We show that unbounded fan-in boolean formulas of depth d + 1 and size s have average sensitivity O ( 1 d log s ) d . In particular, this gives a tight 2 ( d ( n 1 d − 1)) lower bound on the size of depth d + 1 formulas computing the PARITY function. These results strengthen the corresponding 2 ( n 1 d ) and O ( log s ) d bounds for circuits due to Håstad (1986) and Boppana (1997). Our proof technique studies a random process where the Switching Lemma is applied to formulas in an efficient manner.