Interesting post. Well thought out, with an original angle.
In the direction of constructive feedback, consider that the concept of sample size—while it seems to help with the heuristic explanation—likely just muddies the water. (We’d still have the effect even if there were plenty of points at all values.)
For example, suppose there were so many people with extreme height some of them also had extreme agility (with infinite sample size, we’d even reliably have that the best players we’re also the tallest.) So: some of the tallest people are also the best basketball players. However, as you argued, most of the tallest won’t be the most agile also, so most of the tallest are not the best (contrary to what would be predicted by their height alone).
In contrast, if average height correlates with average basketball ability, the other necessary condition for a basketball player with average height to have average ability is to have average agility—but this is easy to satisfy. So most people with average height fit the prediction of average ability.
Likewise, the shortest people aren’t likely to have the lowest agility, so the correlation prediction fails at that tail too.
Some of the ‘math’ is that it is easy to be average in all variables ( say, (.65)^n where n is the number of variables) but the probability of being standard deviations extreme in all variables is hard (say, (.05)^n to be in the top 5 percent.) Other math can be used to find the theoretic shape for these assumptions (e. g., is it an ellipse?).
We’d still have the effect even if there were plenty of points at all values.
Are you talking about relative sample sizes, or absolute? The effect requires that as you go from +4sd to +3sd to +2sd, your population increases sufficiently fast. As long as that holds, it doesn’t go away if the total population grows. (But that’s because if you get lots of points at +4sd, then you have a smaller number at +5sd. So you don’t have “plenty of points at all values”.)
If you have equal numbers at +4 and +3 and +2, then most of the +4 still may not be the best, but the best is likely to be +4.
I don’t believe we disagree on anything. For example, I agree with this:
If you have equal numbers at +4 and +3 and +2, then most of the +4 still may not be the best, but the best is likely to be +4.
Are you talking about relative sample sizes, or absolute?
By ‘plenty of points’… I was imagining that we are taking a finite sample from a theoretically infinite population. A person decides on a density that represents ‘plenty of points’ and then keeps adding to the sample until they have that density up to a certain specified sd.
Interesting post. Well thought out, with an original angle.
In the direction of constructive feedback, consider that the concept of sample size—while it seems to help with the heuristic explanation—likely just muddies the water. (We’d still have the effect even if there were plenty of points at all values.)
For example, suppose there were so many people with extreme height some of them also had extreme agility (with infinite sample size, we’d even reliably have that the best players we’re also the tallest.) So: some of the tallest people are also the best basketball players. However, as you argued, most of the tallest won’t be the most agile also, so most of the tallest are not the best (contrary to what would be predicted by their height alone).
In contrast, if average height correlates with average basketball ability, the other necessary condition for a basketball player with average height to have average ability is to have average agility—but this is easy to satisfy. So most people with average height fit the prediction of average ability.
Likewise, the shortest people aren’t likely to have the lowest agility, so the correlation prediction fails at that tail too.
Some of the ‘math’ is that it is easy to be average in all variables ( say, (.65)^n where n is the number of variables) but the probability of being standard deviations extreme in all variables is hard (say, (.05)^n to be in the top 5 percent.) Other math can be used to find the theoretic shape for these assumptions (e. g., is it an ellipse?).
Are you talking about relative sample sizes, or absolute? The effect requires that as you go from +4sd to +3sd to +2sd, your population increases sufficiently fast. As long as that holds, it doesn’t go away if the total population grows. (But that’s because if you get lots of points at +4sd, then you have a smaller number at +5sd. So you don’t have “plenty of points at all values”.)
If you have equal numbers at +4 and +3 and +2, then most of the +4 still may not be the best, but the best is likely to be +4.
(Warning: I did not actually do the math.)
I don’t believe we disagree on anything. For example, I agree with this:
If you have equal numbers at +4 and +3 and +2, then most of the +4 still may not be the best, but the best is likely to be +4.
By ‘plenty of points’… I was imagining that we are taking a finite sample from a theoretically infinite population. A person decides on a density that represents ‘plenty of points’ and then keeps adding to the sample until they have that density up to a certain specified sd.