secondsmallest

Secondsmallest, often written as the second smallest or as the second order statistic, denotes the second element in the sorted arrangement of a sample or set. In statistics, if X1, X2, ..., Xn are observations, the second smallest value is X_(2) when the sample is ordered nondecreasingly. If the smallest value occurs more than once, X_(2) may equal X_(1).

Many programming problems distinguish between two interpretations: the second smallest value (allowing duplicates) and the second smallest distinct value (requiring a value strictly greater than the minimum). The former returns the second element in a nondecreasing sort, while the latter returns the smallest value that is greater than the minimum.

To compute it, you can sort and take the second element, or use a one-pass method that tracks the smallest and the second smallest values seen so far. For the distinct version, you track the smallest and the next distinct value, ignoring duplicates of the minimum. Handling negative numbers or large datasets does not change the basic approach, but it can affect space and time complexity choices.

Edge cases include when the sample size n < 2, or when all values are equal and no distinct second smallest exists. In such cases the second smallest is undefined or may be reported as not present, depending on the convention used.

Example: data [5, 1, 3, 1, 2] yields a second order statistic of 1 (sorted: 1, 1, 2, 3, 5); the second smallest distinct value is 2.

Applications appear in descriptive statistics, robust estimators, and algorithms that rely on order statistics. Related concepts include minimum, maximum, and the median.