The Angry Statistician

For years the kilogram has been defined with respect to a platinum and iridium cylinder, but this is now no longer the case . Here's a puzzle about kilograms that's easy to state and understand, but the answer is very, very surprising. I've always had a fascination with really large numbers. First 100 when I was really little, and as I got older and more sophisticated numbers like a googol and the smallest number that satisfies the conditions of the Archimedes cattle problem . When I was an undergraduate I interviewed for a summer internship with an insurance company as an actuarial student. They gave me the following puzzle - what's the smallest number that when you move the last digit to the front it multiplies by 2? I calculated for a little while, then said "This can't be right, my answer has 18 digits!". It turns out that the smallest solution does, indeed, have 18 digits. We can see this by letting our \((n+1)\)-digit number \( x = 10 m + a\),...

The following math problem, which appeared on a Scottish maths paper, has been making the internet rounds . The first two parts require students to interpret the meaning of the components of the formula \(T(x) = 5 \sqrt{36+x^2} + 4(20-x) \), and the final "challenge" component involves finding the minimum of \( T(x) \) over \( 0 \leq x \leq 20 \). Usually this would require a differentiation, but if you know Snell's law you can write down the solution almost immediately. People normally think of Snell's law in the context of light and optics, but it's really a statement about least time across media permitting different velocities. One way to phrase Snell's law is that least travel time is achieved when \[ \frac{\sin{\theta_1}}{\sin{\theta_2}} = \frac{v_1}{v_2},\] where \( \theta_1, \theta_2\) are the angles to the normal and \(v_1, v_2\) are the travel velocities in the two media. In our puzzle the crocodile has an implied travel velocity of 1/5 in t...

When you start doing more advanced sports analytics you'll eventually starting working with what are known as hierarchical, nested or mixed effects models . These are models that contain both fixed and random effects . There are multiple ways of defining fixed vs random random effects , but one way I find particularly useful is that random effects are being "predicted" rather than "estimated", and this in turn involves some "shrinkage" towards the mean. Here's some R code for NCAA ice hockey power rankings using a nested Poisson model (which can be found in my hockey GitHub repository ): model The fixed effects are year , field (home/away/neutral), d_div (NCAA division of the defense), o_div (NCAA division of the offense) and game_length (number of overtime periods); offense (strength of offense), defense (strength of defense) and game_id are all random effects. The reason for modeling team offenses and defenses as random vs fixed effec...

Elo's rating system became famous from its use in chess, but it and variations are now used in sports like the NFL to eSports like League of Legends . It also was infamously used on various "Hot or Not" type websites, as shown in this scene from the movie "Social Network": Of course, there's a mistake in the formula in the movie! What is the Elo rating system? As originally proposed, it presumes that if two players A and B have ratings \(R_A\) and \(R_B\), then the expected score of player A is \[\frac{1}{1+10^{\frac{R_B-R_A}{400}}}.\] Furthermore, if A has a current rating of \(R_A\) and plays some more games, then the updated rating \({R_A}'\) is given by \({R_A}' = R_A + K(S_A-E_A)\), where \(K\) is an adjustment factor, \(S_A\) is the number of points scored by A and \(E_A\) was the expected number of points scored by A based on the rating \(R_A\). Now, the expected score formula given above has the same form as a logistic regression model...

One of the simplest and most common power ranking models is known as the Bradley-Terry-Luce model , which is equivalent to other famous models such the logistic model and the Elo rating system . I'll be referring to "teams" here, but of course the same ideas apply to any two-participant game. Let me clarify what I mean when I use the term "power ranking". A power ranking supplies not only a ranking of teams, but also provides numbers that may be used to estimate the probabilities of various outcomes were two particular teams to play a match. In the BTL power ranking system we assume the teams have some latent (hidden/unknown) "strength" \(R_i\), and that the probability of \(i\) beating \(j\) is \( \frac{R_i}{R_i+R_j} \). Note that each \(R_i\) is assumed to be strictly positive. Where does this model structure come from? Here are three reasonable constraints for a power ranking model:  If \(R_i\) and \(R_j\) have equal strength, the probabil...

There's lot of confusion about how best to get started doing baseball analysis. It doesn't have to be difficult! You can start doing it right away, even if you don't know anything about R, Python, Ruby, SQL or machine learning (most GMs can't code). Learning these and other tools makes it easier and faster to do analysis, but they're only part of the process of constructing a well-reasoned argument. They're important, of course, because they can turn 2 months of hard work into 10 minutes of typing. Even if you don't like mathematics, statistics, coding or databases, they're mundane necessities that can make your life much easier and your analysis more powerful. Here are two example problems. You don't have to do these specifically, but they illustrate the general idea. Write up your solutions, then publish them for other people to make some (hopefully) helpful comments and suggestions. This can be on a blog or through a versioning control platform...

Some potentially useful SQL resources - explanations, visualizations, exercises, games, classes. A Visual Explanation of SQL Joins Datamonkey Introduction to Database Management Systems SQL Island Adventure Game PostgreSQL Exercises Public Affairs Data Journalism SQL Teaching's GitHub repo (if you're curious) Stanford's Self-Paced Database MOOC Hackr.io's SQL Section (good to check occasionally) Practical skills of SQL language SQL Teaching (learn SQL in your browser) SQLZOO - Interactive SQL Tutorial The Schemaverse: a space-based strategy game implemented entirely within a PostgreSQL database Treasure Data: Learn SQL by Calculating Customer Lifetime Value