central limit theorem

简明释义

中心极限定理

英英释义

例句

1.Understanding the central limit theorem (中心极限定理) is essential for data scientists when they analyze large datasets.

对于数据科学家来说，理解中心极限定理 (中心极限定理) 在分析大型数据集时至关重要。

2.In statistics, the central limit theorem (中心极限定理) allows us to assume that the sampling distribution of the sample means will be approximately normally distributed, even if the population distribution is not.

在统计学中，中心极限定理 (中心极限定理) 允许我们假设样本均值的抽样分布将近似正态分布，即使总体分布不是。

3.The central limit theorem (中心极限定理) explains why the mean of a large number of independent random variables tends to be normally distributed.

中心极限定理 (中心极限定理) 解释了为什么大量独立随机变量的均值趋向于正态分布。

4.When conducting experiments, researchers rely on the central limit theorem (中心极限定理) to justify using normal distribution for small sample sizes.

在进行实验时，研究人员依赖于中心极限定理 (中心极限定理) 来证明对于小样本大小可以使用正态分布。

5.The central limit theorem (中心极限定理) is crucial in quality control processes, as it helps in determining if a manufacturing process is stable.

在质量控制过程中，中心极限定理 (中心极限定理) 是至关重要的，因为它有助于确定制造过程是否稳定。

作文

The central limit theorem is one of the most important concepts in statistics and probability theory. It states that when independent random variables are added together, their normalized sum tends to follow a normal distribution, even if the original variables themselves are not normally distributed. This remarkable property allows statisticians to make inferences about population parameters based on sample statistics. To understand the central limit theorem, let’s consider an example. Imagine we have a fair six-sided die. If we roll this die once, the outcome can be any number from 1 to 6, each with equal probability. However, if we roll the die multiple times and calculate the average of those rolls, the distribution of these averages will start to resemble a normal distribution as the number of rolls increases. This happens regardless of the fact that the individual outcomes of rolling a die do not follow a normal distribution. The central limit theorem is crucial because it enables researchers to use the properties of the normal distribution to make predictions and decisions based on sample data. For example, if a researcher wants to know the average height of adult males in a city, they do not need to measure every single person. Instead, they can take a random sample, calculate the average height, and apply the central limit theorem to estimate the population average with a certain level of confidence. Moreover, the central limit theorem has practical applications in various fields such as economics, psychology, and healthcare. In quality control processes, manufacturers often rely on this theorem to ensure that their products meet specific standards. By taking samples of items produced and analyzing their characteristics, they can infer whether the entire production batch meets quality requirements. The implications of the central limit theorem extend beyond just theoretical statistics; they have real-world significance. For instance, in political polling, the central limit theorem assures pollsters that the average opinion expressed by a sample of voters will closely reflect the average opinion of the entire electorate, provided the sample is sufficiently large and randomly selected. Despite its power, the central limit theorem does have some limitations. It applies primarily to independent random variables, meaning that if the variables are dependent, the theorem may not hold. Additionally, while the theorem guarantees that the distribution of the sample mean approaches normality as the sample size increases, it does not provide information about the rate of convergence. In practice, larger sample sizes yield better approximations, but the required size can vary depending on the underlying distribution. In conclusion, the central limit theorem is a foundational element of statistical theory that facilitates the understanding of how sample data relates to population parameters. Its ability to bridge the gap between sample statistics and population distributions makes it an invaluable tool for researchers and practitioners alike. By applying the central limit theorem, we can derive meaningful insights from data and make informed decisions in various domains, reinforcing the importance of statistics in our everyday lives.

中心极限定理是统计学和概率论中最重要的概念之一。它指出，当独立随机变量相加时，它们的标准化和趋向于遵循正态分布，即使原始变量本身并不遵循正态分布。这一显著特性使得统计学家能够根据样本统计量对总体参数进行推断。为了理解中心极限定理，我们可以考虑一个例子。想象我们有一个公平的六面骰子。如果我们掷这个骰子一次，结果可以是1到6之间的任何数字，每个数字的概率相等。然而，如果我们多次掷骰子并计算这些掷骰子的平均值，那么随着掷骰子次数的增加，这些平均值的分布将开始类似于正态分布。这种情况发生在不考虑单次掷骰子的结果并不遵循正态分布的情况下。中心极限定理至关重要，因为它使研究人员能够利用正态分布的特性基于样本数据进行预测和决策。例如，如果研究人员想知道某城市成年男性的平均身高，他们不需要测量每一个人。相反，他们可以随机抽取一个样本，计算平均身高，并应用中心极限定理以一定的置信水平估计总体平均身高。此外，中心极限定理在经济学、心理学和医疗保健等多个领域具有实际应用。在质量控制过程中，制造商通常依赖这一理论确保其产品符合特定标准。通过对生产的样品进行分析，他们可以推断整个生产批次是否满足质量要求。中心极限定理的影响不仅限于理论统计；它在现实世界中具有重要意义。例如，在政治民调中，中心极限定理使得民调者确信，从选民样本中表达的平均意见将与整个选民的平均意见密切相关，前提是样本足够大且随机选择。尽管中心极限定理功能强大，但它也有一些局限性。它主要适用于独立随机变量，这意味着如果变量是相关的，该定理可能不成立。此外，虽然该定理保证样本均值的分布随着样本大小的增加而接近正态性，但它并没有提供关于收敛速度的信息。在实践中，更大的样本大小会产生更好的近似值，但所需的大小可能因基础分布而异。总之，中心极限定理是统计理论的基础元素，它促进了样本数据与总体参数之间关系的理解。它能够弥合样本统计量与总体分布之间的差距，使其成为研究人员和从业者不可或缺的工具。通过应用中心极限定理，我们可以从数据中得出有意义的见解，并在各个领域做出明智的决策，进一步加强了统计学在我们日常生活中的重要性。