Just definitions today, folks. We’ll need them later. You’d think a lot of these would be obvious. Experience shows they are not. This article is solely about equality of outcomes and not equality used in any other sense.
A philosophically contradictory, but the most desired, definition of equality is with respect to outcomes. Applied to individuals unconditionally, equality fails immediately for the obvious reason that no two individuals can be identical, and therefore they can never be equal. For example, and trivially, they cannot occupy the same space simultaneously.
Sensing, but not explicitly acknowledging this, equality mavens shift the burden from individuals to groups of individuals. Equality, or the lack of it, then becomes a statistical phenomenon, and logically possible, however unlikely.
Suppose we are interested in the equality of outcome for some specific measure, say, a score on a test. Equality of individuals is now logically possible. For example, all individuals can receive an identical score.
The difference is obvious: unconditional equality is impossible—not just unlikely, understand, but impossible. Conditional equality—where it is defined with respect to some finite, quantifiable, measure—is logically possible, even though, in practice, such an outcome is unlikely.
Again, equality mavens are not strictly concerned with this modification and still speak of equality with respect to groups. For test scores, it is, of course, logically possible for two groups to be identical. One way is as just mentioned: all individuals, regardless of group, receive an identical score, an outcome which guarantees all groups will attain equality. But this example is not what mavens usually seek (though they might secretly desire it).
If all we know about an individual is that he belongs to Group A, then we can use probability to quantify the chance that he receives a particular score on a measure of interest. That is, we can compute things like this statement: “Given that an individual is a member of Group A, then the chance he scores above 50 on the test is X”. We can do the same for members of Group B, Group C, and so forth.
One possible definition of statistical equality is the condition where, regardless of Group, each individual has the same X in sentences like this: “Given that an individual is of the Group of interest , then the chance he scores above Y on the test is X.” That is, equality means all groups are equal in X, for each possible value of Y. (Y can only take the [finite] values possible on the measure of interest.) This is predictive equality.
This does not, of course, imply that all individuals in all groups must receive identical scores; it is only a statement about our ability to predict individuals’ scores.
Statistical equality can also be the condition where, regardless of Group, each individual has the same X in sentences like this: “Given a previously observed set of scores and a member in the Group of interest, X-percent of individuals scored about Y.” Again, equality means all groups are equal in X, for each possible value of Y. This is observational equality.
It is of crucial importance to understand the differences between these two definitions: predictive and observational.
Suppose we had only two individuals, one from Group A and one from Group B. “A” scored 40 out of 100 (where higher is better), and “B” scored 60. In this case we do not have observational equality.
If we only had one member of each Group, they must all have the same score for observational equality to hold. And it should be obvious that if our observational data consists of small numbers of individuals, or large numbers of groups, or both, it is very likely that observational inequality will result.
If we liked, having seen the scores from our example, we could write a paper on the “observed discrepancy” and claim the two groups were not equal. And we would be on solid ground as long we resisted the temptation to extrapolate—which is the same as making a prediction—to other individuals in the two groups, whose test scores as not yet known.
This is because it might not be true that predictive inequality exists between the two groups. That is, our calculations might show that individuals from any group have the same probabilities for scores, given additional information about the historical performance of the groups on the measure of interest.
There are several other caveats. Approximate observational or predictive equality can exist, even when actual equality does not, and approximate equality might be “close enough” for equality seekers. A lot depends on what is meant by “approximate”, of course. As long as it unambiguously defined, and that definition is held to, then no problem exits.
Equality, particularly observational, but also predictive since it depends on observations, is variable: a sample of individuals now might show an inequality that does not obtain later, and vice versa.
Inequality, either observational or predictive, approximate or exact, might exist in one dimension, but equality might be present in others. Those who delight in discovering and reporting inequalities should be compelled to report all the dimensions for which inequalities were searched. For example, there are those who search for inequalities in measures of health, and only report the one (or those) which show the largest negative discrepancy with respect to their favorite group. The dimensions which favor their group, or are those which show equality, are frequently left unreported. This naturally exaggerates actual differences.