Normalized Discounted Cumulative Gain (nDCG)
Also known as: nDCG, Discounted Cumulative Gain, DCG/IDCG Normalization, Cumulated Gain Evaluation
Normalized Discounted Cumulative Gain (nDCG) is the standard metric for evaluating ranked retrieval and recommendation when relevance comes in grades rather than a simple relevant/non-relevant binary. Introduced by Kalervo Järvelin and Jaana Kekäläinen in their 2002 ACM Transactions on Information Systems paper on cumulated gain-based evaluation, nDCG rewards a system for placing highly relevant documents near the top of the ranking. It accumulates the graded relevance ('gain') of each retrieved item, discounts that gain by how far down the list the item sits, and normalizes the total against the best possible ordering so that scores fall on a comparable 0-to-1 scale across queries. Because it handles multi-level relevance and is rank-sensitive, nDCG has become the dominant effectiveness measure for web search, learning-to-rank, and academic-search evaluation.
Key highlights
- Handles multi-level (graded) relevance, rewarding highly relevant results more than marginally relevant ones.
- Rank-sensitive: the logarithmic discount strongly rewards placing the best results at the top of the list.
- Normalization by the ideal ranking yields a 0-to-1 score that is comparable and averageable across queries with different relevance pools.
- Has become the de facto standard for web search and learning-to-rank, enabling direct comparison with the published literature.
Intuition
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How it works
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When to use it
Use nDCG whenever you evaluate a ranked list and relevance is naturally graded rather than binary, and where the position of good results matters: web and academic search evaluation, recommender-system ranking, learning-to-rank model selection, and any retrieval task with multi-level relevance judgments. It is the right choice when you want a single rank-sensitive number per query that is comparable across queries with different numbers of relevant items, and when top-of-list quality is what users experience. nDCG is less appropriate when you only have binary relevance and care about recall over the whole list (where MAP is more natural), when relevance judgments are too sparse or noisy to support fine grades, or when the cutoff k and the gain and discount conventions are not held fixed — differing conventions make reported nDCG values incomparable.
Strengths & limitations
- Handles multi-level (graded) relevance, rewarding highly relevant results more than marginally relevant ones.
- Rank-sensitive: the logarithmic discount strongly rewards placing the best results at the top of the list.
- Normalization by the ideal ranking yields a 0-to-1 score that is comparable and averageable across queries with different relevance pools.
- Has become the de facto standard for web search and learning-to-rank, enabling direct comparison with the published literature.
- Requires graded relevance judgments, which are more expensive and more subjective to collect than binary labels.
- Sensitive to the choice of gain function and log base; the exponential 2^rel - 1 gain and base-2 discount are conventions, and alternatives change scores.
- With incomplete judgments, unjudged documents are often treated as non-relevant, which can bias the metric against systems that surface novel relevant items.
- A single cutoff k can hide differences below it, and IDCG can be ill-defined for queries with no relevant documents.
Common pitfalls
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Applications
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Frequently asked
Why normalize DCG instead of just reporting DCG?
Raw DCG depends on how many relevant documents a query has and how relevant they are, so DCG values are not comparable across queries — a query with many highly relevant documents can produce a large DCG even from a mediocre ranking. Dividing by the ideal DCG (the best achievable ordering for that query) rescales every query to a 0-to-1 range, where 1 is the perfect ranking. This makes nDCG comparable across queries and meaningful to average over a query set.
Which gain and discount formula should I use?
Two conventions are common: a linear gain equal to the relevance grade, and an exponential gain of 2^rel - 1 that more sharply rewards higher grades, both divided by a base-2 logarithmic discount of (rank + 1). The exponential form is widely used in web-search and learning-to-rank work. The key practical rule is to fix one convention and apply it consistently, because nDCG values computed under different gain or discount choices are not directly comparable.
How does nDCG differ from Mean Average Precision (MAP)?
MAP is built for binary relevance and rewards a system for ranking all relevant documents highly across the full list, averaging precision at each relevant retrieval. nDCG is built for graded relevance and is typically reported at a top cutoff k, emphasizing that the most relevant items belong at the very top. Use MAP when relevance is binary and overall ranking of all relevant documents matters; use nDCG when relevance has degrees and top-of-list quality is the priority.
Sources
- 1.Järvelin, K., & Kekäläinen, J. (2002). Cumulated gain-based evaluation of IR techniques. ACM Transactions on Information Systems, 20(4), 422-446.
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Cite this page
ScholarGate. (2026, June 23). Normalized Discounted Cumulative Gain (nDCG). ScholarGate. https://scholargate.app/bibliometrics/ndcg-evaluation