📚 Exercise 22I: Normal Distribution Calculations | 练习22I：正态分布计算
In IB Mathematics, the normal distribution is one of the most important continuous probability models. Exercise 22I typically focuses on using a graphing calculator (GDC) to compute probabilities, find boundaries, and solve real-world problems involving normally distributed variables. Mastery of the normal cumulative distribution function (normalcdf) and inverse normal function (invNorm) is essential for success in the exam.
在 IB 数学中，正态分布是最重要的连续概率模型之一。练习 22I 通常侧重于使用图形计算器 (GDC) 来计算正态分布的概率、寻找边界值，并求解涉及正态变量的实际问题。掌握正态累积分布函数 (normalcdf) 和逆正态函数 (invNorm) 对考试成功至关重要。
1. Understanding the Normal Distribution | 理解正态分布
The normal distribution, often called the Gaussian distribution, is a symmetric, bell-shaped curve defined by two parameters: the mean μ and the standard deviation σ. The total area under the curve equals 1, representing 100% of the possible outcomes. Data values cluster around the mean, and the spread is measured by σ.
正态分布通常称为高斯分布，是一条对称的钟形曲线，由两个参数定义：均值 μ 和标准差 σ。曲线下方的总面积等于 1，代表所有可能结果的 100%。数据集中在均值附近，分散程度由 σ 衡量。
A random variable X that follows a normal distribution is written as X ~ N(μ, σ²). The curve's peak lies at x = μ, and its points of inflection occur at μ ± σ. Roughly 68% of observations fall within one standard deviation of the mean, 95% within two, and 99.7% within three.
服从正态分布的随机变量 X 记作 X ~ N(μ, σ²)。曲线的峰值位于 x = μ，拐点出现在 μ ± σ 处。大约 68% 的观测值落在均值的一个标准差范围内，95% 落在两个标准差内，99.7% 落在三个标准差内。
2. Parameters: Mean μ and Standard Deviation σ | 参数：均值μ与标准差σ
The mean μ determines the location of the centre of the distribution. Changing μ shifts the curve left or right without altering its shape. The standard deviation σ controls the spread: a smaller σ produces a taller, narrower bell; a larger σ gives a flatter, wider curve. Both parameters are always positive real numbers, with σ > 0.
均值 μ 决定了分布中心的位置。改变 μ 会使曲线向左或向右平移而不改变形状。标准差 σ 控制分散程度：较小的 σ 产生更高、更窄的钟形；较大的 σ 产生更扁平、更宽的曲线。两个参数都是正实数，且 σ > 0。
In exam questions, μ and σ are often given directly, or they must be deduced from the context. Always identify them before performing any calculations.
在考试题目中，μ 和 σ 通常会直接给出，或需要从上下文中推断。在执行任何计算之前，务必先确定这两个参数。
3. The Standard Normal Distribution Z ~ N(0,1) | 标准正态分布Z ~ N(0,1)
ial case is the standard normal distribution, where μ = 0 and σ = 1. Any normal variable X ~ dardised using the z-score formula: z = (x − μ) / σ. This converts any normal curve into Z ~ ssible to use standard normal tables, though the GDC has rendered tables largely obsolete.
一种特殊情况是标准正态分布，其中 μ = 0 且 σ = 1。任何正态变量 X ~ N(μ, σ²) 都可以通过 z 分数公式进行标准化：z = (x − μ) / σ。这会将任何正态曲线转化为 Z ~ N(0,1)，使得可以使用标准正态表，尽管 GDC 已使表格基本过时。
Understanding z-scores is still crucial: a z-score of +1.5 means the value is 1.5 standard deviations above the mean; a z-score of −2 means it is 2 standard deviations below the mean.
理解 z 分数仍然至关重要：z 分数为 +1.5 表示该值高于均值 1.5 个标准差；z 分数为 −2 表示低于均值 2 个标准差。
4. Calculating Probabilities with GDC (Normal CDF) | 使用GDC计算概率（正态CDF）
The normal cumulative distribution function on a GDC computes P(a < X < b) for X ~ N(μ, σ). The syntax is typically normalcdf(lower bound, upper bound, μ, σ). For an upper tail, use a very large number such as 10⁹⁹ for the upper bound; for a lower tail, use −10⁹⁹ as the lower bound.
GDC 上的正态累积分布函数用于计算 X ~ N(μ, σ) 的 P(a < X < b)。语法通常为 normalcdf(下界, 上界, μ, σ)。对于上尾概率，上界使用一个非常大的数，例如 10⁹⁹；对于下尾概率，下界使用 −10⁹⁹。
- Lower tail: P(X < k) = normalcdf(−10⁹⁹, k, μ, σ)
- Upper tail: P(X > k) = normalcdf(k, 10⁹⁹, μ, σ)
- Interval: P(a < X < b) = normalcdf(a, b, μ, σ)
- 下尾：P(X < k) = normalcdf(−10⁹⁹, k, μ, σ)
- 上尾：P(X > k) = normalcdf(k, 10⁹⁹, μ, σ)
- 区间：P(a < X < b) = normalcdf(a, b, μ, σ)
Always sketch a quick bell curve and shade the required area. This visual check helps avoid confusing the tails and selecting incorrect bounds.
始终快速画出钟形曲线并涂色所需区域。这种视觉检查有助于避免混淆尾部并选择错误的边界。
5. Inverse Normal Calculations (InvNorm) | 逆正态计算（InvNorm）
area to the left, μ, σ). This returns the value x such that P(X ≤ x) = the specified area.
当给定概率 (面积) 需要求对应的 x 值时，使用逆正态函数。GDC 的语法为 invNorm(左侧面积, μ, σ)。这将返回 x 值，使得 P(X ≤ x) = 指定面积。
If the question gives a right-tail area p, the left-tail area is 1 − p, so invNorm(1 − p, μ, σ) should be entered. For a central area required to find symmetric boundaries, you must work with half of the remaining area in each tail.
如果题目给出右尾面积 p，则左侧面积为 1 − p，因此应输入 invNorm(1 − p, μ, σ)。若需要求出对称边界对应的中心面积，则必须处理每个尾部剩余面积的一半。
6. Working with Standardised Scores (z-scores) | 标准化分数（z值）
ains relevant for comparing values from different normal distributions. A GDC can compute z-scores directly, but you may also be asked to find probabilities using Z ~ N(0,1) with normalcdf lower, upper, 0, 1.
z 分数公式 z = (x − μ) / σ 在比较来自不同正态分布的值时依然有效。GDC 可直接计算 z 值，但也可能要求使用 Z ~ N(0,1) 结合 normalcdf(下界, 上界, 0, 1) 来求概率。
·σ, which is useful when you know the z-score and need the raw score. This relationship underpins the invNorm function.
重新整理公式可得 x = μ + z·σ，当已知 z 分数而需要原始分数时非常有用。这一关系是 invNorm 函数的基础。
7. Finding Boundaries for Given Probabilities | 给定概率求边界值
Typical Exercise 22I tasks include 'Find the value of a such that a given percentage of observations lie above or below it.' For instance, to find the top 10% cut-off, use invNorm(0.90, μ, σ) because 90% of data are below that point. For a symmetric central 95% interval, the boundaries are invNorm(0.025, μ, σ) and invNorm(0.975, μ, σ).
典型的练习 22I 任务包括 '求 a 的值，使得给定百分比的观测值高于或低于它'。例如，要找出前 10% 的分界值，使用 invNorm(0.90, μ, σ)，因为 90% 的数据位于该点以下。对于对称的 95% 中心区间，边界为 invNorm(0.025, μ, σ) 和 invNorm(0.975, μ, σ)。
Always confirm whether the given area is the left-tail, right-tail, or both tails combined. Misinterpreting this is a common source of error.
务必确认给定面积是左尾、右尾还是一条双尾组合。误解这一点是常见的错误来源。
8. Solving Real-World Problems: Example 1 | 实际应用题：示例1
= 20 g. Find the probability that a randomly chosen apple weighs between 140 g and 170 g.
果园苹果的质量服从正态分布，μ = 150 g，σ = 20 g。求随机选取一个苹果的质量介于 140 g 到 170 g 之间的概率。
malcdf(140, 170, 150, 20). The calculator returns approximately 0.5328, so there is a 53.3% chance.
使用 GDC：P(140 < X < 170) = normalcdf(140, 170, 150, 20)。计算器返回约 0.5328，因此概率为 53.3%。
Interpretation: Just over half of the apples will fall within this mass range. Extracting such probabilities is the first step towards making predictions about quality control or pricing.
解释：略多于一半的苹果会落在这个质量范围内。计算此类概率是对质量控制或定价做出预测的第一步。
9. Solving Real-World Problems: Example 2 | 实际应用题：示例2
Using the same apple distribution, the orchard wants to label the heaviest 5% of apples as 'premium'. Above what mass should an apple be classified as premium?
利用相同的苹果分布，果园希望将最重的 5% 苹果标记为 "特级"。苹果的质量应超过多少克才能被归类为特级？
The top 5% corresponds to a right-tail area of 0.05, so the left-tail area is 0.95. Input invNorm(0.95, 150, 20). The GDC returns approximately 182.9 g. Therefore, an apple must weigh above 182.9 g to be premium.
最重的 5% 对应右尾面积 0.05，因此左尾面积为 0.95。输入 invNorm(0.95, 150, 20)。GDC 返回约 182.9 g。因此，苹果质量必须超过 182.9 g 才能成为特级。
Real-world interpretability depends on meaningful rounding. In this context, reporting 183 g is more practical, but for mathematical accuracy retain at least three significant figures during intermediate steps.
实际的解释性取决于有意义的四舍五入。在此上下文中，报告 183 g 更实际，但在中间步骤中，为了数学准确性保留至少三位有效数字。
10. Common Mistakes and Tips | 常见错误与提示
| Common Mistake | 常见错误 | Tip |
| Using σ instead of σ² in syntax | 在语法中使用 σ 而不是 σ² | The normal distribution is N(μ, σ²), but normalcdf and invNorm require σ |
| Confusing left-tail and right-tail areas | 混淆左尾和右尾面积 | Draw a diagram and label the area you know |
| Forgetting to adjust bounds for infinite tails | 忘记为无穷尾部调整边界 | Use ±10⁹⁹ or ±1E99 to represent ±∞ |
| Using invalid μ and σ values (σ ≤ 0) | 使用无效的 μ 和 σ 值 (σ ≤ 0) | Always check that σ > 0 |
| Rounding too early in inverse problems | 在逆问题中过早四舍五入 | Keep full calculator precision until the final answer |
Practice with a variety of contexts, such as test scores, manufacturing tolerances, and biological measurements. Familiarity with these models will make the GDC steps second nature during the exam.
在各种背景下练习，如考试成绩、加工公差和生物测量。熟悉这些模型将使 GDC 步骤在考试时成为第二天性。
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