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  • Sampling Distribution of the Sample Mean|Central Limit Theorem

    7.3 The Sampling Distribution of the Sample Mean

    population:1000;Scale are normally distributed with mean 100 and standard deviation 16

    sample:4;可以得到样本均值的分布图如下:

    与通过公式计算得到的mean 和 标准差一致:μx¯ = μ = 100 and σx¯ = σ/√n = 16/√4 = 8;

     由图可知The histogram is shaped roughly like a normal curve (with parameters 100 and 8)

    所以:

    由此得到:

    即在大数据量的情况下,虽然变量可能不是正态分布的,但是该变量的mean值一定是正态分布的,也就是中心极限定理:

    Usually, however, a sample size of 30 or more (n ≥ 30) is large enough

    example:

    统计每户房子占有人数:可知该变量属于右偏分布:

    household size is far from being normally distributed; it is right skewed. Nonetheless, according to the central limit theorem, the sampling distribution of the sample mean can be approximated by a normal distribution when the sample size is relatively large. Use simulation to make that fact plausible for a sample size of 30

    可以计算得到该样本mean的均值和方差:

     

    We simulated 1000 samples of 30 households each, determined the sample mean of each of the 1000 samples, and obtained a histogram (Output 7.2) of the 1000 sample means.

     从1000个样本中抽出30个样本,计算这三十个样本的均值,得到上图(即样本均值分布图,验证了中心极限定理,即该分布也是正态分布的)

    变量分布/变量mean 分布(在n逐渐变大的趋势下)/

     

    可见,SE也在逐渐变小

    所以,取样越大,数据越集中在均值附近,相应的SE越小。

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  • 原文地址:https://www.cnblogs.com/yuanjingnan/p/11250348.html
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