Reply #47: actually, standard deviation increases [View All]

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but the relative width of the distribution gets smaller (compared
to the sample size)

More precisely, in a random distribution standard deviation
(which measures the width of the bulk of the distribution) grows
like the square root of the sample size: for N=1000, the standard
deviation is the square root of 250, which is about 16. Hence
a deviation of one hundred from the mean is more than six standard
deviations away from the mean.

You'll often hear six sigma mentioned in big business: this refers,
for example, to getting a manufacturing process to be so accurate that
random deviations would have to be more than 6 standard deviations away
from the mean for something bad to happen --- the point being that
this is a tremendously unlikely event: odd are about 1/10^9.