How Grade Probabilities Are Calculated

1. Setting Up the Scale

Each grade (level) has a cut-off score — the raw score boundary that separates it from the next grade up. These raw cut-offs are converted onto a logit scale (a mathematical scale used in measurement theory) using two parameters fetched from the database: a mean and a scale factor.

delta = (cutOff - mean) / scale

The result, called delta, is the grade boundary expressed in logit units. Think of it as repositioning the boundary onto a common ruler that all students' scores can be compared against.

2. Each Student's True Score

Every student also has a true score on the same logit scale. This is their estimated underlying ability, as opposed to their raw or scaled score.

3. Cumulative Probabilities — "At Least This Grade"

For each grade boundary, the code asks: "What's the probability this student is at or above this grade?"

This uses the logistic function:

P(grade ≥ k) = 100 × exp(trueScore − delta_k) / (1 + exp(trueScore − delta_k))

• If a student's true score is well above a boundary, this probability approaches 100%.
• If it's well below, it approaches 0%.
• If it's right at the boundary, it's exactly 50%.

The lowest grade is a special case — every student is guaranteed to be at or above the bottom grade, so its cumulative probability is always 100%.

4. Discrete Probabilities — "Exactly This Grade"

Cumulative probabilities tell us "at least grade k", but we want "exactly grade k". The conversion is simple subtraction:

P(grade = k) = P(grade ≥ k) − P(grade ≥ k+1)

So the probability of being in a specific grade is the gap between its cumulative probability and the one above it. The top grade has no grade above it, so its discrete probability equals its cumulative probability directly.

Summary

The end result for each student is a probability distribution across all grades — rather than a single definitive grade — which reflects the uncertainty in measurement.

Updated on: 16/04/2026