📚 IB Mathematics: Core Concepts of the Normal Distribution | IB数学：正态分布核心概念解析
The normal distribution is one of the most important probability distributions in IB Mathematics, appearing in both Analysis & Approaches (AA) and Applications & Interpretation (AI). It models continuous data that clusters around a central mean, such as heights, test scores, and measurement errors. Mastering its core concepts is essential for exam success.
正态分布是IB数学中最重要的概率分布之一，在分析与方法（AA）和应用与解释（AI）中都会出现。它用于建模围绕中心均值聚集的连续数据，例如身高、考试成绩和测量误差。掌握其核心概念对于考试成功至关重要。
1. Definition and Parameters | 定义与参数
A continuous random variable X follows a normal distribution with mean μ and standard deviation σ. We write X ~ N(μ, σ²). The probability density function (pdf) is symmetric about the mean and has a characteristic bell shape.
连续随机变量X服从均值为μ、标准差为σ的正态分布，记作X ~ N(μ, σ²)。其概率密度函数关于均值对称，具有典型的钟形曲线。
f(x) = 1 / (σ√2π) × e^(−(x−μ)² / (2σ²)), −∞ < x < ∞
The parameter μ determines the center of the distribution, while σ determines the spread. A larger σ produces a wider, flatter curve; a smaller σ produces a narrower, taller curve.
参数μ决定分布的中心位置，σ决定分布的离散程度。σ越大，曲线越宽越扁；σ越小，曲线越窄越高。
2. Properties of the Normal Curve | 正态曲线的性质
The curve is symmetric about x = μ, so the mean, median, and mode are all equal.
曲线关于x = μ对称，因此均值、中位数和众数三者相等。
The total area under the curve is exactly 1, representing total probability.
曲线下的总面积为1，代表总概率。
The curve approaches the horizontal axis asymptotically as x → ±∞, but never touches it.
当x → ±∞时，曲线渐近地接近横轴，但永远不会触及横轴。
Points of inflection occur at x = μ − σ and x = μ + σ.
拐点出现在x = μ − σ和x = μ + σ处。
3. The Empirical Rule (68-95-99.7 Rule) | 经验法则（68-95-99.7法则）
For any normal distribution, approximately 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
对于任何正态分布，约68%的数据落在距均值一个标准差范围内，95%落在两个标准差范围内，99.7%落在三个标准差范围内。
| 区间 | 概率 |
| μ ± 1σ | ≈ 68% |
| μ ± 2σ | ≈ 95% |
| μ ± 3σ | ≈ 99.7% |
This rule provides a quick way to estimate probabilities without a calculator and helps identify unusual values in a dataset.
该法则提供了一种无需计算器即可快速估算概率的方法，并有助于识别数据中的异常值。
4. Standard Normal Distribution and Z-Scores | 标准正态分布与Z分数
The standard normal distribution has mean 0 and standard deviation 1, denoted Z ~ N(0, 1). Any normal variable X can be transformed into Z using the z-score formula:
标准正态分布的均值为0，标准差为1，记作Z ~ N(0, 1)。任何正态变量X都可以通过Z分数公式转换为Z：
Z = (X − μ) / σ
The z-score measures how many standard deviations a value is above or below the mean. A positive z-score indicates a value above the mean, while a negative z-score indicates a value below the mean.
Z分数衡量一个值高于或低于均值多少个标准差。正的Z分数表示该值高于均值，负的Z分数表示该值低于均值。
5. Calculating Probabilities Using the Normal Distribution | 使用正态分布计算概率
n use the standard normal table or a GDC (graphical display calculator). For example, if X ~ N(100, 15²), then P(X < 120) = P(Z < (120 − 100)/15) = P(Z < 1.333).
要计算正态分布中的P(X < a)，首先将a转换为Z分数，然后使用标准正态分布表或图形计算器（GDC）。例如，若X ~ N(100, 15²)，则P(X < 120) = P(Z < (120 − 100)/15) = P(Z < 1.333)。
tation includes P(X ≤ a), P(X > a) = 1 − P(X ≤ a), and P(a < X < b) = P(X < b) − P(X < a).
常见的概率记号包括P(X ≤ a)、P(X > a) = 1 − P(X ≤ a)以及P(a < X < b) = P(X < b) − P(X < a)。
6. Inverse Normal Calculations | 逆正态计算
orresponding to a given cumulative probability. For example, find the 90th percentile of X ~ N(100, 15²), i.e. find x such that P(X ≤ x) = 0.90.
逆正态问题要求给定累积概率下对应的x值。例如，求X ~ N(100, 15²)的第90百分位数，即求x使得P(X ≤ x) = 0.90。
onding to 0.90 (z ≈ 1.2816) and solving x = μ + zσ, we get x ≈ 100 + 1.2816 × 15 ≈ 119.22.
使用GDC上的逆正态函数，或查找0.90对应的Z分数（z ≈ 1.2816）并解x = μ + zσ，可得x ≈ 100 + 1.2816 × 15 ≈ 119.22。
7. Sampling Distributions and the Central Limit Theorem | 抽样分布与中心极限定理
sample of size n from a population with mean μ and standard deviation σ, the sample mean X̄ ≥ 30), regardless of the population distribution. This is the Central Limit Theorem (CLT).
当从均值为μ、标准差为σ的总体中抽取容量为n的随机样本时，如果n足够大（通常n ≥ 30），样本均值X̄近似服从正态分布，无论总体分布是什么。这就是中心极限定理（CLT）。
The sampling distribution of X̄ has mean μ and standard deviation σ/√n. This allows us to make inferences about population means from sample data.
样本均值X̄的抽样分布均值为μ，标准差为σ/√n。这使我们能够从样本数据对总体均值进行推断。
8. Normal Approximation to the Binomial Distribution | 二项分布的正态近似
When n is large and p is not too close to 0 or 1, the binomial distribution B(n, p) can be approximated by a normal distribution with mean np and variance np(1 − p). A common rule is that both np and n(1 − p) should be at least 5 (or 10, depending on the textbook).
当n很大且p不太接近0或1时，二项分布B(n, p)可以用均值为np、方差为np(1 − p)的正态分布来近似。一个常用规则是np和n(1 − p)都应至少为5（或10，取决于教材）。
ed. For example, P(X ≤ k) is approximated by P(X ≤ k + 0.5) using the normal distribution.
由于二项分布是离散的而正态分布是连续的，需要进行连续性修正。例如，P(X ≤ k)用正态分布近似为P(X ≤ k + 0.5)。
9. Common Exam Pitfalls | 常见考试陷阱
Forgetting to use the continuity correction when approximating a binomial distribution with a normal distribution.
在使用正态分布近似二项分布时忘记进行连续性修正。
Confusing σ with σ². Always check whether the problem gives the standard deviation or the variance.
混淆σ与σ²。始终检查题目给出的是标准差还是方差。
Using P(X < a) instead of P(X ≤ a) when the question uses "at least" or "at most". Note that for continuous distributions, P(X = a) = 0, so P(X < a) = P(X ≤ a).
当问题使用"至少"或"至多"时，错误使用P(X < a)代替P(X ≤ a)。注意对于连续分布，P(X = a) = 0，因此P(X < a) = P(X ≤ a)。
Misreading the z-table: always confirm whether the table gives P(Z < z) or P(0 < Z < z).
误读Z表：始终确认表格给出的是P(Z < z)还是P(0 < Z < z)。
10. Practice Question | 练习题
The heights of students at a school are normally distributed with mean 170 cm and standard deviation 8 cm. (a) Find the probability that a randomly chosen student is taller than 180 cm. (b) Find the height that separates the top 10% of students from the rest.
某学校学生的身高服从均值170 cm、标准差8 cm的正态分布。(a) 求随机选择一名学生身高超过180 cm的概率。(b) 求将最高的10%学生与其余学生分开的身高临界值。
Solution: 6. (b) The z-score for the 90th percentile is 1.2816, so h = 170 + 1.2816 × 8 ≈ 180.25 cm.
解答：(a) z = (180 − 170)/8 = 1.25。P(Z > 1.25) = 1 − 0.8944 = 0.1056。(b) 第90百分位数的Z分数为1.2816，因此h = 170 + 1.2816 × 8 ≈ 180.25 cm。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图