Statistics 简明教程
Statistics - Central limit theorem
如果样本抽取的总体是正态总体，则 sample means 等于总体均值， sampling distribution 为正态。当更多的总体为偏态时，就像图示中说明的那样，只要样本很大（即大于30）， sampling distribution 会趋向于更接近正态分布。
If the population from which the sample has a been drawn is a normal population then the sample means would be equal to population mean and the sampling distribution would be normal. When the more population is skewed, as is the case illustrated in Figure, then the sampling distribution would tend to move closer to the normal distribution, provided the sample is large (i.e. greater then 30).
根据 Central Limit Theorem ，对于样本量大于30的充分大的样本， sampling distribution 的形状将变得越来越像 normal distribution ，而与母体群体的形状无关。这个定理解释了 population distribution 和 sampling distribution 的关系。它强调了一个事实，即如果样本集足够大，则均值的 sampling distribution 接近 normal distribution 。理查德·I·莱文用以下文字总结了中心极限定理的重要性：
According to Central Limit Theorem, for sufficiently large samples with size greater than 30, the shape of the sampling distribution will become more and more like a normal distribution, irrespective of the shape of the parent population. This theorem explains the relationship between the population distribution and sampling distribution. It highlights the fact that if there are large enough set of samples then the sampling distribution of mean approaches normal distribution. The importance of central limit theorem has been summed up by Richard. I. Levin in the following words:
