Question re dyslexic representation in pupil sample

Sorry, did not know where to post this, but I know we have some pretty good mathematicians here and I was wondering if one of you could help.

I am going to be representing my son next month, arguing that a school's entry procedures actively discourage applications from pupils with dyslexia.

This school received 684 applications from pupils. According to national statistics, about 7% of pupils aged up to 11 have been diagnosed as dyslexic. The school received one application from a dyslexic pupil (my son).

Is it possible to calculate the odds that this situation could have arisen 'by chance'?

I realise the reasons for such a low representation could be many and varied, but I just wanted to get across the point that 'something is going on' before exploring what the problems are.

Originally posted by @rank-outsider
Sorry, did not know where to post this, but I know we have some pretty good mathematicians here and I was wondering if one of you could help.

I am going to be representing my son next month, arguing that a school's entry procedures actively discourage applications from pupils with dyslexia.

This school received 684 applications from pupils. Acc ...[text shortened]... ted to get across the point that 'something is going on' before exploring what the problems are.

Just a quick look says there should have been 47 diagnosed with dyslexia in your group. So it sounds like there are kids in your group not diagnosed. It would be accurate if all the kids had been diagnosed with or not having dyslexia. My son had it very bad but was able to work through it and graduated with MA at Bryn Mawr university. My wife has it also and was clocked at having an IQ of 156, our son, about 130. I wonder if there is a higher proportion having dyslexia in the higher IQ types? I have a funny version, when I read a word, I subconsciously try to work it out backwards, understanding full well the content but more like a mental exercise for no good purpose

assuming that national statistics of 7% accurately indicate the percentage of pupils with dyslexia, there is 0.07 probability that, if you randomly pick any pupil in the country with equal probability of picking any one, he/she will be dyslexic.

I assume what you really want is the answer to this specific question;

If you randomly picked 684 pupils out of the whole country and if 7% of them are dyslexic, what is the probability of randomly picking NOT more than one dyslexic pupil? (so either pick only just 1 or none)

This requires using the mathematics of what is called the binomial distribution ( see https://en.wikipedia.org/wiki/Binomial_distribution ) and, more specifically, what is called the cumulative distribution (CDF) function for the binomial distribution.

The equation for the CDF for that is;

Pr(X ≤ k) = ∑[i=0, k] C(n, i) p^i (1 – p)^(n – i)
where
n = number of samplings (called "trials" ) which in your application equals 684.
k = number of 'successes' which in your application equals 1.
p = probability of success which in your application equals 0.07.

If you plug in those figures you get;

Pr(X ≤ 1) = ∑[i=0, 1] C(684, i) 0.07^i (1 – 0.07)^(684– i)
≈ 1.453 × 10^-20

which is a minute chance!
So there is less than one in a billion billion chance of randomly picking not more than one dyslexia out of 684 pupils i.e. that much chance of picking either ONLY one or EXACTLY zero dyslexics.

Incidentally, I am a dyslexic and did terribly at school, disastrously in fact because couldn't do the exams, as a child as a direct result but later kind of 'catched up' and did very well at university as an adult. Now most of the time you would find it hard to tell that I am naturally dyslexic. In fact, my Englidelicious is now perfect

Originally posted by @humy
assuming that national statistics of 7% accurately indicate the percentage of pupils with dyslexia, there is 0.07 probability that, if you randomly pick any pupil in the country with equal probability of picking any one, he/she will be dyslexic.

I assume what you really want is the answer to this specific question;

If you randomly picked 684 pupils out ...[text shortened]... find it hard to tell that I am naturally dyslexic. In fact, my Englidelicious is now perfect

That verifies my assumption that those students of the 600+ were not tested for dyslexia or a poor test/tester.

Here is one study result:

http://www.austinlearningsolutions.com/blog/38-dyslexia-facts-and-statistics.html

It says about 1 in 5 school kids are dyslexic. 1 in 10 for the general population. If the 20% number is true then the one in a billion billion would be even further out of whack.

My guess is they are hiding something so they don't have to spend money on special ed teachers. Call my cynical but that's how it looks to me. They are cooking the books. We need to be booking the cooks!

Originally posted by @rank-outsider
Sorry, did not know where to post this, but I know we have some pretty good mathematicians here and I was wondering if one of you could help.

I am going to be representing my son next month, arguing that a school's entry procedures actively discourage applications from pupils with dyslexia.

This school received 684 applications from pupils. Acc ...[text shortened]... ted to get across the point that 'something is going on' before exploring what the problems are.

By what criteria do they select students?

If it is by academic achievement, then of course those with learning disabilities will be put at a disadvantage. It is only natural.

If not, then the school may be trying to save money by keeping out special needs kids who would require more money spent on them with promise of lower outcomes.

Originally posted by @humy
assuming that national statistics of 7% accurately indicate the percentage of pupils with dyslexia, there is 0.07 probability that, if you randomly pick any pupil in the country with equal probability of picking any one, he/she will be dyslexic.

I assume what you really want is the answer to this specific question;

If you randomly picked 684 pupils out ...[text shortened]... find it hard to tell that I am naturally dyslexic. In fact, my Englidelicious is now perfect

Was this a randomly selected group?

Originally posted by @rank-outsider
Sorry, did not know where to post this, but I know we have some pretty good mathematicians here and I was wondering if one of you could help.

I am going to be representing my son next month, arguing that a school's entry procedures actively discourage applications from pupils with dyslexia.

This school received 684 applications from pupils. Acc ...[text shortened]... ted to get across the point that 'something is going on' before exploring what the problems are.

I sympathise with you.

A healthy percentage of dyslexics can be helped with some really low-tech solutions
such as "reading windows" (just a piece of card with a window of two word length), or
coloured acetate film, or combination of both.

Nessy is a great software package (8 years ago - probably more now) that helps
dyslexics.

And finally, humy's probability is based on a uniform distribution of dyslexia across
ability which is probably not the case, but who cares about probabilities when it is
your child?

Dyslexia should be viewed as a disability, but one that can be easily circumvented.

Good Luck.

Originally posted by @eladar
Was this a randomly selected group?

I didn't imply it was and I made no implicit assumptions hence my use of the words "assuming..." and "if..." in that post. Are you trying to make a point? If so, I am not a mind reader, so just say it.

Originally posted by @humy
I didn't imply it was and I made no implicit assumptions hence my use of the words "assuming..." and "if..." in that post. Are you trying to make a point? If so, I am not a mind reader, so just say it.

Whoa
Not trying to make any point, just trying to help.

Wear steel toe-caps next time in case I accidently step on your delicate toes.

Tolerance doesn't guarantee reason but intolerance guarantees none.

Originally posted by @wolfgang59
Not trying to make any point, just trying to help.

But I was speaking to Eladar, not to you!
I am the one that is supposed to have dyslexia here!

Tolerance doesn't guarantee reason but intolerance guarantees none

That is one of my favourite quotes I once made up but it is only strictly correct in the right social context.
Do you like it?
My other favourite ones are;

The only good ways to fight an idea is either with a better one or prove it nonsense (my quote)

What can be asserted without evidence can be dismissed without evidence (Christopher Hitchins)

Originally posted by @humy
I didn't imply it was and I made no implicit assumptions hence my use of the words "assuming..." and "if..." in that post. Are you trying to make a point? If so, I am not a mind reader, so just say it.

Funny how your post had little to do with being biased, simply what would be true if random.

I am still waiting to see exactly what it is about their selection process which is biased.

Originally posted by @humy
But I was speaking to Eladar, not to you!
I am the one that is supposed to have dyslexia here!

Tolerance doesn't guarantee reason but intolerance guarantees none

That is one of my favourite quotes I once made up but it is only strictly correct in the right social context.
Do you like it?
My other favourite ones are;

The only good ways ...[text shortened]...
What can be asserted without evidence can be dismissed without evidence (Christopher Hitchins)

Sincere apologies to you.


edit: And I do like the quote!

Originally posted by @eladar
Funny how your post had little to do with being biased, simply what would be true if random.

If that is supposed to be criticism of my given analysis, you really need to explain that one to me;
In this particular case, how else can I (or anyone for that matter) currently rationally judge the chances of their selection process being biased (whether purely unintentionally so or otherwise) other than via the only relevant statistical knowledge currently known and available to us?
Have you got some other relevant knowledge, either unknown to me or which is known to me but which I erroneously failed to use in my given analysis, about this particular case that I should have but didn't used to better judge the chances of their selection process being biased? If so, tell me this knowledge and its source and show me the (and your) alternative maths I should have used so you can correct me.

I am still waiting to see exactly what it is about their selection process which is biased.

Since, in this particular case we're not sure exactly what their selection process is and we may never know, you might have to wait forever.
We can only ever currently judge anything (including probability) by only whatever finite and limited information we currently have available to us. We may then seek new information to then better judge but that takes time and often effort and there is often no guarantee we will be able to find that necessary new information. If you have a suggestion for a way around that, I really like to hear it.

Originally posted by @sonhouse
Just a quick look says there should have been 47 diagnosed with dyslexia in your group. So it sounds like there are kids in your group not diagnosed. It would be accurate if all the kids had been diagnosed with or not having dyslexia. My son had it very bad but was able to work through it and graduated with MA at Bryn Mawr university. My wife has it also a ...[text shortened]... wards, understanding full well the content but more like a mental exercise for no good purpose

Not quite the question I was trying to ask, but I see that Humy has addressed it!

Originally posted by @humy
assuming that national statistics of 7% accurately indicate the percentage of pupils with dyslexia, there is 0.07 probability that, if you randomly pick any pupil in the country with equal probability of picking any one, he/she will be dyslexic.

I assume what you really want is the answer to this specific question;

If you randomly picked 684 pupils out ...[text shortened]... find it hard to tell that I am naturally dyslexic. In fact, my Englidelicious is now perfect

Yes that is exactly what I wanted to know. I thought it would be in the millions but that is something of a different order!

I will come back a bit later when I have more time, but thanks for now.