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Average-case analysis measures an algorithm's expected cost when its inputs follow a specified probability distribution. If $T(x)$ is the number of counted operations on input $x$ and $p_n(x)$ is the probability of a size-$n$ input, thenThe expected value is meaningful only after specifying both the input model and the counted operation. It is not an average over arbitrary anecdotes or necessarily the cost of a typical individual run.

For a successful linear search through $n$ distinct entries, suppose the sought entry is equally likely to occupy any position. Searching from the beginning uses $i$ comparisons if the entry is at position $i$, soFor $n=9$, this is five comparisons, whereas worst-case analysis gives nine. If the entry is absent with probability $q$ and otherwise has the same equal-position model, the expected count becomes $qn+(1-q)(n+1)/2$. Changing the workload therefore changes the answer without changing the algorithm. Amortized analysis instead bounds the cost of a sequence of operations and need not assume an input probability distribution.

Michael L. Fredman co-developed Fibonacci heaps with Robert Tarjan. Their 1987 paper uses deferred restructuring and amortized analysis to obtain constant amortized time for operations including insertion and decrease-key, with logarithmic amortized time for delete-min.^[ref]

This matters in graph algorithms that repeatedly lower tentative priorities: reducing the cost of many decrease-key operations can improve a whole algorithm's bound. A single operation can still be expensive. The guarantee concerns the total cost of a sequence, so confusing amortized time with the worst-case analysis of each individual operation loses the point of the construction.

Worst-case analysis measures the largest resource use of a specified algorithm among inputs of the same size. If $T(x)$ counts operations on input $x$, the worst-case function isThe cost model and meaning of input size $n$ are part of the definition. A big O notation bound describes growth of this function, rather than a claim about how often the difficult input occurs.^[ref]

For linear search of $n$ entries, looking for an absent item requires $n$ equality tests; finding the first item needs just one. Thus the worst-case comparison count is $n$, even if many actual queries finish earlier. This supplies a guarantee for every allowed input, while an average-case analysis requires a stated distribution over inputs.