How your mastery score works

This is a relatively short explanation of what is going on behind your mastery score: what it means, how it is calculated, what contributes to it, and how to get the most out of it.

Why is it here?

When I first built the site, I had no idea how many of you would actually use the exam simulator, so there was little reason to invest a huge amount of time in building a proper progression system. It turns out that many of you are using it very seriously, collectively completing thousands of questions each day. At that point, I felt compelled to build something better.

Fortunately, when I began researching the problem, I discovered something unexpected: some genuinely interesting mathematics. As someone who used to play all sorts of competitive online games with Elo rating systems, I found the problem particularly fascinating. I therefore spent considerably longer on it than I had originally planned, to the point that my wife and sons complained that I was coding on the beach while on holiday—which, to me, was perhaps the most enjoyable kind of holiday...

At last, I have a first version that I am happy to release.

The aim of the progression system

The purpose of the progression system is to answer a simple question:

How strong are you at TMUA right now?

A raw mark on its own cannot answer that reliably. Scoring 14 out of 20 on a very difficult paper may indicate greater ability than scoring 16 out of 20 on an easier one. Even within the same paper, answering a difficult question correctly should provide more evidence about your ability than answering an easy one correctly.

The progression system therefore looks beyond your total score. It takes into account the difficulty of the papers and questions you have attempted, then combines your results into a single estimate of your current ability, which I call your mastery.

This estimate is placed on a scale anchored to the official TMUA score-conversion tables for historical past papers. These tables provide a common reference point for both mastery levels and question difficulties, allowing results from different papers to be compared on a consistent basis.

The core idea

At its heart, the system places two things on the same scale: how strong you are, represented by your mastery score, and how difficult each question is, represented by its difficulty rating. Whether you answer a question correctly depends largely on the gap between the two. You are more likely to answer a question below your level correctly, less likely to answer one above your level correctly, and have roughly a fifty-fifty chance of answering a question at your level correctly. These are among the model’s basic assumptions. By examining the pattern of questions you answer correctly and incorrectly, the system estimates where you currently sit and continually updates that estimate as you improve.

This is based on well-established methods in educational measurement, particularly Item Response Theory and the Rasch model. UAT-UK uses a Rasch model in the current TMUA scoring process: since 2024, candidates have sat slightly different versions of the test, so the model is used to place results from those versions onto a common, comparable scale. The machinery behind your mastery score therefore draws on the same family of methods used by the exam board; I have simply adapted it for the questions and papers on this site.

If you are familiar with the Elo rating system used in chess and competitive games, the intuition is similar: your rating depends not simply on how often you succeed, but on the difficulty of what you succeed against. Although your mastery is updated only when you submit a complete paper attempt, the calculation is based on your responses to individual questions. The mathematics behind the model has been studied and tested for decades. I will continue refining my implementation as more data becomes available, but I believe it is now robust enough to provide a genuinely useful picture of your current level. A few references are included at the bottom for anyone who would like to explore the ideas further, and to learn some useful statistics!

What the number means

Your mastery is shown on a scale designed to resemble the TMUA score scale and is anchored to the official score-conversion tables for historical past papers. That is the important point: a mastery of, say, 7.5 is intended to represent roughly the same level of ability as a score of 7.5 on the actual TMUA, rather than being a private rating that makes sense only within this website.

The official TMUA score is capped at 9.0. Our mastery scale, however, extends to a maximum of 10.0, allowing us to distinguish more meaningfully between the very strongest students.

The historical TMUA scale provides the most reliable benchmark available because the papers were published together with their answer keys and score-conversion tables. The scoring system changed in 2024/25, and the exam board no longer publishes raw-score conversion tables. Unless otherwise stated, mastery and ability values on this site are therefore calibrated on the pre-2024 scale. I also provide an inferred post-2024 TMUA score equivalent as a guide to where your performance may fall on the current scale. However, this necessarily involves substantial inference, since the exam board has not published the underlying raw-score conversion tables.

Your mastery is shown both as a single figure and as a range. The single figure is the system’s current best estimate of your ability, while the range shows the uncertainty around that estimate—for example, a current mastery of 7.8 might be accompanied by a range of 7.6–8.1. No test can determine your ability to a perfect decimal from only a handful of questions. The more papers you complete, the more evidence the system has, and the narrower the range should generally become. A wide range is not a fault; it simply means that the system needs more evidence before it can estimate your level more precisely.

What counts towards your mastery calculation (and what doesn't)

I have made deliberate choices about what counts towards your mastery and what does not:

Exam-mode attempts on any paper count towards mastery, including repeated exam-mode attempts on the same paper, subject to the caveat below.

Everything else does not count, including practice mode and corrections.

The caveat is that if you repeat an exam-mode attempt on a paper you have completed before, the weight given to the new attempt in your mastery calculation is reduced according to how much time has passed since your previous attempt. The longer the gap, the more the new attempt will count towards your mastery.

Why repeated attempts are discounted

If you re-sit a question you attempted yesterday, you will almost certainly remember it. Getting it right again therefore provides very little new evidence about your ability, so the system gives that result very little weight. Another way to think about this is that memorising a question does not demonstrate increased ability; allowing it to count fully would distort your mastery score.

Memory fades, however. If you encounter the same question again after a sufficiently long gap, it becomes a fairer test and therefore counts for more. The weight given to a repeated question depends on how long it has been since you last saw it. As a rough guide, repeating a paper around two weeks after you last sat it should count substantially towards your mastery.

These rules are subject to change as the system is refined. Whenever a change is made, the system will retrospectively recalculate your mastery from day one. This is one of the core design goals of the system.

Does that mean there is no value in doing corrections or using practice mode?

No, not at all. My advice to my own students is to complete corrections as soon as possible. There is a significant difference between understanding a written solution and being able to reproduce the reasoning correctly yourself. The latter is essential for addressing gaps in your knowledge or technique, as well as weaknesses in particular question types that may be unique to you.

As for practice mode, my own view is that it should be used sparingly, mainly at the beginning when you are becoming familiar with the style of the questions or with how the simulator works.

Blanks

I have noticed that a significant number of students left many questions blank in some of their earlier attempts. In the actual TMUA, a blank answer is always marked as incorrect, so you should never leave a question unanswered. However, I suspect that some students historically left blanks because they were simply experimenting with the simulator rather than completing a genuine exam attempt.

I have therefore decided that:

Blank answers submitted before 00:00 GMT on 8 August 2026 will not count towards mastery. Any questions left blank before that time will simply be excluded from your mastery calculation.

Going forward, if you want the system to assess your current ability honestly, complete exam-mode attempts properly and do not leave any questions blank, as they will be treated as incorrect.

The topic breakdown

For now, this section should be treated with caution and regarded as a placeholder. I have included a preliminary version, but I have not yet developed the underlying theoretical model in substantial depth or properly validated it. I may choose to work on this next, but that has not yet been decided.

Getting the most out of it—and a word on honesty

Most students preparing for the TMUA are perfectly capable of understanding how their actions can influence a system like this, so I will simply offer one gentle reminder:

This tool can only give you useful information if you use it honestly.

There is little value in trying to inflate your mastery score. The solutions are readily available, and using them during an exam-mode attempt would only make the resulting estimate less useful to you. A higher number achieved in that way would simply make the system less useful for its intended purpose.

Used honestly, the system should do something genuinely valuable: give you a realistic indication of where you stand against the TMUA standard and allow you to see your progress over time.

If you'd like to read more

The ideas behind this system draw on a well-established body of work in educational measurement and rating systems. A few useful starting points are:

This is a living feature that I will continue to refine. If anything here is unclear, or if you spot something that looks wrong in your own numbers, please let me know—feedback genuinely helps.

— Joe