I went to the BLS website, and got some salary statistics for different careers. The really interesting thing I see from the statistics is the disparity between what people think are average salaries, and what the average salaries actually are. Most people would assume that lawyers make more than engineers, but in reality the average salary for a law school graduate isn't all that high. The thing is that law school graduates have a much higher standard deviation in their salaries, and most people only focus on the lawyers who make the most money.

I'd imagine that people do the same thing when they're thinking about going to college or grad school. They always think that they're going to be above average, even though in reality the average person is average.

The only other interesting thing to point out is that in China, all of the major leaders are engineers. Hu Jintao, Jiang Zhemin, Wen Jiabao - all engineers. In contrast, most American leaders are lawyers or businessman. Usually Republicans are the businessman, and Democrats are the lawyers, but there are exceptions. Bush Sr. was an economist, and Reagen was an actor (which, in my opinion, made them, respectively, the best and worst presidents of the later half of the 20th century).

The General Social Survey gave people a vocabulary test (often considered a very rough proxy for education and intelligence) called Wordsum and also asked people for their political affiliation. Wordsum is scored from 0 to 10, with those scores corresponding roughly to normal scores (note that I calculated this by using percentiles from the data, so the percentiles should be seen as being with respect to the people who were surveyed, and not the population at large. There are probably some people left out of the survey for some reason or another):

0 - lower than 62
1 - 62 - bottom 1%
2 - 70 - bottom 3%
3 - 76 - bottom 6%
4 - 82 - bottom 12%
5 - 89 - bottom 23%
6 - 96 - bottom 40%
7 - 104 - top 40%
8 - 110 - top 24%
9 - 116 - top 14%
10 - 123 - top 6%

Below I have the data graphed out (the scale is a little cut off, light blue means moderate Republican, pink means strong republican, and brown means independent). I personally think that the results are fascinating.

1. Political affiliation does not correlate very strongly with vocabulary. I calculated a coefficient of 0.111 (where 1 is a perfect correlation, and 0 is no relationship).

2. Democrats seem to dominate the very top and most of the bottom of the distribution. Republicans tend to be in the upper middle sections.

3. Third party affiliations correlate very well with vocabulary. It's likely that people who are uneducated (and have small vocabularies) don't study politics as much, and therefore don't seriously consider third parties.

4. Independents tend to dominate the bottom of the distribution. This is probably because independents lose out on participating in the primary, and uneducated people don't realize that this is the case. It's also possible that uneducated people feel so disenfranchised with the system that they are unwilling to endorse a political party.

Your mileage may vary, but I think that one of the funnest math problems out there is the Problem of Josephus.

--The Problem--

You have N (where N can be any number) people standing around in a circle. Starting with an arbitrary person and going around clockwise, every other living person in the circle is killed until one person remains. For example, if you have 5 people in the circle, the 1st person is killed, then the 3rd, then the 5th, then the 4th (the 1st and 3rd people are dead, and we skip the 2nd), and the 2nd person survives.

For any circle of N people, where would you want to stand in the circle to be the last person standing.

--The Solution--

(Note: when I started working on this problem, I would draw 1, 2, 3, 4, 5... in a circle, and then cross them out while moving around the circle and mumbling "kill skip. kill skip. kill skip". For the purpose of not looking like a homicidal maniac, I strongly advise against this approach).

It's pretty simply to brute force the solution to this problem for small N. For example, if N = 1, it's obvious that there is no solution and no one survives (this is really significant, because it tells us that shifting the domain of the problem can lead to no solutions, suggesting perhaps that there is a logarithm somewhere in our answer). For N = 2, the 2nd person is the survivor. For N = 3, the 2nd person is still the survivor. For ease of reading I create this table:

2 - 2
3 - 2
4 - 4
5 - 2
6 - 4
7 - 6
8 - 8
9 - 2

There are a few interesting patterns here. The first neat one is that if N is 2 raised to an integer power, then the Nth person is the survivor (for example 2, 4, 8). Another neat thing is that the position of the survivor slowly rises until it becomes the last spot in the circle, and then suddenly depresses to the 2nd person. It also rises at a rate of 2 people for every increase in N.

If we say that k is the integer value for which (2^k) is as large as possible while still being smaller than N (so for example, if N = 9, k = 3 because 2^3 is the largest value of 2^k smaller than 9), then we can write our solution quite neatly:

p(N, k) = 2(n - 2^k)

p(2, 0) = 2
p(3, 1) = 2
p(4, 1) = 4
p(5, 2) = 2
p(6, 2) = 4
p(7, 2) = 6
p(8, 2) = 8
p(9, 3) = 2
p(10, 3) = 4

This fits perfectly with our old data, so its pretty safe to say that our formula is right. But just to make the formula more complete, we ought to find k in terms of N. We know that k is just the the largest integer that 2 can be raised to, such that 2^k is smaller than N. Writing this out a little bit more formally:

2^k < N
k is an integer
k is as large as possible, given the two previous statements

Well, if we rearrange the first statement, we get

2^k < N
log2(2^k) < log2(N)
k < log2(N)

How do we ensure that k is an integer? Simple, you ensure that N is 2 raised to some integer power. So for example if N is 9, then k is \\log2(9)// (where \\2.5// is 2, and the slashes mean floor to the greatest integer less than what's inside the slashes). The only odd thing with this is that for this to work, we have to treat \\5// as 4, while the floor function would normally treat \\5// as 5. If anyone knows what you'd actually call the greatest smaller integer function I'm describing, don't hesitate to tell me.

So substituting in k = \\ log2(N) //, we get:

p(n) = 2(n - 2 ^ \\log2(n)// )

If you try this with various suicide circles, it should become apparent that it works as intended.

It's also interesting to note that this formula can easily be generalized to describe the case where you skip more than one person at a time as you move around the suicide circle. I don't feel like trying it now, but I'm pretty sure it just becomes:

p(n, m) = m(n - m ^ \\logm (n)//)

The last few news posts I've made have been about statistics or mathematics or other boring things like that. So now I'm going to write about something different - Radiohead songs!

Here's what I consider to be the top 5:

1. Karma Police
2. Let Down
3. Weird Fishes
4. Airbag
5. Street Spirit

But I'd have to say that my favorite song (by any artist) is Toccata and Fugue by Bach.

Today, in the US education system kids are taught addition, subtraction, multiplication, division, and (for advanced students) basic algebra in Elementary School. In a way this makes sense. It provides everyone with the basic tools that they need to get through life. It lets them add times together, so they can organize their daily lives. It lets them balance a checkbook and calculate tips at restaurants.

But what if we started teaching kids advanced math concepts in elementary school? Vectors could be introduced at the same time as numbers. For example, you could teach that "5" represents 5 turtles, and [1 5] represents one turtle, and 5 apples. And that a vector like:

[1]
[5]

represents one turtle, followed by 5 turtles at another location.

Eventually you could introduce matrices:

[1 3]
[5 7]

And could visualize it as one area with one turtle and 3 apples, and another area with 5 turtles and 7 apples.

Then when you introduce addition, you could do it for matrices, and then just say that adding single numbers is like adding two 1 X 1 matrices.

Multiplying matrices is kind of complicated, so I don't know if you could teach it in elementary school, and even if you could, I don't know if they'd understand why it works:

[1 3] X [4 2] = [1*4 + 3*8 1*2 + 3*6] = [28 20]
[5 9] [8 6] [5*4 + 9*8 5*2 + 9*6] [92 64]

But it also makes multiplication very dynamic and interesting. And it allows one to look at a matrix as a transformation, and not a data set.

Using very advanced regression analysis, I've discovered that the probability of one being a Republican is directly proportional to the amount that a person's douchiness exceeds their intelligence.

Mitt Romney would be plotted at the top right of the graph, while George Bush would be plotted at the center on y-axis and the far left side of the x-axis.

From one of my electrical engineering professors.

Well, I have nothing else to do, so I'm going to prove Euler's formula for all of you Newgrounder's benefit.

What's really interesting about Euler's formula is that it's so counter intuitive, that it is not uncommon for people to present it as proof of God's existence. It says that the natural constant, e, multiplied by itself x times the square root of negative times is equal to the square root of negative one times the ratio of a right triangle's opposite side to its hypotenuse if the triangle's angle is x, plus the ratio of a right triangle's adjacent side to its hypotenuse if the triangle's angle is x.

Or in math: e^(i * t) = i * sin(t) + cos(t)

Anyway, unto the proof:

i = sqrt( -1 ) and t be some variable:

y(t) = e^(i * t)

y'(t) = i * e^(i * t)

y''(t) = -1 * e^(i * t)

y(t) + y''(t) = 0

Taking the laplace transform of both sides:

L[ y(t) ] + L[ y''(t) ] = L[ 0 ]

y(s) + s * y'(s) - y'(0) = 0
y(s) + s^2 * y(s) - s * y(0) - y'(0) = 0

Alright, now going back to our equations

y(0) = 1
y'(0) = i
y''(0) = -1

Plugging those in we get

y(s) + s^2 * y(s) - s - i = 0

y(s) * (1 + s^2) - s - i = 0

y(s) = s / (1 + s^2) + i / (1 + s^2)

Taking the inverse Laplace transform of both sides (which is usually done through tables, since the transform itself is really ugly), we find that:

y(t) = cos(t) + i * sin(t)

y(t) = e^(i * t)

e^(i * t) = cos(t) + i * sin(t)

So there you have it, no magic, just maths.

As many of you who've met me know, I'm a statistics whore, and I calculate correlations on everything. Well, if you're wondering about correlations, wonder no more!

If the correlation between two variables X and Y is 1, then increasing the value of X always leads to a proportional increase in Y. If the correlation is .5, then there is a 50% chance of an increase in X leading to a proportional increase in Y. If the correlation is 0 then the data is random. For example, there is probably a correlation of 1.0 between earning 100k+ per year and driving a Ferrari, but probably a 0 correlation between wearing glasses and driving a Ferrari.

It is often said that "Correlation is not causality", but this isn't quite true. Correlation is a necessary, but not sufficient precondition for causation. If a correlation exists between A and B, there are 3 possibilities:

A causes B
B causes A
A and B share a common cause

Note that for there to be any of the relationships shown above between two variables, there MUST be a correlation.

It seems like, unless Johns Hopkins pulls me off of their waitlist (which looks to be exceedingly unlikely), I'll be going to the Worcester Polytechnic Institute this Fall 2008 for Electrical Engineering. Anyone gone there?

Here's the schedule for my first 2 quarters:

Quarter 1:

Introduction to Electrical Engineering
Environmental Policy and Law
Numerical Analysis I
Discrete Mathematics I

Quarter 2:

Linear Algebra
Principles of Physics: Electricity and Magnetism
Numerical Analysis II
Discrete Mathematics II