On the Mathematical Expectation of the Sample Variance in Simple Sampling Technique

Drawing random samples is the core of modern life jobs. In manufacturing, it is important to inspect ‎deficiencies ‎by ‎only ‎sampling ‎items‎ from a production line, to meet quality worldwide standards and to maintain sufficient statistical quality control‎. Furthermore, in today's survey research, the theory of sampling technique is foundational to ensure that all‎ inquired ‎and ‎essential‎ information is gathered. In effect, one may name thousands of practical applications that rely on taking samples, like climatic studies, industry, ecology, and so on. In effect, many studies were designed and proposed in searching for an effective sampling technique. It is, in fact, both an art and a robust science. So many strategies and considerations were plotted to determine the proper sample size ‎and the proper sampling technique, like simple sampling and stratified sampling. This paper is a brief study focusing on the behaviour of the mathematical expectation of the sample variance in sampling without replacement and in sampling with replacement‎. ‎Formally‎, ‎we show that when sampling is with replacement‎, ‎there exists a crucial difference between the two situations‎, ‎namely‎, ‎distinct samples and indistinct samples‎. Namely, by a series of simulation studies and a famous historical example, it will be shown that there is a faulty fact concerning the unbiasedness of sample variance when drawing indistinct samples with replacement.