Editing Order statistic (section)

=== A small-sample-size example ===

The simplest case to consider is how well the sample median estimates the population median.

As an example, consider a random sample of size 6. In that case, the sample median is usually defined as the midpoint of the interval delimited by the 3rd and 4th order statistics. However, we know from the preceding discussion that the probability that this interval actually contains the population median is {{Clarify|reason=In which way does the "preceding discussion" explain that the probability is that one?|date=September 2021}}

:<math>{6\choose 3}(1/2)^{6} = {5\over 16} \approx 31\%.</math>

Although the sample median is probably among the best distribution-independent [[point estimate]]s of the population median, what this example illustrates is that it is not a particularly good one in absolute terms. In this particular case, a better confidence interval for the median is the one delimited by the 2nd and 5th order statistics, which contains the population median with probability

:<math>\left[{6\choose 2}+{6\choose 3}+{6\choose 4}\right](1/2)^{6} = {25\over 32} \approx 78\%.</math>

With such a small sample size, if one wants at least 95% confidence, one is reduced to saying that the median is between the minimum and the maximum of the 6 observations with probability 31/32 or approximately 97%. Size 6 is, in fact, the smallest sample size such that the interval determined by the minimum and the maximum is at least a 95% confidence interval for the population median.