## Sampling error in sports and politics

Laura McLay asks why is it so easy to forecast the Presidential election and so hard to forecast the NCAA basketball tournament? Nate Silver famously predicted the winner of all 50 states; but if you look at the NCAA basketball tournament, it’s difficult to get much above the low-70-percent range in predictive accuracy. (Silver himself has pointed this out.)

One thing that’s not mentioned, though, is that a basketball game is simply a smaller sample than voting. The basic unit of basketball analysis is the possession; a typical Division I college basketball game might include 150 or so possesions. (Averages per team are at team rankings.) If you let two basketball teams go at it for hundreds of thousands or even millions of possessions, the chance that the better team would win the game would be much higher.

In short, basketball games are subject to sampling error; voting is not.

## Sum of powers of i

What is $1 + i + i^2 + i^3 + i^4 + i^5 + \cdots$?

Of course it’s $1/(1-i)$, right, by the usual formula for summing a geometric series? But this says that

$1 + z + z^2 + \cdots = {1 \over 1-z}$

when $|z|<1$.  And $|i| = 1$, so it doesn’t work here. But who cares? Start taking partial sums. The sum is (after simplifying using $i^2 = -1, i^4 = 1$):

$1 + i - 1 - i + 1 + i - 1 - i + \cdots$

and we can write down partial sums: $1, 1+i, i, 0, 1, 1+i, i, 0, \cdots$ — and the average of this series is $(1+i)/2$, which is $1/(1-i)$. It’s a complex version of Grandi’s series ($1 + 1 - 1 + 1 - 1 + 1 \cdots = 1/2)$, and indeed the argument I’ve outlined here is Cesaro summation.)

## Weekly links for March 25

John Cook has an incomplete post about sphere volumes for which he asked for some help in recognizing some familiar formulas.

Andrew Gelman writes for the New York Times on how fast we slow down running longer distances and comments on his blog on where one might get the data.

Peter Cameron has an extended series on Fibonacci numbers: one, two, three, four, five, six, seven, eight.

How to add up quickly, from Plus magazine, on accelerating series convergence.

From Michael Trick, the indiegogo fundraiser of the traveling salesman movie.

On the distribution of time-to-proof of mathematical conjectures, by Ryohei Hisano and Didier Sornette. (I learned about this paper from Samuel Arbesman‘s book The Half-life of Facts: Why Everything We Know Has an Expiration Date.)

Numberphile on statistics on match day as collected by Opta Sports.

The New York Times on Mayor Bloomberg’s geek squad.

Oscar Boykin at the Northeast Scala Symposium gives a talk Programming isn’t math.

Are the Oxbridge bumps races the longest running Markov Chain Monte Carlo simulation in the world?

How deep is a tennis tournament compared to March Madness?

From the Wall Street Journal: a print article about the use of natural language processing in reinventing the smartphone keyboard and an accompanying interview with Ben Medlock, chief technology officer of SwiftKey.

From the BigML blog, bedtime for boosting.

Chris Wilson of Yahoo Research blogs about social network analysis based on Senate votes. Also at the Washington Post. (Democrats are more cohesive than Republicans.)

Jeff Rosenthal spoke in 2010 at Harvard on How to discuss statistics on live television; this was the inaugural Pickard memorial lecture, which was recently posted on youtube.

## Two plus two is four

The stereotypical or canonical very simple arithmetic problem is “two plus two is four”, I think.

But why isn’t it “one plus one is two”? Is it that that’s just counting, and you have to get into twos to really have what feels like arithmetic?

## Pi day

Here’s a roundup of pi-related links.

A poetic proof of the irrationality of pi.

Liz Landau on Daniel Tammet and pi.

A baker’s dozen of pie chart of pie recipes.

The Bayesian Biologist’s pi day special: estimating π using Monte Carlo.

Pi approximation day: “a holiday for people who are GOOD ENOUGH, just not transcendental”.

John Cook has five posts on computing π.