More fun with statistics

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More fun with statistics

Postby Upton_O_Goode » Mon Jun 12, 2017 11:12 am

Bobbo and others introduced the fascinating topic of Bayesian statistics in the "Geezer Health Care" thread here, and that led me to think of another popular seeming paradox about statistics.

Consider a university with two departments, say A and B. (Call it University X.) Now suppose 200 women and 200 men apply to go there. Further suppose that: 160 men and 40 women apply to Department A, and 120 men and 32 women are admitted. Thus 75% of the men and 80% of the women are admitted. The other 40 men and 160 women apply to Department B, and 8 men and 40 women are admitted, that is, 20% of the men and 25% of the women. Both departments have overall admission rates for women that are higher than for men.

Yet, for University X as a whole, you get 128 men and only 72 women admitted out of the 200 of each who applied, so men have a 64% admission rate and women a 36% admission rate.

This paradox is known inaccurately and informally as Simpson's paradox (No, not Bart Simpson). It actually did manifest itself in the 1970 admissions figures at the University of California at Berkeley, where there were 101 departments.

Thus, the conditional probability of Event E given each of a set of mutually exclusive and exhaustive possibilities may be larger than the conditional probability of Event F on each of those hypotheses, and yet Event E may have, overall, a smaller probability than F.

Curious, no?
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Re: More fun with statistics

Postby bobbo_the_Pragmatist » Mon Jun 12, 2017 11:34 am

I don't see any paradox at all. The 200 Males and 200 Females may be the red herring (Edit: that is making common sense fail and a so called paradox apppear?). Only the number applying is relevant.

Its like some process that is experiencing a doubling each year in growth but after 5 years still has the lowest absolute number of whatever. You start with a lower base, and it takes a while to catch up. You see this all the time when talking economics and everyone else compared to the USA...... unfortunately now, including our Debt.
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Re: More fun with statistics

Postby Upton_O_Goode » Mon Jun 12, 2017 4:10 pm

bobbo_the_Pragmatist wrote:I don't see any paradox at all.


That's why I called it a seeming paradox. It refutes a proposition one would intuitively think to be true, namely the following (false) proposition.

Suppose one of events A, B, C, and D must occur and no more than one can occur. Suppose also that the conditional probabilities of events E and F given each of those events satisfy P(E|A) > P(F|A), P(E|B) > P(F|B), P(E|C) > P(F|C) and P(E|D) > P(F|D). Then P(E) > P(F).

Proof: Since one of the events A, B, C, and D, must occur, without loss of generality, assume that A has occurred. Then, revising probabilities, P(E) = P(E|A) > P(F|A) = P(F). QED.

(Yes, I'm sure you see the ambiguity in the use of the symbols P(E) and P(F) that invalidates the proof. But at first blush, it looks like a slam-dunk.)

It has nothing to do with exponential growth. Statisticians have been calling it Simpson's Paradox for over 60 years, so who are we to tell them it's only an oddity, not a real paradox? It does contradict naive common sense.

The real-world occurrence of it is here.
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Re: More fun with statistics

Postby bobbo_the_Pragmatist » Mon Jun 12, 2017 4:36 pm

So........I'm assuming THIS TIME the paradox does involve Bart Simpson?

If there is no paradox, there is no seeming paradox either. This flows from non exponential Y = WUT?
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Re: More fun with statistics

Postby Nikki Nyx » Mon Jun 12, 2017 4:49 pm

I've never been great with math, mostly because my Algebra I teacher failed to provide me with a good foundation. Oddly, she didn't think girls could do math. 'Murika. So please don't throw equations at me, but it seems to me that, in the initial situation, the three statistical universes are unrelated to one another, and that the so-called paradox is artificial, created by wrongly relating the three statistical universes to each other. Would that be a fair statement in non-mathematical terms? If not, could one of you explain it in non-mathematical terms?
What are the facts? Again and again and again-what are the facts? Shun wishful thinking, ignore divine revelation, forget what “the stars foretell,” avoid opinion, care not what the neighbors think, never mind the unguessable “verdict of history”--what are the facts, and to how many decimal places? You pilot always into an unknown future; facts are your single clue. Get the facts!
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Re: More fun with statistics

Postby bobbo_the_Pragmatist » Mon Jun 12, 2017 5:01 pm

WUT Three Universes? I only see one. Set Theory works that way....for the most part.
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Re: More fun with statistics

Postby Nikki Nyx » Mon Jun 12, 2017 5:24 pm

Ok, perhaps I worded it badly. I'm asking for an explanation of the so-called paradox in non-mathematical terms, because I don't savvy equations, yet it doesn't appear to be a paradox to me.
What are the facts? Again and again and again-what are the facts? Shun wishful thinking, ignore divine revelation, forget what “the stars foretell,” avoid opinion, care not what the neighbors think, never mind the unguessable “verdict of history”--what are the facts, and to how many decimal places? You pilot always into an unknown future; facts are your single clue. Get the facts!
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Re: More fun with statistics

Postby bobbo_the_Pragmatist » Tue Jun 13, 2017 1:01 am

Well....its Upton's example but my view is that a quick sloppy inattentive read of the example could be misinterpreted as showing that women can be selected more than men for all available positions even though their numbers are equal to men. A fallacy==>not a paradox.

Paradox is like irony, is like the use of hyphenated words. More misused than correctly used. Its why I split my use of the hyphen so I'm only wrong 50% of the time...rather than risk being wrong all the time.

Now......do you see another Simpson's Paradox........or just the fallacy?

Spoiler:
Odds of being right or wrong are not 50/50 reflecting the choices given, but rather are equal to the odds of the underlying choices. This is a fallacy, not a paradox.
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Re: More fun with statistics

Postby Upton_O_Goode » Tue Jun 13, 2017 10:44 am

bobbo_the_Pragmatist wrote:Now......do you see another Simpson's Paradox........or just the fallacy?

Spoiler:
Odds of being right or wrong are not 50/50 reflecting the choices given, but rather are equal to the odds of the underlying choices. This is a fallacy, not a paradox.


Certainly a fallacy. That's why I included my hypothetical example of a "proof" of a false proposition. You have to have a fallacy to get that. I'm sure we agree that what seems plausible is very liable to contain a fallacy. The false proposition I stated becomes plausible only because of a certain conflation of things in the human mind. If I replaced "given that" by "and", the proposition would be true, viz:

Suppose one and only one of conditions A, B, C, and D must occur. Suppose events E and F satisfy P(E & A) > P(F & A), P(E & B) > P(F & B), P(E & C) > P(F & C), and P(E & D) > P(F & D). Then P(E) > P(F).

Proof: P(E) = P(E & A) + P(E & B) + P(E & C) + P(E & D) > P(F & A) + P(F & B) + P(F & C) + P(F & D) = P(F).


If we strip away the context of probability and write this proposition as a statement about numbers, we can see easily that it is wrong, and no one commits the fallacy. To wit,

Suppose given ordered sets of non-negative numbers (w, x, y, z) and (w',x',y',z') and also ordered sets of positive numbers (p,q,r,s) and (p',q',r',s') satisfying p+q+r+s= 1 = p'+q'+r'+s'. Then (false proposition) if a/p > a'/p', b/q > b'/q', c/r > c'/r' and d/s > d'/s', it follows that a + b + c + d > a' + b' + c' + d'.

This proposition has much less appearance of being obvious than the equivalent statement in terms of probability, and before proving it, one would probably test it with sample numbers, thereby finding out it was wrong. One would also find out it was true in the case when p = q = r = s = p' = q' = r' = s' = 1/4, which is the true proposition with "and" replacing "given that."
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Re: More fun with statistics

Postby bobbo_the_Pragmatist » Tue Jun 13, 2017 10:52 am

I got out of school before "set theory" became a thing. My Dad was getting his college degree after I graduated from Middle School and he wrestled with set theory and I think "Mathematical Logic" which wasn't the name but it was putting every day issues into formulas and seeing if Math might present any possible solutions. Then as now..... I'm glad I wasn't my younger sisters who also had to put up with that stuff...... and no dodge ball.

Good times. Y = WUT
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Re: More fun with statistics

Postby Upton_O_Goode » Tue Jun 13, 2017 10:53 am

LunaNik wrote:I've never been great with math, mostly because my Algebra I teacher failed to provide me with a good foundation. Oddly, she didn't think girls could do math. 'Murika. So please don't throw equations at me, but it seems to me that, in the initial situation, the three statistical universes are unrelated to one another, and that the so-called paradox is artificial, created by wrongly relating the three statistical universes to each other. Would that be a fair statement in non-mathematical terms? If not, could one of you explain it in non-mathematical terms?



The short answer, in the example, was that men tended heavily to apply to the department that was easy to get into and women applied heavily to the department that was hard to get into. Both departments slightly advantaged women over men, but there were not enough of them applying where it was easy to get in to overcome the handicap caused by the huge number of women applying where it was hard to get in.
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Re: More fun with statistics

Postby Upton_O_Goode » Tue Jun 13, 2017 10:56 am

bobbo_the_Pragmatist wrote:I got out of school before "set theory" became a thing. My Dad was getting his college degree after I graduated from Middle School and he wrestled with set theory and I think "Mathematical Logic" which wasn't the name but it was putting every day issues into formulas and seeing if Math might present any possible solutions. Then as now..... I'm glad I wasn't my younger sisters who also had to put up with that stuff...... and no dodge ball.

Good times. Y = WUT


My God! You must be older than the hills! When I was in high school in the 1950s, my teachers singled me out for some "new math" that one of them referred to as Bo-o-le-an algebra (four syllables---it's actually pronounced Boo-le-an, of course---three syllables).
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Re: More fun with statistics

Postby bobbo_the_Pragmatist » Tue Jun 13, 2017 10:58 am

No.....maybe just a retarded school system.......or I simply didn't pay attention? Ha, ha. I should remember if I was exposed to it though. I enjoyed algebra...little word puzzles. I got thrown by calculus....I even recall the issue: the area under a rolling cone.

Who thinks that {!#%@} up?
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Re: More fun with statistics

Postby Nikki Nyx » Tue Jun 13, 2017 3:28 pm

bobbo_the_Pragmatist wrote:Well....its Upton's example but my view is that a quick sloppy inattentive read of the example could be misinterpreted as showing that women can be selected more than men for all available positions even though their numbers are equal to men. A fallacy==>not a paradox.

Paradox is like irony, is like the use of hyphenated words. More misused than correctly used. Its why I split my use of the hyphen so I'm only wrong 50% of the time...rather than risk being wrong all the time.

Now......do you see another Simpson's Paradox........or just the fallacy?

Spoiler:
Odds of being right or wrong are not 50/50 reflecting the choices given, but rather are equal to the odds of the underlying choices. This is a fallacy, not a paradox.

Thank you, that makes sense to me. Also, I agree with your conclusion on the use of the words paradox and irony. Add the word "literally." :wgrin: I know where I misuse the hyphen: with adverbs. I tend to write "well-made" instead of "well made." @#$% English with its exceptions to the rules!

I'm not a complete loss at math. Recently, I played an adventure game that had a fatal flaw. One of its puzzles presented the player with a device that included five parts and a container of five ingredients. The instructions specified that the goal was to insert each ingredient into one of the parts until the device was full, then press the start button to check if the combination was correct...and keep doing this until successful. So, randomly guess which one belonged where. With 120 permutations? No thanks. Apparently, the developer failed to see the inanity.
What are the facts? Again and again and again-what are the facts? Shun wishful thinking, ignore divine revelation, forget what “the stars foretell,” avoid opinion, care not what the neighbors think, never mind the unguessable “verdict of history”--what are the facts, and to how many decimal places? You pilot always into an unknown future; facts are your single clue. Get the facts!
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Re: More fun with statistics

Postby Upton_O_Goode » Tue Jun 13, 2017 3:56 pm

LunaNik wrote:
bobbo_the_Pragmatist wrote:Well....its Upton's example but my view is that a quick sloppy inattentive read of the example could be misinterpreted as showing that women can be selected more than men for all available positions even though their numbers are equal to men. A fallacy==>not a paradox.

Paradox is like irony, is like the use of hyphenated words. More misused than correctly used. Its why I split my use of the hyphen so I'm only wrong 50% of the time...rather than risk being wrong all the time.

Now......do you see another Simpson's Paradox........or just the fallacy?

Spoiler:
Odds of being right or wrong are not 50/50 reflecting the choices given, but rather are equal to the odds of the underlying choices. This is a fallacy, not a paradox.

Thank you, that makes sense to me. Also, I agree with your conclusion on the use of the words paradox and irony. Add the word "literally." :wgrin: I know where I misuse the hyphen: with adverbs. I tend to write "well-made" instead of "well made." @#$% English with its exceptions to the rules!

I'm not a complete loss at math. Recently, I played an adventure game that had a fatal flaw. One of its puzzles presented the player with a device that included five parts and a container of five ingredients. The instructions specified that the goal was to insert each ingredient into one of the parts until the device was full, then press the start button to check if the combination was correct...and keep doing this until successful. So, randomly guess which one belonged where. With 120 permutations? No thanks. Apparently, the developer failed to see the inanity.



Veering a bit off topic here, but this interests me. I have my own hyphenation rule. Use the hyphen when the phrase is used attributively, not when it is used predicatively. Thus: "This is a well-made car" vs. "This car is well made."
"A general conversion among the boys was once effected by the late excellent Mr. Fletcher: one poor boy only excepted, who unfortunately resisted the influence of the Holy Spirit, for which he was severely flogged; which did not fail of the desired effect, and impressed proper notions of religion on his mind."

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Re: More fun with statistics

Postby bobbo_the_Pragmatist » Tue Jun 13, 2017 5:19 pm

Upton_O_Goode wrote: Veering a bit off topic here, but this interests me. I have my own hyphenation rule. Use the hyphen when the phrase is used attributively, not when it is used predicatively. Thus: "This is a well-made car" vs. "This car is well made."

Well, I'm gonna stick with MY RULE because its based on Math. You are NOT using the rule of hyphenation which basically is for non-standard word combo's...so ... mostly arbitrary. My random choice should get me at an average of 50% correct.... but your-rule is going to be almost totally wrong?

Why do that?...................... I know why "I" would. The self-definition/aggrandizement of it all demonstrated by my use of ...... and //////. Why has the hypen achieved the status it has?===>(THERES another one!!) Picked it up way before Gawd.
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Re: More fun with statistics

Postby Nikki Nyx » Tue Jun 13, 2017 7:50 pm

Upton_O_Goode wrote:Veering a bit off topic here, but this interests me. I have my own hyphenation rule. Use the hyphen when the phrase is used attributively, not when it is used predicatively. Thus: "This is a well-made car" vs. "This car is well made."

See, to me, your first sentence appears correct, but the exception to the hyphenation rule claims the hyphen shouldn't be there.
The major exception is when the compound adjective begins with an adverb ending in -ly. In that case, since a misreading is unlikely, the hyphen is unnecessary. If the -ly adverb is part of a larger compound adjective, use a hyphen.

This is a poorly produced movie.

He followed up with a not-so-poorly-produced sequel.

Hmmm...on second look, they don't claim the hyphenated version is wrong, only that the hyphen is unnecessary, which is subjective. Gord!!!
What are the facts? Again and again and again-what are the facts? Shun wishful thinking, ignore divine revelation, forget what “the stars foretell,” avoid opinion, care not what the neighbors think, never mind the unguessable “verdict of history”--what are the facts, and to how many decimal places? You pilot always into an unknown future; facts are your single clue. Get the facts!
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Re: More fun with statistics

Postby Upton_O_Goode » Tue Jun 13, 2017 7:58 pm

bobbo_the_Pragmatist wrote:
Upton_O_Goode wrote: Veering a bit off topic here, but this interests me. I have my own hyphenation rule. Use the hyphen when the phrase is used attributively, not when it is used predicatively. Thus: "This is a well-made car" vs. "This car is well made."

Well, I'm gonna stick with MY RULE because its based on Math. You are NOT using the rule of hyphenation which basically is for non-standard word combo's...so ... mostly arbitrary. My random choice should get me at an average of 50% correct.... but your-rule is going to be almost totally wrong?

Why do that?...................... I know why "I" would. The self-definition/aggrandizement of it all demonstrated by my use of ...... and //////. Why has the hypen achieved the status it has?===>(THERES another one!!) Picked it up way before Gawd.


Hey, like Dick Cheney, I make my own reality.
"A general conversion among the boys was once effected by the late excellent Mr. Fletcher: one poor boy only excepted, who unfortunately resisted the influence of the Holy Spirit, for which he was severely flogged; which did not fail of the desired effect, and impressed proper notions of religion on his mind."

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Re: More fun with statistics

Postby Nikki Nyx » Wed Jun 14, 2017 4:28 am

Upton_O_Goode wrote:
bobbo_the_Pragmatist wrote:
Upton_O_Goode wrote: Veering a bit off topic here, but this interests me. I have my own hyphenation rule. Use the hyphen when the phrase is used attributively, not when it is used predicatively. Thus: "This is a well-made car" vs. "This car is well made."

Well, I'm gonna stick with MY RULE because its based on Math. You are NOT using the rule of hyphenation which basically is for non-standard word combo's...so ... mostly arbitrary. My random choice should get me at an average of 50% correct.... but your-rule is going to be almost totally wrong?

Why do that?...................... I know why "I" would. The self-definition/aggrandizement of it all demonstrated by my use of ...... and //////. Why has the hypen achieved the status it has?===>(THERES another one!!) Picked it up way before Gawd.


Hey, like Dick Cheney, I make my own reality.

Based on that, I'm sure I didn't make my reality, because it wouldn't include Dick Cheney, along with quite a large number of other people.
What are the facts? Again and again and again-what are the facts? Shun wishful thinking, ignore divine revelation, forget what “the stars foretell,” avoid opinion, care not what the neighbors think, never mind the unguessable “verdict of history”--what are the facts, and to how many decimal places? You pilot always into an unknown future; facts are your single clue. Get the facts!
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Re: More fun with statistics

Postby Upton_O_Goode » Wed Jun 14, 2017 10:26 am

LunaNik wrote:
Upton_O_Goode wrote:
Hey, like Dick Cheney, I make my own reality.

Based on that, I'm sure I didn't make my reality, because it wouldn't include Dick Cheney, along with quite a large number of other people.


:lol: :lol: :lol:
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James Lackington, Memoirs of the First Forty-five Years of the Life of James Lackington, the Present Bookseller


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