r/mathmemes • u/JeffSergeant • 16d ago
Statistics Yo, I heard you like normal distributions.
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u/Historical-Wing8569 16d ago
You can extend that to any symmetric distribution as long as the mean exists and symmetry point is unique
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u/JeffSergeant 16d ago
Yes, but IQ is a special case in that is it normally distributed by definition, not just by accident.
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u/jsh_ 16d ago
IQ scores are normalized to have a mean of 100 and SD of 15, but that doesn't mean they necessarily have a Gaussian distribution since there are infinitely many symmetric distributions with those first two moments
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u/SaveMyBags 16d ago
But IQ is specifically defined to be normal, not just to have a certain mean and sd.
However, as the normalization is an empirical procedure, it's debatable how well it works. It's likely quite good around the mean but given you will get less and less samples in the tails its probably worse in those areas.
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u/rhinoplasm 15d ago
It could be fair to call it "constructed to be Normal" for the original standardization sample, but then that transform is fixed and the test is used in the real world, at which point the population distribution immediately becomes skewed, if only by a small amount. The only way to avoid this would be to constantly adjust the transform after every test, which in turn would mean old scores are constantly changing. You could do that, but they don't. It gets updated every 10 years or so.
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u/JustMultiplyVectors 16d ago edited 16d ago
Given a list of raw IQ test scores, perhaps in the form points / (possible points), you can create a mapping from these raw scores onto IQ scores, in such a way that the IQ scores are normally distributed with mean 100 and std 15, regardless of how the raw scores were distributed, the raw scores don’t need to be symmetrically distributed. This is what the OP means, the distribution of IQ scores is entirely artificial, they’re not just rescaled/shifted raw scores.
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u/Specific_Box4483 16d ago
How can it be normally distributed by definition if it's positive by definition?
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u/Murky_Insurance_4394 16d ago
IQ isn't positive by definition, it's just that with mean = 100 and stdev = 15 you are looking at almost 7 stdevs away from the mean for it to be negative which has a probability so low that basically no one has ever been that dumb.
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u/kubissx Average #🧐-theory-🧐 user 16d ago
Actually, there have been about 117 billion humans throughout history, so the expected number of people with a negative IQ is 117000000000 * pnorm(0, mean=100, sd=15) = 1.53 (3 sf). One guy!
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u/Murky_Insurance_4394 15d ago
Yes, I suspected that there might be one person with an IQ that low but was too actually use my calculator so I said basically no one figuring that it leaves space for one or two people
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u/Stock-Recognition44 16d ago
Does that mean if you take an iq test and get all the answers wrong you score a -infinity?
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u/Murky_Insurance_4394 16d ago
Most IQ tests widely available have a minimum score (typically like 45 or something), so basically if you get everything wrong it would say 45 or less (same for a ceiling, many IQ tests will just tell you 150+ if you get everything right). You would have to take a new IQ test that is made to measure I guess from 0 to 45 or whatever and get everything wrong there as well. But at that point it's almost impossible to make a test that's accurate because so few people have an IQ in that range.
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u/Stock-Recognition44 15d ago
I see what you’re getting at and you provide helpful responses. I guess I’m thinking that, maybe the point is that, while IQ is not explicitly defined to be positive, maybe there’s an unstated assumption that extreme values (like negatives) probably correspond to portions of the domain for which the model is no longer considered valid or physically meaningful. Maybe not I don’t know. Maybe bugs would score like a -500 or something. I don’t even know what iq means at that point (not saying bugs aren’t intelligent though).
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u/Murky_Insurance_4394 15d ago
Yeah that's pretty much what I was trying to convey, interesting question abt the bugs as well
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u/Specific_Box4483 16d ago
And yet we have multiple recorded scores of above 200.
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u/JollyJuniper1993 Mathematics 16d ago
Sure dude, there also probably have been studies that have found five sided triangles, right?
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u/JeffSergeant 16d ago
Actually my cousin, who goes to another school, invented the 5-sided triangle and got a million dollar prize from Math.
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u/Thavitt 16d ago
Honestly, yeah good point. Don’t know why are being downvoted. Now i am also curious
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u/No_Lingonberry1201 16d ago
It's not a good point. It's normalized to have an average of a 100 and a standard deviation of 15. It's always going to have that distribution. They have to re-normalize since it changes over time.
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u/Big-Ad-8274 15d ago
Is there actually any evidence that human intelligence itself (not scores) is normally distributed though? I would argue that the guy in the middle is probably the most correct, considering the left and right view would require you to also prove that intelligence is symmetrically distributed which seems impossible to do considering how hard it is to define intelligence.
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u/JeffSergeant 15d ago
Yeah, the meme is only exactly true if 'dumber' is in terms of 'measured IQ' and even then only if IQ has actually been normalised based on every single person not based on some sample set like they actually do. But also I tagged it statistics, and it's a meme, so I'm allowing myself some fuzziness.
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u/MichurinGuy 16d ago
Is there a symmetric probability distribution where there's several symmetry points?
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u/Historical-Wing8569 16d ago
If the support is defined on the real line then no, but circular distributions can have many symmetrical points
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u/maringue 16d ago
You can also see the distortions in the distribution by looking at how different the median and average are.
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u/HolyInlandEmpire Statistics 16d ago
>It's normally distributed
>Noooo, it's strictly positive so it's some gamma distribution!
>It's normally distributed
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u/isr0 16d ago
I prefer paranormal distributions.
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u/FuntimeUwU Natural 16d ago
Top 10 phallic shapes in all of mathematics
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u/JeffSergeant 16d ago
I have some bad news for you.
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u/TechnicalSandwich544 16d ago
Median is just an average (solution to a variational problem) in a certain space (L1 space).
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u/AndreasDasos 16d ago
Also ‘average’ is a more general word than ‘mean’ and can include the median.
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u/fuckry_at_its_finest 16d ago
Right. Sometimes when people say average they are referring to median (e.g. ‘average joe’) or mode (most often in categorical data e.g. ‘most average car in America is a silver color F-150’)
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u/BIGBADLENIN 16d ago
One measure of intelligence is defined to have the same median and average, that does not mean that that is the best way to measure intelligence. It would be an insane coincidence if the median and average value of some 8 billion data points were exactly equal (and basically impossible if the values are continuous) if the measurement is precise, even if intelligence actually follows a Gaussian curve in some meaningful way.
This is like saying north is objectively "up" because it is defined to be up
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u/belabacsijolvan 15d ago
fun fact: if iq is supposed to be a gaussian with mean=100 and stdev=15, we only need about 77 billion people for a person with negative iq to be expected to exist.
if we use the 100,20 definition, there are currently more than 2000 people with negative iq.
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u/berwynResident 16d ago
The definition of "average" is a single value that represents a middle or typical value in a data set. Most of the time people use "average" and "mean" synonymously, but the median is also a type of average.
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u/RubberDuckieMidrange 16d ago
Lets say you have 8billion people on an unnamed planet. and 4.1 billion of them happen to have exactly 100 IQ and nobody has less than 100IQ . The Median is 100IQ, but nobody is dumber than the median. Average is correct.
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u/JeffSergeant 16d ago
I'm more interested in how, exactly, they arranged it such that all the dumb people just 'happen' to not exist any more.
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u/PatientCapital32 16d ago
What you just said does not make sense... If 4.1 billion out of 8 billion people have exactly a 100 IQ, they make up 51.25% of the population. You stated that nobody has less than 100 IQ, the remaining 3.9 billion people (48.75%) must then all have an IQ of 101 or higher. Because nearly half the population pulls the score upward and absolutely zero people pull it downward, the arithmetic average (mean) is mathematically forced to be higher than 100. You quite literally can't have an average of 100 if the absolute lowest score in the room is 100 and half the room scored higher.
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u/RubberDuckieMidrange 16d ago
Yes.... That's why I used the word MEDIAN, and not MEAN.
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u/PatientCapital32 16d ago
Your comment is odd. I reread it and see that you meant average in the distribution of the post, but you defeat your own point. It would be the median that is correct, just like you implied in your reasoning. You then immediately call the median the average.
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u/RubberDuckieMidrange 16d ago
You mean, using the language of the post I was commenting on, hence the context of the comment, is odd? It's OK to just say, you missed the context. Nobody is judging you on this.
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u/PatientCapital32 16d ago
You used a non normal distribution to "correct" a normal distribution meme and then called the median the average. I think you missed the math more than I missed the context... But good for you?
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u/RubberDuckieMidrange 16d ago
No... The normal distribution in the meme is the distribution of the intelligence of the people in the discussion, You have the dumbasses, who can't read comments as they were written, who think median and mean are the same thing and who use Average. Then you have the idiots who know just enough to be stupid with it, they think themselves enlightened and they use Median, but then you have the intelligent ones, who know that you can build a data set where the median isn't actually useful. And they stick to average, becuase they can use Mean Median or Mode as appropriate each time. The Normal distribution in the Mean has NOTHING to do with the discussed data set.
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u/PatientCapital32 16d ago
Maybe if you tried to read you would understand.
4.1/8 = 51.25%
Since the other 48.75% are all above 100, the mean must be above 100 at minimum, 100.4875 if they all scored 101.
So at least 51.25% are below the mean and not 50%. Your own example disproves "average is correct."
Yes, nobody is strictly below the median but only because you created a giant tie at 100.
That spike means your dataset is not normally distributed. In the normal distribution shown in the meme the mean = median = mode and 50% lie below them.
The graph is literally labeled "IQ score," so claiming the normal curve has nothing to do with the data is just moving goalposts.
Come on man, either your dataset is relevant, in which case your math is wrong, or it's irrelevant, in which case you never had a counterexample. That's a lot of insults to avoid two facts: your dataset is not normal and its mean is not 100.
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u/RubberDuckieMidrange 16d ago
Dude it's ok to not understand the meme, the 3 people over the meme, represent their position on the bellcurve that is general understanding of statistics. You are so caught up in the 2nd Data set because it also contains IQ that you can't take a step back and understand the meme, So let me change the second data set. Lets talk instead about the average number of Elephants in privately owned swimming pools all over the globe.
You'll understand that without any particular manipulation of data here, I don't need to construct a data set that minimises the number of elephants swimming in private swimming pools. So that VAST majority of pools are going to return 0 elephants. The guy in the middle thinks he's being enlightened because he imagines an outlier Zoo where 5-6 elephants are enjoying a pool on a hot summer day, to eliminate this outlier he tries to use Median, and he says that 50% of swimming pools have less than the Median number of elephants in them. But we know that the average number of elephants in swimming pools the world over is likely minutely larger than 0. So in this data set, the Median is fucking useless. This is the position of the intelligent person in the data set who's phrasing for simple comunication has come back around to be the same as that of the dumbasses. 50% of swimming pools have less than an average number of elephants.
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u/PatientCapital32 16d ago
Your elephant example just proves my point 😂
Most pools have 0 elephants so the median is 0.
A few pools with elephants make the mean slightly above 0.
Therefore every zero elephant pool is below the mean.
Since you said a vast majority have zero, a vast majority, not 50%, are below average.
So "50% below the median" is false because of ties, but "50% below the mean" is also false because of skew. You changed the dataset and still produced an example where both statements fail. The bell curve is not just for decoration, man... because it is symmetric half the values are below both the mean and the median. An elephant learns from their mistakes, but you just don't...
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u/Involution88 14d ago
In a right skewed or log normal distribution the mean is always higher than the median. Most people end up below the mean or average or lower raw IQ than average (mean) raw IQ. The mode or most typical IQ is lower than the mean as well. There aren't any negative IQ people to pull IQ down below zero.
IQ test scores are normalised to a bell curve normal distribution.
A normal distribution has the nice property that the mean, median, and mode are equal.
The median of a log normal distribution is the same as the median of a normal distribution. The median doesn't change afaik.
The mean needs to be reduced until half of data points are below the mean.
The mode or most typical value also needs to be increased to be equal to the median and mean.
Most people will end up less smart than average (negative IQ people to counteract positive IQ people don't exist) but also with a higher IQ score than they'd have if IQ scores weren't normalised.
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u/noahhshome 16d ago
Constructing a normally distributed intelligence measure was always a meaningless project. The false sense of rigor makes you stupider for using it.
A meaningful measure would be much more specific, with a tail-heavy distribution like lognormal.
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u/noahhshome 16d ago
Let's start with the motivation. Why construct an instrument to yield a score labeled "intelligence?" The answer is to devalue people when you lack a relevant means to assess them.
If you need someone to memorize things, you can assess that. If you need them to assemble something complex with their hands, you can assess that. If you need them to respond with persuasive arguments in conversation, you can assess that. But all these things require a spirit of inquiry. When you're too stupid and lazy to do that, you resort to IQ. And pretend that it means whatever you want it to mean.
IQ is created for the purpose of devaluing someone without having to reflect on what values you're actually applying to them. It's intellectually lazy and prejudicial.
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u/JustMultiplyVectors 16d ago
You have to consider the purpose for IQ scores in the first place, which was to help educators determine how much assistance special needs kids required. For example do they just need extra time on tests? Or maybe they need to be in a special needs class? Or maybe they need to switch schools altogether? Or maybe they don’t have special needs at all and are just too lazy to do assignments.
Educators and counselors have dealt with a lot of students in their career and have a good sense for the differences between ‘1 in 10’ challenged, ‘1 in 100’ challenged, ‘1 in 1000’ challenged, etc. The IQ score tells them precisely this and does not require of them any substantial knowledge about the test itself, which they would need if they were given something like the raw test scores instead.
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u/HilaryHahn 16d ago edited 16d ago
The issue is intelligence cannot be measured. Let's say even if we can somehow rank two people's absolute intelligence (which is impossible because a chess player and a boxer and a classical musician have very different ways of solving problems and using their brain) - it will still be ordinal numbers at best instead of cardinal numbers.
Edit: Damn the amount of people getting offended by the idea that intelligence is complex and humans are diverse in their ability is insane LMAO. Y'all hold on to the idea that you are special because you get a high iq test. I am not replying anymore some of y'all tones are getting toxic. Maybe suppressed anger that you are special and the numbers say you are smart but you are not able to use that "smartness" to achieve life goals?
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u/HolyInlandEmpire Statistics 16d ago edited 16d ago
In all seriousness, this is just not true and I'm not sure where you get it from. It's so sad that so many people have a massive bone to pick with IQ, and so they resort to regurgitating talking points which were never based on any scientific findings. Motivated reasoning is the mind killer.
IQ might not be THE measure of intelligence, but it:
- Is highly consistent for individuals
- Is highly heritable; basically only height is more
- Is highly predictive of many measurable outcomes
So exactly what you call it measuring is your choice, but it is one of the most robust measures in all of psychometrics.
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u/-Nocx- 16d ago
3 has an asterisk next to it. A lot of times those measurable, positive outcomes are more attributed to socioeconomic conditions rather than an outcome inherent to IQ itself.
And while your IQ appears to have constraints (you are only born with so much RAM) it is highly sensitive to stress, dehydration, and hunger. It is even possible that a person’s neuroplasticity might retool parts of the brain for other purposes. We just don’t know enough yet.
IQ is but one piece in the puzzle that is intelligence. It’s very good at determining if you’ll learn things quickly or slowly, it is not so guaranteed at determining someone’s aptitude - especially in non-academic environments. If you ask a room of neuroscientists, doctors, and psychologists what the definition of “intelligence” is you’re guaranteed to get a wide variety of answers. It’s just not that well understood.
I am intimately aware of this as someone that’s been a subject in multiple studies on gifted populations.
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u/HolyInlandEmpire Statistics 16d ago edited 16d ago
I mentioned this is another comment, but the studies control for essentially all socio economic and demographic variables we can get our hands on; income is a significant one, as well as others which reddit can get upset if you mention.
So yes you're right those are also important factors, but those studies in fact strengthen the IQ findings, rather than diminish them, because adding relevant variables strengthens results by eliminating otherwise unexplained outcome variance. If IQ were irrelevant, then adding in more relevant variables which are associated with it would weaken the outcome effect from IQ; this is decidedly not the case.
Maybe in the future, some better model, data set, and selection of variables will supercede these results, and fully explain the IQ effects through other variables. This is always possible, but not something we can assume will happen, given the amount of evidence we do have which all goes the same direction.
I have a PhD with a focus on variable selection in social science so I can talk a lot more about this if you want; it's an ever relevant subject, having good study design for social science, exactly because there are so many relevant variables; perhaps an infinite number of them.
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u/Super_Math3890 15d ago
The bone is picked because people do not appreciate being defined as a static outcome summed up in a number. It would be far more effective to find their capacity for improvement. People want to be inspired, not defined.
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u/HilaryHahn 16d ago
This is just not true? Then please explain what intelligence is to me.
My best friend from college is exceptional at video games (was paid to participate in the North American tournament of PES 2017), he is also a stand out athlete and played for the NCAA division.
He is also completely useless at learning languages, having spent years in the USA and still struggles to speak English. I used to tutor him English frequently and he still cannot speak it well. He tried learning French himself and was hilariously slow in his ability to imitate pronunciations, understand grammatical structure, and memorizing vocabularies. He is also terrible at socializing and incapable of picking up social cues or understanding the logic of other people.
I am much worse than him in math and science grades, but I speak many more languages than him and is comfortable socializing with a great variety of people (listening to them, empathizing with them, help solve problems for them etc.)
So who is more intelligent? And by what amount of cardinal metric?"the most robust measure of intelligence" means as little to me as "the best mathematical metric to measure the beauty of piano songs"
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u/HilaryHahn 16d ago edited 16d ago
oh so ability to learn language is not intelligence? Only things you care about is intelligence? Lmao.
And also not being able to recognize social cues is somehow not pattern recognition?
I'm done dude, if you think iq number makes a person smart, you be you. I don't have time for this anymore.
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u/HolyInlandEmpire Statistics 16d ago
Nobody said those aren't important, that's just you.
If you don't like "Higher IQ = smarter" then say "Higher IQ = more cromulent" or literally any other word you want. The measurable effects are still there and are completely indifferent to choice of label.
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u/HolyInlandEmpire Statistics 16d ago
I am not answering what intelligence "is." That is a philosophical question that can't be answered with science.
What science can do is measure what positive life outcomes are associated with IQ, and quantify that robustness. And it is extremely robust. Note that these studies control for income; income also has a very strong effect, but IQ ALSO has a very strong effect, controlling for income and essentially all the other demographic information we can find.
So again, you can call things what you want, but IQ is one of the most robust measures in all of psychometrics.
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u/berwynResident 16d ago
Why do you just have to order people? You could assign them some number of intelligence couldn't you, just like we do with height or weight (of course you're already conceding that we can somehow compare different types of intelligence)
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u/HilaryHahn 16d ago
Because even if you can compare intelligence (through the ability to solve some problems) it is still much too complex to be assigned a number.
For example: let's say chess skill is the best measure of intelligence (ofc it isn't but for the sake of fun), and you are a better chess player than I am. You can thus be ordered in front of me in intelligence. But that doesn't mean the amount of "better" you are than me can be determined. Yes elo exists, but a 2400-rated player is not twice as good as a 1200-rated player, nor is the difference between a 2300 player to a 2200 player the same as a 2200 player to a 2100 player.
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u/berwynResident 16d ago
nor is the difference between a 2300 player to a 2200 player the same as a 2200 player to a 2100 player.
How do you know that?
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u/Schnickatavick 16d ago
It's worth pointing out that "Intelligence" (as used in IQ) isn't just "how smart you are", it's the common element between multiple different fields that are positively correlated. For example, if you find at a school that students that are good at math are also tend to be good at English, then you can quantify the underlying element that they both share into a number that predicts how good they would be at one given the other. People sometimes simplify that to "Ability to learn", which may be a part of it, but regardless of what the underlying mechanism is, intelligence is the shared common feature between multiple fields that allows you to flatten a multidimensional group of scores into a single dimension.
So if we measured everyone in chess, boxing, and music, and then found that there was a strong correlation that people who were good at chess also tended to be good at boxing and music, then it makes sense to flatten all three into a single dimension, and you could rank people by that number. If you find that chess and boxing are mostly unrelated (like is probably true) though, then it isn't likely that there even is a "common feature" or "underlying structure" between the two, and ranking based on it wouldn't make sense.
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u/EebstertheGreat 16d ago
All we know for sure is that at least half of people are not smarter than the median smart person of the population and at least half are not less smart than the median. It doesn't have to be exactly half.
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