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# Grade Probabilities

# 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

| Step | What it does |
| ---- |
| Convert cut-offs to deltas | Puts grade boundaries on the logit scale |
| Logistic function | Converts the gap between a student's ability and each boundary into a probability |
| Cumulative → Discrete | Subtracts adjacent cumulative probabilities to get the chance of landing in each specific grade |

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.