Question

I was asked a question by a friend, and I believe it is impossible, but I have a few people who claim it could be possible. What do you guys think?

"Let's say there is Town A and Town B. If you swapped one person in Town A for one person in Town B, would it be possible that the mean IQ in both towns goes up?"

So basically, if you have two sets of numbers and you swapped one number from set 1 for one number in set 2 is it possible for mean of both set 1 and set 2 to go up? The amount of numbers in set 1 and set 2 can be any amount.

Nope

yes, it entirely depends on how many people are in each city

its possible you fucking aspie

It is impossible

Number of people stays constant

Please give an example how this could happen. It's not possible.

The mean of group 1 plus the mean of group 2 must always add up to the same number. If you swap two people, then either the means will stay the same or one will decrease and the other will increase by the exact same amount.

>would it be possible that the mean IQ in both towns goes up?
Sure it's possible
Let's set both towns' populations at 2, in each there is a person with iq 110 and a person with iq 90. Let's denote Town A's smartest person as Person A, and similarly denote Person B. These two call each other to coordinate to move out of their respective towns at the same time, leaving both towns with population count of 1 and mean IQ of 90. When Person A arrives at Town B and Person B arrives at Town A (therefore completing the swap) the mean IQ in both towns goes up to 100, as per the question's requirement.

It's also trivially possible if both people improve their IQs to the point of being higher than the other person's _original_ IQ while travelling through better nutrition, exercise and other means

Town A
>Sum of IQs = 9432
>number of people doe snot matter

Town B
>sum of IQs = 8763
>number of people does not matter

Case 1
>typical Veeky Forums user leaves town A and goes to town B
>-105 for town A; +105 for town B
> /mlp/ fag leaves town B and goes to town A
> +135 for A; - 135 for B

You can do the rest of the math. You can also play around with other combinations of peoples.

Yes, I think, but only if in both towns the vast majority of people are near median intelligence.

My gut says "no", but my brain keeps spinning its wheels. Let me get some paper.

Thats not what is meant by this question you snarky asshole

If you take 110 and 90 as your set and swap both 110s, the average stays the same. If your sets are thousands of numbers the same will stay true.

after this entire process was over B would be overall dumber than it started out

Stop saying "it's possible" if you're not going to give a SPECIFIC example, you brainlets. It's a trick question that sounds possible but isn't

I was proving that it did not work.

Sorry for not posting my hypothesis

Let the IQs of Town A be {100,80}
Mean of A = 180/2 = 90

Let the IQs of Town B be {100,150,80}
Mean of B = 330/3 = 110

Mean of A + Mean of B = 200

Now swap the IQ of 80 in A with the IQ of 150 in B to get

New A = {100,150}
Mean A = 250/2 = 125

New B ={100,80,80}
Mean B = 260/3 = 86.6...

Mean A + Mean B = 211.6...

Thus it has increased.

Exercise for the reader: Prove that if B and A are towns with the exact same amount of people, this could not happen.

if Sa and Sb are sums of iqs of town A and town B populations respectively than the desired outcome can be represented by two inequalities:
Sa - IQ1 + IQ2 > Sa
Sb - IQ2 + IQ1 > Sb, or:
Sa - (IQ1 - IQ2) > Sa (implying IQ1 - IQ2 is negative)
Sb + (IQ1 - IQ2) > Sb (implying IQ1 - IQ2 is positive)
both can't be true so it's impossible.

Wait a minute OP

When you say "the mean of both set 1 and set 2 go up" do you mean "the mean of set 1 goes up. Also, the mean of set 2 goes up" or do you mean "the mean of the set which includes the numbers in both set 1 and set 2 goes up"? Or maybe even "the mean of set 1 added to the mean of set 2"? Those are 3 entirely different questions that I've seen people give "answers" for in this thread already

Okay, I posted the question. The question is NOT the total means of both towns raising. Is it possible for the individual mean of both sets raising? So Town A's mean IQ going up. And Town B's mean IQ going up.

Ah, then it is obviously no. Seriously what the fuck is wrong with you?

Literally get a pen, do some tests, and then write a general proof. Holy fuck.

Sir Robert Muldoon, Prime Minister of New Zealand from 1975-1984, once famously said that New Zealanders moving to Australia “raised the IQ of both countries”.

Not the same. In that case, no citizens were swapped. Some citizens simply moved.

That is entirely different and possible

No. The mean is directly proportional to the total IQ, since the number of people doesn't change. Since one town's IQ will have to go up and the other will have to go down, so will the means. If they have the same IQ then nothing happens of course.

is that a McChicken Sandwich?