## The Significance of Underlying Variance for Social Outcomes

8 August 2024Measures — quantities to which some arithmetic can be meaningfully applied — can be fitted to some human attributes, even if not to others. When attempting to *compare* some populations to others, an assumption is made that the properties of the individuals within these populations are subject to quantification of some sort, and that the quantities are *commensurable* across populations. But usually these assumptions are implicit and unrecognized, and even those who have some awareness that they are dealing with quantities very often don't have a proper grasp of elementary issues.

Very often, people try to understand distinct populations in terms of some notion of *averages*. If each and every member of every population were exactly average in every regard, then averages would be perfect measures of the populations as such. More generally, if for any two populations the share of that population deviating from average by some specific amount were the same, averages would be sufficient for any comparison of the attributes of populations, except for population sizes. But if the overall variance from average in one population is different from that in another, then thinking in terms of averages can go very, very wrong.

Here are hypothetic distributions for some attribute within two populations, each population having the same number of members:[1] For both Population `A` and Population `B`, the *median*[2] of the attribute is the same, but Population `B` has more variance from the arithmetic mean than does Population `A`. Even though each of these two populations have the same median, *more* members of the population of greater variance are *below* some some measure, and more members of that *same* population are *above* some measure. Population `B`_{2} has the same variance as Population `B`, but the same *arithmetic mean* (rather than median) as Population `A`. Again, even though the *center* is, by some measure, the same for both populations, *more* members of the population of greater variance are *below* some some measure, and more members of that *same* population are *above* some measure.

Even if a population has a higher center than Population `A`, if it has a greater variance then it will dominate the *lower* range of measures below some measure. And even if a population has a lower center than Population `A`, if it has a greater variance then it will dominate the *higher* range of measures beyond some measure.

If a measurable attribute correlates positively with social success, then ceteris paribus, a population of higher variance is going to dominate *both* social winners beyond some level *and* social losers below some level. If a population *generally* has greater variance amongst its attributes, then — *discarding* the assumption of ceteris paribus — that population is going to dominate *both* social winners beyond some level *and* social losers below some level; even if it has the same median or same mean or even a lower center than another populations.

In fact, though I cannot readily *graph* the cases in which attributes are only partially ordered and not measurable, the reader should see that the underlying point does not depend upon the measurability of the attributes, but only upon one population having greater propensity for variance than another.

But

- If one is only looking at the losers and thoughtlessly assuming that their numbers are
*explained*by averages, then one is going inappropriately to infer that the population is generically inferior. - If one is only looking at the losers, while thoughtlessly assuming that the averages are the same and that nothing else about the population itself can explain the difference in outcomes, then one is going inappropriately to infer that the population are victims of systemic bias.
- If one is only looking at the winners and thoughtlessly assuming that their numbers are
*explained*by averages, then one is going inappropriately to infer that the population is generically superior. - If one is only looking at the winners, while thoughtlessly assuming that the averages are the same and that nothing else about the population itself can explain the difference in outcomes, then one is going inappropriately to infer that any
*rival*population are victims of systemic bias.

In each case, if we look at the *other* end of the distribution, the thoughtless conclusion falls apart.

When the last of these errors is made, an attempt may be undertaken to *offset* illusory bias, by putting an institutional thumb on the scales to shift Population `A` generally forward, until the the number of social winners *at every level* is *at least* the same. But notice what is then really happening as the *relative* outcomes for *most* members of the population of greater variance fall increasingly below those of the population of less variance — at previously targetted levels the population of lower variance comes to enjoy *greater* social success than does the population of greater variance. And notice that the population of greater variance necessarily *still* dominates above some value, albeït that the value increases as the institutional thumb comes down ever harder in a misguided attempt to match the upper tails of the distribution.

Only *actual* systemic bias can bring the number of social winners across populations into equality above any given level of social success beyond the center; and, the larger the population, the greater the required bias for such an outcome, and the more that *most* of the population of greater variance are victimized.

If the two populations are *not* equal in size, then the foregoing analysis would simply need to entail talk of proportionality. But one might as well speak and write of two populations of the same size, because the real-world application involves two populations that are very close to the same size in most first-world nations. The greater variance of one of those two populations is a consequence of the greater chaos in the formation of one of that population's chromosomes and of a lack of redunancy for another.

Unfortunately, most of the attempts to analyze what has been happening has entailed ham-fisted theorizing about differing averages.

[1] A population with finite membership cannot perfectly conform to a continuous distribution function; but, the larger the population, the less the necessary non-conformance.

[2] The *median* for each population is the point such that as many members of the population are above it as are below it.

Tags: averages, central tendencies, glass ceilings, means, medians, sex, sexual differences, standard deviation, statistical moments, statistics, variance

CDF/central value's derivation depends upon the convolution of independent trials. One suspects that performance measures in the real world are not necessarily independent, in a great deal of trials.

In Vegas, they frequently switch out the dice on the craps table, due to obvious things, like the potential for players swapping in their own dice, but also for less obvious things, like the sweat/skin of the players hands biasing the dice outcome. One imagines that with the current level of academic fraud, the "dice throwing" of testing students would be akin to physically changing the dimensions of the craps table while the dice are in-flight; thereby also altering the physical placement of the prior "bets".

Beyond the points that using some average as-if a

sufficientstatistic results in a miscalculation of the PDF and that differences in tails are not sufficient to arrive at differences in central tendencies, the challenges of accurately computing statistical moments and probability functions aren't relevant to the issues discussed in my entry.