Solution
The Range as a Median Interval
Show the problem again
Six independent observations are drawn from the same continuous distribution (otherwise unknown). The interval [smallest, largest] is used as a confidence interval for the population median. What is its exact coverage probability, that is, the probability that the median lies between the smallest and largest of the six observations? Give a reduced fraction.
Worked solution
Each observation falls below the median with probability 1/2, independently. The interval misses the median only if all six fall on the same side: probability 2 × (1/2)^6 = 1/32. The coverage is 1 - 1/32 = 31/32 ≈ 96.9%, whatever the continuous distribution, so this is a distribution-free interval at a bit over 95% confidence from just six points.
Source: Classic distribution-free confidence interval for the median via order statistics; folklore in nonparametric statistics. Statement written for AxiomIQ.