Sat, Jul 11 · 2:00 PM CDT
On June 27, 2026, our group of four found ways to build confidence to assert a conclusion while exploring Steven Pinker’s Rationality. We also found that all measurements of an object or phenomenon are not the same. Within a group of skilled carpenters, there is variation in the length of boards cut to a specification. Even with precision tooling, there are small differences between manufactured parts, but a designer specifies an allowable range that manufacturers know how much a part can deviate, a tolerance that does not compromise an assembly's function. Are there statistical tools that account for variation?
If we look at the Scholastic Aptitude Test scores of seniors in a high school, we can group the test scores by tens on the x-axis and the number of students with those scores on the y-axis. This creates a histogram that looks like a ziggurat that peaks and then tails off. If we include the scores of students from other high schools, the histogram smooths out into a bell-shaped curve known as the Gaussian or Normal Distribution. The mathematician Gauss recognized that the bell-shaped curve revealed a mathematical property that showed the relationship between noise and physical reality.
We can analyze the normal distribution by looking at the mean (the average score), the median (the score of the middle-ranked student), and the mode (the score with the largest number of students). In a symmetrical normal distribution, the mean, median, and mode have the same value. A right-skewed distribution has a long, tapering tail stretching toward the higher, positive numbers on the right-hand side, and the high values drag the mean upward, changing the median and mode, where the mode < median < mean. A left-skewed distribution has a long, tapering tail stretching out toward the lower, smaller numbers on the left side, and the extreme low values drag the mean downward, while the mode stays anchored, where the values of the mean < median < mode.
Instead of looking at SAT scores, we can apply this analysis to the distribution of returns from a radar beam that shows up as a blip on a screen. The radar beam can be reflected off either an airplane or a flock of birds, but how can the operator distinguish between the two? Suppose that birds give a distribution of returns with values lower than the returns from a plane. It becomes obvious that the return signals for birds can easily be distinguished from the return signals of planes. However, what if the distributions overlapped? Is there a way to determine which return is from a bird or a plane? A Beta, a line perpendicular to the distributions that gives a high signal of planes to the noise of bird ratio at a certain point where they overlap. The formula for Beta gives the level of certainty, a probability that the radar blip was from a plane and not a bird.
What if you were commanding the radar station at the Opana Radar Site, located on the North Shore of Oahu, Hawaii, that detected planes on December 7, 1941, near Kawela Bay, north of Pearl Harbor? The privates operating the radar detected a massive echo on their oscilloscope, but their warning was dismissed by an officer who thought it was a flight of B-17s scheduled to arrive from Los Angeles. Does it matter where to put a Beta in wartime rather than in peacetime footing?
Standard deviation (SD) is calculated from the square root of the variance from the mean, where ±1 SD contains 68.2% of all data points, ±2 SD contains 95.4% of all data points, and ±3 SD contains 99.7% of all data points. A 95% confidence limit is near ±2 SD, and most published scientific papers use a 95% confidence limit on the variation of their data. There are two errors in the conclusions drawn from the data. Type 1 Error is the False Positive, where you conclude that an effect is real, but it was actually noise. Type 2 Error is the False Negative when you conclude that the data is noise, but there was actually a true effect. To prevent False Positives, science sets a strict significance level of p < 0.05. Is the entire scientific enterprise weaving a story with a thread this thin? Perhaps—but it beats weaving a picture of reality out of wishful thinking.
This exact trade-off governs our justice system. Jurist William Blackstone famously stated: "It is better that 10 guilty persons escape than to let one innocent person suffer." This sets an incredibly high Beta. To meet Blackstone's ratio, evidence must clear a 3-SD level of certainty. Punishing the innocent should shock the public conscience, distinguishing a regime of justice from a regime of terror. Yet, when a heinous crime occurs, do our availability and confirmation biases trick us into shifting that Beta back toward a reign of terror?
In our next meeting, we will dive into how to use game theory to make optimal decisions as we continue to discuss Steven Pinker’s Rationality: What It Is, Why It Seems Scarce, Why It Matters, BF441.P56 2021, on July 11, 2026, from 2:00 PM to 4:00 PM.
