📚 The Normal Distribution | 正态分布
The normal distribution is one of the most fundamental continuous probability distributions in statistics. It models a wide range of natural phenomena and measurement errors, making it a cornerstone of the IB Mathematics course. Often called the Gaussian distribution, its bell-shaped curve is defined by two parameters: the mean (μ) and the standard deviation (σ). Understanding how to calculate probabilities, find unknown boundaries, and apply the distribution to real-world contexts is essential for success in the exam.
正态分布是统计学中最基本的连续概率分布之一。它模拟了众多自然现象和测量误差，是IB数学课程的核心内容。这条钟形曲线通常被称为高斯分布，由均值 (μ) 和标准差 (σ) 两个参数决定。掌握如何计算概率、求解未知边界并将其应用于实际问题，是考试成功的关键。
1. Introducing the Normal Distribution | 正态分布简介
A continuous random variable X is said to follow a normal distribution with mean μ and variance σ², written as X ~ N(μ, σ²). The graph of the probability density function is symmetric about the mean, unimodal, and bell-shaped. The total area under the curve equals 1. The distribution is fully characterised by μ and σ, which determine the centre and spread, respectively.
若连续随机变量X服从均值为μ、方差为σ²的正态分布，记作 X ~ N(μ, σ²)。其概率密度函数图像关于均值对称，单峰且呈钟形。曲线下的总面积等于1。该分布完全由μ和σ决定，它们分别确定了中心位置和分散程度。
Many real-world variables, such as heights, IQ scores, and manufacturing tolerances, follow an approximately normal pattern. The normal distribution also arises naturally from the Central Limit Theorem when independent random variables are summed.
许多现实变量，如身高、智商分数和制造公差，均近似服从正态分布。当独立的随机变量相加时，根据中心极限定理，正态分布也会自然出现。
2. Probability Density Function (PDF) | 概率密度函数
The probability density function of a normal distribution N(μ, σ²) is given by:
f(x) = (1/(σ √2π)) exp( − (x − μ)² / (2σ²) ) , for all real x.
Unlike discrete distributions, the probability that X equals a specific value is zero. Probabilities are found by calculating the area under the curve between two bounds. The curve's shape is controlled by σ: smaller σ gives a taller, narrower peak; larger σ gives a flatter, wider spread.
正态分布 N(μ, σ²) 的概率密度函数为：
f(x) = (1/(σ √2π)) exp( − (x − μ)² / (2σ²) )，x 为任意实数。
与离散分布不同，X取某一特定值的概率为零。概率需通过计算曲线下两个边界之间的面积获得。曲线的形状由σ控制：σ越小，峰越高越窄；σ越大，形状越扁平分散。
+ a) for any a > 0. The maximum value of the density is attained at x = μ and equals 1/(σ √2π).
由于对称性，对于任意 a > 0 均有 f(μ − a) = f(μ + a)。密度函数的最大值在 x = μ 处取得，等于 1/(σ √2π)。
3. Parameters μ and σ | 参数 μ 和 σ
coincide. The curve has inflection points at μ ± σ, where the curvature changes direction.
均值μ是位置参数，改变μ会使整条曲线左右平移而不改变形状。标准差σ是尺度参数，它衡量数据的离散程度。对于任何正态分布，均值、中位数和众数均重合。曲线在μ ± σ处存在拐点，此处弯曲方向发生改变。
The variance σ² is never negative, and a larger σ² indicates greater dispersion. In exam questions, always check whether you are given the variance or the standard deviation to avoid misusing the parameter in GDC commands.
方差σ²非负，σ²越大表示数据越分散。在考试题目中，注意区分给出的是方差还是标准差，避免在GDC指令中误用参数。
4. Standard Normal Distribution | 标准正态分布
The standard normal distribution is a special case with μ = 0 and σ = 1, denoted as Z ~ N(0, 1). Any normal variable X can be standardised using the transformation:
Z = (X − μ) / σ.
s of Φ(z) can be found using a GDC or the standard normal table. Standardisation allows us to compare values from different normal populations and compute probabilities for any normal distribution.
标准正态分布是μ = 0、σ = 1 的特殊情形，记作 Z ~ N(0, 1)。任意正态变量X都可通过下列变换标准化：
Z = (X − μ) / σ。
Z的累积分布函数记为Φ(z) = P(Z ≤ z)。Φ(z)的值可利用GDC或标准正态表求得。标准化使得我们能够比较来自不同正态总体的数值，并计算任意正态分布的概率。
ry of Z means that Φ(−z) = 1 − Φ(z). This property is frequently used when only positive z‑values are tabulated.
Z的对称性意味着 Φ(−z) = 1 − Φ(z)。当表格只提供正z值时，这一性质经常被用到。
5. Calculating Probabilities Using GDC or Tables | 使用GDC或表格计算概率
In the IB exam, you are expected to use your graphical display calculator (GDC) to find normal probabilities and inverse values. Common commands include normalcdf (lower bound, upper bound, μ, σ) and invNorm (area to the left, μ, σ). For the standard normal distribution, you may use normalcdf with μ = 0, σ = 1, or use the Z‑ a bell curve, shade the required region, and label boundaries to avoid careless mistakes.
在IB考试中，需使用图形计算器 (GDC) 计算正态分布的概率和反查值。常用指令包括 normalcdf（下限，上限，μ，σ）和 invNorm（左侧面积，μ，σ）。标准正态分布可直接令μ = 0, σ = 1 使用 normalcdf，或使用提供的Z值表。务必画出一条钟形曲线，标出阴影区域并标注边界值，以避免粗心错误。
For example, if X ~ N(100, 15²), to find P(85 ≤ X ≤ 115) you would input normalcdf(85, 115, 100, 15) and obtain about 0.6827. Always double-check that you have entered the standard deviation, not the variance.
例如，若 X ~ N(100, 15²)，要求 P(85 ≤ X ≤ 115)，可输入 normalcdf(85, 115, 100, 15)，结果约为0.6827。务必确认输入的是标准差而非方差。
6. Inverse Normal: Finding Boundaries | 反向正态：求边界值
Sometimes you are given a probability and need to find the corresponding x‑value (or z‑). For example, find x such that P(X ≥ x) = 0.10. First, express the probability as a left‑tail area: P(X ≤ x) = 0.90. Then use the invNorm function with area = 0.90, μ and σ. If using a table, locate the z‑ is skill is crucial for setting grade boundaries, quality control, and hypothesis testing.
有时已知某个概率值，需要求对应的x值（或z值）。例如，求满足P(X ≥ x) = 0.10 的x。首先将概率转化为左尾面积：P(X ≤ x) = 0.90。然后使用 invNorm 函数，输入面积=0.90及μ、σ。若使用表格，先找到满足Φ(z) = 0.90 的z分数，再反推：x = μ + z σ。这项技能对于划定分数线、质量控制和假设检验至关重要。
When the given probability lies in the right tail, always convert to a left‑tail area before applying invNorm. Misreading the direction of the tail is a common exam pitfall.
当给定概率位于右尾时，务必先转化为左尾面积再使用 invNorm。弄错尾部方向是考试中的常见陷阱。
7. The Empirical Rule (68–95–99.7) | 经验法则 (68–95–99.7)
For any normal distribution, approximately 68% of the data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. This rule provides a quick mental check of your answers. It also underpins risk analysis and the central limit theorem. Remember that these percentages are approximate; exact probabilities must be computed from the distribution.
对于任何正态分布，大约68%的数据落在距均值1σ的范围内，95%落在2σ内，99.7%落在3σ内。该法则可用于快速检验答案的合理性，同时也是风险分析和中心极限定理的基础。请记住这些百分比均为近似值，精确概率必须通过分布计算得出。
2σ, μ ± 3σ. This visual guide often reveals whether a calculated probability is plausible.
运用该法则的一个好方法是画出数轴并标出 μ、μ ± σ、μ ± 2σ、μ ± 3σ。这种直观的方法常能揭示计算出的概率是否合理。
8. Normal Approximation to the Binomial Distribution | 二项分布的正态近似
When a binomial random variable X ~ extremely useful for simplifying calculations before computers were widespread, and it remains a key concept in the IB syllabus. However, remember to apply a continuity correction.
当二项随机变量 X ~ B(n, p) 的试验次数很大时，其分布趋近于正态形状。经验法则是要求 np > 5 且 nq > 5（其中 q = 1 − p）。近似正态分布的均值 μ = np，方差 σ² = npq。该近似在计算机普及之前极大地简化了计算，至今仍是IB课程中的重要概念。但必须记得进行连续性校正。
When these conditions are met, the binomial probability can be estimated by a normal probability over an appropriate interval. The approximation works best when p is close to 0.5; for very skewed binomials a larger n is needed.
当满足这些条件时，二项概率可用适当区间内的正态概率来估计。当 p 接近0.5时近似效果最佳；对于非常偏斜的二项分布则需要更大的 n。
9. Continuity Correction | 连续性校正
ignificantly improves the accuracy of the normal approximation. Always draw both the discrete bars and the overlaid normal curve to visualise the adjustment.
由于二项分布是离散的而正态分布是连续的，我们需要通过±0.5来调整边界。例如，要近似二项分布的P(X > 20)，应求正态概率 P(X ≥ 20.5)。近似 P(X = 18) 时则用区间 P(17.5 ≤ X ≤ 18.5)。该项校正显著提高了正态近似的准确性。建议同时绘出离散柱状图和叠加的正态曲线以直观理解调整。
Examiners often expect you to state both the original binomial expression and the corrected normal statement before using the GDC. Skipping the continuity correction can lead to marks being lost even if the numerical answer is close.
阅卷人通常要求你先写出原始的二次项表达式和修正后的正态语句，再使用GDC。跳过连续性校正即使数值接近也可能丢分。
10. Applications and Extended Topics | 应用与扩展主题
Normal distributions appear in countless real‑life situations: heights, test scores, measurement errors, and more
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