What does this equation mean?

Normalised discounted cumulative gain generalises this to graded relevance and to the whole list rather than only the first hit: $\mathrm{DCG@}k = \sum_{i=1}^{k} \frac{2^{\,\mathrm{rel}_i}-1}{\log_2(i+1)}, \qquad \mathrm{nDCG@}k = \frac{\mathrm{DCG@}k}{\mathrm{IDCG@}k}$. where $\mathrm{rel}_i$ is the graded relevance of the result at position i and $\mathrm{IDCG@}$k is the DCG of the ideal ordering, so the ratio is bounded near one regardless of how many relevant documents exist for a given query. The logarithmic discount encodes a specific judgement about attention: a relevant document at position one is worth far more than the same document at position ten, and nDCG is the standard way the information-retrieval literature makes that judgement quantitative. Heterogeneous retrieval benchmarks built to compare retrievers across many domains at once — BEIR evaluated ten lexical, sparse, dense, late-interaction and re-ranking systems across eighteen public datasets and found BM25 a robust baseline that dense retrievers frequently underperformed out of domain — rely on exactly these rank-based metrics to make that comparison possible without ever generating an answer [ 2 ] .