24 Jun 2025
Tags: stats
I recently received some junk mail from a national insurance company, which touted: “Customers who save when they switch save an average of over $800”. OK, so how many didn’t save money, and how much more they were charged? At least it’s not as meaningless as the GEICO slogan: “15 minutes could save you 15% or more.” It could save you nothing. It could cause frogs to rain from the sky. Anything could happen in 15 minutes.
Seriously, though, here’s a toy example. Let’s say out that \(n\) people received an insurance quote that saved \(x\), and \(m\) people received a quote that was more expensive by \(y\). Let’s understand \(x\) to be positive and \(y\) to be negative. Regardless of how huge \(m\) or \(y\) might be, the expected value over the positive values is still \(x\).
Let’s explore a scenario that’s a little more generous. Assume the random variable \(X\) has a standard normal distribution with a mean of zero and a standard deviation of 1. What’s the expected value over the positive part of the distribution only? Symbolically, that’s \(E(X\|X>0)\), but it’s a little easier to just set up a new random value Y on the positive reals and figure out \(E(Y)\). Fortunately, that already has a name: the (standard) half-normal distribution. I could bore you with the derivation anyways, but let’s not: the mean of that distribution is \(\sqrt{\frac{2}{\pi}}\), about 0.8. One wonders if Geico’s 15% is where 0.8 standard deviations lies on the distribution of savings.
(Hi! I’m not dead, I just got busy with the kind of work that actually pays me.)