problems involving standardisation, probability calculations, inverse look-ups, and real-world contexts. This article breaks down the core techniques needed to solve every type of question you will meet, from reading Z-tables to using your calculator efficiently and avoiding common pitfalls.
在IB数学中，正态分布是应用最广泛的连续概率模型之一。练习26C要求你超越简单的辨识，解决各类涉及标准化、概率计算、逆向查表以及现实情境的问题。本文分解了解决你将遇到的每一类题目所需的核心技巧，从查Z表到高效使用计算器，再到避免常见错误。
1. Understanding the Normal Curve | 理解正态曲线
The normal distribution is a bell-shaped, symmetric curve defined by two parameters: the mean (μ) and the standard deviation (σ). The total area under the curve equals 1, representing the entire probability space. The peak is at μ, and the spread is controlled by σ; a larger σ gives a flatter, wider curve.
正态分布呈钟形且对称，由两个参数定义：均值（μ）和标准差（σ）。曲线下总面积为1，代表整个概率空间。峰值位于μ处，分布宽度由σ控制；σ越大，曲线越扁平、越宽。
For any normal distribution, about 68% of data fall within 1σ of μ, 95% within 2σ, and 99.7% within 3σ. This empirical rule helps you estimate probabilities quickly before doing precise calculations.
对于任何正态分布，大约68%的数据落在μ的±1σ内，95%落在±2σ内，99.7%落在±3σ内。这种经验法则可以帮助你在精确计算之前快速估计概率。
2. The Standard Normal Distribution & Z-Scores | 标准正态分布与Z分数
To solve Exercise 26C problems, you must first convert any normal variable X ~ N(μ, σ²) into a standard normal variable Z ~ N(0, 1²). The Z-score measures how many standard deviations a value is away from the mean.
要解决练习26C中的问题，你必须首先将任意正态变量X ~ N(μ, σ²) 转换成标准正态变量 Z ~ N(0, 1²)。Z分数衡量某个值距离均值多少个标准差。
z = (x − μ) / σ
can use the standard normal table or a calculator to find probabilities. For example, if X ~ , then z = (130 − 100)/15 = 2. This tells you 130 is 2 standard deviations above the mean.
得到z后，你可以使用标准正态分布表或计算器来求概率。例如，若X ~ N(100, 15²) 且 x = 130，则 z = (130 − 100)/15 = 2。这表明130比均值高2个标准差。
3. Reading the Z-Table for 'Less Than' Probabilities | 使用Z表计算“小于”概率
Most Z-tables give P(Z < z), the cumulative probability up to z. Here is a mini excerpt from a typical positive Z-table:
大部分Z表给出P(Z < z)，即直到z的累积概率。以下是一个典型正向Z表的局部摘录：
| 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 |
, use symmetry: P(Z < −z) = 1 − P(Z < z). For example, P(Z < −1.00) = 1 − 0.8413 = 0.1587.
当z = 1.00时，我们读出P(Z < 1.00) = 0.8413。如果z为负值，利用对称性：P(Z < −z) = 1 − P(Z < z)。例如，P(Z < −1.00) = 1 − 0.8413 = 0.1587。
4. Calculating 'Greater Than' Probabilities | 计算“大于”概率
x) = 1 − P(X < x). After standardising, find P(Z < z) from the table and subtract from 1.
练习26C经常要求P(X > x)。因为总面积为1，我们使用补集规则：P(X > x) = 1 − P(X < x)。标准化后，从表中查出P(Z < z)，并用1减去该值。
Example: X ~ (X > 65). z = (65 − 50)/10 = 1.5. P(Z < 1.5) = 0.9332, so P(X > 65) = 1 − 0.9332 = 0.0668.
例如：X ~ N(50, 10²)，求P(X > 65)。z = (65 − 50)/10 = 1.5。P(Z < 1.5) = 0.9332，因此P(X > 65) = 1 − 0.9332 = 0.0668。
Always sketch the curve, shade the area, and label the x or z value. This visual check prevents sign errors.
务必画出曲线、涂阴影区域并标出x或z值。这种可视化检查能防止符号错误。
5. Probabilities Between Two Values | 两个值之间的概率
For P(a < X < b), compute P(X < b) − P(X < a). Standardise both a and b to get z₁ and z₂, then find the difference of their cumulative probabilities.
对于P(a < X < b)，计算P(X < b) − P(X < a)。将a和b分别标准化得到z₁和z₂，然后求它们累积概率的差值。
Suppose X ~ 0.8413, P(Z < −1.0) = 0.1587. The probability is 0.8413 − 0.1587 = 0.6826, matching the 68% rule.
假设X ~ N(120, 20²)，求P(100 < X < 140)。z₁ = (100−120)/20 = −1.0，z₂ = (140−120)/20 = 1.0。P(Z < 1.0) = 0.8413，P(Z < −1.0) = 0.1587。概率为0.8413 − 0.1587 = 0.6826，与68%法则吻合。
6. Inverse Normal: Finding X from a Given Probability | 逆正态：由概率求X
When a question gives a probability and asks 'find the value k such that P(X < k) = p' corresponds to p in the Z-table (or using a calculator). Then transform back: k = μ + zσ.
当题目给出概率并问“求k使得P(X < k) = p”时，你正在执行逆正态计算。首先在Z表中（或用计算器）找到对应p的z值，然后反变换：k = μ + zσ。
Example: X ~ table, the closest z is about 1.28 (since 0.8997 ≈ 0.90). Then k = 75 + 1.28 × 8 = 85.24.
例如：X ~ N(75, 8²)，求第90百分位数。我们需要z使得P(Z < z) = 0.90。查表得最近z约为1.28（因为0.8997 ≈ 0.90），则 k = 75 + 1.28 × 8 = 85.24。
areful with the direction: if the tail probability is given, convert it to a cumulative probability first.
有些问题要求求包含一定百分比的对称均值区间，例如P(μ − d < X < μ + d) = 0.95，此时z = ±1.96。注意方向：若给的是尾部概率，需先转化为累积概率。
7. Solving Real-World IB Problems | 解决IB现实问题
IB exam questions often embed the normal distribution in practical scenarios: battery life, test scores, heights, and weights. The key is to identify μ, σ, and the required probability or boundary value from the wording.
IB考试题目常将正态分布嵌入实际情景：电池寿命、考试成绩、身高、体重等。关键在于从文字中识别出μ、σ以及所需的概率或边界值。
For instance: ' 0 hours and standard deviation 40 hours. What proportion of bulbs last more than 870 hours?' 0, σ = 40, find P(X > 870). Standardise, compute, and convert the decimal to a percentage.
例如：“某灯泡寿命服从正态分布，均值为800小时，标准差为40小时。有多少比例的灯泡寿命超过870小时？”这里μ = 800，σ = 40，要求P(X > 870)。标准化、计算，再将小数转为百分比。
Always present your answer in the required form: as a decimal, percentage, or number of items. Read the question carefully to see if a 'more than', 'less than', or 'between' phrasing is used.
答案一定要按照题目要求的形式给出：小数、百分比或物品数量。仔细读题，看清使用的是“多于”、“少于”还是“介于”的措辞。
8. Using the Calculator Effectively | 有效使用计算器
IB allows graphing calculators, and you should use the built-in normal functions to save time and avoid table errors. The common functions are:
IB允许使用图形计算器，你应当利用内置的正态分布函数来节省时间、避免查表错误。常用函数有：
normCdf(low, high, μ, σ) – gives the probability between two values (use −1E99 for −∞ and 1E99 for ∞).
normCdf(下界, 上界, μ, σ) – 给出两个值之间的概率（用−1E99代表−∞，1E99代表∞）。
invNorm(area, μ, σ) – finds the x-value that gives a left-tail area equal to 'area'.
invNorm(面积, μ, σ) – 求出使左尾面积等于给定值的x值。
Example: To find P(X > 120) for N(100, 15²), use normCdf(120, 1E99, 100, 15). For the 95th percentile, invNorm(0.95, 100, 15). Always double-check whether your calculator expects the area to the left; some models have inverse settings for centre or right tail – know your tool.
例如：对于N(100, 15²)求P(X > 120)，使用normCdf(120, 1E99, 100, 15)。求第95百分位数用invNorm(0.95, 100, 15)。务必确认你的计算器是否要求左尾面积；某些型号有中心或右尾逆运算设置——要熟悉自己的工具。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
These pitfalls appear regularly in Exercise 26C and real exams:
以下陷阱在练习26C和真实考试中经常出现：
ded for standardisation, not the variance. If the problem gives variance = 25, then σ = 5.
混淆σ与σ²：标准化需要的是标准差（σ），而非方差。如果题目给的是方差=25，那么σ=5。
Forgetting to standardise: You cannot use the standard normal table directly on X values without converting to z first.
忘记标准化：不能直接将X值代入标准正态表，必须先转换为z。
Sign errors with negative z: P(Z < −a) is a small left-tail area; sometimes students misread the table and take the complement incorrectly. Always draw the curve.
负z的符号错误：P(Z < −a)是左尾小面积；有些学生误读表并错误地取补集。一定要画曲线。
Using the wrong tail in inverse problems: 'Find the value that 10% exceed' means P(X > k) = 0.10, so P(X < k) = 0.90 – use 0.90 in invNorm.
逆问题中选错尾部：“求10%超过的值”意味着P(X > k) = 0.10，因此 P(X < k) = 0.90——在invNorm中使用0.90。
10. Exercise 26C Sample Problems & Step-by-Step Solutions | 练习26C样题与分步解答
Let's work through two problems typical of Exercise 26C.
我们来解练习26C中两道典型题目。
Problem 1: The masses of apples from an orchard are normally distributed with mean 150 g and standard deviation 20 g. Find the probability that a randomly selected apple has a mass between 140 g and 170 g.
问题1：某果园苹果重量服从均值为150 g、标准差为20 g的正态分布。求随机选取一个苹果重量介于140 g到170 g之间的概率。
quired probability = 0.8413 − 0.3085 = 0.5328. Or use normCdf(140, 170, 150, 20) = 0.5328.
解答：将两个边界标准化：z₁ = (140 − 150)/20 = −0.5，z₂ = (170 − 150)/20 = 1.0。P(Z < 1.0) = 0.8413，P(Z < −0.5) = 0.3085，所求概率 = 0.8413 − 0.3085 = 0.5328。或使用normCdf(140, 170, 150, 20) = 0.5328。
Problem 2: The scores on an IB exam are normally distributed with mean 62 and standard deviation 9. The top 8% of students receive a grade 7. What is the minimum score needed for a grade 7?
问题2：某IB考试分数服从均值为62、标准差为9的正态分布。前8%的学生获得7分。获得7分所需的最低分数是多少？
Solution: Let k be the cut-off. P(X > k) = 0.08 ⇒ core is about 75 (rounding appropriately). Using calculator: invNorm(0.92, 62, 9) ≈ 74.65.
解答：设k为分数线，P(X > k) = 0.08 ⇒ P(X < k) = 0.92。求z使P(Z < z) = 0.92。查表或invNorm(0.92,0,1)，得z ≈ 1.405。则 k = μ + zσ = 62 + 1.405 × 9 ≈ 74.6。最低分数约为75（适当四舍五入）。使用计算器：invNorm(0.92, 62, 9) ≈ 74.65。
11. Interpreting Your Results in Context | 在语境中解释结果
IB examiners expect you to state conclusions clearly. After computing a probability of 0.023, say: 'There is a 2.3% chance that a randomly chosen ...' or, for inverse problems, 'The minimum passing score is 47.' Always link back to the scenario and use appropriate units.
IB考官希望你清楚地陈述结论。算出概率0.023后，应该说：“随机选择一个……有2.3%的可能性”，对逆问题则是“最低及格分数为47。”始终与情境相联系，并使用恰当的单位。
When working with sampling distributions later, the same standardising and inverse logic applies, but with adjusted standard error. Mastering the single distribution problems in Exercise 26C provides a solid foundation for those extended topics.
当之后学习抽样分布时，同样的标准化和逆运算逻辑仍然适用，但需调整标准误差。精通练习26C中的单一分布问题将为这些扩展主题打下坚实基础。
12. Summary of Key Steps for Exercise 26C | 练习26C关键步骤总结
To tackle any normal distribution problem systematically:
要系统地解决任何正态分布问题：
Write down X ~ N(μ, σ²) and extract μ, σ from the text.
写下X ~ N(μ, σ²)并从文本中提取μ和σ。
Draw a labeled curve and shade the area of interest.
画出带标注的曲线，并给感兴趣的区域涂上阴影。
Decide whether you need a probability (forward) or a boundary (inverse).
确定你需要的是概率（正向）还是边界（逆向）。
Standardise or use calculator functions correctly, checking tail directions.
正确标准化或使用计算器函数，检查尾部方向。
State your final answer in context, with sensible rounding.
在语境中给出最终答案，合理四舍五入。
With consistent practice, Exercise 26C will sharpen your normal distribution skills to IB mastery level.
通过持续的练习，练习26C将把你的正态分布技能提升到IB精通水平。
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