Point well taken. I was considered the data from a raw standpoint, not normalized. And to add to what you said, so too can the mode fit with the mean and median (though that would be rare).
In IQ distributions (which are normalized to be Gaussian), ~68% within 1 standard deviation of the mean set at 100.
YES!
But..... that standard deviation in relation to the expected deviation from the mean, (kurtosis or "peakedness" ) is a BIG DEAL. In fact, other bell shaped distributions such as the Pearson distribution are more flexible in that they do not conform to a pre-determined "peakedness" as the normal distribution does.
If IQs were distributed as the figure on the top, the difference between the "smart" and the "not so smart" would not be very pronounced. Furthermore as you said, IQ distributions are NORMALIZED form their raw percentile scores:
they are forced to approximate a particular normal distribution but they are not necessarily distributed as such in reality. This is done for mathematical and estimation convenience. Of course there are tests to check for this (i.e: Jarque-Bera or Kolmogorov-Smirnov, but they assume that the parameters of the normal <the average and standard deviation> are known with certainty).
Moreover the calculation of IQ scores (by psychologists
) has always been controversial, with recoding and "re-designing" of results that do not fit, and convenient reparametrization of data. Psychology is not my area, but it is my understanding that IQs are not useful for comparisons among individuals or races (which by the way the Chinese LOVE to do), but they are more useful when comparing the same individual with himself/herself at different points in time, specially before and after some significant event (death of a loved one, PTSD, etc).