Information gain (entropy)

A feature $A$ splits the examples $S$ into subsets $S_i$, each of these subsets is a new branch in the decision tree which have their own entropy. We can measure how ‘good’ $A$ is by the reduction in entropy of $S$ and entropy of all of the $S_i$, this difference is called the information gain:

$$\text{Gain}(S,A)=H(S)-\sum _i\frac{|S_i|}{|S|}H(S_I)$$

So the entropy of $S_i$ collectively is the weighted sum of their entropies.