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Sampling with replacement and without replacement


Simple random sampling is the technique of drawing a sample from a population in such a way that each unit has an equal chance of being included in the sample as compared to the others. There are two different methods for choosing our sample – with replacement and without replacement.

Random sampling with replacement:

In random sampling with replacement whenever we include a unit from the population into our sample, the chosen unit is once again placed back into the population. This means that the particular unit could be chosen once again as part of our sample. So our sample may contain repeated units.

Suppose we want to study the heights of a population. We choose one person randomly and note down their height. The person is once again placed back into the population. So if we were to choose that person randomly once again, his height would be taken down twice as part of our sample values. This method is generally not used in statistical surveys because we do not wish to include the same information in our sample twice. Hence, the most commonly used method is sampling without replacement which is explained below.

Random Sampling without replacement:

In this method, once a unit is included in our sample it is removed from the population and hence no particular unit can occur more than once in the sample. Let the population size be N. Then the probability of a unit being included in the sample is 1/N. At the second step, the probability of a unit being included in the sample is 1/(N-1) because the size of the population decreased by one when the first unit was chosen.

This method has the important property that the probability of a particular unit being chosen in a particular draw (such as the fifth draw) is the same as the probability of it being chosen in the first draw. The hypergeometric distribution can be used to calculate probabilities when we draw a sample without replacement.

This method is the method most commonly used in statistical surveys. We can use a lottery system or random number tables to select our required random sample without replacement.

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