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Equation 11 · Part 8 · Measuring What a RAG System Retrieves, Not Just What It Answers

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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}
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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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