常见分布中心极限定理适用样本量研究

Research of the applicable sample size of common distribution central limit theorem

中心极限定理指出在满足独立同分布且方差有限条件下,当样本量充分大时,样本均值可视为服从正态分布,但定理并未给出样本量充分大的标准.以常见分布为研究对象,通过计算机模拟的方法研究不同分布中心极限定理成立的适用样本量,并根据控制变量法,研究了期望、方差、偏度、峰度和分布类型对适用样本量的影响.结果表明,样本量n≥30的大样本标准仅对部分分布适用,在所选取的代表分布中,期望和方差对适用样本量几乎无影响,偏度和峰度越接近零,适用样本量越小,同等条件下离散型分布所需的适用样本量多于连续型分布.

The central limit theorem states that under the conditions of independent identical distribution and limited variance,when the sample size is sufficiently large,the sample mean can be regarded as obeying a normal distribution,but the theorem does not give a standard for the sufficiently large sample size.Taking common distributions as the research object,computer simulation methods was used to study the applicable sample sizes for the establishment of the central limit theorem of different distributions,and the effects of expectations,variance,skewness,kurtosis,and distribution types on the applicable sample size was studied based on the controlled variable method.The conclusion shows that the large sample standard of sample size n≥30 is only applicable to part of the distribution.In the selected representative distribution,expectation and variance have almost no effect on the applicable sample size.The closer the skewness and kurtosis are to zero,the smaller the applicable sample size is.Under the same conditions,the suitable sample size of discrete distribution is more than that of continuous distribution.