*** There is fluid language and understanding of errors, and how they are handled, in medicine, social sciences, and more precise sciences like physics and engineering[i].
- Types of errors and their explications
*** A first setting the stage: statistics differs from ‘pure’ math/arithmetic/calculus in how strict/sharp vs. ‘loose’ the equality/equivalence relation is: we accept that 1 2; but statistics on the other hand uses modified relational rules, which allow one to sometimes declare
1 2 (when 2 is an average[ii] falling less than two standard errors further away than (the average) 1). This adds another layer of uncertainty, which makes statistics more challenging[iii].
***The labeling entices some strong ideological debates, e.g. reactions to calling the unexplained variability in an outcome ‘residual error’ instead of the ‘proper’ disturbance[iv].
[i] One can rank order sciences along how strong causal statements about the studied phenomena they can make, which can also be alternatively formulated as the ‘percent error’ left in their explicating theories; compare e.g. the A1c(BMI) functional relation in medicine, at one endpoint, to the Cobb-Douglas production function in economics (Y = A * K^α * L^β, where Y is output, K is capital, L is labor, A is a constant representing total factor productivity, and α and β are parameters representing the share of output attributable to each input), and, at the other endpoint, to the ‘pressure*volume ~= temperature’ law of the ‘ideal gas’ in physics (or chemistry).
[ii] The ‘typical’ or average American’ appears to be a fictional entity, statistically generated (the statement “The average American owns 1.02 cars” is “about the average American”, [1] p. 301), of which there are plenty, some which exist, other which don’t: “I intend to use the word exists so that it encompasses exactly those objects that orthodox philosophers hold to exist. In particular, it includes all the ordinary physical objects that we normally take to exist, and it does not include unicorns, gold mountains, winged horses, round squares (round square things), Pegasus, or Sherlock Holmes. The theory given below will say that there are unicorns, there is such a thing as Pegasus, etc., but that none of these exist. “ [2] p. 11 & “If we forget or inhibit our philosophical training for the moment, we are all prepared to cite examples of nonexistent objects: Pegasus, Sherlock Holmes, unicorns, centaurs, . . . . Those are all possible objects, but we can find examples of impossible ones, too; Quine’s example of the round square cupola on Berkeley College will do. It is an impossible object, and it certainly doesn’t exist, so it seems to be an example of an impossible nonexistent object. With so many examples at hand, what is more natural than to conclude that there are nonexistent objects-lots of them [2], p. 2
[iii] The logic of hypothetical reasoning is an old topic in logic, [3] and is involved in both the classical ‘hypothesis testing’ scientific procedure, and the more modern causal inference advances, based on ‘what-if’ contrary-to-fact ‘potential outcomes’ reasoning.
[iv] Judea Pearl says “u’s stand for omitted factors” in [4]