The Kangaroo Maths contest (which admittedly isn't a graded test, but the first example I can think of) involves a multiple choice test, where you lose points for wrong answers, and just get zero points for a blank.
The number of negative points it calculated to give you 0 points on average for blind guessing. If you can eliminate a few possibility and guess between the rest, your average score goes up.
The SAT is similar. Quoting from Wikipedia: "For each correct answer, one raw point is added. For each incorrect answer one-fourth of a point is deducted. ... This ensures that a student's mathematically expected gain from guessing is zero."
Or at least from guessing if your knowledge is no better than a uniform random distribution. It would be interesting to design a test, where you could give your distribution directly. For a multiple choice test with exactly one single right answer, that would be easy to do: Give a percentage (adding up to 100%) for each possible answer. (Or to make it more foolproof, allow people to write down any positive number, and normalize by dividing all answers of a question by their sum.)
For multiple choice questions with a variable number of right answers, it becomes harder.
The Kangaroo Maths contest (which admittedly isn't a graded test, but the first example I can think of) involves a multiple choice test, where you lose points for wrong answers, and just get zero points for a blank.
The number of negative points it calculated to give you 0 points on average for blind guessing. If you can eliminate a few possibility and guess between the rest, your average score goes up.