In this paper we consider the following problem: given π independent samples π = (π 1 , . . . , π π ) from an unknown distribution P over passwords ππ€π 1 , ππ€π 2 , . . . can we generate high confidence upper/lower bounds on the guessing curve π πΊ πΊ π=1 π π where π π = Pr[ππ€π π ] and the passwords are ordered such that π π β₯ π π+1 . Intuitively, π πΊ represents the probability that an attacker who knows the distribution P can guess a random password ππ€π β P within πΊ guesses. Understanding how the guessing curve π πΊ increases with the number of guesses πΊ can help quantify the damage of a password cracking attack and inform password policies. Despite an abundance of large (breached) password datasets upper/lower bounding π πΊ remains a challenging problem. We introduce several statistical techniques to derive tighter upper/lower bounds on the guessing curve π πΊ which hold with high confidence. We apply our techniques to analyze 9 large password datasets finding that our new lower bounds dramatically improve upon prior work. Our empirical analysis shows that even state-of-the-art password cracking models are significantly less guess efficient than an attacker who knows the distribution. When πΊ is not too large we find that our upper/lower bounds on π πΊ are both very close to the empirical distribution Ξ»πΊ which justifies the use of the empirical distribution in settings where the guessing number πΊ is not too large i.e., πΊ βͺ π closely approximates π πΊ . The analysis also highlights regions of the curve where we can, with high confidence, conclude that the empirical distribution significantly overestimates the real guessing curve π πΊ . Our new statistical techniques yield substantially tighter upper/lower bounds on π πΊ though there are still regions of the curve where the best upper/lower bounds diverge significantly.