order statistics
Take a sample of several random measurements and line them up from smallest to largest. The sorted values — the smallest, the second smallest, on up to the largest — are the order statistics. They turn an unordered bag of numbers into a ranked sequence, and they are exactly what you need to talk about the minimum, the maximum, the median, percentiles, and the range.
Write the sorted sample as X_(1) at most X_(2) at most ... at most X_(n). Here X_(1) is the minimum and X_(n) the maximum. Even when the original values are independent and identically distributed, the order statistics are dependent on each other (the second smallest cannot be below the smallest) and each has its own distribution. The cdf of the k-th order statistic comes from counting: X_(k) is at most x exactly when at least k of the n values are at most x, and the number of values below x is binomial. That counting gives a clean formula for the distribution of any rank.
Order statistics are the backbone of practical statistics: the sample median is the middle order statistic, quartiles and percentiles are order statistics, and the spread between extremes is the sample range. They also drive extreme-value theory — floods, record temperatures, insurance losses — where the behaviour of the maximum over a large sample is the whole question.
Sample five exam scores: 71, 88, 64, 92, 80. Sorted, the order statistics are X_(1) = 64 (min), X_(2) = 71, X_(3) = 80 (the median), X_(4) = 88, X_(5) = 92 (max). The range is X_(5) - X_(1) = 28. Even if the scores were drawn independently, these ranked positions are tied together.
Sorting a sample produces order statistics — the min, the median, the max, and everything between.
Order statistics from an iid sample are NOT independent of each other, even though the original observations were. Their joint behaviour is constrained by the ordering itself.