I've recently been seeing quite a few misconceptions online about various topics in mathematics.
First, Benford's law applies to distributions whose pushforward along the map is approximately equidistributed. For instance, dynamics that induce irrational rotations of the circle, such as the powers of 2, satisfy Benford's law. Furthermore, broad distributions that span multiple orders of magnitude, such as uniforms over , also satisfy Benford's law. If for some reason you happen to be looking at distributions of votes over multiple precincts, whose populations could very well be within a few orders of magnitude, whether or not your data satisfy Benford's law probably says more about your choice of precincts than anything else.
Second, let's say that there are two kinds of people in a population, short people and tall people, and you poll them on whether they prefer gymnastics or basketball. Short people are more likely to prefer gymnastics, and tall people are more likely to prefer basketball, but it isn't guaranteed that a short person prefers gymnastics, and it isn't guaranteed that a tall person prefers basketball. Suppose that 90% of short people prefer gymnastics, but 10% of short people prefer basketball. Similarly, suppose that 90% of tall people prefer basketball, but 10% of tall people prefer gymnastics. When we plot the percentage of gymnastics lovers with respect to the percentage of short people in the population, we get a curve that looks like , which has a slope that is almost one, but not quite. When you subtract out , what you get is . Now some PhD in bioengineering who clearly knows nothing about what he's talking about sees the slope of says that this is a feature. Indeed, it is a feature! It's a measurement of the correlation between being short and liking gymnastics.
Pic unrelated.
First, Benford's law applies to distributions whose pushforward along the map is approximately equidistributed. For instance, dynamics that induce irrational rotations of the circle, such as the powers of 2, satisfy Benford's law. Furthermore, broad distributions that span multiple orders of magnitude, such as uniforms over , also satisfy Benford's law. If for some reason you happen to be looking at distributions of votes over multiple precincts, whose populations could very well be within a few orders of magnitude, whether or not your data satisfy Benford's law probably says more about your choice of precincts than anything else.
Second, let's say that there are two kinds of people in a population, short people and tall people, and you poll them on whether they prefer gymnastics or basketball. Short people are more likely to prefer gymnastics, and tall people are more likely to prefer basketball, but it isn't guaranteed that a short person prefers gymnastics, and it isn't guaranteed that a tall person prefers basketball. Suppose that 90% of short people prefer gymnastics, but 10% of short people prefer basketball. Similarly, suppose that 90% of tall people prefer basketball, but 10% of tall people prefer gymnastics. When we plot the percentage of gymnastics lovers with respect to the percentage of short people in the population, we get a curve that looks like , which has a slope that is almost one, but not quite. When you subtract out , what you get is . Now some PhD in bioengineering who clearly knows nothing about what he's talking about sees the slope of says that this is a feature. Indeed, it is a feature! It's a measurement of the correlation between being short and liking gymnastics.
Pic unrelated.
