Normalized discounted cumulative gain

The discounted cumulative gain (DCG) measures the quality of the results in a ranked list, where items in that list are graded in some way. Typically the ranked list in question is the query results of an IR system, and the grading is a relevance judgment for each result. DCG accumulated at a particular rank position $p$ is defined as:

$\mathrm {DCG_{p}} =rel_{1}+\sum _{i=2}^{p}{\frac {rel_{i}}{\log _{2}i}}$

Normalized DCG (nDCG) is a way to calculate this measure across many independent queries, the result lists for which might all be of different length. nDCG is computed as:

$\mathrm {nDCG_{p}} ={\frac {DCG_{p}}{IDCG{p}}}$

The nDCG values for all queries can be averaged to obtain a measure of the average performance of a search engine's ranking algorithm over many queries.