Correlation?

[jimdmcevoy]

This is an interesting statistical situation. I have cross posted in the politics forum as I am not sure if this has more to do with politics or mathematics.

So, let's suppose we find the average IQ of each American state. Now lets assign a number to each state, 1 for highest IQ, 2 for second highest, 3 for third highest, and so on.

Now, suppose that the number of states that Kerry won is x. Let's define K the total number of 'points', where you sum all the points from each state he won. So If Kerry only won two states, and say these states ranked 23th and 45th in terms of average IQ, then K would equal 68.

If there was no correlation between average IQ and which presidential candidate won the state, then the probability distribution for K would be:

P(K<x*(x+1)/2) = 0
P(K=x*(x+1)/2) = 1/(50 choose x)
P(K=x*(x+1)/2+1) = 1/(50 choose x)
P(K=x*(x+1)/2+2) = 2/(50 choose x)
P(K=x*(x+1)/2+3) = 3/(50 choose x)
P(K=x*(x+1)/2+4) = 4/(50 choose x)
P(K=x*(x+1)/2+5) = 6/(50 choose x)
P(K=x*(x+1)/2+6) = 7/(50 choose x)
P(K=x*(x+1)/2+7) = 10/(50 choose x)
P(K=x*(x+1)/2+8) = 11/(50 choose x)
P(K=x*(x+1)/2+9) = 16/(50 choose x)

and so on

Now according to this source http://www.mindfully.org/Reform/2004...ion-IQ2004.htm

x=19

Therefore:

P(K<190) = 0
P(K=190) = 1/(50 choose 19)
P(K=191) = 1/(50 choose 19)
P(K=192) = 2/(50 choose 19)
P(K=193) = 3/(50 choose 19)
P(K=194) = 4/(50 choose 19)
P(K=195) = 6/(50 choose 19)
P(K=196) = 7/(50 choose 19)
P(K=197) = 10/(50 choose 19)
P(K=198) = 11/(50 choose 19)
P(K=199) = 16/(50 choose 19)

and so on

Which therefore makes the probability that K<200 :

P(K<200) = 61/30405943383200
= .000000000002006...

But according to this source, K=199

Now one of three things must be true:

1. It is a miraculous fluke that it has happened this way
2. My assumption that there is no correlation is incorrect
3. The data from my source is incorrect

I'm going with number 2. Even if number 3 is the case, the data would have to be totally way off before number 2 does not have to be the case anymore.

So here is my open question, why the correlation?

For those interested in how I found out the probability distribution, I will show if you want, I find it interesting. It brings rise to the series (1 1 2 3 4 6 7 10 11 16 17 23 24 33 34 44 45 60 61 77 78 .... )

Here is the data I used.





AVERAGE
POP PRESIDENT
LIST STATE IQ ELECT

1 Connecticut 113 John Kerry
2 Massachusetts 111 John Kerry
3 New Jersey 111 John Kerry
4 New York 109 John Kerry
5 Rhode Island 107 John Kerry
6 Hawaii 106 John Kerry
7 Maryland 105 John Kerry
8 New Hampshire 105 John Kerry
9 Illinois 104 John Kerry
10 Delaware 103 John Kerry
11 Minnesota 102 John Kerry
12 Vermont 102 John Kerry
13 Washington 102 John Kerry
14 California 101 John Kerry
15 Pennsylvania 101 John Kerry
16 Maine 100 John Kerry
17 Virginia 100 George Bush
18 Wisconsin 100 John Kerry
19 Colorado 99 George Bush
20 Iowa 99 George Bush
21 Michigan 99 John Kerry
22 Nevada 99 George Bush
23 Ohio 99 George Bush
24 Oregon 99 John Kerry
25 Alaska 98 George Bush
26 Florida 98 George Bush
27 Missouri 98 George Bush
28 Kansas 96 George Bush
29 Nebraska 95 George Bush
30 Arizona 94 George Bush
31 Indiana 94 George Bush
32 Tennessee 94 George Bush
33 North Carolina 93 George Bush
34 West Virginia 93 George Bush
35 Arkansas 92 George Bush
36 Georgia 92 George Bush
37 Kentucky 92 George Bush
38 New Mexico 92 George Bush
39 North Dakota 92 George Bush
40 Texas 92 George Bush
41 Alabama 90 George Bush
42 Louisiana 90 George Bush
43 Montana 90 George Bush
44 Oklahoma 90 George Bush
45 South Dakota 90 George Bush
46 South Carolina 89 George Bush
47 Wyoming 89 George Bush
48 Idaho 87 George Bush
49 Utah 87 George Bush
50 Mississippi 85 George Bush

IO numbers are from IQ And The Wealth Of Nations, although not in the current edition. Tests and data by Raven's APT, and The Test Agency, one of the UK's leading publishers and distributors of psychometric tests. Data was published in The Economist and the St Petersburg Times, though this does not mean it should be taken as fact.

[TomCollins]

This "survey" is bogus. There is no such thing as determining the IQ of states. No such studies exist.

[jimdmcevoy]

How do you know it's bogus? (sincere question)

I also found this which seems to be in the neighborhood of the original survey:

http://www.sq.4mg.com/IQ-States.htm

If you like this data better I can rework the problem with the new data.

[emonrad87]

[ QUOTE ]
This "survey" is bogus. There is no such thing as determining the IQ of states. No such studies exist.

[/ QUOTE ]


The data used to calculate the average IQ of the states was calculated from average SAT scores. It's debated whether this is a valid comparison, as SAT test results have NOT been proven to be highly indicative of intelligence (which is what IQ measures).

[jimdmcevoy]

[ QUOTE ]

SAT test results have NOT been proven to be highly indicative of intelligence


[/ QUOTE ]

There is still a correlation to something, so the question remains, why so?

[gaming_mouse]

[ QUOTE ]
There is still a correlation to something, so the question remains, why so?

[/ QUOTE ]

If the correlation is to SAT scores, the hidden variable could be something like "amount spent on high-school education." Just guessing... I'm sure there are other plausible explanations as well.

gm

[irchans]

Jim,
I recomputed the numbers myself based on your original data and got similar results:

P(K<190) = 0
P(K=190) = 1/(50 choose 19)
P(K=191) = 1/(50 choose 19)
P(K=192) = 2/(50 choose 19)
P(K=193) = 3/(50 choose 19)
P(K=194) = 5/(50 choose 19)
P(K=195) = 7/(50 choose 19)
P(K=196) = 11/(50 choose 19)
P(K=197) = 15/(50 choose 19)
P(K=198) = 22/(50 choose 19)
P(K=199) = 30/(50 choose 19)

P(K<=199) = 97/(50 choose 19) ~= 3.2 * 10^(-12)

[jimdmcevoy]

[img]/images/graemlins/blush.gif[/img] You are quite right, I made a mistake which I only now see. Even so I came across an interesting sequence, if a(n) is the nth term:

a(n) = a(n/2) + a(n-2) if n is even
a(n) = a(n+1) - 1 if n is odd

I think this is not the way you are formally supposed to define a recursive relation, so if you are picky this is equivilent to saying:

a(n) = a(n-1) + 1 if n is even
a(n) = a(n/2+1/2) + a(n-1) - 1 if n is odd

Anyway the sequence starts with

a(0) = 1
a(1) = 1

Would you happen to know a general formula for this sequence, as in solve for a(n) just in terms of n?

[senjitsu]

It is most likely a spurious correlation. More affluent people tend to do better on standardized tests (that is, in fact, one of the most often cited arguments against the validity of those tests), and the states that voted for kerry tend to be more affluent (ie, new england states, california). I would think that, if you looked up median incomes for each of these states, you would find that that statistic correlates well with the ave IQ.

[ QUOTE ]
This is an interesting statistical situation. I have cross posted in the politics forum as I am not sure if this has more to do with politics or mathematics.

So, let's suppose we find the average IQ of each American state. Now lets assign a number to each state, 1 for highest IQ, 2 for second highest, 3 for third highest, and so on.

Now, suppose that the number of states that Kerry won is x. Let's define K the total number of 'points', where you sum all the points from each state he won. So If Kerry only won two states, and say these states ranked 23th and 45th in terms of average IQ, then K would equal 68.

If there was no correlation between average IQ and which presidential candidate won the state, then the probability distribution for K would be:

P(K<x*(x+1)/2) = 0
P(K=x*(x+1)/2) = 1/(50 choose x)
P(K=x*(x+1)/2+1) = 1/(50 choose x)
P(K=x*(x+1)/2+2) = 2/(50 choose x)
P(K=x*(x+1)/2+3) = 3/(50 choose x)
P(K=x*(x+1)/2+4) = 4/(50 choose x)
P(K=x*(x+1)/2+5) = 6/(50 choose x)
P(K=x*(x+1)/2+6) = 7/(50 choose x)
P(K=x*(x+1)/2+7) = 10/(50 choose x)
P(K=x*(x+1)/2+8) = 11/(50 choose x)
P(K=x*(x+1)/2+9) = 16/(50 choose x)

and so on

Now according to this source http://www.mindfully.org/Reform/2004...ion-IQ2004.htm

x=19

Therefore:

P(K<190) = 0
P(K=190) = 1/(50 choose 19)
P(K=191) = 1/(50 choose 19)
P(K=192) = 2/(50 choose 19)
P(K=193) = 3/(50 choose 19)
P(K=194) = 4/(50 choose 19)
P(K=195) = 6/(50 choose 19)
P(K=196) = 7/(50 choose 19)
P(K=197) = 10/(50 choose 19)
P(K=198) = 11/(50 choose 19)
P(K=199) = 16/(50 choose 19)

and so on

Which therefore makes the probability that K<200 :

P(K<200) = 61/30405943383200
= .000000000002006...

But according to this source, K=199

Now one of three things must be true:

1. It is a miraculous fluke that it has happened this way
2. My assumption that there is no correlation is incorrect
3. The data from my source is incorrect

I'm going with number 2. Even if number 3 is the case, the data would have to be totally way off before number 2 does not have to be the case anymore.

So here is my open question, why the correlation?

For those interested in how I found out the probability distribution, I will show if you want, I find it interesting. It brings rise to the series (1 1 2 3 4 6 7 10 11 16 17 23 24 33 34 44 45 60 61 77 78 .... )

Here is the data I used.





AVERAGE
POP PRESIDENT
LIST STATE IQ ELECT

1 Connecticut 113 John Kerry
2 Massachusetts 111 John Kerry
3 New Jersey 111 John Kerry
4 New York 109 John Kerry
5 Rhode Island 107 John Kerry
6 Hawaii 106 John Kerry
7 Maryland 105 John Kerry
8 New Hampshire 105 John Kerry
9 Illinois 104 John Kerry
10 Delaware 103 John Kerry
11 Minnesota 102 John Kerry
12 Vermont 102 John Kerry
13 Washington 102 John Kerry
14 California 101 John Kerry
15 Pennsylvania 101 John Kerry
16 Maine 100 John Kerry
17 Virginia 100 George Bush
18 Wisconsin 100 John Kerry
19 Colorado 99 George Bush
20 Iowa 99 George Bush
21 Michigan 99 John Kerry
22 Nevada 99 George Bush
23 Ohio 99 George Bush
24 Oregon 99 John Kerry
25 Alaska 98 George Bush
26 Florida 98 George Bush
27 Missouri 98 George Bush
28 Kansas 96 George Bush
29 Nebraska 95 George Bush
30 Arizona 94 George Bush
31 Indiana 94 George Bush
32 Tennessee 94 George Bush
33 North Carolina 93 George Bush
34 West Virginia 93 George Bush
35 Arkansas 92 George Bush
36 Georgia 92 George Bush
37 Kentucky 92 George Bush
38 New Mexico 92 George Bush
39 North Dakota 92 George Bush
40 Texas 92 George Bush
41 Alabama 90 George Bush
42 Louisiana 90 George Bush
43 Montana 90 George Bush
44 Oklahoma 90 George Bush
45 South Dakota 90 George Bush
46 South Carolina 89 George Bush
47 Wyoming 89 George Bush
48 Idaho 87 George Bush
49 Utah 87 George Bush
50 Mississippi 85 George Bush

IO numbers are from IQ And The Wealth Of Nations, although not in the current edition. Tests and data by Raven's APT, and The Test Agency, one of the UK's leading publishers and distributors of psychometric tests. Data was published in The Economist and the St Petersburg Times, though this does not mean it should be taken as fact.

[/ QUOTE ]

[irchans]

This sequence should match the the probability figures for

b(i) = P(K = i +189)* (50 choose 19)

when i = 1, 2, 3, ..., or 19 but it does not mach after 19.

I tried to solve your a(n) recurrence problem, but I could not. Maybe next time.

Cheers, irchans