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Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes—how likely they are—when we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

Assume we have a total of \({\rm{n}}\) observations. We define the low fourth as the median of the smallest half and the upper fourth as the median of the biggest half by sorting the observations from smallest to largest and splitting them into two halves (smallest half/largest half; if \({\rm{n}}\) is odd, the median \({\rm{\tilde x}}\) is included in both halves). The fourth spread, \({{\rm{f}}_{\rm{s}}}\), is defined as follows:

\({{\rm{f}}_{\rm{s}}}{\rm{ = upper fourth - lower fourth}}{\rm{. }}\)

The fourth spread contains the statistic of interest. It is simple to calculate the value of the fourth spread given data. We must first simulate the data in order to compare the sampling distribution of the fourth spread for specified sample sizes \({\rm{n = 5,10,20,40}}\).

For example, using a uniform distribution with stated bounds \({\rm{A}}\) and\({\rm{B}}\), use a computer to create samples of the specified sizes. Take the same number of replications for each sample (it does not have to be really big number, \({\rm{1000}}\)should be enough).

Calculate the fourth spread for each replication as follows:

\({{\rm{f}}_{\rm{s}}}{\rm{ = upper fourth - lower fourth}}{\rm{. }}\)

For each sample size, this results in fourth spreads. Use any method to visualize the data to compare the sample distributions. For example, make a distinct histogram for each sample distribution and compare them.