IB Mathematics: Distribution of Sample Sum and Sample Mean | IB数学：样本和与样本均值的分布

📚 IB Mathematics: Distribution of Sample Sum and Sample Mean | IB数学：样本和与样本均值的分布

Sampling distributions are a central topic in IB Mathematics Analysis & Approaches HL and Applications & Interpretation HL. When we draw a random sample from a population, the sample statistics we compute — such as the sample mean x̄ and the sample total T — are themselves random variables with distributions of their own. Understanding these sampling distributions allows us to make probabilistic statements about sample results and forms the essential foundation for confidence intervals and hypothesis testing.

抽样分布是IB数学分析与方法HL以及应用与解释HL的核心专题。当我们从总体中抽取随机样本时，所计算的样本统计量——如样本均值x̄和样本总和T——本身也是随机变量，具有各自的分布。理解这些抽样分布使我们能够对样本结果作出概率陈述，并为置信区间和假设检验奠定必要的基础。

1. What Is a Sampling Distribution? | 什么是抽样分布？

A statistic is any function of the sample data. Because the sample is randomly selected, every statistic — including the sample mean x̄ and the sample total T — is a random variable. The probability distribution of a statistic is called a sampling distribution. It describes how the statistic behaves across all possible random samples of the same size from the population.

统计量是样本数据的任意函数。由于样本是随机抽取的，每一个统计量——包括样本均值x̄和样本总和T——都是随机变量。统计量的概率分布称为抽样分布。它描述了从总体中抽取相同容量的所有可能随机样本时，该统计量的行为特征。

Suppose a population has mean μ and variance σ². We take a random sample of size n: X₁, X₂, …, Xₙ. The sample mean is x̄ = (X₁ + X₂ + … + Xₙ)/n, and the sample sum is T = X₁ + X₂ + … + Xₙ. Both x̄ and T vary from sample to sample; understanding their distributions is the key to solving exam questions on this topic.

设某总体的均值为μ，方差为σ²。我们从中抽取容量为n的随机样本：X₁, X₂, …, Xₙ。样本均值为x̄ = (X₁ + X₂ + … + Xₙ)/n，样本总和为T = X₁ + X₂ + … + Xₙ。x̄和T都随样本的不同而变化；理解它们的分布是解答此类考试题目的关键。

2. The Sampling Distribution of the Sample Mean | 样本均值的抽样分布

If the population itself is normally distributed with mean μ and variance σ², then the sample mean x̄ is exactly normally distributed. This is a result of the reproductive property of the normal distribution under linear combinations. The mean of x̄ equals the population mean μ, and its variance is σ² divided by the sample size n.

如果总体本身服从均值为μ、方差为σ²的正态分布，那么样本均值x̄精确服从正态分布。这是正态分布在线性组合下可加性（再生性）的结论。x̄的均值等于总体均值μ，其方差为σ²除以样本容量n。

x̄ ∼ N(μ, σ²/n)

This result holds for any sample size when the population is normal. Two essential ideas follow: first, the expected value of x̄ is exactly μ, so x̄ is an unbiased estimator of the population mean; second, as n increases, the variance σ²/n decreases, meaning the sample mean becomes more tightly concentrated around μ.

当总体为正态分布时，这一结论对任意样本容量均成立。由此可得两个核心要点：第一，x̄的期望值恰好等于μ，因此x̄是总体均值的无偏估计量；第二，随着n增大，方差σ²/n减小，意味着样本均值越来越紧密地集中在μ附近。

3. The Sampling Distribution of the Sample Sum | 样本和的抽样分布

The sample sum T = X₁ + X₂ + … + Xₙ also obeys a precise distribution. Using the linearity of expectation and the fact that the sample observations are independent, we obtain the mean and variance of T. The expected total is n times the population mean, and the variance is n times the population variance.

样本总和T = X₁ + X₂ + … + Xₙ也具有精确的分布。利用期望的线性性质以及样本观测值之间相互独立的事实，我们得到T的均值和方差：期望总和是总体均值的n倍，方差是总体方差的n倍。

T ∼ N(nμ, nσ²)

Note that the standard deviation of the sum is √n × σ, not n × σ. This distinction is a frequent source of exam errors. When the population distribution is normal, T is exactly normally distributed for any sample size.

注意，总和的标准差是√n × σ，而不是n × σ。这一区别是考试中常见的错误来源。当总体分布为正态时，T对任意样本容量都精确服从正态分布。

4. The Central Limit Theorem | 中心极限定理

What if the population is not normally distributed? The Central Limit Theorem (CLT) states that for a sufficiently large sample size n, the sample mean x̄ and the sample total T are approximately normally distributed regardless of the underlying population distribution. This powerful result justifies the use of normal calculations even when the population is skewed or unknown.

如果总体不是正态分布怎么办？中心极限定理（CLT）指出，当样本容量n足够大时，无论总体服从何种分布，样本均值x̄和样本总和T都近似服从正态分布。这一强有力的结论保证了即使总体偏态或未知，我们仍可使用正态计算。

In the CIE and IB context, the conventional threshold is n ≥ 30. If the population is known to be

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）