NDCG

Normalized Discounted Cumulative Gain (NDCG)

NDCG

NDCG is an evaluation metric for ranked retrieval that handles graded relevance judgments (not just binary relevant/non-relevant). It measures how well the ranking places highly relevant documents at the top.

DCG and NDCG

$\text{DCG}@k={\sum }_{i=1}^k\frac{2^{re{l}_i}-1}{{\log }_2(i+1)}$

$\text{NDCG}@k=\frac{\text{DCG}@k}{\text{IDCG}@k}$

where:

• $re{l}_i$ — relevance grade of the document at position $i$
• $\text{IDCG}@k$ — ideal DCG (DCG of the perfect ranking)
• NDCG ∈ [0, 1], where 1 = perfect ranking

What It Captures

• Graded relevance: A doc with relevance 3 is worth more than relevance 1 (exponential gain: $2^{rel}-1$)
• Position discount: Higher-ranked results matter more ($1/{\log }_2(i+1)$ decay)
• Normalization: Dividing by IDCG makes scores comparable across queries

Comparison with Other Metrics

Metric	Graded?	Position-aware?	Normalized?
Precision	❌ Binary	❌	❌
Precision at K	❌ Binary	Partially (cutoff)	❌
MAP	❌ Binary	✅	✅
NDCG	✅ Graded	✅	✅

Appears In

• IR-L04 - Evaluation