Simpson’s Paradox: A Tale of Two Carpenters

Once upon a time, in a small town, there were two carpenters: Bob and Tom. Bob was an honest, hardworking carpenter with a good reputation. Tom was a somewhat incompetent carpenter with shady business practices. At one point in time, Bob and Tom had each completed 100 carpentry jobs. Bob’s customers were satisfied with 90 out of his 100 jobs. Tom’s customers were satisfied with 95 out of his 100 jobs.

Average Logarithms and the Geometric Mean

I recently came across a surprising connection between the geometric mean and logarithms. At least, it was surprising to me, but isn’t that surprising once you see the derivation (a lot of things in math are this way). The connection is this. Take a set of observations \(y_1, y_2, \dots, y_n\). Assume that these are all positive numbers. Then \[ \langle \log(y) \rangle = \log(G).\] Here the angle brackets indicate the mean (expectation value), and \(G\) is the geometric mean of the observations.

Approximating the Geometric Mean

In my last post, I discussed the geometric mean and how it relates to the more familiar arithmetic mean. I mentioned that the geometric mean is often useful for estimation in physics. Lawrence Weinstein, in his book Guesstimation 2.0, gives a mental algorithm for approximating the geometric mean. Given two numbers in scientific notation \[ a \times 10^x \quad \text{and}\quad b \times 10^y, \] where the coefficients \(a\) and \(b\) are both between 1 and 9.

The Geometric Mean

Given two positive numbers \(a\) and \(b\), the most well-known way to average them is to take the sum divided by two. This is often called the average or the mean of \(a\) and \(b\), but to distinguish it from other means it is called the arithmetic mean: \[ \text{arithmetic mean} = \frac{a + b}{2}. \] Another common, and useful, type of mean is the geometric mean. For two positive numbers \(a\) and \(b\), \[ \text{geometric mean} = \sqrt{ab}.

Symmetric Difference

I was recently flipping through a real analysis textbook, and came across a notation that I had not seen before. First, the set difference of two sets \(A\) and \(B\) (which I had seen before) was defined as \[ A \setminus B = A \cap B^c. \] Here \(B^c\) is the set compliment of \(B\), or all elements not in \(B\). You can equivalently write the set difference as \[ A \setminus B = \{x \, |\, x \in A\, \text{and}\, x \notin B\} \]

Lines in the Complex Plane

What is the equation of a straight line in the complex plane? There are many different forms, but I want to look at some of the simplest ones. If you know the slope \(m \in \mathbb{R}\) and intercept \(b \in \mathbb{R}\) of the line, you can write an equation in parametric form \[ z = x + i (m x + b) ,\] where \(x \in \mathbb{R}\) and \(z \in \mathbb{C}\).