Wikipedia:Reference desk/Archives/Mathematics/2007 July 7

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July 7

How can I get my budgie to take a bath?

I've got a bird bath to hook onto the side of his cage, but he just cocks his head and looks at it as though to say "what the hell is that thing?". He won't go into the water at all. I've tried putting a lettuce leaf in the bath to tempt him in but all he does is stand on the lip of the bath and lean forwards, stretching out to his full length to grab the edge of it, then pulls it out onto the floor of the cage. I've tried showering him with a plant mister a few times but he absolutely HATES that. Any tips? --84.68.70.40 07:32, 7 July 2007 (UTC)

Have you tried taking its reciprocal? - Rainwarrior 07:45, 7 July 2007 (UTC)

What about a dust bath? Perhaps you could make a scaled down version. Lanfear's Bane

This would have been an excellent question for the Science Desk. BTW, what makes you think he needs a bath ? StuRat 03:30, 11 July 2007 (UTC)

Percentiles

Can someone explain and/or help me understand the following concept about percentiles. Thank you. Let's say that John takes an exam, and let's just say that it is the SAT exam. By definition, a percentile is the percentage of examinees that score at or below John's score. So, if John's scaled score has a 75th percentile, that means that 75% of the population of examinees scored worse/lower than John or equal to John. That is my base understanding and the premise of my question. As such, a percentile (by definition) can never be 100% (because John himself is a part of the population of examinees and John cannot score lower than John). So, the "highest" that a percentile can be is 99.999999999% or so, but it can never actually reach 100%. We normally think of exam scores as 0 to 100, and we normally think of percentages as 0% to 100%. While exam scores and percentages can be higher than 100 or 100%, nonetheless, they are based on the value 100. So, I guess my question is ... why are percentiles not defined as going from 0 to 100 (i.e., actually including the 100 value)?

Scenario A: John scores 88% and there are nine other examinees that all have scored 75% on the exam. In rank order, the scores look like this: 75 - 75 - 75 - 75 - 75 - 75 - 75 - 75 - 75 - 88. Thus nine examinees out of ten (i.e., 90% of the examinees) have scored lower than or equal to John. Thus, John's percentile score is 90. In other words, in plain English ... out of everyone who took the exam, 90% did worse than John.

Scenario B: Use the same data as Scenario A. Why can't percentiles be defined such that nine examinees out of nine (i.e., 100% of the examinees) have scored lower than or equal to John? So that John's percentile score is 100. In other words, in plain English, everyone else scored below John ... (since, obviously, John did not and cannot score below John).

So, in Scenario A, the ratio or fraction is 9/10 = 90%. In Scenario B, the ratio or fraction is 9/9 = 100%. So, in other words, why can't the denominator be every examinee except John (n-1) instead of every examinee including John (n) ...? I don't know if I have explained this clearly, but I hope that someone gets the gist of my question. Why can't percentiles go up to and include the 100%, which would seem to make much more common sense or intuitive sense ...? Thanks. (JosephASpadaro 18:28, 7 July 2007 (UTC))

I guess percentile is defined the way it is as it fits with what you would expect for the 25th and 50th percentile, using senario B your would not have 50% below the 50th percentile. You can have the 100th percentile which is everybody. --Salix alba (talk) 19:19, 7 July 2007 (UTC)

Just observe that the current system allows for the 0th percentile, while disallowing the 100th; your proposal would allow the 100th but disallow the 0th. Tesseran 22:52, 7 July 2007 (UTC)

Tesseran - why does my system not allow the 0th percentile? Example: John gets a score of 80% on the test and the other nine people in the class get a score of 90%. The data is: 80 - 90 - 90 - 90 - 90 - 90 - 90 - 90 - 90 - 90. Therefore, zero out of the nine other examinees (excluding John) scored below John. Thus, 0/9 = 0% = 0th percentile. No? Am I missing some concept? Thanks. (JosephASpadaro 01:07, 8 July 2007 (UTC))

I'm sorry, you're correct. I misunderstood your suggestion. I agree with Salix alba's comment. (By the way, you repeatedly use "lower than or equal to John" above (for example) where you mean "lower than".) Tesseran 06:13, 8 July 2007 (UTC)

Yes, my method can range from 0 to 100 (including the 100), as would be "intuitively" expected, whereas the current definition disallows the actual 100 value. That is pretty much my whole point. You are correct also about my mis-statements regarding "lower than or equal to". Thanks. (JosephASpadaro 16:33, 8 July 2007 (UTC))

The reason for using < rather than ≤ comes down to dealing with those middle values. It ends up being a bit more complicated than simple counting with cases like the 75 75 75 75 90 instance (since there is some assumed error in each term so that 75 really means in the range [74.5,75.5). Social science stats texts generally provide a decent gentle introduction to what's going on with this. Donald Hosek 04:16, 9 July 2007 (UTC)

Thanks for the comments -- I appreciate the input. (JosephASpadaro 00:39, 9 July 2007 (UTC))

Note that percentiles are really only appropriate for cases where there are thousands of people being compared, like the SATs. For example, with two people taking a test, would you say the lower has 0% and the upper has 100%, even if the two had only a one question answered differently ? In the case of thousands, the precise definition is less important, as a single top scorer will get 100% by one method and 99.99% by another, which will round to 100%, anyway. StuRat 03:25, 11 July 2007 (UTC)

True, that's a good point, StuRat. Thanks. (JosephASpadaro 06:39, 12 July 2007 (UTC))

Large Numbers

Due to the concept of infinity, there is no "biggest" or "largest" number that exists. Since we can always simply add one to any number, there are infinitely many numbers ... i.e., numbers continue to infinity. My question is: what is the largest named number? By that, I mean verbally (words), not numerically. In other words, if we use numbers, we can say that a very big number is 100 raised to the 100 power. And we can do that with any numbers we so choose. However, we also verbally "name" these numbers with words (i.e., hundreds, thousands, millions, billions, etc.). Using the system of words, what is the largest number that actually has a verbal "name"? Thanks. Also, when we get to very large numbers, is there simply some type of systematic naming scheme -- such that there are infinitely many names (similar to what they do in naming "more new" elements on the Periodic Table)? Thank you. (JosephASpadaro 18:47, 7 July 2007 (UTC))

While I can't say anything about "biggest," you might find Names of large numbers and Graham's number interesting. J Elliot 19:02, 7 July 2007 (UTC)

According to a record book of mine, the largest named number is 'milli-millillion', which equals $\ 10^{3000003}$. 65.31.80.94 22:36, 7 July 2007 (UTC)

PS check out Knuth -yllion. 65.31.80.94 23:29, 7 July 2007 (UTC)

Largest named number is a googolplex . :$10^{\scriptscriptstyle 10\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000}>10^{3000003}$ 211.28.121.144 07:15, 8 July 2007 (UTC)

See also Graham's number. --CiaPan 07:59, 8 July 2007 (UTC)

Yeah, as far as largest named number with an application goes, I think Graham's number takes the cake. But the question of the largest number where every number less than it is also named is an interesting question. J Elliot 08:22, 8 July 2007 (UTC)

See also Interesting number paradox. —Keenan Pepper 08:47, 8 July 2007 (UTC)

Thanks, all, for the input -- and for the relevant links. All very interesting. Thank you. (JosephASpadaro 00:40, 9 July 2007 (UTC))

Incenters with integer coordinates

Do there exist triangles in the Cartesian plane with integer vertices such that the incenter of the triangle also has integer coordinates? If so, can you give an example? Dr. Sunglasses 21:45, 7 July 2007 (UTC)

What about the triangle with vertices (0,0), (3,0) and (0,4)?  --Lambiam^Talk 22:32, 7 July 2007 (UTC)

Yeah, that works. (I looked at this triangle, too (well, with (4,0) and (0,3))...) Dr. Sunglasses 22:57, 7 July 2007 (UTC)