Are we all log-normal deviates?

[Chris Lloyd]

Some simple human characteristics are reasonably close to normal – like height or length of index finger. But more complex human abilities are strongly positively skew. Below is a cute little 90 second talk from Angela Duckworth, a well-known academic pyschologist at the University of Pennsylvania



Click HERE to hear the short address.

I do not know what the data actually say, but the basic argument seems sound. Complex human abilities depend on several contingent abilities that intereact. For instance, tennis requires hand-to-eye, athleticism, fitness and psychological hardness. And it sounds more reasonable that the final achievement would be a product of these abilities rather than a sum. Similarly. skills like communication and organization don’t just contribute additively as they would on a report card; they are multipliers that amplify your effectiveness in other areas.

But to get the log-normal distribution for the final performance, the basic abilities to be multiplied also have to be log-normal. Why would the basic abilities be log-normal? Because variation on the log-scale is variation on the proportional scale. It seems reasonable that you are as likely to be twice as good as average as twice as bad as average. Not to mention that most traits have to be positive.

On the other hand, IQ tests are, I believe, deliberately scaled to be normal and heights within a racial-gender group are close to normal but with heavy tails, due to genetic abnormalities.

That got me to thinking. Normals are additive. Log-normals are additive on the log scale and so are multiplicative. In between, you could also have additivity on the Box-Cox scale. This is fairly standard for regression but I have never heard a name for the 3-parameter family of distributions where

Y=(Xlambda-1)/lambda

is normal - in other words X has the distribution of (Y lambda+1)1/lambda.



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