Saturday, January 14, 2012

The Normal Distribution and You

The normal distribution crops up so often in our daily experience and applies to so many different phenomena that a tremendous amount of mental energy has been spent in its study.  Furthermore, we talk about those aspects of reality so frequently here at the Chariot of Reaction that it is probably useful to make a utility post on what we mean when we speak of sigmas.

Normally when we're talking about sigmas, for instance, saying +1 or +2 sigmas,  what we are referring to is the rarity of that level of talent or capability within the population as a whole.  Generally here at the Chariot, we speak almost exclusively of positive sigmas, but the distribution is symmetric, so the frequency estimates apply just as well for negative sigmas.  What I find particularly interesting is how this is embedded into the language and how even such prosaic metrics like the rating of attractiveness on a 1-10 scale are obviously using something akin to a z-score (basically, the number of sigmas from the mean, NOBODY says 10% of all persons are 10's, for instance).

In terms of frequency within a population, here's what you get:
+1 standard deviation or more is approximately 1 in 6.3.  This means that about one in every 6 people is at or above this level in a given attribute or quality. Linguistically, this usually translates to 'Joe is strong' or 'Joe is smart' or 'Joe is good looking' or the like. Sometimes when the speaker himself has a high attribute in the quality being described, he'll apply a hedge like 'pretty strong'...'fairly smart' or the like. You (the reader), almost certainly know a fair number of people with this level of capability in any attribute you can think of. Good examples include almost anyone with a real degree (intelligence), people who have no debt and live within their means (fiscal discipline, this one is really uncommon in the US as recent events demonstrate), people who tend to lead their small social circle (charisma), and anyone you'd describe honestly as 'athletic'.

+2 standard deviations is about 1 in 44. So for every 40 or 50 people, you generally have someone in this category. In language, we typically apply some sort of amplifier to our description---for instance...Joe is very strong or Joe is very smart. Almost nobody puts a linguistic hedge around their descriptor here. Typical examples here for the physical are noteworthy athletes at the high school level---some of the larger high schools might well have people with this level of physical prowess on their starters on the sports that they emphasize.

This level of intelligence is typical of the 'talented and gifted' and a lot of doctors, lawyers, engineers, and honestly, national level politicians are at or slightly below this level. If your social circle is at all typical, you probably know at least one person at this level for each attribute (albeit not usually the same person).

+3 SD is about 1 in 740. You usually get one, maybe two people at this level per 1000 people. In the US, for instance, you'd expect about 405K people at this level for each attribute. It is very exceptional, and people here usually get the superlatives hauled out when describing them. In a small town, at this level, you might well be 'the strongest man' or 'the smartest guy' in town. This is the bare minimum for 'only the obsessed need apply' sorts of competitions, such as the NFL. At this level in athletics, if you're doing all the other stuff, you've got a very thin chance of going beyond college level. With a broad social circle, you probably know at least one person at this level, although probably not for many different attributes.

+4 SD is about 1 in a little more than 30K. So for every 100,000 people, there are 3 at this level. In the US, you'd expect about 10,000 people at or above this level. So, Mr. Smart guy with the 160 IQ, there are probably 10000 people in the US smarter than you are, more if the US is disproportionately brain-draining other nations. This is the typical level for the NFL or the NBA in their relevant attributes.  Linguistically, people break out the words like 'genius', 'incredible', and the like.

+5 SD is about 1 in 3.5M. So for every 10 Million people, there are probably 3 at this level. The US, for instance, with @300M population would have about 90 people at this level. Really noteworthy guys in the NFL are probably at this level. This is also the level of the 1st string Olympian in sports that your nation takes seriously.

+6 SD is 1 in a billion. Probably 5-10 people like this exist in the entire world. Athletes at this level frankly EXPECT to medal.

+7 SD is past the limits of most of the tables you'll find. Human history Might include one person at this attribute level in each attribute.


Hail said...

Do you believe the conjectures coming from some quarters that different ethnic groups have different trait variances?

E.g., that Europeans have a wider distribution of trait-frequencies than, e.g. East-Asians. So, while the Northeast-Asian IQ-mean is higher than the Northwest-European IQ-mean, given equal size populations there are still more 'very smart' NW-Europeans than NE-Asians (as well as more 'very dumb').

Jehu said...

I believe that it is likely that different groups may have different variances for different attributes. I don't think we really have high enough quality data though to make the statement you mention though (the quality of the data on NE Asians outside of the West is nowhere near as good as that inside, say, the US).