Sampling Distribution of the Mean and its convergence towards the Center

Sampling is a very common practice in empirical research. Due to cost, time and effort required to work with the population, samples provides a realistic opportunity to get the research done. Ideally, a sample should have all the characteristics of the population.

While the cost and effort associated in computing population may be enormous, the estimate from the population will be precise, bar some few recording errors. We also want the sample to be as precise as the population estimate. It is necessary that the sample is not-biased and should neither over-represent nor under-represent a feature of the population.

One of the most used tools in statistics is the average or the mean value. Most often, the population and the samples are best described by their mean. Mean is also important for many analytics techniques such as least-squares regression, hypothesis testing, tree-based regression, etc. Therefore, an important question that can be raised is that if the mean value from the sample is a good estimate of the population?

In order to investigate the question, it is helpful to start with a practical example. This investigation will show how the sample mean has a significant probability of correctly estimating the population mean.

Suppose that there is a game where the value of the dice rolled is the prize money. A contestant may have a guess on what the outcome can be but does not know for certain what value…