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。