在独立样本设计中，每一个总体都有一个均值分布，这样对两个总体来说，我们就要处理两个不同的均值分布。我们从第一个总体的均值分布中随机选择一个均值，再从第二个总体的均值分布中随机选择一个均值。将抽取的两个均值相减，得到一个均值差异的分数，多次重复这一过程，我们就可以得到一个均值差异的分布。

如果虚无假设是正确的，那么两个总体的均值是相等的，所以我们得到的均值差异的分布的均值是0。这一分布的方差总体取决于估计的总体方差，因此我们可以把它看作一个t分布来进行检验。我们先利用样本方差来估计总体方差，从而估计总体样本分布的方差，然后再结合两个样本均值分布的方差构造一个新的估计值，用于描述样本均值分布的变异。

In an independent sample design, each population has a mean distribution. In this way for two population we have to deal with two different mean distributions. We randomly select a mean from the mean distribution of the first population, and then we randomly select a mean from the mean distribution of the second population as well. Subtracting the two extracted means gives us a fraction of the difference in means, and repeating this process many times we obtain a distribution of differences in means.

If the null hypothesis is correct, then the means of two populations are equal, so the mean of difference distribution is 0. The variance of this distribution is decided by the variances of the populations to be estimated, so we can test it as a t-distribution. Using the variance of sample to estimate the variance of distribution, that to estimate the variance of the distribution of population, then combine the variance of two sample mean distributions to make a new estimated value, to describe the variation of the distribution of the mean of the population.