Normal Distribution
nothing is called 95% until it has been counted
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The thread: The tail is where it is read

A p-value, a control limit and a risk figure are all tail statements, and the tail is exactly where every approximation in this subject is worst. The central limit theorem converges in the middle long before it converges where anyone looks.
sample sizerelative error10204080160320640128010%1%at the mediantwo sigma outthree sigma outexact binomial against its normal approximationthe tail converges last The distribution itself

The tail converges last

The central limit theorem is usually shown as a shape arriving. What the demonstration leaves out is the rate — and the rate is wildly different in the middle and in the tail, which is where every approximation in the subject is actually read.

 00.1000.2000.3000.400-4-202standard deviations from the meandensity68.27%95.45%99.73%the bands are integrated, not recalledsigma = 1.00 The distribution itself

The shape, and where its mass is

68, 95, 99.7 is recited more often than any other set of numbers in the subject. They are integrals of a specific curve, they are worth computing rather than remembering, and the third one is the one people misuse.

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