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Equation 11 · Part 12 · Measuring What a RAG System Retrieves, Not Just What It Answers

Denominator: log_2(i+1)

DCG@k=∑i=1k2 reli−1log⁡2(i+1),nDCG@k=DCG@kIDCG@k\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}
log⁡2(i+1)\log_2(i+1)

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

log2(i+1)g_2(i+1) occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

Normalised discounted cumulative gain generalises this to graded relevance and to the whole list rather than only the first hit: DCG@k=∑i=1k2 reli−1log⁡2(i+1),nDCG@k=DCG@kIDCG@k\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 reli\mathrm{rel}_i is the graded relevance of the result at position i and IDCG@\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…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.