Minima and maxima (introduction)

This is an entry-level discussion of an advanced topic. For more comprehensive and rigorous treatment, see minima and maxima or extreme value.

The minimum of a set of numbers is the smallest value in that set. So, for the set of numbers {3, 7, 11, 30, 68, 121} the minimum is 3. The maximum of a set is the largest value it that set; for the set above, the maximum is 121.

In common usage, the terms "minimum" and "maximum" are used to communicate restrictions. For example, if a sale item is marked "maximum 3 per customer", it is restricting the number a single shopper may purchase. Similarly, a fairground ride with a sign saying "you must be this tall to ride" is giving a minimum size limit on who can ride safely.

Minimum and maximum values are also used in descriptive statistics—along with the mean, median, and standard deviation—to provide a clear overview of a set of measurements without needing to quote the entire set. For example, consider the following two sets of numbers: set A, which is all the numbers from 0 to 100; and set B, which is a set of the number 50, 101 times. Both of these sets have an average of 50, so from that statistic alone, they appear identical. When the minima and maxima of the two sets are given as well, the difference becomes clear: for set A, the minimum is 0 and the maximum is 100, while for set B, both the minimum and maximum are 50.

The concept of minima and maxima also occurs in line graphs, where the minimum is the lowest point of the line and the maximum is the highest. If a point is the lowest everywhere on the line, it is called the global minimum; if it's only the lowest in a specific area, for example the lowest point of a dip in the line, it's called a local minimum. The mathematical field of calculus investigates methods for finding both minima and maxima of any function.