📚 E - Distributions of the Sample Mean and the Central Limit Theorem | E - 样本均值的分布与中心极限定理
In computer science, data-driven decision-making, simulation, and machine learning rely heavily on statistical inference. Understanding the behaviour of the sample mean and the Central Limit Theorem (CLT) is fundamental for designing robust algorithms, analysing runtime variability, and assessing model performance.
在计算机科学中，数据驱动的决策、仿真和机器学习严重依赖统计推断。理解样本均值的性质和中心极限定理（CLT）是设计鲁棒算法、分析运行时变异性以及评估模型性能的基础。
1. What Is a Sampling Distribution? | 什么是抽样分布？
A sampling distribution is the probability distribution of a given statistic based on repeated random samples from a population. For the sample mean x̄, it is the distribution of means obtained from all possible samples of a fixed size n drawn from the same population.
抽样分布是指从总体中重复随机抽样时，某一统计量的概率分布。对于样本均值 x̄ 而言，它就是从同一总体中抽取的所有固定样本量 n 的可能样本的均值所构成的分布。
x̄ = (1/n) Σ xi
The sample mean is a random variable because it varies from sample to sample.
样本均值是一个随机变量，因为它会因样本不同而变化。
2. Mean and Variance of the Sample Mean | 样本均值的均值与方差
e population has mean μ and variance σ², then the expected value of the sample mean is E(x̄) = μ, and its variance is Var(x̄ ws that the sample mean is an unbiased estimator and becomes less variable as n increases.
如果总体的均值为 μ，方差为 σ²，那么样本均值的期望值 E(x̄) = μ，方差为 Var(x̄) = σ²/n。这表明样本均值是一个无偏估计量，并且随着 n 的增大，其变异性减小。
E(x̄) = μ, Var(x̄) = σ²/n
The standard deviation of x̄, called the standard error of the mean, is SE = σ/√n.
样本均值的标准差称为均值的标准误，SE = σ/√n。
3. The Central Limit Theorem (CLT) | 中心极限定理
The Central Limit Theorem states that, regardless of the shape of the population distribution, the sampling distribution of the sample mean x̄ will be approximately normally distributed with mean μ and variance σ²/n, provided the sample size n is sufficiently large.
中心极限定理指出，无论总体分布的形状如何，只要样本量 n 足够大，样本均值 x̄ 的抽样分布就会近似服从均值为 μ、方差为 σ²/n 的正态分布。
x̄ ~ N(μ, σ²/n) as n → ∞
This theorem bridges descriptive statistics and inferential statistics, enabling hypothesis tests and confidence intervals even when the population is not normal.
这一定理是描述统计与推断统计之间的桥梁，使得即使总体非正态时，我们也能进行假设检验和构建置信区间。
4. Conditions and Assumptions | 条件与假设
d populations a larger n may be required. When sampling without replacement from a finite population, the finite population correction factor should be considered if n > 5% of the population.
中心极限定理在两个主要条件下成立：(1) 观测值必须独立（例如通过简单随机抽样获得）。(2) 样本量 n 必须足够大。常见的经验法则是 n ≥ 30，但对于高度偏态的总体，可能需要更大的 n。当从有限总体中无放回抽样且 n 超过总体的 5% 时，需要考虑有限总体校正因子。
For computer simulations that generate pseudo-random numbers, independence is achieved by using a good random number generator and ensuring samples do not overlap inadvertently.
对于生成伪随机数的计算机仿真，独立性可通过使用良好的随机数生成器并确保样本不会无意间重叠来实现。
5. The Role of Sample Size n | 样本量 n 的作用
As n increases, the variance σ²/n decreases, making the distribution of x̄ narrower and more concentrated around μ. This increased precision is why larger datasets provide more reliable estimates. In simulation studies, increasing the number of Monte Carlo iterations reduces variability in the estimated mean, directly reflecting the CLT.
随着 n 增大，方差 σ²/n 减小，使得 x̄ 的分布更窄且更集中在 μ 周围。这种更高的精度正是更大的数据集能提供更可靠估计的原因。在仿真研究中，增加蒙特卡洛迭代次数可降低估计均值的变异性，这直接体现了中心极限定理。
For example, when measuring the average runtime of an algorithm over many runs, the standard error of the mean runtime decreases with the square root of the number of runs.
例如，在多次运行中测量算法的平均运行时间时，平均运行时间的标准误随着运行次数的平方根而减小。
6. Standard Error and Precision | 标准误与精度
The standard error SE = σ/√n quantifies how much the sample mean is expected to fluctuate from one sample to another. In practice, σ is often unknown and is estimated by the sample standard deviation s, giving an estimated standard error s/√n
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