The Miniature Population

We are well familiar with the terms ‘population’, ‘sample’ etc. The sample is mainly a part of the population. Sampling is classified into two categories mainly, (1) Subjective or purposive sampling and (2) Objective sampling. When the statistician picks up elements from the population trusting his or her own judgment or for a purpose, that kind of sampling is called subjective sampling. On the other hand objective sampling is also classified into three categories, (a) Probabilistic sampling, (b) Non- probabilistic sampling, and (c) Mixed sampling. We are bothered about the probabilistic sampling. Here each element has a definite preassigned probability of being selected. Suppose we have 40 identical but distinguishable balls in an urn. We blindly pick one ball and that ball has a probability of 1/40 of being selected. If we put the ball back and again draw one, that ball also has a probability of 1/40 of being selected. Here the urn of 40 balls is the population and each element has a preassigned probability of being selected. This type of probabilistic sampling is known as random sampling.

The concept of random sampling is quite easy but it is tough to apply practically. From common sense, if a statistician is given a population of 300 students and supposes he has to choose 50 students from that. Practically he is seeing the students and choosing them one by one. The students don’t appear random to him. The students are not identical. Naturally, the sampling becomes subjective. This scenario can be explained better if we consider if someone lets those 300 students move in a field at a high speed so that one cannot differentiate one from the other and the statistician picks up 50 students from that field. This will be a random sampling but the scenario is totally hypothetical. To get rid of this problem, if we make identical tickets carrying the names of the students and shuffle them in a jar. Then the statistician picks up 50 tickets. This situation is feasible. Tickets could be anything like balls, cards, etc. Again, consider we want to select 200 people from the whole of India. We will select 200 Aadhar numbers randomly from the national database instead of considering the people actually. This is the concept of a miniature population. When we cannot draw samples randomly from the actual population due to some constraints, we create another population whose elements are identical but carries some identification mark of the elements from the actual population. The population we created is called a miniature population. Each element of the miniature population represents the respective elements of the actual population in such a way so that random sampling can be easily done.

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Reference

1.?????Introduction to Survey Sampling: Graham Kalton

2.?????Fundamental of Statistics Vol II: Goon, Gupta, Dasgupta?