What entropy actually measures
Entropy, measured in bits, quantifies how many possible passwords exist given a character pool and length — more bits means exponentially more possible combinations an attacker would need to try in a brute-force guessing attack. Entropy (bits) = length × log2(pool size), where pool size is the total count of distinct characters available (uppercase letters, lowercase letters, numbers, symbols — however many of those categories are actually used).
A worked example showing length beats complexity
A 12-character password using uppercase, lowercase, and numbers (a pool of 62 characters, no symbols) has about 71.45 bits of entropy. An 8-character password using all four character types including symbols (a pool of 80 characters) has only about 50.58 bits of entropy — meaningfully LESS, despite "looking" more complex with its symbols. The longer, simpler password is genuinely harder to brute-force than the shorter, more symbol-decorated one.
Why length has such an outsized effect
Entropy grows linearly with length but only logarithmically with pool size — adding one more character to a password multiplies the total possible combinations by the full pool size, while adding one more character CATEGORY (like symbols) only multiplies the pool size itself by a modest factor. This is why extending a password by a few characters typically strengthens it more than adding symbols to an already-fixed length.
A third worked example — combining both
A 16-character password using all four character types (the same 80-character pool as the 8-character example) has about 101.15 bits of entropy — exactly double the 8-character version's 50.58 bits, since doubling length at a fixed pool size exactly doubles entropy. It's also meaningfully more than the 12-character, 3-type password's 71.45 bits, showing that combining greater length with full character diversity beats either improvement applied alone.
What entropy doesn't capture
This calculation assumes a genuinely random password drawn uniformly from the specified pool — it does not account for predictable patterns (a dictionary word, a keyboard pattern like "qwerty," a name plus a birth year) that make a password far easier to guess than its raw entropy number suggests, since attackers commonly try likely patterns before resorting to pure brute force. A randomly generated password (like the kind produced by a password generator tool) genuinely achieves its calculated entropy; a human-chosen password following a memorable pattern typically does not, even at the identical length and character pool.