NDCG

CG

cumulative gain

for each query result, it has a relevant score ,

DCG

discounted cumulative gain

take the position of each result into account,

it is calculated as following in industral environment

where , the numerator using exprenation seems to winden the gap between top scores and low scores, and minus one is to reback to [0, 1]. Add 1 is to set the value of denominator into

NDCG

normalized discounted cumulative gain

The results size of each query is not the same, which means the max DCG of each query is not the same. So, DCG needs normalizing.

where the (ideal) is the best search result by human sorting.