To give DD school (and the SATs agency) some basic statistical literacy?

It's not a normal distribution. The majority of children meet or exceed the "expected level" (100).

Here's an overview of this year's scores, and scores over time:
www.tes.com/magazine/news/general/sats-results-2025-improvement-in-all-subjects

As others have said, that’s not how it works.

It’s just a way of standardising test scores which might differ from year to year, so they they can compare across years.

So there is a score they need to achieve to be hitting age-related expectation, which is always equated to 100.

There’s another checkpoint that relates to 110 which is greater depth.

The score changes every year but it’s no different from saying “The pass mark is 34 out of 50” except it’s translated into a standardised score.

I think your school have confused you with their communication.

For SATs, however, 100 is not the median, nor the mean. It is the bottom of the range of standardised scores that the Government has set such that c.25% fall below this range.

My DC haven’t done SATs yet so I’ve got that to look forward to, but why on earth is it so complicated?? Who is that meant to help? Our school got really good results looking at the final percentages, but I can’t get my head around the different ‘pass/fail’ numbers. Is this all explained to parents in Yr6?

Why can’t we just let teachers blooming well teach?!

Ok, OP, here is a rough explanation of your mistaken though process.

You are confusing repeated measurements of the same thing with multiple measurements of different related things in a population.

So if I measure a length of wood 100 times, I'll get 100 different numbers. These will probably roughly follow a normal distribution. Lets say I get a figure of 1007.1mm and the sd is 2.3mm.

I measure another piece of wood and get a figure of 1004.8. That is just outside 1sd, so we can say that there is a 68% (roughly) chance it is statistically a different length. Most sciences require a minimum of 2sd as proof of difference (so roughly 95% confidence it is different) - in this case it would not be considered different at the 2sd confidence interval.

In your case, you would need your daughter to do many SATS tests, marked and adjusted in the same way in order to assess the precision and accuracy of her score, and to be able to assess whether her score is statistically the same or different from another persons

And if you actually deal with children and their learning, it's very obvious that a child who has a Standardised Average Score of 85 is a very different child with very different educational needs to a child with one of 110.

If you then look at other modalities of testing, such as CAT4s or the New Group Reading Test, they can help to give a broader/clearer picture - they can highlight areas where there is higher ability or areas of difficulty, such as where literacy levels affect decoding questions or where their spatial reasoning is very high and go some way into identifying reasons or approaches that may be more effective in enhancing understanding of the areas measured in SATs.

Rather than the OP stressing about statistics in a vacuum, it's better to focus upon what the data says about their specific child. It's not an insult, it's not a time to start lecturing staff about it, it's a time to look at what needs the child has to be able to reach their full potential.

It's not a stretch, it's statistical nonsense. Assuming a normal distribution (which as others have suggested, isn't reasonable), 1sd just means that ~68% of results fall within that bracket, not that they are indistinguishable.

I'm not sure what subject OPs PhD is in, but I really hope it isn't a science. I'll take a punt on something social sciencey as its a field that has been known to have a fairly dubious relationship with stats - although to be fair I've encountered enough science PhDs who couldn't stat their way out of a wet paper bag.

As a PhD in a subject requiring statistics I know that 85 would be one standard deviation below the mean (100)

This is where your assumptions have created your error in understanding- 100 is NOT the mean. The mean is actually significantly higher than 100, though it is not a figure disclosed by the Government, iirc.

As a PhD in a subject requiring statistics
Well yes - doesn't make you the expert of standardised SATs scores though. You're coming across as rude and arrogant.

It is better for the student to just miss rather than just scrape a pass. If they fail they may get extra help at secondary school

Just reiterating - to say that 85 and 115 are statistically indistinguishable in a set of scores where the minimum is 80 and the maximum 120 , and 100 is artificially set at ‘a politically set number of around 75% pf children are above this level’ is obviously false.

You have to take into account the full range of scores available (so a much wider range than 80-120) and whether 100 is genuinely ‘the middle of the normal distribution’.

The government's figures for this year's SATS are that approximately three quarters of children reached required levels in reading and maths, slightly less for writing. Children's attainment does not fit a standard bell curve.; It's skewed.
Also there's a scaling model on the results, to ensure fairness, and the size of the population would support the result being statistically accurate. 120 is the highest score possible. 80 is the lowest score you can achieve, below which the student doesn't receive a score, they get an 'N'. So it's just not true to say that marks between 85 and 115 are statistically indistinguishable from 100.

I understand that you are disappointed but this model was designed by people with experience in the field, it wasn't sketched up on the back of a fag packet by your DC's headteacher. Unless you have reviewed the full data and the model, you should focus your energy on your daughter's hard work, and congratulate her on her progress not her position relative to her peers.

The raw scores will have a normal(ish) distribution (not completely normal, as those who would have very low scores do not take the tests and are assessed in an alternative way).

However, the standardised scores do not. The lowest possible standardised score is c. 80, the highest 120, and the 100 is artificially forced to create the right percentage of ‘expected standard’ - ie that the right number of children get 100-109; then the right number 110-120.

It’s a similar process for GCSE and A levels, where artificial ‘lines’ are drawn on the normal distribution of marks to give top and bottom of each grade.

Probably better to think of it - if you want to think of ‘expected is a range’, that the range of expected is 100-109. The actual peak of the normal distribution will be somewhere within that range.

Even if it were a normal distribution, the assertion that scores between 85 and 115 are statistically indistinguishable is a stretch.

There clearly is a vast difference between a score of 85 and 115 in any scenario

As an ex teacher who is reasonably mathematically literate,I have had this conversation with senior leaders many times.
It's very disappointing how mathematically illiterate many SLT and so called data managers are. A combination of poorly written assessments and poorly designed algorithems often lead to an output of total garbage, but as long as it prints a report with lots of pretty colours many school leadership teams are satisfied.
Ask them about the degree of uncertainty on the algorithem they are using and the size of cohort needed to give statistical significance.
I hate the way dodgy data is used to label children

Standardised scores aren’t spread across a bell curve. They take their raw scores and adapt it to allow for age differences within a year group because there’s a big difference between a child born in August and one in July taking the same test - standardised scores even this indifference out.

A standardised score of 100 shows average level intelligence. 115 is children who are performing very well. 85 less so. The school isn’t wrong, you’re applying your statistical background inaccurately within this context. There’s a big difference between those scores, sorry.

I think 100 is just the 'scaled score' to meet the 'expected standard' - whatever that standard is judged to be. The 'scaling' adjusts raw scores for small differences in test difficulty between years.

So, no I don't think you're right when you say that half of children will underperform.

However, it's right to highlight some of the problems with testing. For instance, if you re-tested all the children who fell below 100 one week later, many would improve without any extra help. Explanation: because you're selecting the lowest performers, some of whom may have had a 'bad day' or not been feeling well or just randomly underperformed on the original test.

All testing is imperfect system.

No, you are incorrect. It's not 50/50 at all. Between 70-75% of pupils passed each subject last year meaning they scored 100 or over in the scaled score(75% in reading, 74% in maths, 72% in writing and 73% in grammar and spelling)