Methodology

This page describes the statistical model behind the grade prior calculator. The approach uses a latent ability factor model fitted to 15 years of Oxford PPE examiners' report data, with Monte Carlo simulation to estimate classification probabilities.

1. The model

Each paper's mark is modelled as a draw from:

$$\text{mark}_i = \mu_i + \lambda_i \cdot \theta + \varepsilon_i$$

The key idea is a variance decomposition: the total spread in marks on any paper ($\sigma_i^2$) is split into two components:

• Ability component ($\lambda_i^2 = \sigma_i^2 \rho$): the part of variance explained by how good you are overall. Papers with higher total spread are assumed to be more discriminating.
• Noise component ($\sigma_i^2(1-\rho)$): residual randomness — exam-day luck, marker variation, topic lottery.

The parameters:

• $\mu_i$ — paper-specific mean mark.
• $\sigma_i$ — paper-specific standard deviation (total spread). Both $\mu_i$ and $\sigma_i$ are fitted jointly by MLE.
• $\theta$ — standardised latent ability. The ability slider sets $\theta = \Phi^{-1}(\text{percentile})$, where $\Phi^{-1}$ is the standard normal quantile function. So the 75th percentile → $\theta \approx 0.67$.
• $\lambda_i = \sigma_i \sqrt{\rho}$ — ability loading, proportional to paper spread.
• $\varepsilon_i \sim \mathcal{N}(0,\; \sigma_i^2(1-\rho))$ — residual exam-day noise, independent across papers.

Why truncated normal?

Marks are bounded in $[0, 100]$ and typically clustered in the 55–75 range. An ordinary normal would place implausible probability mass below 0 or above 100. The truncated normal respects these bounds and better fits the observed compression of marks near class boundaries.

Fitting $\mu_i$ and $\sigma_i$

The examiners' reports give band counts: for each paper, the number of candidates scoring 70+, 60–69, 50–59, 40–49, 30–39, and <30. We fit a truncated normal $\mathcal{N}(\mu_i, \sigma_i^2)$ truncated to $[0, 100]$ by maximising the multinomial log-likelihood of these bin counts. Data is pooled across all available years (2017–2022, 2024–2025) excluding 2020.

Calibration of $\rho$

The inter-paper correlation $\rho \approx 0.196$ is calibrated so that the model reproduces the observed ~23% first-class rate when averaged across the full ability distribution (integrating over $\theta \sim \mathcal{N}(0,1)$). This was done via binary search on 500k simulations with the 8 most popular papers.

Note that at $\theta = 0$ specifically (the median student), the First rate is only ~11%. The population average is pulled up by the right tail — analogous to how mean income exceeds median income.

2. Classification rules

PPE uses conjunctive classification — candidates need both an average threshold and a minimum count of papers at the relevant mark level:

These conjunctive rules create non-linear interactions with paper variance — for instance, a single mark below 50 vetoes a First regardless of average, making high-$\sigma$ papers risky even when their mean is above 70.

3. Simulation

For a given set of 8 papers and ability percentile, the tool draws $N = 50{,}000$ independent exam sittings. Each draw:

1. Computes the shifted mean for each paper: $\tilde{\mu}_i = \mu_i + \sigma_i \sqrt{\rho} \cdot \theta$
2. Draws $\varepsilon_i \sim \mathcal{N}(0, \sigma_i^2(1-\rho))$ independently for each paper
3. Clips $\text{mark}_i = \max(0, \min(100, \tilde{\mu}_i + \varepsilon_i))$
4. Classifies the 8 marks using the conjunctive rules above

The reported probability for each class is the empirical frequency across all $N$ draws. Uncertainty from finite simulation is negligible ($\lt 0.1\text{pp}$ at $N = 50{,}000$); the reported $\pm 3\text{pp}$ uncertainty reflects model limitations rather than Monte Carlo error.

4. Data

Source data is extracted from Oxford PPE Final Honour School internal examiners' reports, 2011–2025. Mark distributions come from band data: the percentage of candidates achieving each class-level mark on each paper. 79 papers are fitted in total — 63 via MLE on band data, 16 via method-of-moments from reported mean and standard deviation.

Two years receive special treatment:

• 2020 (COVID): excluded from all fitting. The first-class rate doubled (~40%) despite paper-level marks barely shifting (+0.3 on average). The anomaly was caused by modified classification conventions: the lowest two passing results were dropped as a safety net, while the marking scale was unchanged.
• 2023 (marking boycott): no per-paper data available; excluded by data absence.

5. Limitations

• Uses aggregate band data, not individual marks — the true joint distribution across papers for any one candidate is unobservable.
• Assumes a single latent ability factor with constant $\rho$ across all paper pairs. In reality, within-subject correlations are likely higher than cross-subject correlations.
• No conditioning on prior performance — the ability slider is a self-assessment, not a calibrated prediction from collections data.
• Temporal trends exist for some papers (e.g. Microeconomic Analysis: +1.65 marks/year) but are ignored in simulation, which uses pooled estimates.
• Estimates are priors — they describe what has happened historically for similar paper combinations, not what will happen to any individual candidate.