TMUA score conversion: what your raw mark out of 40 is actually worth

Your TMUA result is one number between 1.0 and 9.0, covering both papers together.

There is no total out of 40 on your results slip, no pass mark, no grade boundaries, and no published table that turns your raw mark into that number.

That last one is not the exam board being secretive.

It is because you and the person sitting next to you did not take the same paper.

Several versions of the test run across the testing window so that people can sit on different days.

Some versions land slightly harder than others, so 27 out of 40 on one paper is not the same achievement as 27 on another.

Your mark is converted afterwards to account for that, which is why nobody can hand you a single fixed table.

Three things follow, and between them they cover most of what people are really asking when they search for boundaries.

Papers 1 and 2 are scaled together. There is no separate Paper 1 score, so there is no per-paper boundary for one to sit on.

There is no appeals process. UAT-UK state plainly that they do not accept appeals of test scores.

And your score is not a percentage. A 4.5 does not mean you answered half the questions correctly.

It means you landed where the middle of the October cohort landed, whatever raw mark that took.

What a TMUA score is, precisely, is a rank. It tells a university where you sat relative to everyone else who took the test that cycle.

The rest of this page is the useful version of all that: what your mark is roughly worth, whether any of it is fair, and how much of your score is luck.

If what you actually want to know is whether a particular number is good enough for the course you are applying to, that is a different question with its own page: what is a good TMUA score.

And if you have landed here before working out whether you even need to sit the test, start with the complete TMUA guide instead.

Here is the thing almost nobody mentions. A TMUA score conversion table does exist.

UAT-UK build one every year.

In their words, once the testing window closes "the item difficulties are used to generate a raw score to theta (or ability) table", and that ability figure becomes your 1.0 to 9.0 score.

The catch is that there is one of those tables for every version of the test, in every sitting. Each is built after you have already sat it, and none of them are ever published.

So the question is not whether a conversion exists. It is that yours is calculated privately, afterwards, for a paper only some candidates sat.

Which leaves the honest version below: the same table built three times, because your mark is worth more on a hard version than an easy one, and nobody outside the exam board knows which one you had.

Here's how that breaks down:

Find your mock mark on the left and read across.

A raw 28 out of 40 is a 5.5 if the version was gentle, a 6.6 if it was average, and a 7.5 if it was hard. That is the difference between an ordinary score and a top-10% one, on exactly the same performance.

Which is why any TMUA score calculator you find online is running an estimate much like this one, whether it says so or not.

The real conversion for your paper is never released, so a tool that hands you one confident decimal is hiding a guess, not avoiding one.

How many marks is one TMUA point worth?

Whichever column you were in, the gaps between the rows stay the same. Moving up the table always costs about the same number of marks.

Roughly four extra marks out of 40 is worth one scaled point.

That is the genuinely useful takeaway, because it turns your practice scores into something meaningful. Eight marks better than last month is about two points.

It also tells you what a target is worth in marks. The gap from the middle candidate at 4.5 to the top 10% at 7.0 is 2.5 points, which is about ten raw marks out of 40.

One number in that working is worth pulling out on its own, because it is published rather than derived.

The report puts the measurement noise at 2.58 to 2.71 marks out of 40. So a two-mark or three-mark movement between two practice papers is not an improvement.

It is the instrument wobbling.

Chase eight marks. Ignore two.

The half of the conversion UAT-UK do publish

Once the October results are in, two points on the scale are fixed by decision rather than measured. The middle candidate is placed at 4.5, and the candidate at the top 10% mark is placed at 7.0.

Here's how that breaks down:

A straight line is then drawn between those two points, and UAT-UK publish its numbers.

Here's how that breaks down:

The same constants were used for January, so an October 6.5 and a January 6.5 mean the same thing.

So why can you not just use that on your mock mark?

Because "ability" is not your mark out of 40. Getting from one to the other needs the private table built for the exact version you sat, and that is the half nobody publishes.

You have the second half of the map and not the first.

What to do with a raw mark in the meantime is covered lower down, in what a raw mark on a past paper tells you, along with the conversion tables already circulating online and what they actually are.

The papers themselves are free, with UAT-UK's own worked answers: every official TMUA past paper, 2016 to 2023.

This is the question that actually worries people, usually about ten minutes after they walk out and discover a friend had different questions.

The short answer is yes, it is fair, and correcting for this is the single thing the whole scoring system exists to do.

Here is the process end to end, and none of it happens on the day you sit the test.

Because the table at stage three is built for your version specifically, a 27 on a harder version converts to a higher score than a 27 on an easier one.

So drawing the hard paper does not cost you, and drawing the easy one does not help.

One detail worth getting right, because it is widely misunderstood: it is your total that gets converted, not which particular questions you got right.

Two people who sat the same version and both scored 27 get the same result, even if they got a completely different 27.

There is a price for all this, and you have already paid it.

None of the calculation can start until everybody has finished, which is why no score appears on screen when you click submit, and why results take about four weeks.

It is also why UAT-UK can use questions that have never been pre-tested and still scale them "based on the observed candidate population as opposed to a benchmark population".

In the 2025/26 cycle, 100% of the items used met their statistical quality criteria, against a typical target of 80%.

One thing worth knowing if you are sitting more than one admissions test:

UAT-UK score TARA and ESAT the same way, with the same 1.0 to 9.0 scale and the same conversion built after the window closes.

UAT-UK publish a technical report for the TMUA every year. The current edition is the TMUA Technical Report 2025-26, published July 2026, and it contains the score distribution in full.

These are the numbers that actually answer "where did I come", and they are free.

The previous cycle's report, published 24 September 2025, gives the same breakdown for 2025 entry.

Two things in those tables are worth pulling out, because they are not obvious at a glance.

The full-cohort median is 4.1, not 4.5. The 4.5 anchor is set on the October cohort alone, so once January is folded in, the middle of the whole year sits lower.

That single fact explains most of the confusion online about whether the "average" TMUA score is 4.5, 4.1 or something else. Both numbers are correct; they describe different groups.

Candidate numbers are climbing. 13,855 in 2025 entry became 16,092 in 2026 entry, a rise of roughly 16% in a single cycle.

With Oxford adopting the TMUA and Cambridge adding Mathematics for 2027 entry, the next cohort will almost certainly be larger again.

One number the reports carry is worth knowing before you sit: the median candidate used 2 hours 29 minutes and 37 seconds of the 2 hours 30 minutes available, and 91% used at least 2h 15m.

UAT-UK's own conclusion is that the test "might be speeded", which is their term for a test where time pressure is part of what is being measured.

That is a preparation finding rather than a scoring one, and it is the most useful line in the report: plan on running out of time, because nearly everyone does.

What a given number is worth to the course you are applying to is a separate question, and it has its own page: what is a good TMUA score.

For most people this is decided for you, so check first. Cambridge and Oxford applicants must sit in October.

Everyone else has a choice.

If you do have one, here is the piece of evidence nobody talks about, sitting in plain sight in two consecutive technical reports.

January candidates score substantially lower than October candidates, on a scale that is explicitly built to make the two comparable.

The gap at the top end is just as stark. The 90th percentile was 7.0 in October and 5.9 in January for 2026 entry.

In other words, a score that put you in the top 10% of the January cohort would have put you around the top 25% of the October one.

Now the part that matters, and the part we will not fudge.

The obvious driver is self-selection, and UAT-UK point at part of it directly:

Cambridge applicants must sit in October, which is why October carries the larger volume.

Oxford applicants must sit in October too. So the October cohort is loaded with candidates applying to the most competitive mathematics courses in the country.

January, by contrast, draws more candidates applying to courses where the TMUA is recommended rather than compulsory, and more who decided late.

The cause is not established in any published source, and we are not going to invent one. What is established is the pattern, twice, in the same direction.

It is tempting to read that as "January is easier to stand out in", and we would not push it that far. Sit when your course requires, and when you will actually be ready.

The deadlines for both windows are on the TMUA dates and registration page.

Probably not exactly, and it is worth knowing by roughly how much before you spend a week agonising over a decimal place.

A TMUA score is not exact. It is an estimate, and UAT-UK publish how big the error bar around it is.

The measure is the standard error of measurement, and the 2025-26 report gives it on both scales, which is unusually transparent and very useful.

The reliability figures are genuinely good. Anything above 0.80 is the conventional threshold and every version of the test cleared it.

But the scaled figure is the number to sit with. It is roughly 0.6 of a scaled point.

UAT-UK spell out how to read it: there is about a 68% chance your true score lies within one of those of the score you were given, and about a 95% chance it lies within two.

Translated into something usable:

Which means a candidate on 6.0 and a candidate on 6.5 are, statistically, not reliably distinguishable from one another.

Three things follow, and they are worth more than any target-score advice.

Chasing a decimal is not a strategy. The difference between 6.2 and 6.4 is inside the measurement noise of the instrument, and in raw marks it is barely one question.

The decimal still counts anyway. Universities get your score exactly as calculated, to one decimal place, and nothing stops an admissions team preferring a 6.3 to a 6.0 when they are separating hundreds of similar applicants.

So do not read any of this as permission to relax about small gaps. A 6.0 and a 6.3 are not a real difference in ability, and they can still be a real difference in outcome.

Which is an argument for aiming past your target, not at it. You cannot control which end of that band you land on, so the only protection is to be far enough clear that a bad day still puts you where you need to be.

Universities know all this. It is one reason Imperial and Cambridge decline to publish thresholds, and why Warwick describes its own 5.0 figure as where the majority of offers happened to fall rather than as a cut-off.

And it tells you what to work on. Chasing two marks is chasing noise.

Eight marks is two full points, which is the kind of gap no amount of wobble hides.

This is also why we will not ship a one-number score predictor.

Turning a past-paper raw mark into a single confident 1.0 to 9.0 score would mean pretending to a precision the test itself does not claim, which is exactly why our conversion table shows three columns rather than one.

A lot of TMUA searching attaches a year to the word "boundaries", or to the word "conversion". The answer is the same every time, but what sits behind it changes, so here is each year on its own.

The short version, before the detail: no year has ever had grade boundaries, and no year has a raw-to-scaled score conversion published by the organisation that ran it.

TMUA 2026 grade boundaries

None, and there will be no 2026 score conversion published either. October 2026 and January 2027 are the current cycle, and results are not out yet: the October ones land on 16 November 2026.

The scale will be built the same way it was last year, with the median fixed at 4.5 and the 90th percentile at 7.0, both computed from whoever sits in October.

TMUA 2025 grade boundaries

None. October 2025 and January 2026 make up the 2026-entry cycle, and it is the one the technical report above covers in full.

No conversion was published for it, but the distribution was: median 4.1 across both sittings, 4.5 in October and 3.6 in January. Our estimate of what a raw mark from this cycle is worth is in the conversion table above.

TMUA 2024 grade boundaries

None, and this is the year the ground moved. October 2024 was UAT-UK's first TMUA sitting and the first delivered on computer at Pearson VUE.

It is also where the paper trail stops. No sitting from 2024 onwards has been released, so the free archive ends at 2023.

For the cycle itself: median 4.0 across both sittings, mean 4.20. See the 2024 scale reset for why nothing before it compares.

TMUA 2023 grade boundaries

None, and no 2023 score conversion from anybody official. 2023 was the last Cambridge Assessment sitting and the last on paper.

It is the year people search for most, and the one most likely to turn up a conversion table somewhere on the web.

Those tables come from neither UAT-UK nor Cambridge Assessment, and they describe a scale that no longer exists, which the section below goes through.

The papers themselves are worth having.

They are the closest match to the current test in the whole archive, and both come with official worked answers:

TMUA 2023, paper 1 and paper 2.

TMUA 2022 grade boundaries

None, and no published conversion. Cambridge Assessment era, sat on paper.

One thing makes 2022 worth handling carefully: the official interactive sample test is this sitting, question for question. Do one and you have spent the other, so pick which.

Both papers are free with worked answers.

TMUA 2021 grade boundaries

None, and nothing resembling a conversion table. Cambridge Assessment era.

2021 is the best-supported year in the archive. Alongside the free papers and worked answers, Warwick's maths department has published video walkthroughs of every question on it.

TMUA 2020 grade boundaries

None, and no conversion published for it. Cambridge Assessment era, paper-based.

Squarely mid-archive, and useful volume rather than a format rehearsal. Both papers and the worked answers are free.

TMUA 2019 grade boundaries

None. No boundaries, no pass mark and no conversion table, exactly as in every other Cambridge Assessment year.

The 2019 papers and worked answers are free to download.

TMUA 2018 grade boundaries

None, and no published conversion. Cambridge Assessment era.

Treat this one the way you treat 2022: the official interactive specimen test is the 2018 sitting, so working through the 2018 papers spends your specimen test as well.

TMUA 2017 grade boundaries

None, and no conversion. The second year the test ran, still under Cambridge Assessment.

Both papers come with worked answers, free.

TMUA 2016 grade boundaries

None. The first year the TMUA existed, so there was no precedent to publish boundaries against even if anybody had wanted to.

The oldest papers in the free archive, and the furthest from the current test, but the question style is the part UAT-UK say has not changed.

If somebody has told you what score you need, and they learned it before 2024, they are quoting a number from a scale that no longer exists.

When UAT-UK took over the TMUA for October 2024, they did not simply continue the old scale. The scale was rebuilt.

The clearest published evidence comes from Durham, who state that a 2023 score of 6.5 equates to 5.0 today.

Two structural changes sit underneath that shift, and both matter.

Scores used to be reported per paper. Under Cambridge Assessment you received a Paper 1 score and a Paper 2 score.

Today you receive one combined score, and only one.

That is why old conversion tables have separate columns for each paper. They are not simply out of date.

They describe a reporting structure that no longer exists.

The anchoring changed. The current approach pins the median at 4.5 and the 90th percentile at 7.0 on the October cohort.

Whatever the old scale did, it did not produce the same mapping, or Durham's 6.5 and 5.0 would be the same number.

The practical rule is blunt and worth following.

What did not change is worth stating in the same breath, because it is the reason the old papers are still worth doing.

UAT-UK say directly that "both the content specification and question style are unchanged, so this archive of past papers can be used when preparing for the test".

So the papers transfer. The scores do not.

Results arrive roughly four weeks after your sitting, and the dates for the current cycle are fixed.

Full booking deadlines, access arrangements and the bursary timeline live on the TMUA dates and registration page.

Who sees your score is the part candidates most often get wrong, so here it is plainly.

You do not choose. Results are sent automatically to participating institutions listed in your UCAS application, matched course by course.

There is no mechanism to withhold a score from one university and send it to another. There is no "send my best result" step, because there is only ever one result per cycle.

You cannot resit within a cycle. One sitting per admissions cycle, and Oxford and Cambridge applicants must use the October one.

And there are no appeals. UAT-UK state that they do not accept appeals of test scores.

That is a harder line than most qualifications, and it follows from the scoring model: there is no marker's judgement to contest. The answers are multiple choice and the scaling is applied to the entire cohort by the same formula.

Two universities publish something concrete about what they do with the number, and only two.

Warwick state that for 2025 entry the majority of offers went to applicants scoring 5.0 and above, while also saying the exact requirement will not be published.

Durham use 5.0 as the threshold for automatic eligibility for a reduced offer, and treat lower scores as positive evidence down to a threshold set annually.

Imperial publish an anonymised dashboard of score distributions and offer outcomes by department, which is the richest primary data any university offers.

Cambridge, Oxford, LSE and UCL publish nothing. Any specific threshold you see quoted for those four is somebody's guess.

You have just sat a past paper and scored 27 out of 40. The question is what that means.

Our estimate of what it is worth is in the conversion table above, in three versions, because the conversion depends on the difficulty of the specific version you sat and on the ability distribution of the live cohort who sat it alongside you.

Neither of those exists for a past paper you sat alone at your kitchen table, which is why the answer comes as a range rather than a number.

The conversion tables you will find online

Search for a TMUA conversion table and you will find some. It is worth knowing exactly what they are, because the answer is more interesting than "they're fake".

We checked the two most widely cited sources in August 2026, and what they actually publish is not what people think.

So the picture is clearer than it first appears, and it resolves in a way that is worth stating directly.

UAT-UK publish no raw-to-scaled conversion table, for any cycle. That is consistent across their results page and their technical reports.

The tables circulating elsewhere are Cambridge Assessment-era, carry no primary source, and - by the publisher's own description - describe a per-paper reporting structure that no longer exists.

They are not forgeries. They are archaeology, and they are being read as if they were a map.

Treat any conversion table you meet the same way, ours included: ask which cycle it came from, whether it assumed per-paper scores, and whether it tells you what it had to assume. A table built on a 2021 sitting tells you very little about a 2027 one.

What to track instead

Raw marks are still worth recording. They are just worth recording as raw marks.

Track your mark out of 40 over time. The trend across several papers under real conditions is far more informative than any single conversion would be.

Sit them properly if you want the number to mean anything:

75 minutes per paper, both papers back to back, no calculator, no formulae booklet, and nothing to write on but a whiteboard.

Then compare the shape of your performance, not the score. Are you losing marks late in each paper, where the questions are hardest?

Nearly everyone does.

Our own question-by-question analysis of all 360 questions across the nine official sittings - the 2016 to 2023 archive plus the early specimen set - found difficulty climbs almost without exception with position, from the easiest band at question 1 to the hardest at question 20, on both papers.

UAT-UK confirm this is deliberate: the items are "designed to progressively increase in difficulty throughout each section".

The papers themselves are free, ungated, and come with UAT-UK's own worked answers for every year: every official TMUA past paper, 2016 to 2023.