A good test doesn’t just count right answers — it places every student AND every question on one shared scale of difficulty. The Rasch model says: your chance of getting an item right depends only on how far your ABILITY sits above its DIFFICULTY, on a graceful S-curve. Slide the ability and watch the odds climb.
The Rasch (1PL) model: P(correct) = 1 / (1 + e^(−(θ−b))), where θ is a person’s ability and b an item’s difficulty, placed on the SAME logit scale. When ability equals difficulty the chance is exactly 50%; above it the S-curve rises toward 1, below it falls toward 0. The instrument draws three item curves (easy/medium/hard) and reads off P for the current θ. A fail-loud self-check throws unless P(θ=b)=0.5 exactly and P rises monotonically with ability — the property that lets one test rank students and questions together.
Rasch is the 1-parameter model (difficulty only); richer IRT adds discrimination (2PL) and guessing (3PL). Its strong assumption — all items equally discriminating — is a feature (objective measurement) that real data must be checked against. The logistic and the θ=b→0.5 fact are exact.