I just learned about Deming's Red Bead Experiment. The experiment simulates a business that produces white beads using a somewhat convoluted system (the linked page above describes it well). The idea behind is that individual worker's performance has no impact on the results that worker produces. However, due to randomness and other factors outside of the worker's control. This is simulates a modern business environment. However, good performance is rewarded with pay raises while bad performance is punished, for example by being put on probation. The point is that this is how many businesses are being run. Namely, employees are rewarded or punished due to the workings of the system they are placed within; not due to their performance.

For more:

• Youtube video of the experiment being run.
• Hacker News discussion.

I was reading the (so far) very interesting book, How to Measure Anything by Douglas W. Hubbard, and he mentioned the rule of five. Essentially, if you collect 5 data points at random the median of that distribution will be within the biggest and smallest sample value with a probability of 0.9375. This follows from the definition of the median, i.e., that 50% of the probability mass is on each sides. So there is a \(\frac{1}{2^5} = 0.03125\) probability that all data points are above the median (assuming independence). It is equally probable that all sample points are below the median. This gives you a very quick way to get a rough, back-of-the-napkin style estimate of things like average commuting times (the example from the book) and other things were you can easily get hold of five new data points. This all assumes independence.

I came over a .pdf on linkedin, which I'm unfortunately unable to find again, containing some lessons of investing from Benjamin Graham's mentor¹. One of them was the rule of 72. The rule of 72 is used to find out how long it takes to double your investment given an annual return rate:

\[\frac{72}{\text{Rate of Return}} \approx \text{Years to Double}\]

I never heard about it before, but it works pretty well for a reasonable range of returns. This plot shows how close the curves are:

This made me wonder where this rule came from and how one could come up with such a useful heuristic. It was referenced already in Summa de Arithmetica² by Luca Pacioli in 1494, also applied to investments. The rule is not explained so it is assumed that the rule predates the book. Fascinating!

The number is somehow derived from \(log(2) = 0.693147 \dots\), which is a transcendental number. Wikipedia also refers to the rule of 69.3, which is more exact. However, it is impractical. 72 is preferable over 69 or 69.3 as it is a mental arithmetic heuristic for the simple reason that 72 is divisible by 2, 3, 4, 6, 8, 9, 12 (and 18, 24, 36, but the heuristic comes from a list of tips for the investor; not the speculator…). However, logarithms were not really around in Europe in 1494. They only started being used from the 17th century onwards. So how would one compute \(log(2)\) in those days?