Wikipedia:Reference desk/Archives/Mathematics/2013 January 10

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January 10

How low can you go?

What is the smallest sample size that a researcher can make and still can be useful to infer of the general population? Let's start with a big population of 1000 reindeer on an island. Now, let's say there's a terrible volcano eruption that killed most of the reindeer. Those that survived the volcano eruption swam to a nearby island. (I have to make it as random as possible, so that each participant in the population of 1000 is equally likely to survive, but most didn't and were killed by the volcano eruption by random chance.) Can the survivors be used as a random sample? How low can you go in terms of survivors? Is this scenario random? (I would have thought of a disease one, but that wouldn't be random, because obviously reindeer with a stronger immune system or have a mutation to defend themselves from the disease would survive and pass their genes onto their offspring). With the new small population, what happens if the population is stressed by genetic drift so that reindeer with big horns just happens to survive better? How can this be random? Or am I overcomplicating the situation by introducing too many variables to consider? 75.185.79.52 (talk) 03:38, 10 January 2013 (UTC)

See population bottleneck. However, any mass kill-off will cause an evolutionary leap. In your volcano example, those which detected the volcano first, ran the fastest, swam the farthest, and were able to find the next island the best would survive, so those traits would be passed on. StuRat (talk) 04:07, 10 January 2013 (UTC)

Actually, my main question was the first one about sampling size, but thanks for answering the later ones, though. 75.185.79.52 (talk) 04:31, 10 January 2013 (UTC)

Inference from a sample to the population is somewhat uncertain. If the sample is small the uncertainty is big because you don't know much about the population based on a small sample. If the surviving sample consist of 2 reindeer with big horns and 8 reindeer with small horns, then there were 250±120 reindeer with big horns and 750±120 reindeer with small horns amongst the 1000 reindeer.

   2 8 induce 1000
  249.5   750.5
119.614 119.614

Based on the empty sample you know nothing. All possibilities are equally likely. There were 500±289 reindeer of each kind.

   0 0 induce 1000
    500     500
288.964 288.964

The induce formula is from here. Bo Jacoby (talk) 08:55, 10 January 2013 (UTC).

See Mean#Small sample sizes and Student's t distribution#Sampling distribution and Student's t-test. Duoduoduo (talk) 13:51, 10 January 2013 (UTC)

Also Sample size determination#Required sample sizes for hypothesis tests. Duoduoduo (talk) 14:01, 10 January 2013 (UTC)

Trying to Calculate my Class Rank

The school only reveals two data points: 12.5 percentile is at 3.70 and 37.5 percentile is at 3.40. With these two datapoints do I have enough information to chart a normal distribution and find out the percentile for any other class rank? I don't know the mean (although it's probably 3.2 if I had to guess) nor do I know the standard dev. Can Wolfram Alpha do this? Thanks. 71.221.247.204 (talk) 04:21, 10 January 2013 (UTC)

You do have enough information to chart a normal distribution. But you don't know that the distribution is normal. Bo Jacoby (talk) 09:43, 10 January 2013 (UTC).

If you assume the grades follow a normal distribution (as Bo Jacoby says, this is a big assumption) then this is just a question of retrofitting standard scores and solving two simultaneous equations. So in a normal distribution the 12.5 percentile is about 1.15 standard deviations above the mean and the 37.5 percentile is about 0.32 standard deviations above the mean (assuming your percentiles are going backwards). So we have

Mean + 1.15 SD = 3.7

Mean + 0.32 SD = 3.4

which we can solve to give

Mean = 3.3 and SD = 0.36 to 2 d.p.

Gandalf61 (talk) 11:13, 10 January 2013 (UTC)

That is of course assuming there is a normal distribution and not some people who do the work well, a fair number who get by, and a couple of dossers. The distribution could easily have three peaks. Dmcq (talk) 13:35, 10 January 2013 (UTC)

Indeed. I would be surprised if it were even close to normal unless the marks have been deliberately calibrated to fit a normal distribution. --Tango (talk) 01:03, 11 January 2013 (UTC)

Topics more tougher than calculus

I know there is no limit of mathematics and its each topic is more tougher than other. Earlier I was told trigonometry is tough, I read it thoroughly and in a very detailed way. Then, I was told calculus is tough, I have almost finished it. The toughest topic I know is calculus. Please suggest me some other mathematical topics which I should learn after completing calculus. Show your knowledge (talk) 05:50, 10 January 2013 (UTC)

Grammar. (I just read the heading. And yes, I know grammar isn't mathematics.) HiLo48 (talk) 05:52, 10 January 2013 (UTC)

I don't think any one would disagree if I said after calculus run don't walk to linear algebra and differential equations. - Looking for Wisdom and Insight! (talk) 07:23, 10 January 2013 (UTC)

I've always had trouble understanding why people think linear algebra is difficult. I mean, don't get my wrong, there's plenty of deep results in linear algebra if you really go looking for them, but these are not treated in an introductory course.

Objectively, an introductory course in linear algebra should be much easier than calculus. I think the problem is that it's just a different way of thinking than most students have ever been exposed to at that stage of their studies (mostly, more abstract). If they would just let it be abstract, they would probably find it easy, but because it varies from their expectations, they find it difficult. I think. Anyone who can shed further light on it, I'd be interested to hear it. --Trovatore (talk) 07:54, 10 January 2013 (UTC)

The toughness of a topic is not objective. It depends on the book and the reader. Lack of applications make abstract expositions tough. Bo Jacoby (talk) 09:15, 10 January 2013 (UTC).

Probability and statistics is pretty tough and it has loads of applications. Other areas you might be interested in are number theory or topology or if you like computers there's computational complexity theory. Einstein wrote a good book about Relativity, and differential equations are a nice follow on from calculus. If you can derive the formula in the book for the advance of the perihelion of Mercury you'll know you're getting somewhere. Dmcq (talk) 13:19, 10 January 2013 (UTC)

See Mathematical analysis#Subdivisions. Duoduoduo (talk) 13:57, 10 January 2013 (UTC)

Dirichlet distribution

How do you calculate the mean of the Dirichlet distribution?--AnalysisAlgebra (talk) 10:41, 10 January 2013 (UTC)

As explained in Dirichlet distribution, if you have a Dirichlet distribution with parameters $(\alpha _{1},\ldots ,\alpha _{K})$, the mean is $\left({\frac {\alpha _{1}}{\sum _{i}\alpha _{i}}},\ldots ,{\frac {\alpha _{K}}{\sum _{i}\alpha _{i}}}\right)$.

If you meant how is this result derived: Using the properties of the beta function and some straightforward integration. -- Meni Rosenfeld (talk) 12:51, 10 January 2013 (UTC)

Base 10, 11 and fingers question.

People say we use decimal system because we have 10 fingers or something like that. If the base we use is based on numbers of fingers we have why we use base 10 and not base 11? I mean, we have 10 fingers and not 9.177.40.130.22 (talk) 13:11, 10 January 2013 (UTC)

Because people use one to one correspondence between fingers and the things being counted. If you hold up two open hands with the fingers and thumbs pointing out it is clearly ten. Do that twice and you have twenty. If you want to hire a boat in a country you don't know the language of you understand the price pretty quickly that way! Dmcq (talk) 13:32, 10 January 2013 (UTC)

"If you hold up two open hands with the fingers and thumbs pointing out it is clearly ten."

And thats why, if the base system is based on our number of fingers it would need to be base 11. We have 1 finger, 2 fingers, 3 fingers.., 8 fingers..., 9 fingers and (lets use A to 10) A fingers in our hands. With base 10 we run out of digits after 9 while there is one extra finger to count. — Preceding unsigned comment added by 177.40.130.22 (talk) 15:31, 10 January 2013 (UTC)

You are not the first person to have thought this [1][2][3]. -- Q Chris (talk) 15:45, 10 January 2013 (UTC)

(edit conflict) The fact that we count using fingers, of which we have ten, means that it is natural to express larger quantities as multiples of ten. That’s why human languages tend to express numbers in terms of multiples of ten (or less commonly, twenty). It has worked like this for thousands of years before the positional numeral system was invented, and then it was natural to use base 10 for the positional system, so that counting in multiples of ten was also easy to accomplish with the numerals. How many digits are needed to express the particular counts that can be shown with the fingers of two hands is not directly relevant.—Emil J. 15:50, 10 January 2013 (UTC)

Note I said immediately afterwards "If you do that twice you have twenty". It clearly makes for a base ten way of counting. As said above other people have used base four or five or twenty but they have been based on the same idea. Actually the common system in Europe used to be five and twenty I believe rather than ten. Dmcq (talk) 16:11, 10 January 2013 (UTC)

Fine call ten 'A' your finger based counting system is still base 10. It just goes like this: 1,2,3,4,5,6,7,8,9,A,11,12,13,14,15,16,17,18,19,1A,21,22,23,24,25,26,27,28,29,2A. Note there are still 10 digits between like numbers. This is because either 10 or 0 have to be considered as part of the first digit cohort. You cannot exclude both. As for the argument from fingers, it becomes obvious that 10 is logical when you consider how many fingers are on two peoples hands and then n peoples hands. — Preceding unsigned comment added by 202.65.245.7 (talk) 16:23, 10 January 2013 (UTC)

No it would have 0 before the number 0, so 11 digits. So it would be base 11. Of couse we can't start with zero saying "0 finger in my hand, 1 fingers in my hand,... and stop at 9 or we would have one finger left to count", so this digit would exist, but we wouldnt use it to count the first finger. 177.40.130.22 (talk) 18:26, 10 January 2013 (UTC)

If you have both zero and A to get a base 11 system, then for example thirty would be written 28 (two elevens plus eight). People wouldn't find that useful because 28 doesn't make it obvious that thirty involves holding up both hands exactly three times. Duoduoduo (talk) 18:51, 10 January 2013 (UTC)

No you're still mistaken. Think of it this way, the number 9 is elliptical for 'the value equivalent to 9 fingers' whereas 10 is elliptical for 'the value equivalent to 1 persons fingers and 0 additional fingers'. If you gave a single digit symbolic form to this, say 'A' it would be replace 10 rather than supplementing it. For a proof of this consider 11: 11 is '1 persons fingers plus 1 extra finger' if I deduct one, I have 1 persons fingers, which is the value you named 'A'. Hence your counting system proceeds as

1  2  3  4  5  6  7  8  9  A
11 12 13 14 15 16 17 18 19 1A
21 22 23 24 25 26 27 28 29 2A
...

Regardless of how you choose to notate 0 in your system, the difference between the number 1A and 2A is 10, so it is base 10. — Preceding unsigned comment added by 123.136.64.14 (talk) 01:35, 11 January 2013 (UTC)

Although that number system is internally consistent, it resembles that of a culture which has not yet invented zero, and is thus useless for mathematical operations because it does not form a monoid under normal addition. 72.128.82.131 (talk) 04:45, 11 January 2013 (UTC)

It's not useless – this is the bijective base-10 system. While it does not have a zero, it does have place value, which would already be a large step for that culture. Mathematical operations can still be performed as in decimal. Double sharp (talk) 16:46, 12 January 2013 (UTC)

Precisely, the error was in failing to count 0 as one of the 10 digits representing natural numbers below 10, and consequentially believing that another must be introduced. — Preceding unsigned comment added by 123.136.64.14 (talk) 07:24, 11 January 2013 (UTC)

I was not failing on that, the thing is that zero is nothing and so 0 cant be used to represent/count your first finger. Thats why we would need an extra zero. But, yes DMCq last answer nailed it, and answered my thing. You can interpret "Being based on numbers of fingers" on different ways, because it inst specified enought, and I was thinking about it in an wrong way. — Preceding unsigned comment added by 177.40.130.22 (talk) 15:35, 11 January 2013 (UTC)

Rotation of arbitrary coordinate systems

Let's say you have a Cartesian coordinate system. A rotation of this coordinate system can be defined as any coordinate transformation such that x^2 + y^2 = x'^2 + y'^2, where the primed coordinates are the new coordinates etc. Maybe one or two other conditions are needed, but this captures the main idea. How can a rotation of an arbitrary (ie not necessarily Cartesian) coordinate system be defined? 65.92.6.137 (talk) 18:11, 10 January 2013 (UTC)

What exactly do you mean by an “arbitrary coordinate system”?—Emil J. 18:38, 10 January 2013 (UTC)

I'm guessing he's asking about systems where the points are specified by something other than (x, y) pairs. For example, a polar coordinate system where the specification is by (r,θ) instead. -- To answer, the first point is that your condition is a necessary limit for a rotation around the origin, but not sufficient, even in a Cartesian system. For example (x,y) -> (√(x^2 + y^2 - (x+y)^2/4 ),(x+y)/2) fulfills the condition, but isn't a rotation. To be a rotation, not only do you have the same distance to the fixed point before and after your rotation (this is what your equation is saying - the distance to the origin (fixed point) before the rotation is the same as the distance to the origin (fixed point) after the rotation), you also have to preserve the geometry of the other regions (there's various ways of saying this, one of which is to say that the distance between all points must be conserved). That's what you need to encode in your rotation test (for Euclidean geometries, at least) - are all the distances before and after the same, and is there a fixed point of the transformation. The way you do this algebraically will differ based on the coordinate system used. -- 205.175.124.30 (talk) 19:48, 10 January 2013 (UTC)

Oops - one *major* point I forgot. Rotations preserve orientation (e.g. clockwise vs. counterclockwise) as well. The simple "test for a fixed point & test distances" would incorrectly classify a reflection as a rotation. - As you can see, determining what is or is not a rotation in an arbitrary sense is not necessarily straightforward. -- 205.175.124.30 (talk) 20:03, 10 January 2013 (UTC)

regression for data that is a power function: copied from Miscellaneous desk

I have some data that looks like it is in a Power function. Of course, taking logs and doing a linear regression will give the constants.

1. Is this a good way to do get the constants?

2. If a correlation coefficient is calculated on the line, is it meaningful for the original data? Bubba73 ^{You talkin' to me?} 20:21, 10 January 2013 (UTC)

Copying this to the Maths desk where they really know about these things and ignoramuses like me won't be tempted to hazard an answer. Itsmejudith (talk) 20:25, 10 January 2013 (UTC)

Say $y_{t}=Ax_{t}^{b}U_{t}$ where U is lognormally distributed. Taking logs gives $(\log y)_{t}=\log A+b\cdot \log x_{t}+u_{t}$ where $u_{t}=\log U_{t}$ is normally distributed. Yes, it is standard to run this regression. And the coefficient of determination (square of the correlation of the logged variables) is valid to consider as one measure of the fit. No, the correlation has no direct interpretation in terms of the original variables, but that's okay -- it's fine as an indication of the log-log relationship. Duoduoduo (talk) 20:53, 10 January 2013 (UTC)

(ec)Regression analysis has a number of assumptions (they differ slightly based on the technique used). One of the typical ones that's most relevant to you is that the error in the response variable is identically distributed for all points. That is, if the error of measuring your point at low x is ±0.1, then the error of measuring a point at high x is also ±0.1. For most inputs, this holds. At day 1 you measure the response to be 52±1 mm, and at day 20 you measure the response to be 783±1 mm. In that case, you would want to do (nonlinear) regression on the untransformed values, because if you transformed the values first, the errors would not be equal, resulting in a comparatively poorer fit. However, there are some cases where the errors are not constant - say your measuring device only gives the first two digits of a value that ranges over many orders of magnitude. In that sort of case you can safely transform the data, as long as the transform makes the numerical errors in the transformed response be similarly distributed. - The reason you see a large number of people doing transformations before fitting is because before modern computers, nonlinear fits were hard. Linear graphs also look nicer, but the answer to that is to plot linearly, but fit nonlinearly. 2) It's meaningful in the sense that it describes how well the plot fits. If you're doing something else with it ("fraction of unexplained variance" and the like), I'd be exceedingly cautious. -- 205.175.124.30 (talk) 21:03, 10 January 2013 (UTC)

Basically agreeing with that response -- the logarithm method is good if the errors for the data points are proportional to their true values. If that doesn't hold, then some other method is likely to be better. Looie496 (talk) 21:09, 10 January 2013 (UTC)

For some examples, see Log-log plot#Applications re fractal dimension, and Cobb–Douglas production function. Duoduoduo (talk) 00:13, 11 January 2013 (UTC)

(ec) Thanks. On two data sets of 10 values each, with the log method, I'm getting correlation coefficients > 0.9998, which doesn't seem reasonable. Bubba73 ^{You talkin' to me?} 04:14, 11 January 2013 (UTC)

If zero or negative values of your data doesn't occur and make no sense, then take the logarithm immediately and forget all about the original data. Bo Jacoby (talk) 08:40, 11 January 2013 (UTC).

Bubba, if you're getting correlation coefficients > 0.9998, I strongly suspect that this is just rounding error at some point in the calculations or in the data itself; I suspect the actual correlation is exactly 1.0, and that your data were actually generated jointly based on the power law. That is, whatever generated the x data also generated the y data according to $y_{t}=Ax_{t}^{b}$; then when you run the regression you get a perfect fit except for rounding error. Duoduoduo (talk) 13:49, 11 January 2013 (UTC)

I was thinking that taking logs makes it very close to linear, hiding some of the variability. Bubba73 ^{You talkin' to me?} 22:11, 11 January 2013 (UTC)