Amortized analysis is a method in computer science for analyzing the average time complexity of an algorithm over a sequence of operations, ensuring that the worst-case cost per operation remains low when averaged over all operations. Unlike worst-case analysis, which considers the maximum time an operation can take, amortized analysis provides a more comprehensive view by spreading out the cost of expensive operations over a sequence of cheaper ones.

A common example of amortized analysis is in the dynamic array (or resizable array) data structure. When an element is appended to a dynamic array that is full, the array is resized, typically by doubling its capacity. The resizing operation is costly because it involves copying all elements to the new array. However, this expensive operation doesn’t happen every time an element is appended; it occurs only occasionally.

To analyze the amortized cost, consider that resizing happens every time the number of elements reaches a power of two (e.g., 1, 2, 4, 8,…). If we insert \( n \) elements, the total number of operations includes both the regular insertions and the copying steps during resizing. By spreading the cost of these copying steps across all \( n \) insertions, the amortized cost per insertion remains constant, i.e., \( O(1) \), despite individual insertions occasionally costing \( O(n) \) during resizing.