Interesting mathematical statements (1 Viewer)

[glittergal96]

Well the problem never stated any width. It is a mathematical solution, after all.

It can't be that EACH of the individual faculties has a bias towards women, just that most of them do. At least one faculty must have male bias if the overall bias is male.

 

[InteGrand]

This is the nature of weighted averages.

 

[glittergal96]

Another good one is that there are always two opposite points on Earh with exactly the same temperature. This is proved quickly and accessibly in this video by Dr James Grime:

https://youtube.com/watch?v=5Px6fajpSio

I love intermediate value theorem consequences like this. You can get quite a lot out of such a seemingly common sense result.

Another one is the one that colloquially states that you can always rotate a square four legged table on wobbly ground into a position in which it balances.

 

[Paradoxica]

I love intermediate value theorem consequences like this. You can get quite a lot out of such a seemingly common sense result.

Another one is the one that colloquially states that you can always rotate a square four legged table on wobbly ground into a position in which it balances.

I'd just rather have the triangular legged table in the first place. Stable on almost any readily available terrain.

 

[InteGrand]

I love intermediate value theorem consequences like this. You can get quite a lot out of such a seemingly common sense result.

Another one is the one that colloquially states that you can always rotate a square four legged table on wobbly ground into a position in which it balances.

Haha yeah that table one is really nice. Here's a math / physics paper about it: http://arxiv.org/pdf/math-ph/0510065.pdf

Here's the Numbephile video about it:

 

[Soulful]

I don't have the mathematical vocabulary or ability to precisely describe it but I find Benford's Law fascinating and very strange

https://en.m.wikipedia.org/wiki/Benford's_law

 

[braintic]

It can't be that EACH of the individual faculties has a bias towards women, just that most of them do. At least one faculty must have male bias if the overall bias is male.

Imagine that
(a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
(b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

In both years, Warner's average was higher than Smith's.
Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3


So it doesn't have to be simply a majority. It can be all.

 

[braintic]

I love the "wobbly circle" and "dragoncurve" videos on Numberphile.

 

[braintic]

Haha yeah, it is an interesting result, I just meant that it's not really real-world as braintic wanted.

No, that is good enough for me. Statistical paradoxes are definitely real-world.

I thought someone might give a chaos theory example. That seems to be a topic that is never given the attention it deserves.

 

[InteGrand]

No, that is good enough for me. Statistical paradoxes are definitely real-world.

I thought someone might give a chaos theory example. That seems to be a topic that is never given the attention it deserves.

I was referring to the Kakeya Needle one in those posts.

 

[braintic]

I was referring to the Kakeya Needle one in those posts.

Oops - sorry. Nevertheless, I wonder if someone can come up with an example of the "regression to the mean" fallacy.

 

[InteGrand]

Imagine that
(a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
(b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

In both years, Warner's average was higher than Smith's.
Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3


So it doesn't have to be simply a majority. It can be all.

I think glittergal96 might have thought that the bias thing with men and women was referring to the percentage out of all those accepted who are male. The study actually considered the acceptance rates of women and men applicants separately (like your cricket example basically).

 

[braintic]

By the way, my example with averages partially shows why, when your maths teacher is calculating your yearly mark based on say 4 tests, the individual test marks should NOT be artificially scaled to the same mean before adding.

 

[InteGrand]

By the way, my example with averages partially shows why, when your maths teacher is calculating your yearly mark based on say 4 tests, the individual test marks should NOT be artificially scaled to the same mean before adding.

What about when English teachers calculate your yearly mark?

 

[braintic]

Imagine that
(a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
(b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

In both years, Warner's average was higher than Smith's.
Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3


So it doesn't have to be simply a majority. It can be all.

Actually, let me explain why people have trouble seeing how this works.

People try to compare 30 to 31 and 40 to 41.
Instead, they should be comparing diagonally: 30 to 41 in Warner's favour, and 40 to 31 in Smith's favour.
The differences are 11 and 9 - roughly the same.
But the weightings of the two pairs are 20 to 70.
So the slightly lower difference which is in Smith's favour has a much higher weighting.

So the key is the large discrepancy between the number of innings each year, and the turnaround in those numbers between the two years.

 

[braintic]

What about when English teachers calculate your yearly mark?

They call in the maths staff

 

[Paradoxica]

Oops - sorry. Nevertheless, I wonder if someone can come up with an example of the "regression to the mean" fallacy.

braintic is having trouble with his mobile data connection. He tries walking around the room until the signal strength improves enough for him to be able to view the latest BOS forum posts on his phone. Incidentally, braintic subconsciously picks up on background details, and his subconscious observation is that he is near a pineapple which is there for no apparent reason. A short while later, Integrand enters the room and removes the pineapple for some nefarious purpose. Later braintic is in a pyrotechnics lab, which happens to be the room where Integrand placed the pineapple. It turns out Integrand is making a pineapple bomb. braintic tries to view the latest posts, but is unable to load the page, and walks around the room to get a better signal. The page loads as soon as braintic walks by the pineapple. braintic realises that the pineapple was near his phone when he was able to load the previous time. braintic concludes the pineapple caused his signal to improve, and kills Integrand to prevent him from destroying the pineapple.

 

[Paradoxica]

No, that is good enough for me. Statistical paradoxes are definitely real-world.

I thought someone might give a chaos theory example. That seems to be a topic that is never given the attention it deserves.

Two celestial bodies orbiting each other, assuming there exists no other sources of gravity in their entire universe, are comprised by a perfectly deterministic system where the exact location of each body can be predicted with 100% accuracy infinitely far into the future.

Introduce a third body into the simulation, and it is analytically impossible to determine with any amount of accuracy where any of the bodies will be located infinitely far into the future. This has to do with the resulting calculations giving rise to an integral that cannot be expressed in terms of elementary functions, and can only be resolved numerically.

 

[glittergal96]

Imagine that
(a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
(b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

In both years, Warner's average was higher than Smith's.
Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3


So it doesn't have to be simply a majority. It can be all.

Yeah, sorry about that. Misinterpreted the original post and what kind of bias was entailed. (overall students vs acceptance rates)

 

[leehuan]