central limit theorem

简明释义

中心极限定理

英英释义

例句

1.The central limit theorem 中心极限定理 explains why the distribution of sample means approaches a normal distribution as the sample size increases.

中心极限定理解释了为什么样本均值的分布随着样本大小的增加而接近正态分布。

2.When conducting surveys, researchers utilize the central limit theorem 中心极限定理 to analyze the average responses from different groups.

在进行调查时，研究人员利用中心极限定理分析来自不同组的平均响应。

3.In quality control, manufacturers rely on the central limit theorem 中心极限定理 to ensure that their product measurements are normally distributed.

在质量控制中，制造商依赖中心极限定理来确保他们的产品测量是正态分布的。

4.The central limit theorem 中心极限定理 allows statisticians to make inferences about population parameters based on sample data.

中心极限定理使统计学家能够根据样本数据对总体参数进行推断。

5.The central limit theorem 中心极限定理 is crucial for understanding the behavior of averages in large samples.

中心极限定理对于理解大样本中均值的行为至关重要。

作文

The central limit theorem is one of the most fundamental concepts in statistics and probability theory. It states that the distribution of sample means approaches a normal distribution as the sample size becomes larger, regardless of the shape of the population distribution from which the samples are drawn. This remarkable property allows statisticians to make inferences about population parameters even when the underlying distribution is unknown. Understanding the central limit theorem is crucial for anyone involved in data analysis, research, or any field that relies on statistical methods.To illustrate the importance of the central limit theorem, consider a simple example. Suppose we have a population of students in a school, and their heights are recorded. If we take random samples of students and calculate the average height for each sample, we may find that the sample means vary widely if we only take a few students at a time. However, if we increase the sample size—say, by taking samples of 30 or more students—the distribution of those sample means will start to resemble a normal distribution, even if the original population of heights does not follow a normal distribution.This phenomenon occurs because of the law of large numbers, which states that as the number of trials increases, the average of the results will converge to the expected value. The central limit theorem builds on this idea by providing a framework for understanding how the averages of samples behave. As researchers collect more data, they can rely on the central limit theorem to justify the use of normal distribution models, making it easier to conduct hypothesis testing and construct confidence intervals.Moreover, the implications of the central limit theorem extend beyond just academic statistics. In practical applications, such as quality control in manufacturing, the central limit theorem allows engineers to monitor processes and ensure products meet specifications. By taking random samples of products and analyzing their characteristics, manufacturers can apply the central limit theorem to determine whether the production process is under control or if adjustments are necessary.In addition, the central limit theorem plays a vital role in fields like economics, psychology, and medicine. For instance, in clinical trials, researchers often need to compare the effects of a new drug against a placebo. By applying the central limit theorem, they can analyze the average treatment effects and draw conclusions about the efficacy of the drug, even if the individual responses vary greatly.Understanding the central limit theorem also helps in appreciating the significance of sample size in research. A small sample might lead to unreliable results due to high variability, while a larger sample size tends to yield more stable and reliable estimates. This principle is particularly important when designing experiments or surveys, as researchers must carefully consider how many observations are needed to achieve valid results.In conclusion, the central limit theorem is a cornerstone of statistical theory that enables researchers and practitioners to make informed decisions based on sample data. Its ability to transform the behavior of sample means into a predictable pattern allows for a wide range of applications across various fields. By grasping the essence of the central limit theorem, individuals can enhance their understanding of statistical principles and improve their analytical skills, ultimately leading to better decision-making processes in both research and practical scenarios.

中心极限定理是统计学和概率论中最基本的概念之一。它指出，样本均值的分布在样本容量增大时趋向于正态分布，无论样本来自的总体分布形状如何。这一显著特性使得统计学家能够对总体参数进行推断，即使在不知道基础分布的情况下。理解中心极限定理对于任何涉及数据分析、研究或依赖统计方法的领域的人来说都是至关重要的。为了说明中心极限定理的重要性，考虑一个简单的例子。假设我们有一所学校的学生群体，并记录了他们的身高。如果我们随机抽取学生并计算每个样本的平均身高，如果我们只抽取少数几名学生，我们可能会发现样本均值变化很大。然而，如果我们增加样本量，比如抽取30名或更多学生，样本均值的分布将开始呈现正态分布，即使原始身高的总体并不遵循正态分布。这种现象发生是因为大数法则，它指出随着试验次数的增加，结果的平均值将收敛于期望值。中心极限定理在此基础上提供了一个框架，用于理解样本均值的行为。随着研究人员收集更多数据，他们可以依靠中心极限定理来证明使用正态分布模型的合理性，从而更容易进行假设检验和构建置信区间。此外，中心极限定理的影响超出了学术统计的范畴。在实际应用中，例如制造业的质量控制，中心极限定理使工程师能够监控过程并确保产品符合规格。通过抽取随机样本并分析其特征，制造商可以应用中心极限定理来确定生产过程是否处于控制之中，或者是否需要进行调整。此外，中心极限定理在经济学、心理学和医学等领域也发挥着重要作用。例如，在临床试验中，研究人员通常需要比较新药与安慰剂的效果。通过应用中心极限定理，他们可以分析平均治疗效果并得出关于药物有效性的结论，即使个体反应差异很大。理解中心极限定理还有助于认识样本大小在研究中的重要性。小样本可能由于高度变异导致不可靠的结果，而较大的样本量往往会产生更稳定和可靠的估计。这一原则在设计实验或调查时尤为重要，因为研究人员必须仔细考虑所需的观察数量，以实现有效的结果。总之，中心极限定理是统计理论的基石，使研究人员和从业者能够基于样本数据做出明智的决策。它使样本均值的行为转变为可预测的模式，从而在各个领域的广泛应用成为可能。通过掌握中心极限定理的本质，个人可以增强对统计原理的理解，提高分析技能，从而在研究和实践场景中改善决策过程。